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-rw-r--r--FILES.byname970
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-rw-r--r--fonts/libertinus-type1/README2
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-rw-r--r--macros/latex/contrib/hindawi-latex-template/README43
-rw-r--r--macros/latex/contrib/hindawi-latex-template/hindawi_bib_style.bst982
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-rw-r--r--macros/latex/contrib/hindawi-latex-template/hindawi_latex_template.tex216
-rw-r--r--macros/latex/contrib/numerica/README.txt35
-rw-r--r--macros/latex/contrib/numerica/numerica-basics.pdf13442
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-rw-r--r--macros/latex/contrib/verifiche/Esempio.pdfbin174217 -> 0 bytes
-rw-r--r--macros/latex/contrib/verifiche/README.md9
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-rw-r--r--macros/latex/contrib/verifiche/verifiche.dtx276
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-rw-r--r--macros/luatex/generic/chinese-jfm/README.md8
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-rw-r--r--macros/luatex/generic/chinese-jfm/chinese-jfm.tex8
-rw-r--r--macros/luatex/generic/chinese-jfm/jfm-ja_JP.lua2
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-rw-r--r--systems/texlive/tlnet/tlpkg/texlive.tlpdb75
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-rwxr-xr-xsystems/unix/tex-fpc/MFT/logmacfonts2
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-rw-r--r--systems/unix/tex-fpc/README182
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-rw-r--r--systems/unix/tex-fpc/dvips/README3
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-rwxr-xr-xsystems/unix/tex-fpc/shell/mkfont9
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-rw-r--r--systems/unix/tex-fpc/tex.ch184
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-rw-r--r--systems/win32/miktex/tm/packages/files.csv.lzmabin569553 -> 571234 bytes
-rw-r--r--systems/win32/miktex/tm/packages/next/files.csv.lzmabin569335 -> 571347 bytes
-rw-r--r--systems/win32/miktex/tm/packages/next/pr.ini8
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-rw-r--r--systems/win32/w32tex/ChangeLog36
-rw-r--r--timestamp2
166 files changed, 42525 insertions, 1255 deletions
diff --git a/FILES.byname b/FILES.byname
index 14623a462b..f102b7f519 100644
--- a/FILES.byname
+++ b/FILES.byname
@@ -5087,7 +5087,7 @@
1990/10/02 | 5440 | dviware/vutex/vaxvms.hlp
1990/10/02 | 812 | dviware/vutex/vaxvms.ins
1990/10/02 | 127576 | dviware/vutex/vutex.web
-2021/02/14 | 352411 | FILES.last07days
+2021/02/15 | 363551 | FILES.last07days
2020/08/14 | 104065 | fonts/academicons/academicons.pdf
2020/08/14 | 17247 | fonts/academicons/academicons.sty
2020/08/14 | 6546 | fonts/academicons/academicons.tex
@@ -64835,8 +64835,8 @@
2017/09/29 | 2000 | fonts/libertinust1math/vf/libertinust1-mathsfrm-bold.vf
2017/09/29 | 1992 | fonts/libertinust1math/vf/libertinust1-mathsfrm.vf
2020/11/05 | 5861103 | fonts/libertinust1math.zip
-2020/10/10 | 1114808 | fonts/libertinus-type1/doc/libertinus-samples.pdf
-2020/10/10 | 4450 | fonts/libertinus-type1/doc/libertinus-samples.tex
+2021/02/14 | 1114923 | fonts/libertinus-type1/doc/libertinus-samples.pdf
+2021/02/14 | 4438 | fonts/libertinus-type1/doc/libertinus-samples.tex
2020/12/16 | 285806 | fonts/libertinus-type1/doc/libertinus-type1.pdf
2020/12/16 | 6398 | fonts/libertinus-type1/doc/libertinus-type1.tex
2020/08/22 | 4427 | fonts/libertinus-type1/doc/OFL.txt
@@ -65212,7 +65212,7 @@
2020/08/22 | 3096 | fonts/libertinus-type1/latex/LGRLibertinusSerif-Sup.fd
2020/08/22 | 5461 | fonts/libertinus-type1/latex/LGRLibertinusSerif-TLF.fd
2020/08/22 | 5548 | fonts/libertinus-type1/latex/LGRLibertinusSerif-TOsF.fd
-2020/08/22 | 6666 | fonts/libertinus-type1/latex/libertinus-type1.sty
+2021/02/14 | 6761 | fonts/libertinus-type1/latex/libertinus-type1.sty
2020/08/22 | 2144 | fonts/libertinus-type1/latex/LY1LibertinusMono-Sup.fd
2020/08/22 | 2365 | fonts/libertinus-type1/latex/LY1LibertinusMono-TLF.fd
2020/08/22 | 3912 | fonts/libertinus-type1/latex/LY1LibertinusSans-LF.fd
@@ -65330,7 +65330,7 @@
2020/08/22 | 3096 | fonts/libertinus-type1/latex/TS1LibertinusSerif-TLF.fd
2020/08/22 | 3141 | fonts/libertinus-type1/latex/TS1LibertinusSerif-TOsF.fd
2020/08/22 | 119633 | fonts/libertinus-type1/map/libertinus.map
-2020/12/16 | 3085 | fonts/libertinus-type1/README
+2021/02/14 | 3085 | fonts/libertinus-type1/README
2020/08/22 | 1540 | fonts/libertinus-type1/tfm/LibertinusMono-Bold-lf-ly1--base.tfm
2020/08/22 | 1588 | fonts/libertinus-type1/tfm/LibertinusMono-Bold-lf-ly1.tfm
2020/08/22 | 1232 | fonts/libertinus-type1/tfm/LibertinusMono-Bold-lf-ot1.tfm
@@ -66712,7 +66712,7 @@
2020/08/22 | 1700 | fonts/libertinus-type1/vf/LibertinusSerif-Semibold-tosf-sc-t1.vf
2020/08/22 | 1732 | fonts/libertinus-type1/vf/LibertinusSerif-Semibold-tosf-t1.vf
2020/08/22 | 576 | fonts/libertinus-type1/vf/LibertinusSerif-Semibold-tosf-ts1.vf
-2020/12/16 | 13104174 | fonts/libertinus-type1.zip
+2021/02/15 | 13104278 | fonts/libertinus-type1.zip
2020/05/06 | 17330 | fonts/libertinus.zip
2020/07/27 | 2464 | fonts/librebaskerville/doc/FONTLOG.txt
2020/08/01 | 29079 | fonts/librebaskerville/doc/librebaskerville-samples.pdf
@@ -147660,7 +147660,7 @@
1998/01/30 | 23451 | help/comp-fonts-FAQ/Finding_Fonts_for_Internationalization_FAQ
2006/03/20 | 201672 | help/comp-fonts-FAQ.zip
2004/05/25 | 67001 | help/comp.programming.literate_FAQ
-2020/04/15 | 52882 | help/ctan/CTAN-upload-addendum.html
+2021/02/15 | 52885 | help/ctan/CTAN-upload-addendum.html
1995/09/20 | 827310 | help/ctan/ctan-web.data.gz
2003/09/20 | 34040 | help/de-tex-faq/part1
2003/09/20 | 24919 | help/de-tex-faq/part10
@@ -171823,7 +171823,7 @@
2017/06/16 | 484821 | install/fonts/libertinegc.tds.zip
2020/03/26 | 20354723 | install/fonts/libertine.tds.zip
2020/11/05 | 5870367 | install/fonts/libertinust1math.tds.zip
-2020/12/16 | 13253227 | install/fonts/libertinus-type1.tds.zip
+2021/02/14 | 13253331 | install/fonts/libertinus-type1.tds.zip
2020/08/01 | 871048 | install/fonts/librebaskerville.tds.zip
2016/01/12 | 776281 | install/fonts/librebodoni.tds.zip
2020/07/31 | 1576536 | install/fonts/librecaslon.tds.zip
@@ -185171,7 +185171,7 @@
2013/04/01 | 14776 | macros/context/contrib/context-visualcounter/tex/context/third/visualcounter/t-visualcounter.mkvi
2018/03/05 | 11 | macros/context/contrib/context-visualcounter/VERSION
2018/03/22 | 276789 | macros/context/contrib/context-visualcounter.zip
-2021/02/12 | 119771714 | macros/context/latest/cont-tmf.zip
+2021/02/14 | 119826723 | macros/context/latest/cont-tmf.zip
2016/04/21 | 32846 | macros/cstex/base/cmexb.tar.gz
2016/06/26 | 1728262 | macros/cstex/base/csfonts-t1.tar.gz
2007/03/01 | 187106 | macros/cstex/base/csfonts.tar.gz
@@ -192838,12 +192838,12 @@
2020/09/29 | 1091 | macros/latex/contrib/bxpapersize/LICENSE
2020/09/29 | 6232 | macros/latex/contrib/bxpapersize/README.md
2020/09/30 | 8009 | macros/latex/contrib/bxpapersize.zip
-2020/04/19 | 28418 | macros/latex/contrib/bxpdfver/bxpdfver.sty
-2020/04/19 | 1091 | macros/latex/contrib/bxpdfver/LICENSE
-2020/04/19 | 5859 | macros/latex/contrib/bxpdfver/README-ja.md
-2020/04/19 | 4436 | macros/latex/contrib/bxpdfver/README.md
+2021/02/14 | 29741 | macros/latex/contrib/bxpdfver/bxpdfver.sty
+2021/02/14 | 1091 | macros/latex/contrib/bxpdfver/LICENSE
+2021/02/14 | 5951 | macros/latex/contrib/bxpdfver/README-ja.md
+2021/02/14 | 4525 | macros/latex/contrib/bxpdfver/README.md
2017/02/12 | 644 | macros/latex/contrib/bxpdfver/sample-bxpdfver.tex
-2020/04/19 | 12463 | macros/latex/contrib/bxpdfver.zip
+2021/02/15 | 12997 | macros/latex/contrib/bxpdfver.zip
2018/03/31 | 422874 | macros/latex/contrib/bxtexlogo/bxtexlogo-sample.pdf
2018/03/27 | 2745 | macros/latex/contrib/bxtexlogo/bxtexlogo-sample.tex
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2021/02/13 | 208072 | macros/unicodetex/latex/aalok.zip
2021/02/13 | 203656 | language/japanese/platex/plfonts.dtx
@@ -515,7 +770,6 @@
2021/02/13 | 1500 | language/japanese/platex-tools/LICENSE
2021/02/13 | 1488 | language/japanese/platex/plcls.ins
2021/02/13 | 147796 | language/japanese/platex/platex-en.pdf
-2021/02/13 | 141951880 | systems/win32/w32tex/context.tar.xz
2021/02/13 | 141588 | systems/texlive/tlnet/archive/texlive-msg-translations.r57719.tar.xz
2021/02/13 | 140918 | language/japanese/plautopatch.zip
2021/02/13 | 139885 | language/japanese/pxjahyper.zip
@@ -528,13 +782,11 @@
2021/02/13 | 116383 | language/japanese/pxjahyper/pxjahyper.pdf
2021/02/13 | 1147 | macros/latex/contrib/pdfpc/README.md
2021/02/13 | 114028 | language/japanese/platex/plcore.dtx
-2021/02/13 | 11080032 | systems/win32/w32tex/win64/xetex-w64.tar.xz
2021/02/13 | 1091 | language/japanese/pxjahyper/LICENSE
2021/02/13 | 107870 | language/japanese/platex/platexrelease.pdf
2021/02/13 | 1066 | language/japanese/platex-tools/plextcolortbl.sty
2021/02/13 | 10430 | language/japanese/platex/plnewsc07.tex
2021/02/13 | 10426 | language/japanese/platex/plnews01.tex
-2021/02/12 | 99723028 | systems/win32/w32tex/web2c-lib.tar.xz
2021/02/12 | 96929 | systems/texlive/tlnet/tlpkg/translations/vi.po
2021/02/12 | 96201 | systems/texlive/tlnet/tlpkg/translations/de.po
2021/02/12 | 9589 | macros/latex/contrib/twemojis/pdf-twemojis/1f1e6-1f1e8.pdf
@@ -565,7 +817,6 @@
2021/02/12 | 731041 | macros/latex/contrib/komacv.zip
2021/02/12 | 6994 | macros/latex/contrib/twemojis/pdf-twemojis/1f1e9-1f1f4.pdf
2021/02/12 | 6870 | macros/latex/contrib/twemojis/pdf-twemojis/1f1f9-1f1e6.pdf
-2021/02/12 | 6622592 | systems/win32/w32tex/ptex-ng-w32.tar.xz
2021/02/12 | 643920 | systems/texlive/tlnet/archive/texlive.infra.x86_64-darwin.r57584.tar.xz
2021/02/12 | 63904 | systems/texlive/tlnet/tlpkg/translations/zh_TW.po
2021/02/12 | 6385 | macros/latex/contrib/twemojis/pdf-twemojis/1f1ef-1f1ea.pdf
@@ -603,7 +854,6 @@
2021/02/12 | 5246 | macros/latex/contrib/twemojis/pdf-twemojis/1f9d1-1f3ff-200d-1f9bc.pdf
2021/02/12 | 5246 | macros/latex/contrib/twemojis/pdf-twemojis/1f9d1-1f3fd-200d-1f9bc.pdf
2021/02/12 | 5242 | macros/latex/contrib/twemojis/pdf-twemojis/1f9d1-200d-1f9bc.pdf
-2021/02/12 | 524144 | systems/win32/miktex/tm/packages/easybook.tar.lzma
2021/02/12 | 5238 | macros/latex/contrib/twemojis/pdf-twemojis/1f9d1-1f3fb-200d-1f9bc.pdf
2021/02/12 | 5217 | macros/latex/contrib/twemojis/pdf-twemojis/1f1f0-1f1fe.pdf
2021/02/12 | 5175 | macros/latex/contrib/twemojis/pdf-twemojis/1f6d6.pdf
@@ -3061,7 +3311,6 @@
2021/02/12 | 1715 | macros/latex/contrib/twemojis/pdf-twemojis/1f44c.pdf
2021/02/12 | 1715 | macros/latex/contrib/twemojis/pdf-twemojis/1f355.pdf
2021/02/12 | 1714 | macros/latex/contrib/twemojis/pdf-twemojis/1f64d-1f3fe-200d-2640-fe0f.pdf
-2021/02/12 | 1714616 | systems/win32/w32tex/win64/ptex-ng-w64.tar.xz
2021/02/12 | 1713 | macros/latex/contrib/twemojis/pdf-twemojis/1f64d-1f3ff-200d-2640-fe0f.pdf
2021/02/12 | 1713 | macros/latex/contrib/twemojis/pdf-twemojis/1f4c9.pdf
2021/02/12 | 1712 | macros/latex/contrib/twemojis/pdf-twemojis/2614.pdf
@@ -3910,7 +4159,6 @@
2021/02/12 | 1199 | macros/latex/contrib/twemojis/pdf-twemojis/27a1.pdf
2021/02/12 | 1198 | macros/latex/contrib/twemojis/pdf-twemojis/2b05.pdf
2021/02/12 | 1197 | macros/latex/contrib/twemojis/pdf-twemojis/23cf.pdf
-2021/02/12 | 119771714 | macros/context/latest/cont-tmf.zip
2021/02/12 | 1194 | macros/latex/contrib/twemojis/pdf-twemojis/1f538.pdf
2021/02/12 | 1193 | macros/latex/contrib/twemojis/pdf-twemojis/1f53a.pdf
2021/02/12 | 1191 | macros/latex/contrib/twemojis/pdf-twemojis/1f536.pdf
@@ -3957,10 +4205,10 @@
2021/02/12 | 1144 | macros/latex/contrib/twemojis/pdf-twemojis/1f9b2.pdf
2021/02/12 | 113238 | systems/texlive/tlnet/tlpkg/translations/ru.po
2021/02/12 | 110626 | systems/texlive/tlnet/tlpkg/translations/sr.po
+2021/02/12 | 1008 | macros/latex/contrib/verifiche/README.md
2021/02/12 | 100511 | systems/texlive/tlnet/tlpkg/translations/fr.po
2021/02/11 | 984478 | info/biblio/texbook3.bib
2021/02/11 | 904 | macros/latex/contrib/hyperref/doc/hyperref-doc2.html
-2021/02/11 | 904160 | systems/texlive/tlnet/archive/tex4ht.r57704.tar.xz
2021/02/11 | 89853 | graphics/mcf2graph/mcf2graph.mf
2021/02/11 | 894349 | macros/latex/contrib/hyperref/hyperref.dtx
2021/02/11 | 79310 | macros/latex/contrib/komacv/komacv.dtx
@@ -4007,7 +4255,6 @@
2021/02/11 | 14053 | web/pwebmac.zip
2021/02/11 | 14012 | systems/texlive/tlnet/archive/fontsize.source.r57703.tar.xz
2021/02/11 | 1366178 | graphics/mcf2graph.zip
-2021/02/11 | 136156 | systems/texlive/tlnet/archive/tex4ht.doc.r57704.tar.xz
2021/02/11 | 125339 | macros/latex/contrib/komacv/examples/komacv_example_casual.pdf
2021/02/11 | 118943 | macros/latex/contrib/komacv/examples/komacv_example_oldstyle.pdf
2021/02/11 | 117342 | macros/latex/contrib/hyperref/ChangeLog.txt
@@ -4031,7 +4278,6 @@
2021/02/10 | 4895 | fonts/eczar/README.txt
2021/02/10 | 4788919 | systems/win32/miktex/tm/packages/yathesis.tar.lzma
2021/02/10 | 4479 | macros/latex/contrib/hyperref/README.md
-2021/02/10 | 4356391 | systems/win32/miktex/tm/packages/twemojis.tar.lzma
2021/02/10 | 42877 | fonts/eczar/doc/eczar.pdf
2021/02/10 | 4265 | fonts/eczar/doc/eczar.tex
2021/02/10 | 425054 | macros/latex/contrib/beamer/doc/beamerexample-lecture-print-version.pdf
@@ -4178,62 +4424,3 @@
2021/02/09 | 106690 | macros/unicodetex/latex/texnegar/xepersian-XB.pdf
2021/02/09 | 1051276 | systems/texlive/tlnet/archive/teubner.doc.r57684.tar.xz
2021/02/09 | 10117 | macros/unicodetex/latex/texnegar/doc-ligature-xep.pdf
-2021/02/08 | 9992 | macros/latex/contrib/teubner/teubner.txt
-2021/02/08 | 988527 | macros/latex/contrib/acrotex/doc/eformman.pdf
-2021/02/08 | 95480 | systems/texlive/tlnet/archive/l3experimental.source.r57669.tar.xz
-2021/02/08 | 90292 | macros/latex/contrib/teubner/teubner-doc.tex
-2021/02/08 | 8387500 | systems/win32/w32tex/uptex-w32.tar.xz
-2021/02/08 | 789136 | systems/texlive/tlnet/archive/l3backend.doc.r57669.tar.xz
-2021/02/08 | 784500 | systems/texlive/tlnet/archive/fancyhdr.doc.r57672.tar.xz
-2021/02/08 | 7808 | systems/texlive/tlnet/archive/innerscript.source.r57672.tar.xz
-2021/02/08 | 685712 | systems/texlive/tlnet/archive/pmx.doc.r57672.tar.xz
-2021/02/08 | 6608 | systems/texlive/tlnet/archive/ut-thesis.source.r57672.tar.xz
-2021/02/08 | 651576 | systems/win32/w32tex/win64/pmx-w64.tar.xz
-2021/02/08 | 624428 | macros/latex/contrib/teubner/teubner.pdf
-2021/02/08 | 552828 | systems/texlive/tlnet/archive/l3kernel.source.r57669.tar.xz
-2021/02/08 | 5412 | systems/texlive/tlnet/archive/babel-slovenian.source.r57666.tar.xz
-2021/02/08 | 5319 | macros/luatex/optex/base/pdfuni-string.opm
-2021/02/08 | 526482 | macros/latex/contrib/acrotex/doc/aeb_man.tex
-2021/02/08 | 52214 | fonts/utilities/ps2pk/Makefile.in
-2021/02/08 | 513440 | systems/texlive/tlnet/archive/biblatex-swiss-legal.doc.r57672.tar.xz
-2021/02/08 | 5112 | systems/texlive/tlnet/archive/fancyhdr.r57672.tar.xz
-2021/02/08 | 511088 | systems/texlive/tlnet/archive/oup-authoring-template.doc.r57672.tar.xz
-2021/02/08 | 486609 | macros/latex/contrib/acrotex/examples/webeqtst.pdf
-2021/02/08 | 46044 | systems/texlive/tlnet/archive/fancyhdr.source.r57672.tar.xz
-2021/02/08 | 4574 | macros/latex/contrib/acrotex/README.md
-2021/02/08 | 400279 | macros/latex/contrib/teubner/teubner-doc.pdf
-2021/02/08 | 3970512 | systems/texlive/tlnet/archive/l3experimental.doc.r57669.tar.xz
-2021/02/08 | 3750376 | macros/latex/contrib/acrotex.zip
-2021/02/08 | 3168 | systems/texlive/tlnet/archive/innerscript.r57672.tar.xz
-2021/02/08 | 3161 | macros/latex/contrib/hyperref/bmhydoc.sty
-2021/02/08 | 31172 | systems/texlive/tlnet/archive/l3backend.source.r57669.tar.xz
-2021/02/08 | 29656 | systems/texlive/tlnet/archive/l3experimental.r57669.tar.xz
-2021/02/08 | 28408 | macros/latex/contrib/hyperref/backref.dtx
-2021/02/08 | 279 | macros/latex/contrib/hyperref/ntheorem-hyper.sty
-2021/02/08 | 275 | macros/latex/contrib/hyperref/minitoc-hyper.sty
-2021/02/08 | 2586 | macros/latex/contrib/teubner/README
-2021/02/08 | 2544 | systems/texlive/tlnet/archive/ut-thesis.r57672.tar.xz
-2021/02/08 | 2412 | systems/texlive/tlnet/archive/babel-slovenian.r57666.tar.xz
-2021/02/08 | 23652 | macros/latex/contrib/hyperref/nameref.dtx
-2021/02/08 | 2135045 | macros/latex/contrib/acrotex/doc/aeb_man.pdf
-2021/02/08 | 209809 | macros/latex/contrib/acrotex/web.dtx
-2021/02/08 | 189042 | macros/latex/contrib/acrotex/doc/eformman.tex
-2021/02/08 | 18735756 | systems/win32/w32tex/luatex-exp-w32.tar.xz
-2021/02/08 | 18734744 | systems/win32/w32tex/luatex-dev-w32.tar.xz
-2021/02/08 | 18720624 | systems/win32/w32tex/luatex-w32.tar.xz
-2021/02/08 | 18148 | systems/texlive/tlnet/archive/biblatex-swiss-legal.r57672.tar.xz
-2021/02/08 | 172656 | macros/latex/contrib/teubner/teubner.dtx
-2021/02/08 | 1726052 | systems/win32/w32tex/pmx.tar.xz
-2021/02/08 | 157648 | systems/texlive/tlnet/archive/l3kernel.r57669.tar.xz
-2021/02/08 | 15128 | systems/texlive/tlnet/archive/oup-authoring-template.r57672.tar.xz
-2021/02/08 | 14468 | systems/texlive/tlnet/archive/l3backend.r57669.tar.xz
-2021/02/08 | 13987 | macros/latex/contrib/acrotex/examples/webeqtst.tex
-2021/02/08 | 136132 | systems/texlive/tlnet/archive/babel-slovenian.doc.r57666.tar.xz
-2021/02/08 | 12374 | macros/luatex/optex/base/if-macros.opm
-2021/02/08 | 123238 | macros/latex/contrib/acrotex/insdljs.dtx
-2021/02/08 | 120788 | systems/texlive/tlnet/archive/innerscript.doc.r57672.tar.xz
-2021/02/08 | 116940 | systems/texlive/tlnet/archive/ut-thesis.doc.r57672.tar.xz
-2021/02/08 | 1096110 | macros/latex/contrib/teubner.zip
-2021/02/08 | 10616904 | systems/win32/w32tex/ptex-w32.tar.xz
-2021/02/08 | 10294108 | systems/texlive/tlnet/archive/l3kernel.doc.r57669.tar.xz
-2021/02/08 | 10088 | systems/texlive/tlnet/archive/pmx.r57672.tar.xz
diff --git a/fonts/libertinus-type1/README b/fonts/libertinus-type1/README
index d1c279a9fb..92e10d5901 100644
--- a/fonts/libertinus-type1/README
+++ b/fonts/libertinus-type1/README
@@ -1,4 +1,4 @@
-This is the README for libertinus-type1, version 2020-12-16.
+This is the README for libertinus-type1, version 2021-02-14.
The Libertinus fonts are derived from the Linux Libertine
and Biolinum families of fonts. To use the OpenType fonts
diff --git a/fonts/libertinus-type1/doc/libertinus-samples.pdf b/fonts/libertinus-type1/doc/libertinus-samples.pdf
index f4c9b8c41f..a521df34c5 100644
--- a/fonts/libertinus-type1/doc/libertinus-samples.pdf
+++ b/fonts/libertinus-type1/doc/libertinus-samples.pdf
Binary files differ
diff --git a/fonts/libertinus-type1/doc/libertinus-samples.tex b/fonts/libertinus-type1/doc/libertinus-samples.tex
index 1136fc51f0..296071f831 100644
--- a/fonts/libertinus-type1/doc/libertinus-samples.tex
+++ b/fonts/libertinus-type1/doc/libertinus-samples.tex
@@ -1,6 +1,6 @@
\documentclass{article}
\usepackage[LGR,T2A,T1]{fontenc}
-\usepackage[sb,ScaleTT=0.8]{libertinus}
+\usepackage[sb]{libertinus}
\usepackage{lipsum}
\usepackage{lettrine}
\usepackage{soul}
diff --git a/fonts/libertinus-type1/latex/libertinus-type1.sty b/fonts/libertinus-type1/latex/libertinus-type1.sty
index e7cbdab07e..e9ac53da8c 100644
--- a/fonts/libertinus-type1/latex/libertinus-type1.sty
+++ b/fonts/libertinus-type1/latex/libertinus-type1.sty
@@ -1,5 +1,5 @@
\ProvidesPackage{libertinus-type1}[%
- 2020/08/22 (Bob Tennent) Supports libertinus fonts for latex/pdflatex.]
+ 2021/02/14 (Bob Tennent) Supports libertinus fonts for latex/pdflatex.]
\RequirePackage{ifxetex,ifluatex,xkeyval,textcomp}
\newif\iflibertinus@otf
@@ -31,28 +31,28 @@ use libertinus.sty or libertinus-otf.sty.\MessageBreak
\DeclareOptionX{sans}[true]{%
\@nameuse{if#1}
- \libertinus@sftrue\libertinus@rmfalse
+ \libertinus@sftrue\libertinus@rmfalse\libertinus@ttfalse
\fi
}
\DeclareOptionX{sf}[true]{%
\@nameuse{if#1}
- \libertinus@sftrue\libertinus@rmfalse
+ \libertinus@sftrue\libertinus@rmfalse\libertinus@ttfalse
\fi
}
\DeclareOptionX{ss}[true]{%
\@nameuse{if#1}
- \libertinus@sftrue\libertinus@rmfalse
+ \libertinus@sftrue\libertinus@rmfalse\libertinus@ttfalse
\fi
}
\DeclareOptionX{serif}[true]{%
\@nameuse{if#1}
- \libertinus@rmtrue\libertinus@sffalse
+ \libertinus@rmtrue\libertinus@sffalse\libertinus@ttfalse
\fi
}
\DeclareOptionX{rm}[true]{%
\@nameuse{if#1}
- \libertinus@rmtrue\libertinus@sffalse
+ \libertinus@rmtrue\libertinus@sffalse\libertinus@ttfalse
\fi
}
diff --git a/help/ctan/CTAN-upload-addendum.html b/help/ctan/CTAN-upload-addendum.html
index ecd2cd9d3b..a7a28f2d57 100644
--- a/help/ctan/CTAN-upload-addendum.html
+++ b/help/ctan/CTAN-upload-addendum.html
@@ -10,7 +10,7 @@
-->
<link rel="stylesheet" type="text/css" href="https://www.tug.org/tugstyle.css" />
<meta name="author" content="Petra R&uuml;be-Pugliese" />
- <meta name="date" content="2020-04-15" />
+ <meta name="date" content="2021-02-15" />
<meta http-equiv="content-type" content="text/html; charset=us-ascii" />
<style type="text/css">
.highlight { color:navy;}
@@ -516,8 +516,8 @@
CTAN and TeX Live have a preference for
<a id="hyphenvsunderscore"><span class="highlight">hyphens</span></a>
in package ids, rather than underscores.
- There are at the moment (April 2020)
- 1240 Catalogue entries with a hyphen in their name vs.
+ There are at the moment (February 2021)
+ 1275 Catalogue entries with a hyphen in their name vs.
10 with an underscore.
You will therefore have to give <em>very</em> good reasons
if you want us to accept the 11th package
@@ -1536,6 +1536,6 @@
$LastChangedDate$
</p>
Until then: -->
- Last updated: 2020-04-15
+ Last updated: 2021-02-15
</body>
</html>
diff --git a/macros/latex/contrib/bxpdfver/LICENSE b/macros/latex/contrib/bxpdfver/LICENSE
index 6464d05972..9e57162d4c 100644
--- a/macros/latex/contrib/bxpdfver/LICENSE
+++ b/macros/latex/contrib/bxpdfver/LICENSE
@@ -1,6 +1,6 @@
The MIT License
-Copyright (c) 2014-2020 Takayuki YATO (aka. "ZR")
+Copyright (c) 2014-2021 Takayuki YATO (aka. "ZR")
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
diff --git a/macros/latex/contrib/bxpdfver/README-ja.md b/macros/latex/contrib/bxpdfver/README-ja.md
index 78b35c6da6..5296c49a9c 100644
--- a/macros/latex/contrib/bxpdfver/README-ja.md
+++ b/macros/latex/contrib/bxpdfver/README-ja.md
@@ -45,7 +45,7 @@ bxpdfver パッケージ
+ PDF 出力ã®ã‚¨ãƒ³ã‚¸ãƒ³ã®å ´åˆã¯ã€æ—¢å®šã§é©åˆ‡ãªãƒ‰ãƒ©ã‚¤ãƒãŒé¸æŠžã•ã‚Œã‚‹
ã®ã§ãƒ‰ãƒ©ã‚¤ãƒã‚ªãƒ—ションã¯ä¸è¦ã§ã‚る。
+ `dvipdfmx`: dvipdfmx 用ã®ãƒ‰ãƒ©ã‚¤ãƒã‚’指定ã™ã‚‹ã€‚
- + `disabled`ï¼`nodvidriver`: å…¨ã¦ã®æ©Ÿèƒ½ã‚’無効化ã™ã‚‹ã€‚
+ + `nodvidriver`ï¼`disabled`: å…¨ã¦ã®æ©Ÿèƒ½ã‚’無効化ã™ã‚‹ã€‚
* `lenient`: サãƒãƒ¼ãƒˆã—ãªã„機能ã«å¯¾ã™ã‚‹ã‚¨ãƒ©ãƒ¼ã‚’警告ã«æ ¼ä¸‹ã’ã™ã‚‹ã€‚
* `new-dvipdfmx`: dvipdfmx ã«å¯¾ã™ã‚‹ãƒãƒ¼ã‚¸ãƒ§ãƒ³æ¤œæŸ»ã‚’çœç•¥ã—ã€å¸¸ã«æœ€æ–°ã®
ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã§ã‚ã‚‹ã¨ä»®å®šã™ã‚‹ã€‚
@@ -96,13 +96,15 @@ bxpdfver パッケージ
* 使用ä¸å¯èƒ½ãªæ©Ÿèƒ½ã‚’使ãŠã†ã¨è©¦ã¿ã‚‹ã¨ã‚¨ãƒ©ãƒ¼ãŒç™ºç”Ÿã™ã‚‹ã€‚
* `dvips` ç­‰ã®â€œå…¨ã対応ã—ã¦ã„ãªã„â€è‹¥å¹²ã®ãƒ‰ãƒ©ã‚¤ãƒã‚ªãƒ—ションをèªè­˜
ã™ã‚‹ã€‚ã“ã®å ´åˆã€å…¨ã¦ã®æ©Ÿèƒ½ã®å‘¼å‡ºã§ã‚¨ãƒ©ãƒ¼ãŒç™ºç”Ÿã™ã‚‹ã€‚
- * `disabled` ã¯ç‰¹æ®Šã§ã€ã“れを指定ã—ãŸå ´åˆã¯ã€ã©ã®æ©Ÿèƒ½ã®å‘¼å‡ºã§ã‚‚
+ * `nodvidriver` ã¯ç‰¹æ®Šã§ã€ã“れを指定ã—ãŸå ´åˆã¯ã€ã©ã®æ©Ÿèƒ½ã®å‘¼å‡ºã§ã‚‚
エラーã¯ç™ºç”Ÿã—ãªã„ãŒã€å…¨ã何ã®å‹•ä½œã‚‚è¡Œã‚ãªã„。
更新履歴
--------
+ * Version 0.5a ‹2021/02/14›
+ - æ–°ã—ã„版㮠hyperref ã®ãŸã‚ã®èª¿æ•´ã€‚
* Version 0.5 ‹2020/04/19›
- PDF ãƒãƒ¼ã‚¸ãƒ§ãƒ³ã®å€¤ã¨ã—㦠`2.0` をサãƒãƒ¼ãƒˆã™ã‚‹ã€‚
* Version 0.4 ‹2017/02/11›
diff --git a/macros/latex/contrib/bxpdfver/README.md b/macros/latex/contrib/bxpdfver/README.md
index ec17bff09a..594c9ed8dd 100644
--- a/macros/latex/contrib/bxpdfver/README.md
+++ b/macros/latex/contrib/bxpdfver/README.md
@@ -46,7 +46,7 @@ The available options are:
+ When using a PDF-output engine, you need not give driver options
since the appropriate one is auto-detected.
+ `dvipdfmx`: Uses dvipdfmx driver.
- + `disabled`/`nodvidriver`: Disables all functions of the package.
+ + `nodvidriver`/`disabled`: Disables all functions of the package.
* `lenient`: Turns the errors for unsupported features into warnings.
Note that the options `compress` and `objcompress` mean that this
@@ -95,12 +95,14 @@ More notices:
* The package recognizes some “unsupported†driver options such as
`dvips`; when such drivers are used, use of any feature will cause
an error.
- * When `disabled` is used, use of any feature will do nothing (nor
+ * When `nodvidriver` is used, use of any feature will do nothing (nor
issue an error).
REVISION HISTORY
----------------
+ * Version 0.5a ‹2021/02/14›
+ - Adjustment for the new version of hyperref.
* Version 0.5 ‹2020/04/19›
- Suuport PDF version value `2.0`.
* Version 0.4 ‹2017/02/11›
diff --git a/macros/latex/contrib/bxpdfver/bxpdfver.sty b/macros/latex/contrib/bxpdfver/bxpdfver.sty
index ad472da1b4..826eef3efd 100644
--- a/macros/latex/contrib/bxpdfver/bxpdfver.sty
+++ b/macros/latex/contrib/bxpdfver/bxpdfver.sty
@@ -1,7 +1,7 @@
%%
%% This is file 'bxpdfver.sty'.
%%
-%% Copyright (c) 2014-2020 Takayuki YATO (aka. "ZR")
+%% Copyright (c) 2014-2021 Takayuki YATO (aka. "ZR")
%% GitHub: https://github.com/zr-tex8r
%% Twitter: @zr_tex8r
%%
@@ -10,7 +10,7 @@
%% package declaration
\NeedsTeXFormat{LaTeX2e}
-\ProvidesPackage{bxpdfver}[2020/04/19 v0.5]
+\ProvidesPackage{bxpdfver}[2021/02/14 v0.5a]
\def\bxpv@pkgname{bxpdfver}
%% preparations
@@ -140,6 +140,10 @@
\else \expandafter\@gobble
\fi{\let\ifbxpv@defined\iffalse}
+%% \ifbxpv@old@hook@system
+\newif\ifbxpv@old@hook@system
+\@ifl@t@r\fmtversion{2020/10/01}{}{\bxpv@old@hook@systemtrue}
+
%--------------------------------------- internal dispatchers
%% \bxpv@suppress@compress
@@ -185,6 +189,9 @@
\@onlypreamble\bxpv@begin@document@hook
\let\bxpv@begin@document@hook\@empty
\AtBeginDocument{\bxpv@begin@document@hook}
+\g@addto@macro\bxpv@begin@document@hook{%
+ \bxpv@check@hyperref % must come first
+}
%% 'disabled' driver
\if 0\bxpv@drv@type
@@ -363,6 +370,13 @@
\global\bxpdfverObjCompressionSuppressedtrue
}
+% Note: Priority against hyperref's 'pdfversion' parameter
+% The existing behaviors on PDF mode should be preserved:
+% - If hyperref is loaded but 'pdfversion' is not given, then
+% bxpdfver's setting (explicit request) should win.
+% - If hyperref is loaded and 'pdfversion' is given, then
+% hyperref should win (with a warning).
+
%%<*> \setpdfversion{<version>}
\@onlypreamble\setpdfversion
\newcommand*\setpdfversion[1]{%
@@ -570,11 +584,37 @@
\expandafter\bxpv@tmpa\bxpv@tmpb\bxpv@end
\endgroup
+%% \bxpv@check@hyperref
+\def\bxpv@check@hyperref{%
+ \bxpv@oktrue
+ \ifx\bxpdfverMinorVersion\relax\else
+ \@ifpackageloaded{hyperref}{%
+ \expandafter\ifx\csname ifHy@setpdfversion\expandafter
+ \endcsname\csname iftrue\endcsname
+ \bxpv@okfalse
+ \fi
+ }{}%
+ \fi
+ \ifbxpv@ok\else
+ \PackageWarningNoLine\bxpv@pkgname
+ {Both hyperref and this package try to set the PDF\MessageBreak
+ version; this can cause problems. This package's\MessageBreak
+ setting will be withdrawn for now, but you must\MessageBreak
+ fix your source to avoid it}%
+ % Note: In PDF mode, the real parameters will be overwritten
+ % later by hyperref.
+ \let\bxpdfverMinorVersion\relax
+ \let\bxpdfverMajorVersion\relax
+ \fi
+}
+
%--------------------------------------- dvipdfmx something
\ifx d\bxpv@drv@type % 'dvipdfmx'
%% required packages
+\ifbxpv@old@hook@system
\RequirePackage{atbegshi}[2007/04/19]% v1.2
+\fi
%% variables
\let\bxpv@C@value\relax
@@ -604,16 +644,12 @@
\fi
\ifbxpv@ok
\ifx\bxpdfverMinorVersion\relax\else
- \g@addto@macro\bxpv@page@specials{%
- \special{pdf:minorversion \the\bxpdfverMinorVersion}%
- }%
+ \bxpv@put@special{pdf:minorversion \the\bxpdfverMinorVersion}%
\fi
\ifx\bxpdfverMinorVersion\relax\else
\bxpv@cache@fetch{dvipdfmxver}%
\ifnum\bxpv@cache@@dvipdfmxver<20180217 \else
- \g@addto@macro\bxpv@page@specials{%
- \special{pdf:majorversion \the\bxpdfverMajorVersion}%
- }%
+ \bxpv@put@special{pdf:majorversion \the\bxpdfverMajorVersion}%
\fi
\fi
\fi
@@ -623,9 +659,7 @@
{Cannot suppress compression}%
\ifbxpv@ok
% New dvipdfmx supports this special.
- \g@addto@macro\bxpv@page@specials{%
- \special{dvipdfmx:config z 0}%
- }%
+ \bxpv@put@special{dvipdfmx:config z 0}%
\fi
\fi
% object compression
@@ -673,6 +707,7 @@
% special added to every page
\ifx\bxpv@page@specials\@empty\else
\edef\bxpv@page@specials{\bxpv@page@specials}%
+ \ifbxpv@old@hook@system
\AtBeginShipout{%
\setbox\AtBeginShipoutBox=\vbox{%
\baselineskip\z@skip\lineskip\z@skip\lineskiplimit\z@
@@ -680,6 +715,10 @@
\copy\AtBeginShipoutBox
}%
}%
+ \else
+ \AddToHook{shipout/foreground}{%
+ \hb@xt@\z@{\bxpv@page@specials\hss}}%
+ \fi
\fi
}
diff --git a/macros/latex/contrib/changes/changes.drv b/macros/latex/contrib/changes/changes.drv
index dc9bc65432..22ee08eb90 100644
--- a/macros/latex/contrib/changes/changes.drv
+++ b/macros/latex/contrib/changes/changes.drv
@@ -259,11 +259,11 @@
% document title
\ifENGLISH
\title{The \docpackage{changes}-package}
- \subtitle{Manual change markup --- version 4.0.0}
+ \subtitle{Manual change markup --- version 4.0.1}
\fi
\ifGERMAN
\title{Das \docpackage{changes}-Paket}
- \subtitle{Manuelle Änderungsmarkierung -- Version 4.0.0}
+ \subtitle{Manuelle Änderungsmarkierung -- Version 4.0.1}
\fi
\author{Ekkart Kleinod}
diff --git a/macros/latex/contrib/changes/changes.dtx b/macros/latex/contrib/changes/changes.dtx
index 8206f28c5e..c5f18a6d61 100644
--- a/macros/latex/contrib/changes/changes.dtx
+++ b/macros/latex/contrib/changes/changes.dtx
@@ -1,4 +1,4 @@
-% \CheckSum{1258}
+% \CheckSum{1262}
%
% \iffalse meta-comment
%
@@ -63,6 +63,7 @@
% \changes{v3.2.1}{2019/11/17}{removed obsolete package pdfcolmk}
% \changes{v3.2.2}{2020/06/16}{Merge changes script (pyMergeChanges): Correctly handle multiple different macros in the same line}
% \changes{v4.0.0}{2021/01/28}{new: convenience macros, setlocextension, option commandnameprefix, defaultcolor; removed: xifthen, remark}
+% \changes{v4.0.1}{2021/02/14}{bugfix: Command do undefined}
% \GetFileInfo{changes.dtx}
% \RecordChanges
%
@@ -103,8 +104,8 @@
% \begin{macrocode}
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{changes}
-[2021/01/28 v4.0.0 changes package]
-\typeout{*** changes package 2021/01/28 v4.0.0 ***}
+[2021/02/14 v4.0.1 changes package]
+\typeout{*** changes package 2021/02/14 v4.0.1 ***}
% \end{macrocode}
%
% Package \docpackage{xkeyval} provides options with key-value-pairs.
@@ -153,7 +154,10 @@
% to work if \#2 is a macro, so I used the code below.
%
% Thanks to Ulrike Fischer for the fix of the macro call of \doccommand{dopsvlist} with \doccommand{expandafter}.
+%
+% To make sure \doccommand{do} can be renewed, we define it first, some macros, such as \doccommand{chapter} undefine it.
% \begin{macrocode}
+ \def\do{}%
\renewcommand*{\do}[1]{%
\listadd{\Changes@tmplist}{##1}%
}%
@@ -1699,8 +1703,11 @@
}\\%
% \end{macrocode}
% Compute the relevant sum of changes in order to print "no changes" correctly.
+%
+% To make sure \doccommand{do} can be renewed, we define it first, some macros, such as \doccommand{chapter} undefine it.
% \begin{macrocode}
\numdef{\Changes@InSum}{0}%
+ \def\do{}%
\renewcommand*{\do}[1]{%
\numdef{\Changes@InSum}{\Changes@InSum + \csuse{Changes@In########1}}%
}%
diff --git a/macros/latex/contrib/changes/changes.english.pdf b/macros/latex/contrib/changes/changes.english.pdf
index 6ba651b498..67f536d250 100644
--- a/macros/latex/contrib/changes/changes.english.pdf
+++ b/macros/latex/contrib/changes/changes.english.pdf
Binary files differ
diff --git a/macros/latex/contrib/changes/changes.english.withcode.pdf b/macros/latex/contrib/changes/changes.english.withcode.pdf
index b2a851bc44..d5436b7cb7 100644
--- a/macros/latex/contrib/changes/changes.english.withcode.pdf
+++ b/macros/latex/contrib/changes/changes.english.withcode.pdf
Binary files differ
diff --git a/macros/latex/contrib/changes/changes.ins b/macros/latex/contrib/changes/changes.ins
index f29e4e4417..489d804e35 100644
--- a/macros/latex/contrib/changes/changes.ins
+++ b/macros/latex/contrib/changes/changes.ins
@@ -1,6 +1,6 @@
%% This file creates the needed files from changes.dtx
%%
-\def\filedate{2021/01/28}
+\def\filedate{2021/02/14}
\def\batchfile{changes.ins}
\input docstrip
diff --git a/macros/latex/contrib/changes/changes.ngerman.pdf b/macros/latex/contrib/changes/changes.ngerman.pdf
index fbd71eef6a..4b74cae5ff 100644
--- a/macros/latex/contrib/changes/changes.ngerman.pdf
+++ b/macros/latex/contrib/changes/changes.ngerman.pdf
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index 595a62e67a..59fd765bcd 100644
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index 2b0ab941fd..7d89a89191 100644
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diff --git a/macros/latex/contrib/hindawi-latex-template/README b/macros/latex/contrib/hindawi-latex-template/README
new file mode 100644
index 0000000000..e56ebe944a
--- /dev/null
+++ b/macros/latex/contrib/hindawi-latex-template/README
@@ -0,0 +1,43 @@
+License:
+-------
+ This work in this bundle are distributed under the
+ LaTeX Project Public License (LPPL) ( http://www.latex-project.org/ )
+ version 1.3, and may be freely used, distributed and modified.
+ A copy of the LPPL, version 1.3, is included in the base
+ LaTeX documentation of all distributions of LaTeX released
+ 2003/12/01 or later.
+
+About the bundle:
+----------------
+hindawi_latex_template.zip is a small bundle which contains LaTeX
+templates for authors of the Hindawi journals. The authors can use
+this template for formatting their research articles for the
+submissions.
+
+This package has been created and maintained by the Typeset team.
+Typeset.io is a platform to write and submit better research papers. It
+takes the best part of online LaTeX Editor and Ms-Word Editor
+(https://typeset.io/t/latex/).
+
+The latest version of the package can be found at CTAN.
+
+https://ctan.org/tex-archive/macros/latex/contrib/hindawi_latex_template
+
+Files:
+-----
+
+ README - This file.
+ hindawi_latex_template.tex - LaTeX template
+ hindawi_latex_template.pdf - PDF of the template
+ hindawi_bib_style.bst - BibTeX style file
+
+Version history:
+---------------
+ Version: 1.0 [20210209]
+
+Contact:
+-------
+Bugs and queries may be reported to: shanukus98765@gmail.com
+
+The author is a contributor at Typeset.io (a pesudo LaTeX compiler +
+MS-Word alternative)
diff --git a/macros/latex/contrib/hindawi-latex-template/hindawi_bib_style.bst b/macros/latex/contrib/hindawi-latex-template/hindawi_bib_style.bst
new file mode 100644
index 0000000000..c9dd3faa6a
--- /dev/null
+++ b/macros/latex/contrib/hindawi-latex-template/hindawi_bib_style.bst
@@ -0,0 +1,982 @@
+%% Please refer README file for more details about this document
+%
+
+ENTRY
+ { address
+ author
+ booktitle
+ chapter
+ edition
+ editor
+ howpublished
+ institution
+ journal
+ key
+ month
+ note
+ number
+ organization
+ pages
+ publisher
+ school
+ series
+ title
+ type
+ url
+ volume
+ year
+ }
+ {}
+ { label }
+
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+ #1 'mid.sentence :=
+ #2 'after.sentence :=
+ #3 'after.block :=
+}
+
+STRINGS { s t }
+
+FUNCTION {output.nonnull}
+{ 's :=
+ output.state mid.sentence =
+ { ", " * write$ }
+ { output.state after.block =
+ { add.period$ write$
+ newline$
+ "\newblock " write$
+ }
+ { output.state before.all =
+ 'write$
+ { add.period$ " " * write$ }
+ if$
+ }
+ if$
+ mid.sentence 'output.state :=
+ }
+ if$
+ s
+}
+
+FUNCTION {output}
+{ duplicate$ empty$
+ 'pop$
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.check}
+{ 't :=
+ duplicate$ empty$
+ { pop$ "empty " t * " in " * cite$ * warning$ }
+ 'output.nonnull
+ if$
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+ "\bibitem{" write$
+ cite$ write$
+ "}" write$
+ newline$
+ ""
+ before.all 'output.state :=
+}
+
+FUNCTION {fin.entry}
+{ add.period$
+ write$
+ newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+ 'skip$
+ { after.block 'output.state := }
+ if$
+}
+
+FUNCTION {new.sentence}
+{ output.state after.block =
+ 'skip$
+ { output.state before.all =
+ 'skip$
+ { after.sentence 'output.state := }
+ if$
+ }
+ if$
+}
+
+FUNCTION {not}
+{ { #0 }
+ { #1 }
+ if$
+}
+
+FUNCTION {and}
+{ 'skip$
+ { pop$ #0 }
+ if$
+}
+
+FUNCTION {or}
+{ { pop$ #1 }
+ 'skip$
+ if$
+}
+
+FUNCTION {new.block.checka}
+{ empty$
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.block.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.block
+ if$
+}
+
+FUNCTION {new.sentence.checka}
+{ empty$
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {new.sentence.checkb}
+{ empty$
+ swap$ empty$
+ and
+ 'skip$
+ 'new.sentence
+ if$
+}
+
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+ { pop$ "" }
+ 'skip$
+ if$
+}
+
+FUNCTION {emphasize}
+{ duplicate$ empty$
+ { pop$ "" }
+ { "{\em " swap$ * "}" * }
+ if$
+}
+
+FUNCTION {format.url}
+{ url duplicate$ empty$
+ { pop$ "" }
+ { "[Online]. Available: \url{" swap$ * "}" * }
+ if$
+}
+
+INTEGERS { nameptr namesleft numnames }
+
+FUNCTION {bbl.etal}
+{ "et~al." }
+
+
+FUNCTION {format.names}
+{ 's :=
+ #1 'nameptr :=
+ s num.names$ 'numnames :=
+ numnames 'namesleft :=
+ { namesleft #0 > }
+ { s nameptr "{ff~}{vv~}{ll}{, jj}" format.name$ 't :=
+ nameptr #1 >
+ {
+ nameptr #3
+ #1 + =
+ numnames #5
+ > and
+ { "others" 't :=
+ #1 'namesleft := }
+ 'skip$
+ if$
+ namesleft #1 >
+ { ", " * t * }
+ {
+ " " *
+ s nameptr "{ll}" format.name$ duplicate$ "others" =
+ { 't := }
+ { pop$ }
+ if$
+ t "others" =
+ {
+ " " * bbl.etal *
+ }
+ { " and " * t * }
+ if$
+ }
+ if$
+ }
+ 't
+ if$
+ nameptr #1 + 'nameptr :=
+ namesleft #1 - 'namesleft :=
+ }
+ while$
+}
+
+FUNCTION {format.authors}
+{ author empty$
+ { "" }
+ { author format.names }
+ if$
+}
+
+FUNCTION {format.editors}
+{ editor empty$
+ { "" }
+ { editor format.names
+ editor num.names$ #1 >
+ { ", editors" * }
+ { ", editor" * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.title}
+{ title empty$
+ { "" }
+ { "``" title "t" change.case$ * "''" * }
+ if$
+}
+
+FUNCTION {n.dashify}
+{ 't :=
+ ""
+ { t empty$ not }
+ { t #1 #1 substring$ "-" =
+ { t #1 #2 substring$ "--" = not
+ { "--" *
+ t #2 global.max$ substring$ 't :=
+ }
+ { { t #1 #1 substring$ "-" = }
+ { "-" *
+ t #2 global.max$ substring$ 't :=
+ }
+ while$
+ }
+ if$
+ }
+ { t #1 #1 substring$ *
+ t #2 global.max$ substring$ 't :=
+ }
+ if$
+ }
+ while$
+}
+
+FUNCTION {format.date}
+{ year empty$
+ { month empty$
+ { "" }
+ { "there's a month but no year in " cite$ * warning$
+ month
+ }
+ if$
+ }
+ { month empty$
+ 'year
+ { month " " * year * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.btitle}
+{ title emphasize
+}
+
+FUNCTION {tie.or.space.connect}
+{ duplicate$ text.length$ #3 <
+ { "~" }
+ { " " }
+ if$
+ swap$ * *
+}
+
+FUNCTION {either.or.check}
+{ empty$
+ 'pop$
+ { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+ if$
+}
+
+FUNCTION {format.bvolume}
+{ volume empty$
+ { "" }
+ { "volume" volume tie.or.space.connect
+ series empty$
+ 'skip$
+ { " of " * series emphasize * }
+ if$
+ "volume and number" number either.or.check
+ }
+ if$
+}
+
+FUNCTION {format.number.series}
+{ volume empty$
+ { number empty$
+ { series field.or.null }
+ { output.state mid.sentence =
+ { "number" }
+ { "Number" }
+ if$
+ number tie.or.space.connect
+ series empty$
+ { "there's a number but no series in " cite$ * warning$ }
+ { " in " * series * }
+ if$
+ }
+ if$
+ }
+ { "" }
+ if$
+}
+
+FUNCTION {format.edition}
+{ edition empty$
+ { "" }
+ { output.state mid.sentence =
+ { edition "l" change.case$ " edition" * }
+ { edition "t" change.case$ " edition" * }
+ if$
+ }
+ if$
+}
+
+INTEGERS { multiresult }
+
+FUNCTION {multi.page.check}
+{ 't :=
+ #0 'multiresult :=
+ { multiresult not
+ t empty$ not
+ and
+ }
+ { t #1 #1 substring$
+ duplicate$ "-" =
+ swap$ duplicate$ "," =
+ swap$ "+" =
+ or or
+ { #1 'multiresult := }
+ { t #2 global.max$ substring$ 't := }
+ if$
+ }
+ while$
+ multiresult
+}
+
+FUNCTION {format.pages}
+{ pages empty$
+ { "" }
+ { pages multi.page.check
+ { "pages" pages n.dashify tie.or.space.connect }
+ { "page" pages tie.or.space.connect }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.vol.num.pages}
+{ "vol. " volume * field.or.null
+ number empty$
+ 'skip$
+ { ", no. " number * "" * *
+ volume empty$
+ { "there's a number but no volume in " cite$ * warning$ }
+ 'skip$
+ if$
+ }
+ if$
+ pages empty$
+ 'skip$
+ { duplicate$ empty$
+ { pop$ format.pages }
+ { ", " * pages n.dashify * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+ 'format.pages
+ { type empty$
+ { "chapter" }
+ { type "l" change.case$ }
+ if$
+ chapter tie.or.space.connect
+ pages empty$
+ 'skip$
+ { ", " * format.pages * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.in.ed.booktitle}
+{ booktitle empty$
+ { "" }
+ { editor empty$
+ { "In " booktitle emphasize * }
+ { "In " format.editors * ", " * booktitle emphasize * }
+ if$
+ }
+ if$
+}
+
+FUNCTION {empty.misc.check}
+{ author empty$ title empty$ howpublished empty$
+ month empty$ year empty$ note empty$
+ and and and and and
+ { "all relevant fields are empty in " cite$ * warning$ }
+ 'skip$
+ if$
+}
+
+FUNCTION {format.thesis.type}
+{ type empty$
+ 'skip$
+ { pop$
+ type "t" change.case$
+ }
+ if$
+}
+
+FUNCTION {format.tr.number}
+{ type empty$
+ { "Technical Report" }
+ 'type
+ if$
+ number empty$
+ { "t" change.case$ }
+ { number tie.or.space.connect }
+ if$
+}
+
+FUNCTION {format.article.crossref}
+{ key empty$
+ { journal empty$
+ { "need key or journal for " cite$ * " to crossref " * crossref *
+ warning$
+ ""
+ }
+ { "In {\em " journal * "\/}" * }
+ if$
+ }
+ { "In " key * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.crossref.editor}
+{ editor #1 "{vv~}{ll}" format.name$
+ editor num.names$ duplicate$
+ #2 >
+ { pop$ " et~al." * }
+ { #2 <
+ 'skip$
+ { editor #2 "{ff }{vv }{ll}{ jj}" format.name$ "others" =
+ { " et~al." * }
+ { " and " * editor #2 "{vv~}{ll}" format.name$ * }
+ if$
+ }
+ if$
+ }
+ if$
+}
+
+FUNCTION {format.book.crossref}
+{ volume empty$
+ { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+ "In "
+ }
+ { "Volume" volume tie.or.space.connect
+ " of " *
+ }
+ if$
+ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { series empty$
+ { "need editor, key, or series for " cite$ * " to crossref " *
+ crossref * warning$
+ "" *
+ }
+ { "{\em " * series * "\/}" * }
+ if$
+ }
+ { key * }
+ if$
+ }
+ { format.crossref.editor * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {format.incoll.inproc.crossref}
+{ editor empty$
+ editor field.or.null author field.or.null =
+ or
+ { key empty$
+ { booktitle empty$
+ { "need editor, key, or booktitle for " cite$ * " to crossref " *
+ crossref * warning$
+ ""
+ }
+ { "In {\em " booktitle * "\/}" * }
+ if$
+ }
+ { "In " key * }
+ if$
+ }
+ { "In " format.crossref.editor * }
+ if$
+ " \cite{" * crossref * "}" *
+}
+
+FUNCTION {article}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title "title" output.check
+ new.block
+ crossref missing$
+ { journal emphasize "journal" output.check
+ format.vol.num.pages output
+ format.date "year" output.check
+ }
+ { format.article.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {book}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ new.block
+ format.btitle "title" output.check
+ crossref missing$
+ { format.bvolume output
+ new.block
+ format.number.series output
+ new.sentence
+ publisher "publisher" output.check
+ address output
+ }
+ { new.block
+ format.book.crossref output.nonnull
+ }
+ if$
+ format.edition output
+ format.date "year" output.check
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {booklet}
+{ output.bibitem
+ format.authors output
+ new.block
+ format.title "title" output.check
+ howpublished address new.block.checkb
+ howpublished output
+ address output
+ format.date output
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+ author empty$
+ { format.editors "author and editor" output.check }
+ { format.authors output.nonnull
+ crossref missing$
+ { "author and editor" editor either.or.check }
+ 'skip$
+ if$
+ }
+ if$
+ new.block
+ format.btitle "title" output.check
+ crossref missing$
+ { format.bvolume output
+ format.chapter.pages "chapter and pages" output.check
+ new.block
+ format.number.series output
+ new.sentence
+ publisher "publisher" output.check
+ address output
+ }
+ { format.chapter.pages "chapter and pages" output.check
+ new.block
+ format.book.crossref output.nonnull
+ }
+ if$
+ format.edition output
+ format.date "year" output.check
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title "title" output.check
+ new.block
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.bvolume output
+ format.number.series output
+ format.chapter.pages output
+ new.sentence
+ publisher "publisher" output.check
+ address output
+ format.edition output
+ format.date "year" output.check
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.chapter.pages output
+ }
+ if$
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {inproceedings}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title "title" output.check
+ new.block
+ crossref missing$
+ { format.in.ed.booktitle "booktitle" output.check
+ format.bvolume output
+ format.number.series output
+ format.pages output
+ address empty$
+ { organization publisher new.sentence.checkb
+ organization output
+ publisher output
+ format.date "year" output.check
+ }
+ { address output.nonnull
+ format.date "year" output.check
+ new.sentence
+ organization output
+ publisher output
+ }
+ if$
+ }
+ { format.incoll.inproc.crossref output.nonnull
+ format.pages output
+ }
+ if$
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {conference} { inproceedings }
+
+FUNCTION {manual}
+{ output.bibitem
+ author empty$
+ { organization empty$
+ 'skip$
+ { organization output.nonnull
+ address output
+ }
+ if$
+ }
+ { format.authors output.nonnull }
+ if$
+ new.block
+ format.btitle "title" output.check
+ author empty$
+ { organization empty$
+ { address new.block.checka
+ address output
+ }
+ 'skip$
+ if$
+ }
+ { organization address new.block.checkb
+ organization output
+ address output
+ }
+ if$
+ format.edition output
+ format.date output
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title "title" output.check
+ new.block
+ "Master's thesis" format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.date "year" output.check
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+ format.authors output
+ title howpublished new.block.checkb
+ format.title output
+ howpublished new.block.checka
+ howpublished output
+ format.date output
+ new.block
+ note output
+ format.url output
+ fin.entry
+ empty.misc.check
+}
+
+FUNCTION {phdthesis}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.btitle "title" output.check
+ new.block
+ "PhD thesis" format.thesis.type output.nonnull
+ school "school" output.check
+ address output
+ format.date "year" output.check
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+ editor empty$
+ { organization output }
+ { format.editors output.nonnull }
+ if$
+ new.block
+ format.btitle "title" output.check
+ format.bvolume output
+ format.number.series output
+ address empty$
+ { editor empty$
+ { publisher new.sentence.checka }
+ { organization publisher new.sentence.checkb
+ organization output
+ }
+ if$
+ publisher output
+ format.date "year" output.check
+ }
+ { address output.nonnull
+ format.date "year" output.check
+ new.sentence
+ editor empty$
+ 'skip$
+ { organization output }
+ if$
+ publisher output
+ }
+ if$
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title "title" output.check
+ new.block
+ format.tr.number output.nonnull
+ institution "institution" output.check
+ address output
+ format.date "year" output.check
+ new.block
+ note output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+ format.authors "author" output.check
+ new.block
+ format.title "title" output.check
+ new.block
+ note "note" output.check
+ format.date output
+ format.url output
+ fin.entry
+}
+
+FUNCTION {default.type} { misc }
+
+MACRO {jan} {"January"}
+
+MACRO {feb} {"February"}
+
+MACRO {mar} {"March"}
+
+MACRO {apr} {"April"}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"June"}
+
+MACRO {jul} {"July"}
+
+MACRO {aug} {"August"}
+
+MACRO {sep} {"September"}
+
+MACRO {oct} {"October"}
+
+MACRO {nov} {"November"}
+
+MACRO {dec} {"December"}
+
+MACRO {acmcs} {"ACM Computing Surveys"}
+
+MACRO {acta} {"Acta Informatica"}
+
+MACRO {cacm} {"Communications of the ACM"}
+
+MACRO {ibmjrd} {"IBM Journal of Research and Development"}
+
+MACRO {ibmsj} {"IBM Systems Journal"}
+
+MACRO {ieeese} {"IEEE Transactions on Software Engineering"}
+
+MACRO {ieeetc} {"IEEE Transactions on Computers"}
+
+MACRO {ieeetcad}
+ {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"}
+
+MACRO {ipl} {"Information Processing Letters"}
+
+MACRO {jacm} {"Journal of the ACM"}
+
+MACRO {jcss} {"Journal of Computer and System Sciences"}
+
+MACRO {scp} {"Science of Computer Programming"}
+
+MACRO {sicomp} {"SIAM Journal on Computing"}
+
+MACRO {tocs} {"ACM Transactions on Computer Systems"}
+
+MACRO {tods} {"ACM Transactions on Database Systems"}
+
+MACRO {tog} {"ACM Transactions on Graphics"}
+
+MACRO {toms} {"ACM Transactions on Mathematical Software"}
+
+MACRO {toois} {"ACM Transactions on Office Information Systems"}
+
+MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"}
+
+MACRO {tcs} {"Theoretical Computer Science"}
+
+READ
+
+STRINGS { longest.label }
+
+INTEGERS { number.label longest.label.width }
+
+FUNCTION {initialize.longest.label}
+{ "" 'longest.label :=
+ #1 'number.label :=
+ #0 'longest.label.width :=
+}
+
+FUNCTION {longest.label.pass}
+{ number.label int.to.str$ 'label :=
+ number.label #1 + 'number.label :=
+ label width$ longest.label.width >
+ { label 'longest.label :=
+ label width$ 'longest.label.width :=
+ }
+ 'skip$
+ if$
+}
+
+EXECUTE {initialize.longest.label}
+
+ITERATE {longest.label.pass}
+
+FUNCTION {begin.bib}
+{ preamble$ empty$
+ 'skip$
+ { preamble$ write$ newline$ }
+ if$
+ "\begin{thebibliography}{" longest.label * "}" * write$ newline$
+}
+
+EXECUTE {begin.bib}
+
+EXECUTE {init.state.consts}
+
+ITERATE {call.type$}
+
+FUNCTION {end.bib}
+{ newline$
+ "\end{thebibliography}" write$ newline$
+}
+
+EXECUTE {end.bib}
diff --git a/macros/latex/contrib/hindawi-latex-template/hindawi_latex_template.pdf b/macros/latex/contrib/hindawi-latex-template/hindawi_latex_template.pdf
new file mode 100644
index 0000000000..7979fd3dad
--- /dev/null
+++ b/macros/latex/contrib/hindawi-latex-template/hindawi_latex_template.pdf
Binary files differ
diff --git a/macros/latex/contrib/hindawi-latex-template/hindawi_latex_template.tex b/macros/latex/contrib/hindawi-latex-template/hindawi_latex_template.tex
new file mode 100644
index 0000000000..9b2ce6c809
--- /dev/null
+++ b/macros/latex/contrib/hindawi-latex-template/hindawi_latex_template.tex
@@ -0,0 +1,216 @@
+% Please refer README file for more details about this document
+
+\documentclass[twoside,twocolumn]{article}
+
+\usepackage{tabulary,graphicx,times,caption,fancyhdr,amsfonts,amssymb,amsbsy,latexsym,amsmath}
+\usepackage[utf8]{inputenc}
+\usepackage{url,multirow,morefloats,floatflt,cancel,tfrupee,textcomp,colortbl,xcolor,pifont}
+\usepackage[nointegrals]{wasysym}
+\urlstyle{rm}
+
+\makeatletter
+
+%Etal definition in references
+\usepackage{ifxetex}
+\ifxetex\else
+ \usepackage{dblfloatfix}
+\fi
+
+\@ifundefined{subparagraph}{
+\def\subparagraph{\@startsection{paragraph}{5}{2\parindent}{0ex plus 0.1ex minus 0.1ex}%
+{0ex}{\normalfont\small\itshape}}%
+}{}
+
+\def\URL#1#2{\@ifundefined{href}{#2}{\href{#1}{#2}}}
+
+%%For url break
+\def\UrlOrds{\do\*\do\-\do\~\do\'\do\"\do\-}%
+\g@addto@macro{\UrlBreaks}{\UrlOrds}
+
+\makeatother
+\def\floatpagefraction{0.8}
+\def\dblfloatpagefraction{0.8}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\usepackage[paperheight=11in,paperwidth=8.3in,margin=2.5cm,headsep=.7cm,top=2.5cm]{geometry}
+\usepackage[T1]{fontenc}
+\def\floatpagefraction{0.8}
+\widowpenalty 10000
+\clubpenalty 10000
+
+\renewenvironment{abstract}
+ {\trivlist\item[]\leftskip0pt\par\vskip4pt\noindent
+ \textbf{\abstractname}\mbox{\null}\\}
+ {\par\noindent\endtrivlist}
+
+\def\keywords#1{\par\medskip\par\noindent\textbf{Keywords}: #1\par}
+
+\linespread{1.13} \date{} \emergencystretch 8pt
+
+\captionsetup[figure]{labelfont=normal,skip=1.4pt,aboveskip=1pc}
+\captionsetup[table]{labelfont=normal,skip=1.4pt}
+
+\makeatletter
+\def\author#1{\gdef\@author{\hskip-\tabcolsep%
+ \parbox{\textwidth}{\raggedright\bfseries#1\\[1pc]}}}
+\def\address[#1]#2{\g@addto@macro\@author{\\\hskip-\tabcolsep\parbox{\textwidth}{\raggedright%
+ \normalsize\normalfont\textsuperscript{#1}#2}}}
+\let\addresslink\textsuperscript
+\def\correspondence#1{\g@addto@macro\@author{\\\hskip-\tabcolsep\parbox{\textwidth}{\raggedright%
+ \vspace*{10pt}\normalsize\normalfont~\\#1~\\[12pt]}}}
+\def\email#1{\g@addto@macro\@author{\\\hskip-\tabcolsep\parbox{\textwidth}{\raggedright%
+ \normalsize\normalfont Emails: #1}}}
+
+\def\title#1{\gdef\@title{\vspace*{-30pt}%
+ \raggedright\textbf{\@journaltitle}~\\%
+ \raggedright\bfseries\ifx\@articleType\@empty\vspace*{20pt}\else%
+ \vspace*{20pt}\@articleType\vspace*{20pt}\\\fi#1}}
+\let\@journaltitle\@empty \def\journaltitle#1{\gdef\@journaltitle{{\normalfont\itshape#1}}}
+\let\@articleType\@empty \def\articletype#1{\gdef\@articleType{{\normalfont\itshape#1}}}
+
+\let\@runningHead\@empty \def\RunningHead#1{\gdef\@runningHead{{\normalfont #1}}}
+
+\usepackage{fancyhdr}
+\fancypagestyle{headings}{\renewcommand{\headrulewidth}{0pt}\fancyhf{}
+ \fancyhead[R]{\itshape\@runningHead}
+ \fancyfoot[C]{\thepage}}
+\pagestyle{headings}
+
+\fancypagestyle{plain}{\renewcommand{\headrulewidth}{0pt}%
+ \fancyhf{}\fancyhead[R]{\LaTeX\ template for Hindawi journal articles}
+ \fancyfoot[C]{\thepage}}
+\makeatother
+
+\usepackage[%
+ numbers,sort&compress%
+ %authoryear
+ ]{natbib}
+
+\setcounter{secnumdepth}{0}
+\usepackage{float,xcolor}
+
+\journaltitle{<journal title>}
+\articletype{<article type>} % Research Article/Review Article/Clinical Study
+
+\begin{document}
+
+% Title of the document
+\title{Title goes here}
+
+% Author names
+\author{%
+ A. Author\addresslink{1,2},
+ B. Author\addresslink{2} and
+ C. Author\addresslink{3}
+ }
+
+% Affiliation
+\address[1]{Affiliation one}
+\address[2]{Affiliation two}
+\address[3]{Affiliation three}
+
+% Corresponding author details
+\correspondence{Correspondence should be addressed to
+ <corresponding author's name>: <corresponding author's email>}
+
+% Emails of authors
+\email{A@gmail.com (A. Author), B@gmail.com (B. Author)}%
+
+% Running Head
+\RunningHead{Running head}
+
+\maketitle
+
+% Abstract
+\begin{abstract}
+Abstract text goes here
+
+Abstract text goes here Abstract text goes here Abstract text goes here
+Abstract text goes here Abstract text goes here Abstract text goes
+here Abstract text goes here Abstract text goes here Abstract text goes
+here Abstract text goes here Abstract text goes here Abstract text
+goes here Abstract text goes here Abstract text goes here Abstract text
+goes here
+
+% Keywords - if any
+\keywords{Key1; Key2; Key3}
+\end{abstract}
+
+% First level heading
+\section{Introduction}
+Normal section goes here ...
+
+% Second level heading
+%\subsection{}
+
+% Third level heading
+%\subsubsection{}
+
+% Fourth level heading
+%\paragraph{}
+
+% Fourth level heading
+%\subparagraph{}
+
+% Numbered equation
+%\begin{eqnarray}
+%\end{eqnarray}
+
+% Table
+\begin{table}[!htbp]
+% Use starred version (\begin{table*} \end{table*}) for two column spanned table
+\caption{\label{tbl1}Caption of the table.}
+\centering
+\def\arraystretch{1.15} % To increase the row spacing
+\begin{tabular}{ll}
+\hline
+One & Two \\
+\hline
+Three & Four\\
+\hline
+\end{tabular}
+\end{table}
+
+% Figure
+\begin{figure}[!htbp]
+\centering
+% Uncomment below line to include image
+% \includegraphics{image1}% <image name>
+\caption{\label{fig1}Caption of the figure}
+\end{figure}
+
+% Itemized list
+%\begin{itemize}
+% \item
+% \item
+% \item
+%\end{itemize}
+
+% Numbered list
+%\begin{enumerate}
+ %\item
+ %\item
+ %\item
+%\end{enumerate}
+
+% Bibliography style - if using a .bib file
+ %\bibliographystyle{hindawi_bib_style}
+ %\bibliography{<bib file name>} % without .bib extension
+
+ %Or
+
+ \begin{thebibliography}{00}
+ \bibitem{b1}
+ J. Bruix and M. Sherman, ``Management of hepatocellular carcinoma:
+ an update'', \textit{Hepatology}, vol. 53, no. 3, pp. 1020--1022, 2011.
+
+ % \bibitem{b2}
+ % ...
+
+ % \bibitem{b3}
+ % ...
+
+ \end{thebibliography}
+
+\end{document}
diff --git a/macros/latex/contrib/numerica/README.txt b/macros/latex/contrib/numerica/README.txt
new file mode 100644
index 0000000000..e31e824f7c
--- /dev/null
+++ b/macros/latex/contrib/numerica/README.txt
@@ -0,0 +1,35 @@
+numerica: a package to numerically evaluate mathematical
+expressions in the LaTeX form in which they are typeset.
+
+Andrew Parsloe (ajparsloe@gmail.com) 05 February 2021
+
+This work may be distributed and/or modified under the conditions
+of the LaTeX Project Public License, either version 1.3c of this
+license or any later version; see
+http://www.latex-project.org/lppl.txt
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+This is version 1.0.0 of the numerica bundle
+The LaTeX3 bundles l3kernel and l3packages
+are required; also amsmath and mathtools.
+
+numerica-basics.pdf explains more.
+
+Manifest
+%%%%%%%%
+README.txt this document
+
+numerica.sty LaTeX .sty file
+numerica-plus.def optionally loaded files for use
+numerica-tables.def with numerica.sty
+numerica-lyx.def
+
+numerica-basics.pdf documentation for numerica.sty &
+ numerica-lyx.def
+numerica-plus.pdf documentation for numerica-plus.def
+numerica-tables.pdf documentation for numerica-tables.def
+
+numerica-basics.tex LaTeX documentation source files
+numerica-plus.tex
+numerica-tables.tex
diff --git a/macros/latex/contrib/numerica/numerica-basics.pdf b/macros/latex/contrib/numerica/numerica-basics.pdf
new file mode 100644
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diff --git a/macros/latex/contrib/numerica/numerica-basics.tex b/macros/latex/contrib/numerica/numerica-basics.tex
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@@ -0,0 +1,4743 @@
+%% LyX 2.4.0-alpha1 created this file. For more info, see https://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english,tableposition=top]{report}
+\usepackage{lmodern}
+\renewcommand{\sfdefault}{lmss}
+\renewcommand{\ttdefault}{lmtt}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage[utf8]{inputenc}
+\setcounter{secnumdepth}{3}
+\usepackage{color}
+\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{cprotect}
+\usepackage{wrapfig}
+\usepackage{booktabs}
+\usepackage{framed}
+\usepackage{url}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=2,
+ breaklinks=true,pdfborder={0 0 1},backref=section,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+\providecommand{\LyX}{\texorpdfstring{\ensureascii{%
+ L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}}{LyX}}
+\DeclareRobustCommand*{\lyxarrow}{%
+\@ifstar
+{\leavevmode\,$\triangleleft$\,\allowbreak}
+{\leavevmode\,$\triangleright$\,\allowbreak}}
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+\newenvironment{centred}%
+ {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}}
+\newenvironment{lyxcode}
+ {\par\begin{list}{}{
+ \setlength{\rightmargin}{\leftmargin}
+ \setlength{\listparindent}{0pt}% needed for AMS classes
+ \raggedright
+ \setlength{\itemsep}{0pt}
+ \setlength{\parsep}{0pt}
+ \normalfont\ttfamily}%
+ \item[]}
+ {\end{list}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+\usepackage{numerica}
+
+\newcommand\rel{\,\varrho\;}
+\DeclareMathOperator{\erf}{erf}
+\DeclareMathOperator{\gd}{gd}
+
+\reuse
+
+\usepackage{upquote}
+
+\makeatother
+
+\begin{document}
+\title{\texttt{numerica-basics}}
+\author{Andrew Parsloe\\
+(\url{ajparsloe@gmail.com})\\
+}
+\maketitle
+\begin{abstract}
+The \verb`numerica` package defines a command to wrap around a mathematical
+expression in its LaTeX form and, once values are assigned to variables,
+numerically evaluate it. The intent is to avoid the need to modify
+the LaTeX form of the expression being evaluated. For programs with
+a preview facility like LyX, or compile-as-you-go systems, interactive
+back-of-envelope calculations and numerical exploration are possible
+within the document being worked on. The package requires the bundles
+\verb`l3kernel` and \verb`l3packages`, and the \verb`amsmath` and
+\verb`mathtools` packages. Additional modules define commands to
+iterate and find fixed points of functions of a single variable, to
+find the zeros or extrema of such functions, to calculate the terms
+of recurrence relations, and to create multi-column tables of function
+values (which requires the \verb`booktabs` package).\\
+\\
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\paragraph*{Note:}
+\begin{itemize}
+\item {\normalsize This document applies to version 1.0.0 of }{\normalsize\texttt{numerica.sty}}{\normalsize .}{\small\par}
+\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required.}{\small\par}
+\item {\normalsize The package requires }{\normalsize\texttt{amsmath}}{\normalsize{}
+and }{\normalsize\texttt{mathtools}}{\normalsize .}{\small\par}
+\item {\normalsize I refer many times in this document (especially §\ref{sec:Argument-parsing})
+to }{\normalsize\emph{Handbook of Mathematical Functions}}{\normalsize ,
+edited by Milton Abramowitz and Irene A. Segun, Dover, 1965. This
+is abbreviated to }{\normalsize\emph{HMF}}{\normalsize , often followed
+by a number like 1.2.3 to locate the actual expression referenced.}{\small\par}
+\end{itemize}
+\end{shaded}%
+\end{minipage}
+\end{abstract}
+\begin{center}
+\tableofcontents{}
+\par\end{center}
+
+\chapter{Introduction}
+
+\texttt{numerica} is a \LaTeX{} package offering the ability to numerically
+evaluate mathematical expressions in the \LaTeX{} form in which they
+are typeset.\footnote{\texttt{numerica} evolved from the author's \texttt{calculyx} package
+that was designed for use with the document processor \LyX{} and available
+for download from a link on the \LyX{} wiki website (but not from CTAN). }
+
+There are a number of packages which can do calculations in \LaTeX ,\footnote{A simple search finds the venerable \texttt{calc} in the \LaTeX{} base,
+\texttt{calculator }(including an associated \texttt{calculus} package),
+\texttt{fltpoint}, \texttt{fp} (\emph{fixed} rather than floating
+point), \texttt{spreadtab} (using either \texttt{fp} or \texttt{l3fp}
+as its calculational engine) if you want simple spreadsheeting with
+your calculations, the elaborate \texttt{xint}, \texttt{pst-calculate}
+(a limited interface to \texttt{l3fp}), \texttt{l3fp} in the \LaTeX 3
+kernel, and \texttt{xfp}, the \LaTeX 3 interface to \texttt{l3fp}.
+Other packages include a calculational element but are restricted
+in their scope. (\texttt{longdivision} for instance is elegant, but
+limited only to long division.) } but those I am aware of all require the mathematical expressions
+they operate on to be changed to an appropriate syntax. Of these packages
+\texttt{xfp} comes closest to my objective with \texttt{numerica}.
+For instance, given a formula
+\begin{centred}
+\verb`\frac{\sin (3.5)}{2} + 2\cdot 10^{-3}`
+\end{centred}
+(in a math environment), this can be evaluated using \texttt{xfp}
+by transforming the expression to \verb`sin(3.5)/2 + 2e-3` and wrapping
+this in the command \verb`\fpeval`. In \texttt{numerica} you don't
+need to transform the formula, just wrap it in an \verb`\eval` command
+(for the acutal calculation see §\ref{subsec:introSimple-examples}):
+\begin{centred}
+\verb`\eval{ \frac{\sin (3.5)}{2} + 2\cdot 10^{-3} }`.
+\end{centred}
+\texttt{numerica}, like \texttt{xfp} and a number of other packages,
+uses \texttt{l3fp} (the \LaTeX 3 floating point module in \texttt{l3kernel})
+as its calculational engine. To some extent the main command, \verb`\nmcEvaluate`,
+short-name form \verb`\eval`, is a pre-processor to \texttt{l3fp},
+converting mathematical expressions written in the \LaTeX\texttt{
+}form in which they will be typeset into an `fp-ified' form that
+is digestible by \texttt{l3fp}. The aim is to make the command act
+as a wrapper around such formulas. Ideally, one should not have to
+make \emph{any} adjustment to them, although any text on Fourier series
+suggests that hope in full generality is delusional. Surprisingly
+often however it \emph{is} possible. We will see shortly that even
+complicated formulas like
+
+\[
+\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
+\]
+and
+\[
+\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
+\]
+
+\noindent can be evaluated `as is' (see below, §\ref{subsec:introReassurance}).
+There is no need to shift the position of the superscript $2$ on
+the sines, no need to parenthesize the arguments of $\sin$ and $\cos$,
+no need to insert asterisks to indicate multiplication, no need to
+change the \verb`\frac` and \verb`\tfrac`-s to slashes, \texttt{/},
+no need to delete the \verb`\left` and \verb`\right` that qualify
+the big parentheses (in the underlying \LaTeX ) in the second expression.
+Of course, if there are variables in an expression, as in these examples,
+they will need to be assigned values. And how the result of the evaluation
+is presented also requires specifying, but the aim is always: to evaluate
+mathematical expressions in \LaTeX{} with as little adjustment as possible
+to the form in which they are typeset.
+
+\texttt{numerica} is written in \texttt{expl3}, the programming language
+of the \LaTeX 3 project. It uses the \LaTeX 3 module \texttt{l3fp}
+(part of \texttt{l3kernel}) as its calculational engine. This enables
+floating point operations to 16 significant figures, with exponents
+ranging between $-10000$ and $+10000$. Many functions and operations
+are built-in to \texttt{l3fp} – arithmetic operations, trigonometric,
+exponential and logarithm functions, factorials, absolute value, max
+and min. Others have been constructed for \texttt{numerica }from \texttt{l3fp}
+ingredients – binomial coefficients, hyperbolic functions, sums and
+products – but to the user there should be no discernible difference.
+
+Associated modules provide for additional operations: iteration, finding
+zeros, recurrence relations, mathematical table building. Further
+modules are planned (e.g. calculus).
+
+\section{How to use \texttt{numerica}}
+
+The package is invoked in the usual way:\footnote{I use the angle-bracket notation to indicate optional user input.
+Of course what is input does not include the angle brackets.} put
+\begin{lyxcode}
+\textbackslash usepackage{[}<options>{]}\{numerica\}
+\end{lyxcode}
+\noindent in the \LaTeX{} preamble. \texttt{numerica} requires the
+\texttt{amsmath} and \texttt{mathtools} packages and loads these automatically.
+\texttt{numerica} will also accept use of some relational symbols
+from the \texttt{amssymb} package provided that package is loaded;
+see §\ref{subsec:evalBoolean-output}.
+
+\subsection{Packages and package options}
+
+\label{subsec:introPackagesOptions}Version 1.0.0 of \texttt{numerica}
+has three package options.
+\begin{description}
+\item [{\texttt{plus}}] By calling \texttt{numerica} with the \texttt{plus}
+package option,\\
+{\ttfamily{}%
+\noindent\begin{minipage}[t]{1\columnwidth}%
+~~~~\textbackslash usepackage{[}plus{]}\{numerica\}%
+\end{minipage}}\texttt{}~\\
+the file \texttt{numerica-plus.def }is loaded where a number of additional
+commands: \verb`\nmcIterate`, \verb`\nmcSolve`, \verb`\nmcRecur`
+are defined. These enable the iteration of functions of a single variable\footnote{At this stage!},
+including finding fixed points; the solving of equations of the form
+$f(x)=0$ (or the location of local maxima or minima); and the calculation
+of terms in recurrence relations (like the Fibonacci series or othogonal
+polynomials). See the associated document \texttt{numerica-plus.pdf}.
+\item [{\texttt{tables}}] By calling \texttt{numerica} with the \texttt{tables}
+package option\\
+{\ttfamily{}%
+\noindent\begin{minipage}[t]{1\columnwidth}%
+~~~~\textbackslash usepackage{[}tables{]}\{numerica\}%
+\end{minipage}}\texttt{}~\\
+the file \texttt{numerica-tables.def} is loaded with with the command
+\verb`\nmcTabulate` enabling the creation of multi-column tables
+of function values with a wide variety of formatting options (most
+of those employed in \emph{HMF} in fact). See the associated document
+\texttt{numerica-tables.pdf}.
+\item [{\texttt{lyx}}] By calling \texttt{numerica} with the \texttt{lyx}
+package option,\\
+{\ttfamily{}%
+\noindent\begin{minipage}[t]{1\columnwidth}%
+~~~~\textbackslash usepackage{[}lyx{]}\{numerica\}%
+\end{minipage}}\texttt{}~\\
+the file \texttt{numerica-lyx.def }is loaded with code enabling the
+full use of the \verb`\nmcReuse` command in the document processor
+\LyX{} (along with all other commands of the \texttt{numerica} package).
+Use of \texttt{numerica} in \LyX{} exploits the mini-\LaTeX{} runs of
+the instant preview facility of that program to give immediate feedback
+on calculations without requiring the whole document to be compiled.
+See Chapter~\ref{chap:LyX}.
+\end{description}
+More than one option can be used at a time by separating the options
+with a comma; e.g. \texttt{\textbackslash usepackage{[}plus,tables{]}\{numerica\}}.
+However, apart from Chapter~\ref{chap:LyX}, the present document
+focuses on \texttt{numerica} when called with no options: \texttt{\textbackslash usepackage\{numerica\}}.
+
+\subsection{Simple examples of use}
+
+\label{subsec:introSimple-examples}A simple example of use is provided
+by the document
+\begin{verbatim}
+ \documentclass{minimal}
+ \usepackage{numerica}
+ \begin{document}
+
+ \eval{$ mc^2 $}[m=70,c=299 792 458][8x]
+
+ \end{document}
+
+\end{verbatim}
+\noindent We have a formula between math delimiters: \verb`$ mc^2 $`.
+We have wrapped a command \verb`\eval` around the lot, added an optional
+argument in parentheses specifying numericaal values for the quantities
+\texttt{m} and \texttt{c}, and concluded it all with a trailing optional
+argument specifying that the result should be presented to $8$ places
+of decimals and in scientific notation (the \verb`x`). Running \texttt{pdflatex}
+on this document generates a pdf displaying
+\begin{centred}
+\eval{$ mc^2 $}[m=70,c=299 792 458][8x]
+\end{centred}
+\noindent where the formula ($mc^{2})$ is equated to the numerical
+value resulting from substituting the given values of $m$ and $c$.
+Those values are displayed in a list following the result. The calculation
+is presented to $8$ decimal places in scientific notation. (According
+to Einstein's famous equation $E=mc^{2}$ this is the enormous energy
+content, in joules, of what was once considered an average adult Caucasian
+male. Only a minute fraction is ever available.)
+
+A second example is provided by the formula in earlier remarks:
+\begin{verbatim}
+ \documentclass{minimal}
+ \usepackage{numerica}
+ \begin{document}
+
+ \eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}
+
+ \end{document}
+
+\end{verbatim}
+Running \texttt{pdflatex} on this document produces the result
+
+\eval{\[ \frac{\sin(3.5)}{2} + 2\cdot 10^{-3} \]}
+
+The \verb`\eval` command used in these examples is the main command
+of the \texttt{numerica} package. It is discussed in full in the next
+chapter, but first some preliminaries.
+
+\subsection{Display of the result}
+
+\label{subsec:introResultDisplay}In what follows I shall write things
+like (but generally more complicated than)
+\begin{centred}
+\verb`$ \eval{ 1+1 } $` $\Longrightarrow \eval{ 1 + 1 } $
+\end{centred}
+to mean: run \texttt{pdflatex} on a document containing \verb`\eval{1+1}`
+in the document body to generate a pdf containing the calculated result
+($2$ in this instance). In this case the \verb`\eval` command is
+used \emph{within} a math environment (delimited by the dollar signs).
+It is not limited to this behaviour. The command can also wrap \emph{around}
+the math delimiters (as we saw in the previous examples):
+\begin{centred}
+\verb`\eval{$ 1+1 $}` $\Longrightarrow$ \eval{$ 1+1 $}.
+\end{centred}
+As you can see, the display that results is different.
+\begin{itemize}
+\item When the \verb`\eval` command is used\emph{ within} a math environment,
+only the \emph{result,} followed possibly by the \emph{variable~=~value
+list} (see §\ref{subsec:evalVv-list}) is displayed.
+\end{itemize}
+Environments may include the various AMS environments as well as the
+standard \LaTeX{} inline ( \verb`$ $` ), \texttt{equation} ( \verb`\[ \]`
+) and \texttt{eqnarray} environments. For an example of \verb`\eval`
+within an \verb`align*` environment see §\ref{subsec:introExploring}
+below.
+\begin{itemize}
+\item When the \verb`\eval` command is wrapped\emph{ around} a math environment,
+the result is displayed in the form, \emph{formula = result} (followed
+possibly by the \emph{variable~=~value list}) within that environment,
+\begin{itemize}
+\item If the formula is long or contains many variables then it may be desirable
+to split the display over two lines; see §\ref{subsec:evalChanging-display-format}
+and §\ref{subsec:settings New-line-display},
+\end{itemize}
+\end{itemize}
+the whole presented as an inline expression if \verb`$` delimiters
+are used, or as a display-style expression otherwise. (See the $mc^{2}$
+example for an illustration.)
+
+It is not clear to me that wrapping \verb`\eval` \emph{around} the
+AMS environments, except for \texttt{multline}, makes much sense,
+although it can be done. Here is an example of \verb`\eval` wrapped
+around a \texttt{multline{*}} environment (the phantom is there so
+that the hanging $+$ sign spaces correctly),\medskip{}
+
+\begin{minipage}[t]{0.8\columnwidth}%
+\begin{verbatim}
+\eval{ \begin{multline*}
+ 1+2+3+4+5+6+7+8+9+10+\phantom{0}\\
+ 11+12+13+14+15+16+17+18+19
+ \end{multline*} }
+\end{verbatim}
+%
+\end{minipage}
+
+\medskip{}$\Longrightarrow$ %
+\begin{minipage}[t]{0.8\columnwidth}%
+\vspace{-2ex}\eval{ \begin{multline*}
+ 1+2+3+4+5+6+7+8+9+10+\phantom{0}\\ 11+12+13+14+15+16+17+18+19
+\end{multline*} }%
+\end{minipage}
+\begin{itemize}
+\item It is also possible to dispense with math delimiters entirely, neither
+wrapped within nor wrapped around the \verb`\eval` command, but in
+that case \texttt{numerica} acts as if \verb`\eval` had been used
+within \verb`\[` and \verb`\]` and displays the result accordingly.
+\end{itemize}
+
+\subsection{Exploring}
+
+\label{subsec:introExploring}When working on \texttt{numerica}'s
+predecessor package, I constantly tested it against known results
+to check for coding errors. One test was to ensure that
+\[
+\left(1+\frac{1}{n}\right)^{n}
+\]
+did indeed converge to the number $e$ as $n$ increased.\texttt{ }Let's
+do that here. Try first $n=10$:
+\begin{center}
+\verb`\eval{$ e-(1+1/n)^n $}[n=10][x]` $\Longrightarrow$ \eval{$ e-(1+1/n)^n $}[n=10][x].
+\par\end{center}
+
+\noindent (The default number of decimal places displayed is $6$.)
+The difference between $e$ and $(1+1/n)^{n}$ is about an eighth
+($0.125$) when $n=10$, which is encouraging but hardly decisive.
+The obvious thing to do is increase the value of $n$. I'll use an
+\verb`align*` environment to `prettify' the presentation of the
+results:
+\begin{verbatim}
+ \begin{align*}
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^5][*x],\\
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^6][*x],\\
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^7][*x],\\
+ e-(1+1/n)^{n} & =\eval{e-(1+1/n)^n}[n=1\times10^8][*x].
+ \end{align*}
+\end{verbatim}
+(most of which was written using copy and paste) which produces
+
+\begin{align*}
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{5}][x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{6}][*x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{7}][x],\\
+e-(1+1/n)^{n} & =\eval{e-(1+1/n)^{n}}[n=1\times10^{8}][x].
+\end{align*}
+Clearly $(1+1/n)^{n}$ converges to $e$, the difference between them
+being of order $1/n$, but that is not what catches the eye. There
+is an unanticipated regularity here. 1.35914? Double the number: \verb`$\eval{2\times 1.35914}[5]$`\texttt{
+}$\Longrightarrow\eval{2\times1.35914}[5]$ which is close enough
+to $e$ to suggest a relationship, namely,
+\[
+\lim_{n\to\infty}n\left(e-\left(1+\frac{1}{n}\right)^{n}\right)=\tfrac{1}{2}e.
+\]
+
+This was new to me. Is it true? From the familiar expansion of the
+logarithm
+
+\begin{align*}
+\ln\left(1+\frac{1}{n}\right)^{n} & =n\ln\left(1+\frac{1}{n}\right)\\
+ & =n\left(\frac{1}{n}-\frac{1}{2}\frac{1}{n^{2}}+\frac{1}{3}\frac{1}{n^{3}}-\ldots\right)\\
+ & =1-\frac{1}{2n}\left(1-\frac{2}{3}\frac{1}{n}+\frac{2}{4}\frac{1}{n^{2}}-\right)\\
+ & \equiv1-\frac{1}{2n}E_{n},
+\end{align*}
+say. Since $E_{n}$ is an alternating series and the magnitudes of
+the terms of the series tend to $0$ monotonically, $1>E_{n}>1-2/3n$.
+From this and the inequalities $1/(1-x)>e^{x}>1+x$ when $x<1$ it
+proved a straightforward matter to verify the proposed limit.
+
+\subsection{Reassurance}
+
+\label{subsec:introReassurance}In the course of some hobbyist investigations
+in plane hyperbolic geometry I derived the formula
+\[
+\Phi_{1}(m,n)=\cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)\frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi},
+\]
+for $m=2,3,\ldots$ and integral $n\ge2m+1$. A key concern was: when
+is $\Phi_{1}$ positive? After an embarrassingly laborious struggle,
+I managed to work this expression into the form
+\[
+\Phi_{2}(m,n)=\left(\frac{1-4\sin^{2}\tfrac{m}{3n}\pi}{2\sin^{2}\tfrac{m}{3n}\pi}\right)\sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi,
+\]
+in which the conditions for positivity are clear: with $n\ge2m+1$,
+so that $m\pi/3n<\pi/6$, the first factor is always positive, the
+second is positive for $m\ge2$, and the third is positive for $m\ge4$.
+All well and good, but given the struggle to derive $\Phi_{2}$, was
+I confident that $\Phi_{1}$ and $\Phi_{2}$ really\emph{ }are equal?
+It felt all too likely that I had made a mistake.
+
+The simplest way to check was to see if the two expressions gave the
+same numericaal answers for a number of $m,\thinspace n$ values.
+I wrote \verb`\eval{\[ \]}[m=2,n=5]` twice and between the delimiters
+pasted the already composed expressions for $\Phi_{1}$ and $\Phi_{2}$,
+namely:
+\begin{verbatim}
+ \eval{\[
+ \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+ \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \]}[m=2,n=5]
+ \eval{\[
+ \left(
+ \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \right)
+ \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+ \]}[m=2,n=5]
+\end{verbatim}
+I have added some formatting – indenting, line breaks – to make the
+formulas more readable for the present document but otherwise left
+them unaltered. The \verb`\eval` command can be used for even quite
+complicated expressions without needing to tinker with their \LaTeX{}
+form, but you may wish – as here – to adjust white space to clarify
+the component parts of the formula. Running \texttt{pdflatex} on these
+expressions, the results were
+
+\eval{\[
+ \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+ \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \]}[m=2,n=5]
+
+\eval{\[
+ \left(
+ \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \right)
+ \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+ \]}[m=2,n=5]
+
+\noindent which was reassuring. Doing it again but with different
+values of $m$ and $n$, again the results coincided:
+
+\eval{\[
+ \cos\tfrac{m}{n}\pi-(1-4\sin^{2}\tfrac{m}{3n}\pi)
+ \frac{\sin\tfrac{1}{n}\pi\sin\tfrac{m-1}{n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \]}[m=5,n=13]
+
+\eval{\[
+ \left(
+ \frac{1-4\sin^{2}\tfrac{m}{3n}\pi}
+ {2\sin^{2}\tfrac{m}{3n}\pi}
+ \right)
+ \sin\tfrac{2m-3}{3n}\pi\sin\tfrac{m-3}{3n}\pi
+ \]}[m=5,n=13]
+
+\noindent Thus reassured that there was \emph{not }an error in my
+laborious derivation of $\Phi_{2}$ from $\Phi_{1}$, it was not difficult
+to work back from $\Phi_{2}$ to $\Phi_{1}$ then reverse the argument
+to find a straightforward derivation.
+
+\chapter{\texttt{\textbackslash nmcEvaluate} (\texttt{\textbackslash eval)}}
+
+The main calculational command in \texttt{numerica} is \verb`\nmcEvaluate`.
+Unlike some other commands which are loaded optionally, \verb`\nmcEvaluate`
+is \emph{always} loaded, and therefore always available. Because \verb`\nmcEvaluate`
+would be tiresome to write too frequently,\texttt{ }particularly for
+back-of-envelope calculations, there is an equivalent short-name form,
+\verb`\eval`, used almost exclusively in the following. But note:
+wherever you see the command \verb`\eval`, you can substitute \verb`\nmcEvaluate`
+and obtain the same result.
+
+\verb`\eval` (like other short-name forms of other commands in the
+\texttt{numerica} suite) is defined using \verb`\ProvideDocumentCommand`
+from the \texttt{xparse} package. Hence if \verb`\eval` has already
+been defined in some other package already loaded, it will not be
+redefined by \texttt{numerica}. It will retain its meaning in the
+other package. Its consequent absence from \texttt{numerica} may be
+an irritant, but only that; \verb`\nmcEvaluate` is defined using
+\texttt{xparse}'s \verb`\DeclareDocumentCommand` which would override
+any (freakishly unlikely) previous definition of \verb`\nmcEvaluate`
+in another package and would therefore still be available.
+
+\section{Syntax of \texttt{\textbackslash nmcEvaluate (\textbackslash eval)}}
+
+There are five arguments to the \verb`\nmcEvaluate` (or \verb`\eval`)
+command, of which only one, the third, is mandatory. All others are
+optional. If all are deployed the command looks like
+\begin{centred}
+\noindent \verb`\eval*[settings]{expr.}[vv-list][num. format]`
+\end{centred}
+I discuss the various arguments in the referenced sections.
+\begin{enumerate}
+\item \verb`*` optional switch; if present ensures display of only the
+numerical result (suppresses display of the formula and vv-list);
+see §\ref{subsec:evalVvSuppresList}
+\item \verb`[settings]` optional comma-separated list of \emph{key=value
+}settings for this particular calculation; see §\ref{sec:settingsOption}
+\item \verb`{expr.}` the only mandatory argument; the mathematical expression/formula
+in \LaTeX{} form that is to be evaluated
+\item \verb`[vv-list]` optional comma-separated list of \emph{variable=value
+}items; see §\ref{subsec:evalVv-list}
+\item \verb`[num. format]` optional format specification for presentation
+of the numerical result (rounding, padding with zeros, scientific
+notation, boolean output); see~§\ref{subsec:evalRoundingEtc}
+\end{enumerate}
+Note that arguments 4 and 5 are both square-bracket delimited optional
+arguments. Should only one such argument be used, \texttt{numerica}
+determines which is intended by looking for an equals sign within
+the argument. Its presence indicates the argument is the vv-list;
+its absence indicates the argument is the number format specification.
+
+The vv-list and number-format specification are \emph{trailing} optional
+arguments. There is a possibility that the \verb`\eval` command could
+be followed by a square-bracketed mathematical expression which \texttt{numerica}
+might therefore confuse with one of its trailing arguments. Experience
+using \texttt{numerica} suggests that this will be a (very) rare occurrence
+and is easily prevented by inserting an empty brace pair (\verb`{}`)
+before the offending square-bracketed expression. Allowing spaces
+between the arguments enables complicated expressions and large vv-lists
+to be formatted, in the interests of clarity, with new lines and white
+space – without requiring the insertion of comment characters (\verb`%`).
+
+Recommended practice is to minimise the number of optional arguments
+used in \LaTeX{} commands by consolidating such arguments into a single
+\emph{key=value} list. Although \texttt{numerica} uses such an argument,
+the vv-list does not fit naturally into that scheme. And practice
+suggests that separating out the elements of the number format specification
+(rounding value, padding with zeros, scientific notation, boolean
+output) and placing them in a trailing argument feels natural for
+the kind of back-of-envelope calculations envisaged for \texttt{numerica}.
+
+\section{The variable=value list}
+
+\label{subsec:evalVv-list}To evaluate algebraic, trigonometric and
+other formulas that involve \emph{variables} we need to give those
+variables values. This is done in the \emph{variable=value list} –
+or \emph{vv-list} for short. This is the fourth argument of the \texttt{\textbackslash nmcEvaluate}
+command and is a square-bracket delimited optional argument (optional
+because an expression may depend only on constants and numbers).
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+I was sorely tempted to use parentheses to delimit this argument,
+since then both the placement and delimiters of the vv-list would
+anticipate the way it is displayed in the evaluated result (see the
+$mc^{2}$ example in §\ref{subsec:introSimple-examples} above). But
+there is good reason not to. Parentheses will often occur in expressions
+in the vv-list. With parentheses nested within parentheses it is all
+too easy to get a pairing wrong, which would cause a \LaTeX{} error
+and halt compilation. As it is, using the standard square bracket
+delimiters, unbalanced parentheses cause a \texttt{numerica} error
+(see §\ref{subsec:errorsMismatched-brackets}), which does not halt
+compilation. (Of course unbalanced \emph{square} brackets now will
+cause a \LaTeX{} error, but such brackets are used less often in mathematical
+expressions and are rarely nested within other square-bracketed expressions.)\end{shaded}%
+\end{minipage}
+
+\subsection{Variable names}
+
+\label{subsec:evalVariableNames}In mathematical practice, variable
+names are generally single letters of the Roman or Greek alphabets,
+sometimes also from other alphabets, in a variety of fonts, and often
+with subscripts or primes or other decorations. In \texttt{numerica}
+a variable name is \emph{what lies to the left of the equals sign
+in an item} of the vv-list. Thus variables can be multi-token affairs:
+$x',x'',x^{iv},x_{n},x'_{n},x''_{mn}$, $^{k}C_{n},var,\mathrm{var},Fred,\mathbf{Fred},\mathcal{FRED}\ldots$
+Although variable names start and end with non-space tokens, a variable
+name may contain spaces – for instance \verb`x x` should not cause
+a \verb`numerica` error, but such names are not part of mathematical
+practice. Usually, for the kind of back-of-envelope calculations envisaged
+for \verb`numerica`, and for ease of typing, most variables will
+be single letters from the Roman or Greek alphabets.
+
+Because equals signs and commas give structure to the vv-list, it
+should also be clear that a variable name should not contain a \emph{naked}
+equals sign or a \emph{naked} comma. They can be incorporated in a
+variable name but only when decently wrapped in braces, like \verb`R_{=}`
+displaying as $R_{=}$ or \verb`X_{,i}` displaying as $X_{,i}$.
+
+Note that $x$ and $\mathrm{x}$ will be treated by \verb`numerica`
+as \emph{different} variables since, in the underlying \LaTeX , one
+is \texttt{x} and the other \verb`\mathrm{x}`. Even names that look
+identical in the pdf may well be distinct in \LaTeX . This is true
+particularly of superscripts and subscripts: \verb`x_0` and \verb`x_{0}`
+appear identical in the pdf but in the underlying \LaTeX{} they are
+distinct, and will be treated as distinct variables by \verb`numerica`.
+
+Although multi-token variables are perfectly acceptable, \emph{internally}
+\verb`numerica` expects variables to be single tokens. Hence a necessary
+initial step for the package is to convert all multi-token variable
+names in the vv-list and the formula to single tokens. \verb`numerica`
+does this by turning the multi-token variable names into control sequences
+with names in the sequence \verb`\_nmca`, \verb`\_nmcb`, \verb`\_nmcc`,
+etc., then searches through the vv-list and the formula for every
+occurrence of the multi-token names and replaces them with the relevant
+control sequences. It does this in order of decreasing size of name,
+working from the names that contain most tokens down to names containing
+only two tokens.
+
+The conversion process uses computer resources. Even if there are
+no multi-token variables present, \verb`numerica` still needs to
+check that this is so – unless the user alerts the program to the
+fact. This can be done by making a brief entry \texttt{xx=0 }in the
+settings option (the second optional argument of \verb`\nmcEvaluate`);
+see §\ref{subsec:settingsMultitokSwitch}. If the user never (or hardly
+ever) uses multi-token variables, then a more permanent solution is
+to create a file \verb`numerica.cfg`\texttt{ }with the line \texttt{multitoken-variables
+= false}; see §\ref{sec:settingsDefaults} for this.
+
+\subsection{The vv-list and its use}
+
+A vv-list is a comma-separated list where each item is of the form
+\emph{variable=value}. It might be something simple like
+\begin{lyxcode}
+{[}g=9.81,t=2{]}
+\end{lyxcode}
+or something more complicated like
+\begin{lyxcode}
+{[}V\_S=\textbackslash tfrac43\textbackslash pi~r\textasciicircum 3,V\_C=2\textbackslash pi~r\textasciicircum 2h,h=3/2,r=2{]}.
+\end{lyxcode}
+Spaces around the equals signs or the commas are stripped away during
+processing so that
+\begin{lyxcode}
+{[}g=9.81,t=2{]}~\textrm{and}~{[}~g~=~9.81~,~t~=~2{]}
+\end{lyxcode}
+are the \emph{same} variable=value list.
+
+\subsubsection{Evaluation from right to left}
+
+In these examples, with variables depending on other variables, there
+is an implication: that the list is evaluated \emph{from the right}.
+Recall how a function of a function is evaluated, say\emph{ }$y=f(g(h(x)))$.
+To evaluate $y$, first $x$ is assigned a value then $h(x)$ is calculated,
+then $g(h(x))$ then $f(g(h(x)))=y$. We work from right to left,
+from the innermost to the outermost element. Or consider an example
+like calculating the area of a triangle by means of the formula
+\[
+A=\sqrt{s(s-a)(s-b)(s-c)}.
+\]
+First we write the formula; then we state how $s$ depends on $a,b,c$,
+namely $s=\frac{1}{2}(a+b+c)$, then we give values to $a,b,c$. In
+\texttt{numerica} this is mirrored in the layout of the \verb`\eval`
+command:
+\begin{verbatim}
+ \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
+ [s=\tfrac12(a+b+c),a=3,b=4,c=5]
+\end{verbatim}
+The formula in a sense is the leftmost extension of the vv-list. The
+entire evaluation occurs from right to left.
+
+This means that the rightmost variable in the vv-list can depend only
+on (mathematical) constants and numbers – although it may be a complicated
+expression of those elements. Other variables in the vv-list can depend
+on variables \emph{to their right} but not to their left.
+
+\subsubsection{Expressions in the variable=value list}
+
+Suppose our expression is $\tfrac{4}{3}\pi r^{3}$, the volume $V_{S}$
+of a sphere in terms of its radius $r$, and we want to calculate
+the volume for different values of $r$ to get a sense of how rapidly
+volume increases with radius.
+\begin{centred}
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $` $\Longrightarrow$ $ V_S=\eval{ \tfrac43\pi r^3 }[r=1] $.
+\end{centred}
+Having set up this calculation it is now an easy matter to change
+the value of $r$ in the vv-list:
+\begin{centred}
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=1.5] $` $\Longrightarrow$
+$ V_S= \eval{ \tfrac43\pi r^3 }[r=1.5] $.
+
+\verb`$ V_S=\eval{ \tfrac43\pi r^3 }[r=2] $` $ \Longrightarrow $ $V_S= \eval{ \tfrac43\pi r^3 }[r=2] $.
+\end{centred}
+To compute the volume $V_{C}=\pi r^{2}h$ of a cylinder, we have two
+variables to assign values to:
+\begin{centred}
+\verb`$ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $` $\Longrightarrow$ $ V_C=\eval{ \pi r^2h }[h=4/3,r=1] $.
+\end{centred}
+Although values in the vv-list are generally either numbers or simple
+expressions (like \texttt{4/3}), that is not essential. A little more
+complicated is
+\begin{centred}
+\verb`$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $.
+\end{centred}
+where calculation of the volume of the cylinder has been split into
+two: first calculate the area $A_{C}$ of its circular base and then,
+once that has been effected, calculate the volume.
+
+A second example is provided by Brahmagupta's formula for the area
+of a triangle in terms of its semi-perimeter. In a triangle ABC, the
+sides are $a=3$, $b=4$ and $c=5$. (Of course we know this is a
+right-angled triangle with area $\tfrac{1}{2}ab=6$.) The semi-perimeter
+$s=\tfrac{1}{2}(a+b+c)$ and the area of ABC is \medskip{}
+\begin{verbatim}
+ \eval{$ \sqrt{s(s-a)(s-b)(s-c) $}
+ [s=\tfrac12(a+b+c),a=3,b=4,c=5]
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
+ [s=\tfrac12(a+b+c),a=3,b=4,c=5].
+
+\subsubsection{Constants}
+
+There are five constants built-in to \texttt{numerica}: \verb`\pi`,
+the ratio of circumference to diameter of a circle, \verb`e`, the
+base of natural logarithms, \verb`\gamma`, the limit of $\left(\sum_{1}^{N}1/n\right)-\ln N$
+as $N\to\infty$, \verb`\phi`, the golden ratio, equal to $\tfrac{1}{2}(1+\surd5)$,
+and the utilitarian constant \verb`\deg`, the number of radians in
+a degree.
+\begin{centred}
+\verb`\eval{$ \pi $}` $\Longrightarrow$ \eval{$ \pi $},
+
+\verb`\eval{$ e $}` $\Longrightarrow$ \eval{$ e $},
+
+\verb`\eval{$ \gamma $}` $\Longrightarrow$ \eval{$ \gamma $},
+
+\verb`\eval{$ \phi $}` $\Longrightarrow$ \eval{$ \phi $},
+
+\verb`\eval{$ \deg $}` $\Longrightarrow$ \eval{$ \deg $},
+\end{centred}
+so that \verb`\eval{$ 180\deg $}` $\Longrightarrow$ \eval{$ 180\deg $}
+(as it should).
+
+Let's combine two of these in a formula:
+\begin{centred}
+\verb`\eval{$ e^\pi-\pi^e $}` $\Longrightarrow$ \eval{$ e^\pi-\pi^e $},
+\end{centred}
+which is close-ish to $\tfrac{1}{4}e$: \verb`\eval{$ \tfrac14e $}`
+$\Longrightarrow$ \eval{$ \tfrac14e $}.
+
+\subsubsection{Use of \texttt{\textbackslash pi},\texttt{ e},\texttt{ \textbackslash gamma},\texttt{
+\textbackslash phi} as variables}
+
+In some contexts it may feel natural to use any or all of \verb`\pi`,
+\verb`e`, \verb`\gamma` and \verb`\phi` as variables by assigning
+values to them in the vv-list. \texttt{numerica} does not object.
+The values assigned in this way override the constants' values. For
+example, if the triangle we labelled ABC previously was instead labelled
+CDE then it has sides $c=3,d=4$ and (note!) $e=5$. It's area therefore
+is\medskip{}
+\begin{verbatim}
+ \eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
+ [s=\tfrac12(c+d+e),c=3,d=4,e=5]
+\end{verbatim}
+$\Longrightarrow$
+
+\eval{$ \sqrt{s(s-c)(s-d)(s-e)} $}
+ [s=\tfrac12(c+d+e),c=3,d=4,e=5].\medskip{}
+
+\noindent Since this is the correct area we see that \texttt{e} has
+been treated as a variable with the assigned value $5$, not as the
+constant. But if \texttt{e} (or \verb`\pi` or \verb`\gamma` or \verb`\phi`)
+is not assigned a value in the vv-list then it has, by default, the
+value of the constant.
+
+In the case of \texttt{e}, if you wish to use it as a variable, the
+constant is always available as \verb`\exp(1)`. There is no similar
+alternative available for \verb`\pi`, \verb`\gamma` or \verb`\phi`
+although you can always do something like \verb`[\pi=<new value>,\pi_0=\pi]`
+in the vv-list, so that \verb`\pi_0` now has the constant's value.
+
+\subsection{Display of the vv-list}
+
+By default, the vv-list is displayed with (in fact following) the
+numerical result. That and the format of the display can both be changed.
+
+\subsubsection{Star option: suppressing display of the vv-list}
+
+\label{subsec:evalVvSuppresList}If display of the vv-list is not
+wanted at all, only the numerical result, it suffices to attach an
+asterisk (star) to the \texttt{\textbackslash eval} command:
+\begin{centred}
+\verb`$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval*{ hA_C }[A_C=\pi r^2,h=4/3,r=1] $,
+\end{centred}
+or simply the naked result:
+\begin{centred}
+\verb`\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1]` $\Longrightarrow$
+\eval*{$ hA_C $}[A_C=\pi r^2,h=4/3,r=1].
+\end{centred}
+In the latter case, note that a negative result will display with
+a hyphen for the minus sign unless you, the user, explicitly write
+math delimiters around the \verb`\eval*` command:
+\begin{centred}
+\verb`\eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]` $\Longrightarrow$ \eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]
+\end{centred}
+The \verb`$` signs that \verb`\eval*` wraps around are ignored.
+The star option delivers a number, pure and simple, with no accompaniments.
+
+\subsubsection{Suppressing display of items}
+
+\label{subsec:evalVvSuppressVars}You may wish to retain some variables
+in the vv-list display, but not all. For those variables you wish
+omitted from the display, wrap each variable (but not the equals sign
+or value) in braces. When calculating the volume of a cylinder in
+the previous examples, the base area $A_{C}$ has a different status
+from the `fundamental' variables $r$ and $h$. It is an intermediate
+value, one that we pass through on the way to the final result. To
+suppress it from display enclose the variable in braces:
+\begin{centred}
+\verb`$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $` $\Longrightarrow$
+$ V_C=\eval{ hA_C }[{A_C}=\pi r^2,h=4/3,r=1] $.
+\end{centred}
+As you can see, $A_{C}$ no longer appears in the displayed vv-list.
+Of course the name and its value are still recorded `behind the scenes'
+and can still be used in calculations.
+
+\subsubsection{Changing the display format}
+
+\label{subsec:evalChanging-display-format}In two examples above,
+we have calculated the area of a triangle using Brahmagupta's formula.
+Display of the result is crowded. Two remedies have just been suggested,
+but a third one and preferable in this case would be to force display
+of the vv-list and result to a new line. This can be done through
+the settings option to the \verb`\eval` command, discussed in §\ref{subsec:settings New-line-display}.
+However, if \verb`\eval` is wrapped around an \emph{appropriate}
+environment (like \verb`multline`, but not \verb`equation`) it can
+also be done simply by including \texttt{\textbackslash\textbackslash}
+at the end of the formula.
+
+In the following example I use Brahmagupta's formula for calculating
+the area of a cyclic quadrilateral (of which his formula for a triangle
+is a special case). The cyclic quadrilateral in the example is formed
+by a 45-45-90 triangle of hypotenuse 2 joined along the hypotenuse
+to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
+Adding the areas of the two triangles, the area of the quadrilateral
+is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
+$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
+formula:
+\begin{verbatim}
+ \eval{
+ \begin{multline*}
+ \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
+ \end{multline*}
+ }[s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+\end{verbatim}
+$\Longrightarrow$ \eval{
+ \begin{multline*}
+ \sqrt{(s-a)(s-b)(s-c)(s-d)}\\
+ \end{multline*}
+ }[s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+
+\section{Formatting the numerical result}
+
+\label{subsec:evalRoundingEtc} A result of a calculation is displayed,
+by default, to $6$ decimal places. All our results so far have been
+rounded to this figure, although not all digits are displayed, for
+instance if the sixth one is $0$, or the result is an integer. Like
+other elements of the display, both rounding value and the (dis)appearance
+of trailing zeros can be customized, in this case by means of an optional
+argument following the vv-list (or the formula if there is no vv-list).
+This optional argument may contain up to four juxtaposed items from
+seven possibilities:
+\begin{itemize}
+\item a question mark ?, which gives boolean output, or
+\item an integer, the \emph{rounding value}, positive, negative or zero,
+specifying how many decimal places to display the result to, or
+\item an asterisk {*}, which pads the result with zeros should it not have
+as many decimal places as the rounding value specifies, or
+\item the character \texttt{x} (lower case!) which presents the result in
+`proper' scientific notation (a form like $1.234\times10^{5}$ for
+123450), or
+\item the character \texttt{t} (lower case!) which presents the result in
+a bastardized form of scientific notation useful in tables (a form
+like $(5)1.234$ for 123450), or
+\item a character other than \texttt{?}, \texttt{{*}}, \texttt{x}, \texttt{t}
+or a digit, usually (but not necessarily) one of the letters\texttt{
+e d} \texttt{E D}, which presents the result in scientific notation
+using that character as the exponent mark (a form like $1.234\text{e}5$
+for $123450$), or
+\item a prime
+\begin{itemize}
+\item attached to the character specifying scientific notation, which extends
+that notation to numbers in the interval \texttt{{[}1,10)}, or
+\item attached to a question mark, which changes the format of boolean output.
+\end{itemize}
+\end{itemize}
+If you use \texttt{?} in the same specification as some other text
+character, the \texttt{?} prevails; if you use \texttt{x} in the same
+specification as some other text character except for \texttt{?},
+the \texttt{x} prevails; if you use \texttt{t} in the same specification
+as some other text character except for \texttt{?} or \texttt{x},
+the \texttt{t} prevails.
+
+\subsection{Rounding value }
+
+\label{subsec:evalRounding-value}The rounding value specifies the
+number of decimal places displayed:
+\begin{centred}
+\verb`$ \eval{ 1/3 }[4] $` $\Longrightarrow$ $ \eval{ 1/3 }[4] $
+\end{centred}
+The default rounding value is $6$:
+\begin{centred}
+\verb`$ \eval{ 35/3 } $` $\Longrightarrow$ $ \eval{ 35/3 } $
+\end{centred}
+Following the default behaviour in \texttt{l3fp}, `ties' are rounded
+to the nearest \emph{even} digit. Thus a number ending $55$ has a
+`choice' of rounding to $5$ or $6$ and rounds up to the even digit
+$6$, and a number ending $65$ with a `choice' of rounding to $6$
+or $7$ rounds down to the even digit $6$:
+\begin{centred}
+\verb`$ \eval{ 0.1234555 } $` $\Longrightarrow\eval{0.1234555}$
+
+\verb`$ \eval{ 0.1234565 } $` $\Longrightarrow\eval{0.1234565}$
+\end{centred}
+The calculational engine which \texttt{numerica} uses, \texttt{l3fp},
+works to 16 significant figures and never displays more than that
+number (and often less).
+\begin{itemize}
+\item In the first of the following although I have specified a rounding
+value of $19$ only $16$ decimal places are displayed, with the final
+digit rounded up to $7$;
+\item in the second I have added $10$ zeros after the decimal point, meaning
+that all $19$ decimal places specified by the rounding value can
+be displayed since the 10 initial zeros do not contribute to the significant
+figures;
+\item in the third I have changed the figure \emph{before} the decimal point
+to $1$ so that the $10$ added zeros are now included among the significant
+figures;
+\item and in the fourth, I have added $9$ digits before the decimal point:
+\end{itemize}
+\begin{centred}
+\verb`$ \eval{ 0.1234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 0.1234567890123456789 }[19]$
+
+\verb`$ \eval{ 0.00000000001234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 0.00000000001234567890123456789 }[19]$
+
+\verb`$ \eval{ 1.00000000001234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 1.00000000001234567890123456789 }[19]$
+
+\verb`$ \eval{ 987654321.1234567890123456789 }[19] $` $\Longrightarrow$
+$\eval{ 987654321.1234567890123456789 }[19]$
+\end{centred}
+In all cases, no more than $16$ \emph{significant} figures are displayed,
+although the number of decimal places displayed may exceed $16$ as
+in the second example.
+
+It is possible to use \emph{negative} rounding values. Such a value
+zeroes the specified number of digits \emph{before} the decimal point.
+\begin{centred}
+\verb`$ \eval{ 987654321.123456789 }[-4] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[-4]$
+\end{centred}
+A rounding value of $0$ rounds to the nearest integer:
+\begin{centred}
+\verb`$ \eval{ 987654321.123456789 }[0] $` $\Longrightarrow$ $\eval{ 987654321.123456789 }[0]$
+\end{centred}
+If you wish to change the \emph{default} rounding value from $6$
+to some other value, this can be done by creating or editing a file
+\texttt{numerica.cfg} in a text editor; see §\ref{sec:settingsDefaults}.
+
+\subsection{Padding with zeros }
+
+\label{subsec:evalPadding-with-zeros}A result may contain fewer decimal
+places than the rounding value specifies, the trailing zeros being
+suppressed by default (this is how \texttt{l3fp} does it). Sometimes,
+perhaps for reasons of presentation like aligning columns of figures,
+it may be desirable to pad results with zeros. This is achieved by
+inserting an asterisk, {*}, into the final optional argument of the
+\verb`\eval` command:
+\begin{centred}
+\verb`$ \eval{ 1/4 }[4] $` $\Longrightarrow$ $ \eval{ 1/4 }[4] $,
+
+\verb`$ \eval{ 1/4 }[4*] $` $\Longrightarrow$ $ \eval{ 1/4 }[4*] $.
+\end{centred}
+
+\subsection{Scientific notation }
+
+\label{subsec:evalScientificNotation} The \texttt{l3fp} package can
+output numbers in scientific notation. For example, $1234$ is rendered
+as $\eval{1234}[e]$, denoting $1.234\times10^{3}$ , and $0.008$
+as $\eval{0.008}[e]$, denoting $8\times10^{-3}$. The `e' here,
+the \emph{exponent mark}, separates the \emph{significand} ($1.234$)
+from the \emph{exponent} ($3$). To switch on output in scientific
+notation in \texttt{numerica} enter \texttt{e} in the trailing optional
+argument:
+\begin{centred}
+\verb`$ \eval{ 123456789 }[e] $` $\Longrightarrow$ $ \eval{ 123456789 }[e] $.
+\end{centred}
+The default rounding value $6$ is in play here. In \texttt{numerica},
+when scientific notation is selected rounding takes a different meaning:
+it is the \emph{significand }which is rounded (not the number as a
+whole). One digit precedes the decimal point, at most $15$ follow
+it.
+
+Negative rounding values are pointless for scientific notation. A
+zero might on occasion be relevant:
+\begin{centred}
+\verb`$ \eval{ 987654321 }[0e] $` $\Longrightarrow$ $ \eval{ 987654321 }[0e] $.
+\end{centred}
+Sometimes letters other than `e' are used to indicate scientific
+notation, like `E' or `d' or `D'. With a few exceptions, \texttt{numerica}
+allows any letter or text character to be used as the exponent marker:
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4d]`\texttt{ $\Longrightarrow$} \eval{$ 1/23456789 $}[4d].
+\end{centred}
+But when \texttt{x} is inserted in the trailing optional argument,
+the output is in the form $d_{0}.d_{1}\ldots d_{m}\times10^{n}$ (except
+when $n=0$), where each $d_{i}$ denotes a digit.
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4x]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4x] .
+\end{centred}
+The requirements of tables leads to another form of scientific notation.
+Placing \texttt{t} in the trailing argument turns on this table-ready
+form of notation:
+\begin{centred}
+\verb`\eval{$ 1/23456789 $}[4t]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4t].
+\end{centred}
+This is discussed more fully in the associated document \texttt{numerica-tables.pdf}.
+
+In the next example three options are used in the trailing argument.
+The order in which the items are entered does not matter:
+\begin{centred}
+\verb`\eval{$ 1/125 $}[*e4]` $\Longrightarrow$ \eval{$ 1/125 $}[*e4].
+\end{centred}
+Finally, to illustrate that `any' text character\footnote{Be sensible! An equals sign for instance might confuse \texttt{numerica}
+into thinking the number-format option is the vv-list, and will certainly
+confuse the reader.} save for \texttt{x} or \texttt{t} can be used to distinguish the
+exponent, I use an @ character:
+\begin{centred}
+\verb`\eval{$ 1/123 $}[@4]` $\Longrightarrow$ \eval{$ 1/123 $}[@4].
+\end{centred}
+
+\subsubsection{Numbers in \texttt{{[}1,10)}}
+
+Usually when scientific notation is being used, numbers with magnitude
+in the interval $[1,10)$ are rendered in their normal decimal form,
+$3.14159$ and the like. Occasionally it may be desired to present
+numbers in this range in scientific notation (this can be the case
+in tables where the alignment of a column of figures might be affected).
+\texttt{numerica} offers a means of extending scientific notation
+to numbers in this range by adding a prime to the letter chosen as
+the exponent mark in the trailing optional argument.
+\begin{centred}
+\verb`\eval{$ \pi $}[4t']` $\Longrightarrow$ \eval{$ \pi $}[4t']
+\end{centred}
+
+\subsubsection{\textbackslash eval{*} and scientific notation}
+
+Scientific notation can be used for the numerical result output by
+\verb`\eval*`:
+\begin{centred}
+\verb`\eval*{$ \pi $}[e']` $\Longrightarrow$ \eval*{$ \pi $}[e']
+\end{centred}
+There is one catch: if you substitute \texttt{x} for \texttt{e} here,
+\LaTeX{} will complain about a missing \verb`$`. An \texttt{x} in
+the number-format option produces a \verb`\times` in the output which
+requires a math environment. It is up to you, as the user, to provide
+the necessary delimiters outside the \verb`\eval*` command. (This
+applies even when \verb`\eval*` wraps around math delimiters.)
+
+(Because of the way \texttt{numerica} parses the number-format option,
+entering a prime with neither exponent character nor question mark
+specified will result in scientific output using \texttt{e} as the
+exponent mark. The last example could have been written \verb`\eval*{$ \pi $}[']`.)
+
+\subsection{Boolean output}
+
+\label{subsec:evalBoolean-output}\texttt{l3fp} can evaluate comparisons,
+outputting $0$ if the comparison is false, $1$ if it is true. By
+entering a question mark, \texttt{?}, in the trailing optional argument,
+you can force \texttt{numerica} to do the same depending as the result
+of a calculation is zero or not. The expression being evaluated does
+not need to be a comparison, \verb`$ \eval{\pi}[?] $` $\Longrightarrow$ $ \eval{\pi}[?]$,
+but comparisons are what this is designed for.
+
+Possible comparison relations are \verb`=`, \verb`<`, \verb`>`,
+\verb`\ne`, \verb`\neq`, \verb`\ge`, \verb`\geq`, \verb`\le`,
+\verb`\leq`. Although programming languages use combinations like
+\verb`<=` or \verb`>=`, \texttt{numerica} does \emph{not} accept
+these (they are not part of standard \emph{mathematical} usage) and
+will generate an error. An example where the relation is equality
+exhibits a numerological curiosity:\footnote{The \texttt{{[}p=.{]}} of this and the next example ensures a full
+stop appears in the correct place; see §\ref{subsec:settingsPunctuation}.}
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]
+\end{centred}
+Notice the $5$ alongside the question mark in the trailing argument.
+That is critical. Change the $5$ to a $6$ (or omit it since the
+default rounding value is $6$) and the outcome is different:
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?]
+\end{centred}
+Now the relation is false. Evaluating the fraction to more than $6$
+places, say to $9$, we can see what is going on:
+\begin{centred}
+\verb`\eval{$ 1/0.0123456789 $}[9]` $\Longrightarrow$ \eval{$ 1/0.0123456789 $}[9].
+\end{centred}
+
+\subsubsection{Outputting \texttt{T} or \texttt{F}}
+
+To my eye, outputting $0$ or $1$ in response to a `question' like
+$1/0.0123456789=81$ is confusing. It is easy to change the boolean
+output from $0,1$ to a more appropriate $F,T$, or \texttt{$\texttt{F,\texttt{T}}$}
+by adding a prime or two primes respectively to the question mark
+in the number-format option.
+\begin{centred}
+\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']` $\Longrightarrow$
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']
+\end{centred}
+The default boolean output format is chosen to be $0,1$ in case an
+\verb`\eval*` command is used within another \verb`\eval` command
+(`nesting'– see Chapter~\ref{chap:Nesting}~). The inner command
+needs to output a \emph{numerical} answer.
+
+\subsubsection{Rounding error tolerance}
+
+\label{subsec:evalToleranceRounding}If at least one of the terms
+in a comparison is the result of a calculation, then it's value is
+likely to contain rounding errors. What level of rounding error can
+we tolerate before such errors interfere with the comparison being
+made? \texttt{l3fp} tolerates none. It decides the truth or falsity
+of a comparison to all $16$ significant figures: 1.000 0000 0000
+0000 and 1.000 0000 0000 0001 are \emph{not }equal in \texttt{l3fp}.
+But for most purposes this will be far too severe a criterion.
+
+Suppose our comparison relation is $\varrho$, denoting one of =,
+<, >, \verb`\le`, etc. If $X\rel Y$ then $X-Y\rel Y-Y$, i.e. $X-Y\rel0$.
+This is what \texttt{numerica} does. It takes the right-hand side
+of the relation from the left-hand side and then compares the \emph{rounded}
+difference under $\varrho$ to $0$. The rounding value used is the
+number specified with the question mark in the trailing argument of
+the \verb`\eval` command or, if no number is present, the default
+rounding value (`out of the box' this is $6$). Thus, in a recent
+example, $1/0.0123456789-81$ when rounded to $5$ decimal places
+is \@.$0.00000$, indistinguishable from zero at this rounding value;
+hence the equality $1/0.0123456789=81$ is true. But when rounded
+to $6$ places it is $0.000001$ which \emph{is} distinguishable from
+zero and so the equality is false. Truth or falsity depends on the
+rounding value.
+
+When dealing with numbers generated purely mathematically, rounding
+values of $5$ or $6$ are likely to be too small. More useful would
+be rounding values closer to \texttt{l3fp}'s $16$ – perhaps $14$?
+– depending on how severe the calculations are that generate the numbers.
+However if the numbers we are dealing with come from outside mathematics,
+from practical experiments perhaps, then even a rounding value of
+$5$ or $6$ may be too large.
+
+\subsubsection{Rationale}
+
+Mathematically, the claim that $X=Y$ at a rounding value $n$ is
+the claim that
+\[
+\abs{X-Y}\le5\times10^{-(n+1)}.
+\]
+since this rounds \emph{down} to zero at $n$ places of decimals.
+This gives a more accurate test of equality than doing things in the
+opposite order – rounding each number first and then taking the difference.
+One might, for instance, have numbers like $X=0.12345$, $Y=0.12335$.
+Rounding to $n=4$ places, both round to $0.1234$ and yet the difference
+between them is $0.0001$ – they are distinguishable numbers to $4$
+places of decimals. This is why \texttt{numerica} forms the difference
+\emph{before }doing the rounding.
+
+\subsubsection{And, Or, Not}
+
+For logical And \LaTeX{} provides the symbols \verb`\wedge` and \verb`\land`,
+both displaying as $\land$, but \texttt{numerica} adds thin spaces
+( \verb`\,` ) around the symbol for \verb`\land` (copying the package
+\texttt{gn-logic14.sty}). For logical Or \LaTeX{} provides the symbols
+\verb`\vee` and \verb`\lor`, both displaying as $\lor$, but again
+\texttt{numerica} adds thin spaces around the symbol for \verb`\lor`.
+\begin{centred}
+\verb`\eval{$ 1<2 \wedge 2<3 $}[?'']` $\Longrightarrow$ \eval{$ 1<2 \wedge 2<3 $}[?''],
+
+\verb`\eval{$ 1<2 \land 2<3 $}[?'']` $\Longrightarrow$ \eval{$ 1<2 \land 2<3 $}[?''].
+\end{centred}
+To my eye the second of these with its smidgen more space around the
+wedge symbol displays the meaning of the overall expression better
+than the first. Both And and Or have equal precedence; in cases of
+ambiguity the user needs to parenthesize as necessary to clarify what
+is intended.
+
+\LaTeX{} provides two commands for logical Not, \verb`\neg` and \verb`\lnot`,
+both displaying as $\lnot$ . Not binds tightly to its argument:
+\begin{centred}
+\verb`\eval{$ \lnot A \land B $}[A=0,B=0]` $\Longrightarrow$ \eval{$ \lnot A \land B $}[A=0,B=0].
+\end{centred}
+Here \verb`\lnot` acts only on the $A$; if it had acted on $A\land B$
+as a whole the result would have been $1$.
+
+For a little flourish, I evaluate a more complicated logical statement:\footnote{Quoting from an article in \emph{Quanta Magazine} (August 2020) by
+Kevin Hartnett: `Let’s say you and two friends are planning a party.
+The three of you are trying to put together the guest list, but you
+have somewhat competing interests. Maybe you want to either invite
+Avery or exclude Kemba. One of your co-planners wants to invite Kemba
+or Brad or both of them. Your other co-planner, with an ax to grind,
+wants to leave off Avery or Brad or both of them. Given these constraints,
+you could ask: Is there a guest list that satisfies all three party
+planners?' I have written $C$ for Kemba, $A$ and $B$ for Avery
+and Brad.}
+\begin{verbatim}
+ \eval{$(A\lor\lnot C)\land(C\lor B)\land
+ (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][?'']
+\end{verbatim}
+$\Longrightarrow$ \eval{$(A\lor\lnot C)\land(C\lor B)\land
+ (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][?'']
+
+\subsubsection{Chains of comparisons}
+
+\texttt{numerica} can handle chains of comparisons like $1<2<1+2<5-1$.
+`Behind the scenes' it inserts logical And-s into the chain, $1<2\land2<1+2\land1+2<5-1$,
+and evaluates the modified expression:
+\begin{centred}
+\verb`\eval{$ 1<2<1+2<5-1 $}[?'']` $\Longrightarrow$ \eval{$ 1<2<1+2<5-1 $}[?''].
+\end{centred}
+
+\subsubsection{\texttt{amssymb} comparison symbols}
+
+\label{subsec:evalAmssymb-comparisons}\texttt{numerica} accepts some
+alternative symbols for the basic comparison relations from the \texttt{amssymb}
+package provided that package is loaded, i.e. the preamble of your
+document includes the statement
+\begin{lyxcode}
+\textbackslash usepackage\{amssymb\}
+\end{lyxcode}
+The variants from this package are: \verb`\leqq` ( $\leqq$ ), \verb`\leqslant`
+( $\leqslant$ ), \verb`\geqq` (~$\geqq$~), and \verb`\geqslant`
+( $\geqslant$ ).\footnote{No, that is not \texttt{eggplant}.} There
+are also negations: \verb`\nless` ( $\nless$ ), \verb`\nleq` (~$\nleq$~),
+\verb`\nleqq` ( $\nleqq$ ), \verb`\nleqslant` ( $\nleqslant$ ),
+\verb`\ngtr` ( $\ngtr$ ), \verb`\ngeq` ( $\ngeq$ ), \verb`\ngeqq`
+( $\ngeqq$ ), \verb`\ngeqslant` ( $\ngeqslant$ ).
+
+\section{Calculational details}
+
+\subsection{Arithmetic}
+
+\label{subsec:evalArithmetic}Addition, subtraction, multiplication,
+division, square roots, \emph{$n$}-th roots, and exponentiating (raising
+to a power) are all available.
+
+Multiplication can be rendered explicitly with an asterisk,
+\begin{centred}
+\verb`\eval{$ 9*9 $}` $\Longrightarrow$ \eval{$ 9*9 $},
+\end{centred}
+but that's ugly. More elegant is to use \verb`\times`:
+\begin{centred}
+\verb`\eval{$ 9\times9 $}` $\Longrightarrow$ \eval{$ 9\times9 $}.
+\end{centred}
+\verb`\cdot` is also available and in many cases juxtaposition alone
+suffices:
+\begin{centred}
+\verb`\eval{$ \surd2\surd2 $}` $\Longrightarrow$ \eval{$ \surd2\surd2 $},
+
+\verb`\eval{$ ab $}[a=123,b=1/123]` $\Longrightarrow$ \eval{$ ab $}[a=123,b=1/123].
+\end{centred}
+Division can be rendered in multiple ways too:
+\begin{centred}
+\verb`\eval{$ 42/6 $}` $\Longrightarrow$ \eval{$ 42/6 $},
+
+\verb`\eval{$ 42\div6 $}` $\Longrightarrow$ \eval{$ 42\div6 $},
+\end{centred}
+or by using \verb`\frac` or \verb`\tfrac` or \verb`\dfrac` as in
+\begin{centred}
+\verb`\eval{$ \frac{42}6 $}` $\Longrightarrow$ \eval{$ \frac{42}6 $}.
+\end{centred}
+But note that since juxtaposition means multiplication, it is also
+true that $42\tfrac{1}{6}$\texttt{ }evaluates to $7$ inside an \verb`\eval`
+command rather than denoting `forty two and a sixth'. Hence if you
+want to use `two and a half' and similar values in \texttt{numerica},
+they need to be entered as improper fractions like $\tfrac{5}{2}$
+or in decimal form, $2.5$ (as one does automatically in mathematical
+expressions anyway because of the ambiguity in a form like $2\tfrac{1}{2}$).
+
+\subsubsection{Square roots and $n$-th roots}
+
+\label{subsec:evalSquareRootsEtc}Let us check that 3, 4, 5 and 5,
+12, 13 really are Pythagorean triples (I use \verb`\sqrt` in the
+first, \verb`\surd` in the second):
+\begin{centred}
+\verb`\eval{$ \sqrt{3^2+4^2} $}` $\Longrightarrow$ \eval{$\sqrt{3^{2}+4^{2}}$},
+
+\verb`\eval{$ \surd(5^2+12^2) $}` $\Longrightarrow$ \eval{$ \surd(5^2+12^2) $}.
+\end{centred}
+The \verb`\sqrt` command has an optional argument which can be used
+for extracting $n$-th roots of a number. This notation is generally
+used when $n$ is a small positive integer like $3$ or $4$. This
+practice is followed in \texttt{numerica}: $n$ must be a (not necessarily
+small) \emph{positive integer}:
+\begin{centred}
+\verb`\eval{$ \sqrt[4]{81} $}` $\Longrightarrow$ \eval{$ \sqrt[4]{81} $},
+
+\verb`\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}]` $\Longrightarrow$
+\eval{$ \sqrt[n]{125} $}[n=\floor{\pi}].
+\end{centred}
+If $n$ should not be a positive integer, an error message is generated;
+see §\ref{sec:evalErrors}.
+
+For display-style expressions, the \verb`\sqrt` command grows to
+accommodate the extra vertical height; the surd doesn't. Here is an
+example which anticipates a number of matters not discussed yet. It
+shows \verb`\eval` wrapping around a square root containing various
+formatting commands (negative spaces, \verb`\left` and \verb`\right`
+nested within \verb`\bigg` commands), all digested without complaint
+(see §\ref{subsec:evalFormatting-commands}; and see §\ref{subsec:settingsPunctuation}
+for the \verb`[p=.]`): \medskip{}
+\begin{verbatim}
+ \eval[p=.]
+ {\[
+ \sqrt[3]
+ {\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
+ \]}[A=729,B=81,C=9,D=3]
+\end{verbatim}
+$\Longrightarrow$\eval[p=.]
+ {\[
+ \sqrt[3]
+ {\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
+ \]}[A=729,B=81,C=9,D=3]
+
+As implemented in \texttt{numerica}, $n$-th roots found using \verb`\sqrt[n]`
+are \verb`n=<integer>`\emph{ }roots. This raises an interesting question:
+if the `$n$' of an $n$-th root is the result of a calculation,
+what happens with rounding errors? The calculation may not produce
+an \emph{exact} integer. (This problem also arises with factorials;
+see §\ref{subsec:evalFactorialsBinom}.) The solution employed in
+\texttt{numerica} is to make what is considered an integer depend
+on a rounding value. Most calculations will produce rounding errors
+in distant decimal places. For `int-ifying' calculations, \texttt{numerica}
+uses a rounding value of $14$: a calculation produces an integer
+if, when rounded to $14$ figures, the result is an integer. Since
+\texttt{l3fp} works to $16$ significant figures, a rounding value
+of $14$ allows ample `elbowroom' for rounding errors to be accommodated
+when judging what is an integer and what is not. As a practical matter
+problems should not arise.
+
+\subsubsection{\emph{n}-th roots of negative numbers}
+
+Odd (in the sense of `not even') integral roots of \emph{negative}
+numbers are available with \verb`\sqrt`,
+\begin{centred}
+\verb`\eval{$ \sqrt[3]{-125} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-125} $},
+
+\verb`\eval{$ \sqrt[3]{-1.25} $}` $\Longrightarrow$ \eval{$ \sqrt[3]{-0.125} $}.
+\end{centred}
+
+\subsubsection{Inverse integer powers }
+
+Of course to find an $n$-th root we can also raise to the inverse
+power,
+\begin{centred}
+\verb`\eval{$ 81^{1/4} $}` $\Longrightarrow$ \eval{$ 81^{1/4} $}.
+\end{centred}
+However, raising a \emph{negative} number to an inverse power generates
+an error even when, mathematically, it should not. This matter is
+discussed below in §\ref{subsec:errorsInverse-powers}.
+
+\subsection{Precedence, parentheses}
+
+The usual precedence rules apply: multiplication and division bind
+equally strongly and more strongly than addition and subtraction which
+bind equally stongly. Exponentiating binds most strongly. Evaluation
+occurs from the left.
+\begin{centred}
+\verb`\eval{$ 4+5\times6+3 $}` $\Longrightarrow$ \eval{$ 4+5\times6+3 $},
+
+\verb`\eval{$ 6\times10^3/2\times10^2 $}` $\Longrightarrow$ \eval{$ 6\times10^3/2\times10^2 $},
+\end{centred}
+which may not be what was intended. Parentheses (or brackets or braces)
+retrieve the situation:
+\begin{centred}
+\verb`\eval{$ (4+5)(6+3) $}` $\Longrightarrow$ \eval{$ (4+5)(6+3) $},
+
+\verb`\eval{$ (6\times10^3)/(2\times10^2) $}` $\Longrightarrow$
+\eval{$ (6\times10^3)/(2\times10^2) $}.
+\end{centred}
+Because exponentiating binds most strongly, negative values must be
+parenthesized when raised to a power. If not,
+\begin{centred}
+\verb`\eval{$ -4^2 $}` $\Longrightarrow$ \eval{$ -4^2 $},
+\end{centred}
+which is clearly not $(-4)^{2}$. But
+\begin{centred}
+\verb`\eval{$ (-4)^2 $}` $\Longrightarrow$ \eval{$ (-4)^2 $}.
+\end{centred}
+
+\subsubsection{Command-form brackets}
+
+\label{subsec:evalCommandBrackets}Note that brackets of all three
+kinds are available also in command form: \verb`\lparen \rparen`
+(from \verb`mathtools`) for \verb`( )`, \verb`\lbrack \rbrack`
+for \verb`[ ]`, and \verb`\lbrace \rbrace` for \verb`\{ \}`.
+
+\subsection{Modifiers\texttt{ (\textbackslash left \textbackslash right}, etc.)}
+
+The \verb`\left` and \texttt{\textbackslash right} modifiers and
+also the series of \verb`\big...` modifiers\texttt{ }(\texttt{\textbackslash}\verb`bigl \bigr`,
+\verb`\Bigl \Bigr`, \verb`\biggl \biggr`, \verb`\Biggl \Biggr`)
+are available for use with all brackets (parentheses, square brackets,
+braces):
+\begin{verbatim}
+ \eval[p=.]{\[ \exp\left(
+ \dfrac{\ln2}{4}+\dfrac{\ln8}{4}
+ \right) \]}
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{\[ \exp\left( \dfrac{\ln2}{4}+\dfrac{\ln8}{4} \right) \]}
+
+\texttt{numerica} also accepts their use with \texttt{.} (dot) and
+with \texttt{/} (as noted earlier, the \verb`[p]` and \verb`[p=.]`
+are explained at §\ref{subsec:settingsPunctuation}):
+\begin{centred}
+\verb`\eval[p]{\[ \left.\dfrac{3+4}{2+1}\right/\!\dfrac{1+2}{4+5} \]}`
+$\Longrightarrow$ \eval[p=.]{\[ \left. \dfrac{3+4}{2+1} \right/\!\dfrac{1+2}{4+5} \]}
+\end{centred}
+They can be nested.
+
+\subsection{Other formatting commands}
+
+\label{subsec:evalFormatting-commands}There are many formatting commands
+which change the layout of a formula on the page but do not alter
+its content. These include various spacing commands like \verb`\!`,
+\verb`\quad`, etc., phantoms (\verb`\phantom` etc.), \verb`\mathstrut`
+from \TeX{} and its \verb`mathtools` cousin \verb`\xmathstrut`.
+
+Consider the same package's \verb`\splitfrac` and \verb`\splitdfrac`.
+The \verb`mathtools` documentation gives an example to illustrate
+the use of these last two. I've mangled it to produce a ridiculous
+illustration of their use, and of the modifiers \verb`\left \right`,
+and of the command-form alternatives to parentheses \verb`\lparen \rparen`;
+also the use of \verb`\dfrac`. A little mental arithmetic will convince
+that we are evaluating the square root of $(9\times7)^{2}$ which
+indeed is what we get:\footnote{For the \texttt{{[}p=.,vvd={]}} see §\ref{subsec:settingsPunctuation}
+and §\ref{subsec:settingsVvDisplayChangeLocal}. The first puts the
+concluding full stop in the right place; the second suppresses the
+vv-list.} \medskip{}
+\begin{verbatim}
+ \eval[p=.,vvd=]{\[
+ \sqrt{\left\lparen
+ \frac{ \splitfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ { \dfrac z7}
+ \right\rparen \left\lparen
+ \frac{ \splitdfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ {\dfrac z7}\right\rparen}
+ \]}[x=2,y=5,z=10]
+\end{verbatim}
+$\Longrightarrow$\eval[p=.,vvd=]{\[
+ \sqrt{\left\lparen
+ \frac{ \splitfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ { \dfrac z7}
+ \right\rparen \left\lparen
+ \frac{ \splitdfrac{xy + xy + xy + xy + xy}
+ {+ xy + xy + xy + xy}
+ }
+ {\dfrac z7}\right\rparen}
+ \]}[x=2,y=5,z=10]\texttt{numerica} essentially ignores formatting commands (the ones
+it knows of). They do not alter the mathematical content of a formula,
+only how it looks. But there will undoubtedly be formatting commands
+it does not recognize which will probably trigger an `Unknown token'
+message. Please contact the author in that case.\footnote{ajparsloe@gmail.com}
+
+\subsection{Trigonometric \& hyperbolic functions}
+
+\LaTeX{} provides all six trignometric functions, \verb`\sin`, \verb`\cos`,
+\verb`\tan`, \verb`\csc`, \verb`\sec`, \verb`\cot` and the three
+principal inverses \verb`\arcsin`, \verb`\arccos`, \verb`\arctan`.
+It also provides four of the six hyperbolic functions: \verb`\sinh`,
+\verb`\cosh`, \verb`\tanh`, \verb`\coth`, and \emph{no} inverses.
+\texttt{numerica} provides the missing hyperbolic functions, \verb`\csch`
+and \verb`\sech`, and all missing inverses, the three trigonometric
+and all six hyperbolic: \verb`\arccsc`, \verb`\arcsec`, \verb`\arccot`,
+and \verb`\asinh`, \verb`\acosh`, \verb`\atanh`, \verb`\acsch`,
+\verb`\asech`, \verb`\acoth`. (\emph{HMF} writes $\text{arcsinh}$,
+$\text{arccosh}$, etc. and ISO recommends $\text{arsinh}$, $\text{arcosh}$,
+etc. The first seems ill-advised, the second not widely adopted. At
+present neither is catered for in \texttt{numerica}.)\emph{ }
+\begin{centred}
+\verb`\eval{$ \arctan1/1\deg $}` $\Longrightarrow$ \eval{$ \arctan 1/1\deg $} ,
+
+\verb`\eval{$ \atanh\tanh3 $}` $\Longrightarrow$ \eval{$ \atanh\tanh3 $} .
+\end{centred}
+Inverses can also be constructed using the `$-1$' superscript notation.
+Thus
+\begin{centred}
+\verb`\eval{$ \sin^{-1}(1/\surd2)/1\deg $}` $\Longrightarrow$ \eval{$ \sin^{-1}(1/\surd2)/1\deg $} ,
+
+\verb`\eval{$ \tanh\tanh^{-1}0.5 $}` $\Longrightarrow$ \eval{$ \tanh\tanh^{-1}0.5 $} .
+\end{centred}
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection*{Hyperbolic functions}
+
+Please note that \texttt{l3fp} does not (as yet) provide \emph{any}
+hyperbolic functions natively. The values \texttt{numerica} provides
+for these functions are \emph{calculated} values using familiar formulas
+involving exponentials (for the direct functions) and natural logarithms
+and square roots for the inverses. Rounding errors mean the values
+calculated may not have $16$-figure accuracy. The worst `offenders'
+are likely to be the least used, \verb`\acsch` and \verb`\asech`.
+For instance,
+\[
+\acsch x=\ln\left[\frac{1}{x}+\left(\frac{1}{x^{2}}+1\right)^{1/2}\right],
+\]
+
+\begin{centred}
+\verb`\eval{$ \csch \acsch 7 $}[16]` $\Longrightarrow$ \eval{$ \csch \acsch 7 $}[16].
+\end{centred}
+\end{shaded}%
+\end{minipage}
+
+\subsection{Logarithms}
+
+The natural logarithm \verb`\ln`, base $10$ logarithm \verb`\lg`,
+and binary or base $2$ logarithm \verb`\lb` are all recognized,
+as is \verb`\log`, preferably with a subscripted base:
+\begin{centred}
+\verb`\eval{$ \log_{12}1728 $}` $\Longrightarrow$ \eval{$ \log_{12}1728 $}
+\end{centred}
+If there is no base indicated, base $10$ is assumed. (The notations
+\verb`\ln`, \verb`\lg`, and \verb`\lb` follow ISO 80000-2 recommendation,
+which frowns upon the use of the unsubscripted \verb`\log` although
+only \verb`\ln` appears widely used.) The base need not be explicitly
+entered as a number. It could be entered as an expression or be specified
+in the vv-list:
+\begin{centred}
+\verb`\eval*{$ \log_b c $}[b=2,c=1024]` $\Longrightarrow$ \eval*{$ \log_b c $}[b=2,c=1024],
+\end{centred}
+the log to base $2$ in this case. It is possible to use the unadorned
+\verb`\log` with a base different from $10$; if you wish to do this
+only for a particular calculation see §\ref{subsec:settingsLogBase},
+or see §\ref{sec:settingsDefaults} if you want to make this default
+behaviour.
+
+\subsection{Other unary functions}
+
+Other unary functions supported are the exponential function \verb`\exp`
+and signature function \verb`\sgn` (equal to $-1$, $0$, or $1$
+depending as its argument is $<0$, $=0$, or $>0$).
+
+\subsection{Squaring, cubing, \ldots unary functions}
+
+\texttt{numerica} has no difficulty reading a familiar but `incorrectly
+formed' expression like
+\[
+\sin^{2}1.234+\cos^{2}1.234.
+\]
+You do not have to render it $(\sin1.234)^{2}+(\cos1.234)^{2}$ or
+(heaven forbid) $(\sin(1.234))^{2}+(\cos(1.234))^{2}$. The everyday
+usage is fine:
+\begin{centred}
+\verb`\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234]` $\Longrightarrow$
+\eval{$ \sin^2\theta+\cos^2\theta $}[\theta=1.234] .
+\end{centred}
+Equally \texttt{numerica} has no difficulty reading the `correct'
+but pedantic form
+\begin{centred}
+\verb`\eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234]`
+$\Longrightarrow$ \eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234] .
+\end{centred}
+A hyperbolic identity is confirmed in this example:
+\begin{centred}
+\verb`\eval{$ \sinh 3x $}[x=1]` $\Longrightarrow$ \eval{$ \sinh 3x $}[x=1],\medskip{}
+
+\verb`\eval{$ 3\sinh x+4\sinh^3x $}[x=1]` $\Longrightarrow$ \eval{$ 3\sinh x+4\sinh^3x $}[x=1].
+\end{centred}
+In fact all named unary functions in \texttt{numerica} can be squared,
+cubed, etc., in this `incorrect' but familiar way, although the
+practice outside the trigonometric and hyperbolic context seems (vanishingly?)
+rare.
+
+When the argument of the function is parenthesized and raised to a
+power – like $\sin(\pi)^{2}$ – it is read by \texttt{numerica} as
+the `sine of the square of pi', $\sin(\pi^{2})$, and \emph{not
+}as the `square of the sine of pi', $(\sin\pi)^{2}$:
+\begin{centred}
+\verb`\eval{$ \sin(\pi)^2 $}` $\Longrightarrow$ \eval{$ \sin(\pi)^2 $} .
+\end{centred}
+Things are done like this in \texttt{numerica} above all to handle
+the logarithm in a natural way. Surely $\ln x^{n}=n\ln x=\ln(x^{n})$
+rather than $(\ln x)^{n}$? And if we wish to write (as we do) $\ln(1+1/n)^{n}=n\ln(1+1/n)=1-1/2n+1/3n^{2}-\ldots$
+to study the limiting behaviour of $(1+1/n)^{n}$, then we cannot
+avoid $\ln(x)^{n}=n\ln(x)=\ln(x^{n})$.
+
+\subsection{\emph{n}-ary functions}
+
+The functions of more than one variable ($n$-ary functions) that
+\texttt{numerica} supports are \verb`\max`, \verb`\min` and \verb`\gcd`,
+greatest common divisor. The comma list of arguments to \verb`\max`,
+\verb`\min` or \verb`\gcd` can be of arbitrary length. The arguments
+themselves can be expressions or numbers. For \verb`\gcd`, non-integer
+arguments are truncated to integers. Hence both $y$ and $3y$ are
+independently truncated in the following example – to $81$ and $243$
+respectively:
+\begin{centred}
+\verb`\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3]` $\Longrightarrow$
+\eval{$ \gcd(12,10x^2,3y,y,63) $}[y=1/0.0123456789,x=3] .
+\end{centred}
+(The truncation occurs in the argument of \verb`\gcd`, not in the
+vv-list.)
+
+For $n$-ary functions, squaring, cubing, etc. follows a different
+pattern from that for unary functions. For \verb`\max`, \verb`\min`,
+\verb`\gcd` the argument of the function is a comma list. Squaring
+the argument makes no sense. We understand the superscript as applying
+to the function as a whole. (Consistency is not the point here; it
+is what mathematicians do that \texttt{numerica} tries to accommodate.)
+\begin{centred}
+\verb`\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24]` $\Longrightarrow$
+\eval{$ \gcd(3x,x,\arcsin 1/\deg)^2 $}[x=24] .
+\end{centred}
+
+\subsection{Delimiting arguments with brackets \& modifiers }
+
+Arguments of unary and $n$-ary functions can be delimited not only
+with parentheses, but also with square brackets and braces, both in
+explicit character form and also in the command form of §\ref{subsec:evalCommandBrackets}.
+The brackets, of whatever kind, can be qualified with \verb`\left \right`,
+\verb`\bigl \bigr`, etc.\footnote{See §\ref{subsec:settingsPunctuation} for the \texttt{{[}p=.{]}}
+(which ensures the concluding full stop appears in the correct place.}
+\begin{centred}
+\verb`\eval[p=.]{\[ \sin\left\lbrack \dfrac\pi{1+2+3}\right\rbrack \]}`
+$\Longrightarrow$\eval[p=.]{\[ \sin\left\lbrack\dfrac\pi{1+2+3}\right\rbrack \]}
+\end{centred}
+
+\subsection{Absolute value, floor \& ceiling functions}
+
+It is tempting to use the \texttt{|} key on the keyboard for inserting
+an absolute value sign. \texttt{numerica} accepts this usage, but
+it is deprecated. The spacing is incorrect – compare $|-l|$ using
+\texttt{|} against $\lvert-l\rvert$ using \verb`\lvert \rvert`.
+Also, the identity of the left and right delimiters makes nested absolute
+values difficult to parse. \texttt{numerica} does not attempt to do
+so. Placing an absolute value constructed with \texttt{|} within another
+absolute value constructed in the same way is likely to produce a
+compilation error or a spurious result. \verb`\lvert \rvert` are
+better in every way except ease of writing.\texttt{ }To aid such ease
+\texttt{numerica }provides the \verb`\abs` function (using the \texttt{\textbackslash DeclarePairedDelimiter}
+command of the \texttt{mathtools} package). This takes a mutually
+exclusive star (asterisk) or square bracketed optional argument, and
+a mandatory braced argument. The starred form expands to \verb`\left\lvert #1 \right\rvert`
+where \verb`#1` is the mandatory argument:
+\begin{centred}
+\verb`\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]` $\Longrightarrow$
+\eval[p=.]{\[ 3\abs*{\frac{\abs{n}}{21}-1} \]}[n=-7]
+\end{centred}
+The optional argument provides access to the \verb`\big...` modifiers:
+\begin{verbatim}
+ \eval[p=.]{\[
+ \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}}
+ \]}[A=12,a=-10,C=7,c=-5]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{\[
+ \abs[\Big]{\abs{a-c}-\abs[\big]{A-C}}
+ \]}[A=12,a=-10,C=7,c=-5]
+
+The form without either star or square bracket option dispenses with
+the modifiers altogether:
+\begin{centred}
+\verb`\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].` $\Longrightarrow$
+\eval{$ \tfrac12(x+y)+\tfrac12\abs{x-y} $}[x=-3,y=7].
+\end{centred}
+As noted, the star and square bracketed option are mutually exclusive
+arguments.
+
+\texttt{numerica} also provides the functions \verb`\floor` and \verb`\ceil`,
+defined in the same way, taking a mutually exclusive star or square
+bracketed optional argument and for the starred forms expanding to
+\verb`\left\lfloor #1 \right\rfloor` and \verb`\left\lceil #1 \right\rceil`
+where \verb`#1` is the mandatory argument, and for the square bracket
+option forms replacing the \verb`\left` and \verb`\right` with the
+corresponding \verb`\big` commands. The form without star or square-bracket
+option dispenses with any modifier at all.
+\begin{centred}
+\verb`\eval{$ \floor{-\pi} $}` $\Longrightarrow$ \eval{$ \floor{-\pi} $},
+
+\verb`\eval{$ \ceil{\pi} $}` $\Longrightarrow$ \eval{$ \ceil{\pi} $}.
+\end{centred}
+The floor function, $\lfloor x\rfloor$, is the greatest integer $\le x$;
+the ceiling function, $\lceil x\rceil$ is the smallest integer $\ge x$.
+Like the absolute value, the floor and ceiling functions, can be nested:
+\begin{centred}
+\verb`\eval{$ \floor{-\pi+\ceil{e}} $}` $\Longrightarrow$ \eval{$ \floor{-\pi+\ceil{e}} $}.
+\end{centred}
+
+\subsubsection{Squaring, cubing, \ldots{} absolute values, etc.}
+
+These three functions can be raised to a power \emph{without} extra
+parentheses:
+\begin{centred}
+\verb`\eval{$ \ceil{e}^2 $},` $\Longrightarrow$ \eval{$ \ceil{e}^2 $},
+
+\verb`\eval{$ \abs{-4}^2 $}.` $\Longrightarrow$ \eval{$ \abs{-4}^2 $}.
+\end{centred}
+
+\subsection{Factorials, binomial coefficients}
+
+\label{subsec:evalFactorialsBinom}Factorials use the familiar trailing
+\texttt{!} notation:
+\begin{centred}
+\verb`\eval{$ 7! $}` $\Longrightarrow$ \eval{$ 7! $},
+
+\verb`\eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3]`
+$\Longrightarrow$ \eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3].
+\end{centred}
+The examples illustrate how \texttt{numerica} interprets the argument
+of the factorial symbol:\texttt{ }it `digests'
+\begin{enumerate}
+\item a preceding (possibly multi-digit) integer, or
+\item a preceding variable token, or
+\item a bracketed expression, or
+\item a bracket-like expression – an absolute value, floor or ceiling function,
+\end{enumerate}
+since they delimit arguments in a bracket-like way:
+\begin{centred}
+\verb`\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}` $\Longrightarrow$
+\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}.
+\end{centred}
+The result of feeding the factorial an expression different in kind
+from one of these four cases may give an error message or an unexpected
+result. Use parentheses around such an expression; for example write
+$(3^{2})!$, rather than $3^{2}!$.
+
+Nesting of brackets for factorials is accepted:
+\begin{centred}
+\verb`\eval{$ ((5-2)!+1)! $}` $\Longrightarrow$ \eval{$ ((5-2)!+1)! $}.
+\end{centred}
+The factorials of negative integers or of non-integers are not defined
+in \texttt{numerica}. Again there is the problem met in relation to
+$n$-th roots of what happens if the argument of a factorial is the
+result of a calculation and rounding errors mean it is not an exact
+integer. This problem is unlikely to be of practical concern since
+\texttt{numerica} rounds the result of such a calculation by default
+to $14$ significant figures before offering it to the factorial.
+Since \texttt{l3fp} works to $16$ significant figures, there is ample
+`elbowroom' to accommodate rounding errors before the result of
+a calculation ceases to round to an integer.
+
+\subsubsection{Double factorials}
+
+The double factorial, written $n!!,$ is the product $n(n-2)(n-4)\ldots\times4\times2$
+when $n$ is even and the product $n(n-2)(n-4)\ldots\times3\times1$
+when $n$ is odd.
+\begin{centred}
+\verb`\eval{$ 6!! $}` $\Longrightarrow$ \eval{$ 6!! $},
+
+\verb`\eval{$ n!! $}[n=\sqrt{49}]` $\Longrightarrow$ \eval{$ n!! $}[n=\sqrt{49}],
+\end{centred}
+Since $n!=n!!(n-1)!!$ it follows that
+\[
+n!!=\frac{n!}{(n-1)!!}=\frac{(n+1)!}{(n+1)!!}.
+\]
+Putting $n=0$ in the outer equality shows that $0!!=1$. Now putting
+$n=0$ in the left equality gives $(-1)!!=1$. Double factorials therefore
+are defined for integers $\ge-1$.
+
+\subsubsection{Binomial coefficients}
+
+Binomial coefficients are entered in \LaTeX{} with the \verb`\binom`\textbf{
+}command. It takes two arguments and has a text-style version \verb`\tbinom`
+and a display-style version \verb`\dbinom`. As implemented in \texttt{numerica},
+these are \emph{generalised} binomial coefficients:
+\[
+\binom{x}{k}=\frac{x(x-1)\dots(x-k+1)}{k(k-1)\dots1},\quad(x\in\mathbb{R},~k\in\mathbb{N}),
+\]
+where $x$ need not be a non-negative integer, and where $\binom{x}{0}=1$
+by definition. Although the first (or upper) argument can be any real
+number, the lower argument \emph{must} be a non-negative integer.
+Thus, \verb`\eval{$ \tbinom53 $}` $\Longrightarrow$ \eval{$\tbinom53$},
+\verb`\eval{$ \tbinom70 $}` $\Longrightarrow$ \eval{$\tbinom70$},
+\verb`\eval{$ \tbinom{4.2}3 $}` $\Longrightarrow$ \eval{$\tbinom{4.2}3$},
+but if the second (or lower) argument of \verb`\binom` is \emph{not}
+a non-negative integer, \texttt{numerica} displays a message; see
+§\ref{subsec:errorsIntegerArgs}.
+
+\subsection{Sums and products}
+
+\texttt{numerica} recognizes sums (\verb`\sum` displaying as $\sum$)
+and products (\verb`\prod` displaying as $\prod$), and expects both
+symbols to have lower and upper summation/product limits specified.
+The lower limit must be given in the form \emph{sum/prod variable
+= initial value}; the upper limit requires only the final value to
+be specified (although it can also be given in the form \emph{sum/prod
+variable = final value}). The values may be expressions depending
+on other variables and values but must evaluate to integers (or infinity
+– see §\ref{sec:settingsInfiniteSumsProds}). Evaluating to an integer
+means that they \emph{round} to an integer, using a rounding value
+that is set by default to $14$; (recall that \texttt{l3fp} works
+to $16$ significant figures). If a limit evaluates to a non-integer
+at this `int-ifying' rounding value, an error message results. (To
+change this `int-ifying' rounding value, see §\ref{subsec:defaultsIntifyingRounding}.)
+
+As an example of expressions in the limits, this example uses the
+floor and ceiling functions to convert combinations of constants to
+integers (the \verb`[p]` is explained in §\ref{subsec:settingsPunctuation}),
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}` $\Longrightarrow$\eval[p]{\[ \sum_{n=\floor{\pi/e}}^{\ceil{\pi e}}n \]}
+\end{centred}
+\noindent (which is $\sum_{n=1}^{9}n$).\emph{ }If the upper limit
+is less than the lower limit the result is zero. Notice that there
+is no vv-list. The summation variable does not need to be included
+there unless there are other variables that depend on it. However,
+in the case
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
+\eval[p]{\[ \sum_{k=1}^N\frac1{k^3} \]}[N=100][4]
+\end{centred}
+the upper limit $N$ is necessarily assigned a value in the vv-list.
+
+To the author it seems natural to enter the lower limit first, immediately
+after the \verb`\sum` command (the sum is \emph{from }something \emph{to}
+something), but no problem will accrue if the upper limit is placed
+first (after all, the appearance of the formula in the pdf is the
+same):
+\begin{centred}
+\verb`\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]` $\Longrightarrow$
+\eval[p=.]{\[ \sum^N_{k=1}\frac1{k^3} \]}[N=100][4]
+\end{centred}
+Another example of a sum, using binomial coefficients this time, is
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]`
+$\Longrightarrow$ \eval[p]{\[ \sum_{m=0}^5\binom{5}{m}x^m y^{5-m} \]}[x=0.75,y=2.25]
+\end{centred}
+which is just \verb`\eval{$(x+y)^5$}[x=0.75,y=2.25]` $\Longrightarrow$
+\eval{$ (x+y)^5 $}[x=0.75,y=2.25], or $3^{5}$.
+
+Now let's calculate a product:
+\begin{verbatim}
+ \eval[p]{\[
+ \prod_{k=1}^{100}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[\prod_{k=1}^{100} \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)\]}[x=1][3]
+
+\noindent to be compared with \verb`\eval{$ \sinh 1 $}[3]` $\Longrightarrow$
+\eval{$ \sinh1 $}[3]. Obviously more terms than $100$ are required
+in the product to achieve 3-figure accuracy.
+
+\subsubsection{Infinite sums and products}
+
+There is a strong urge to use $\infty$ in the upper limit of this
+product. Let's do so:
+\begin{verbatim}
+ \eval[p=.]{\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]
+ {\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3]
+
+\noindent Disappointingly, we still get the same result, deficient
+by $1$ in the third decimal place. Obviously \texttt{numerica} has
+not multiplied an infinite number of terms and, just as obviously,
+the finite number of terms it \emph{has} multiplied are too few. How
+\texttt{numerica} decides when to stop evaluating additional terms
+in an infinite sum or product is discussed later, §\ref{sec:settingsInfiniteSumsProds}.
+
+For this particular product the problem is that it converges slowly.
+Any criterion for when to stop multiplying terms or, for an infinite
+sum adding terms, seems bound to fail for some product or series.
+Presumably any stopping criterion must measure smallness in some way.
+But terms of the divergent harmonic series, $\sum(1/n)$ can always
+be found smaller than any value we care to specify. It is not surprising
+that a sufficiently slowly converging product or series falls foul
+of a given criterion.
+
+The default criterion however can be changed. Because this involves
+values assigned in the settings option of the \verb`\eval` command,
+I discuss infinite sums and products in the section discussing that
+optional argument; see §\ref{sec:settingsInfiniteSumsProds}.
+
+Other infinite sums converge more rapidly, and the default settings
+work admirably. For example \verb`\eval{$ (1+0.1234)^{4.321} $}`
+$\Longrightarrow$ \eval{$ (1+0.1234)^{4.321} $}. Using binomial
+coefficients we can express this as an infinite sum:\medskip{}
+\begin{verbatim}
+ \eval[p=.]{\[
+ \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
+ \]}[\alpha=4.321,x=0.1234]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]
+ {\[
+ \sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
+ \]}[\alpha=4.321,x=0.1234]
+
+\section{Error messages }
+
+\label{sec:evalErrors}There are two kinds of error in \texttt{numerica}:
+those in the underlying \LaTeX{} which are reported in the \LaTeX{}
+log, shown on the terminal, and generally halt compilation, and specifically
+\texttt{numerica}-related errors which do not halt compilation and
+produce messages displayed in the pdf where one would expect the result
+of the calculation to be. The original reason for doing things this
+way was to enable \texttt{numerica} to be used effectively with the
+instant preview facility of the document processor \LyX . More philosophically,
+one might view such errors as similar to grammatical errors or spelling
+mistakes in text. It is not clear that they should halt compilation.
+Hence strictly \texttt{numerica}-related errors leave brief messages
+in the pdf at the offending places.
+
+Before discussing specific error messages, note that there is a debug
+facility (of a sort) discussed below in §\ref{subsec:settingsDebug}.
+
+Error messages are in two parts: a \emph{what} part and a \emph{where}
+part.
+
+\subsection{Mismatched brackets}
+
+\label{subsec:errorsMismatched-brackets}An unmatched left parenthesis
+or other left bracket (in this case a missing right parenthesis) usually
+results in a \texttt{numerica} error:
+\begin{centred}
+\verb`$\eval{\sin(\pi/(1+x)}[x=1]$` $\Longrightarrow$ $\eval{\sin(\pi/(1+x)}[x=1]$
+\end{centred}
+For the same error in the vv-list, the what-part remains unchanged
+but the where-part is altered:
+\begin{centred}
+\verb`$\eval{ 1+y }[x=1,y=\sin(\pi/(1+x)]$` $\Longrightarrow$ $\eval{ 1+y }[y=\sin(\pi/(1+x),x=1]$
+\end{centred}
+The \emph{what} message is the same; the \emph{where} is different.
+
+An unmatched right parenthesis or other right bracket (in this case
+a missing \emph{left} parenthesis) usually results in a similar \texttt{numerica}
+error:
+\begin{centred}
+\verb`$\eval{2((x+y)/(y+z)))^2}[x=1,y=2,z=3]$` $\Longrightarrow$
+\eval{2((x+y)/(y+z)))^{2}}[x=1,y=2,z=3]
+\end{centred}
+But note that an unmatched modifier like \verb`\left` or \verb`\right`
+is a \LaTeX{} error and is caught by \LaTeX{} before \texttt{numerica}
+can respond and so results in a terminal and logfile message.
+
+\subsection{Unknown tokens}
+
+An `Unknown token' message can arise in a number of ways. If an
+expression involves a number of variables, some of which depend on
+others, their order in the vv-list matters:
+\noindent \begin{center}
+\verb`$\eval{\tfrac12 vt}[t=2,v=gt,g=9.8]$` $\Longrightarrow$ \eval{\tfrac{1}{2}vt}[t=2,v=gt,g=9.8]
+\par\end{center}
+
+The vv-list is evaluated from the \emph{right} so that in this example
+the variable \texttt{v} depends on a quantity \texttt{t} that is not
+yet defined. Hence the message. The remedy is to move \texttt{t} to
+the right of \texttt{v} in the vv-list.
+
+Similarly, if we use a variable in the formula that has not been assigned
+a value in the vv-list, we again get the `Unknown token' message,
+but this time the location is the formula:
+\begin{centred}
+\verb`$\eval{\pi r^2h}[r=3]$` $\Longrightarrow$ \eval{\pi r^{2}h}[r=3]
+\end{centred}
+The remedy obviously is to assign a value to \texttt{h} in the vv-list\texttt{.}
+
+The same message will result if a mathematical operation or function
+is used that has not been implemented in \texttt{numerica}:
+\begin{centred}
+\verb`$\eval{u \bmod v }[v=7,u=3]$` $\Longrightarrow$ \eval{u\bmod v}[v=7,u=3]
+\end{centred}
+A missing comma in the vv-list will generally result in an unknown
+token message:
+\begin{centred}
+\verb`$\eval{axy}[a=3 y=2,x=1]$` $\Longrightarrow$ \eval{axy}[a=3y=2,x=1]
+\end{centred}
+Because of the missing comma, \texttt{numerica} assumes \texttt{a}
+has the `value' \texttt{3y=2} and has no knowledge of \texttt{y}
+as a variable.
+
+The presence of multi-token variables can also cause this error message
+if the check for such variables is turned off; see §\ref{subsec:settingsMultitokSwitch}.
+
+\subsection{Other vv-list errors}
+
+Other errors that can occur with the vv-list are overlooked value
+assignments to variables, or missing commas. For the first, it is
+essential that we do actually give a value to all variables occuring
+in the vv-list:
+\begin{centred}
+\verb`$\eval{axy}[a=3,y=,x=1]$` $\Longrightarrow$ \eval{axy}[a=3,y=,x=1]
+\end{centred}
+The remedy is obvious – assign a value to \texttt{$y$}.
+
+\emph{Extra} commas in the vv-list should cause no problems:
+\begin{centred}
+\verb`$\eval{axy}[,a=3,,y=2,x=1,]$` $\Longrightarrow$ $\eval{axy}[,a=3,,y=2,x=1,]$
+\end{centred}
+
+\subsection{Inverse powers of negative numbers}
+
+\label{subsec:errorsInverse-powers}Inverse integer powers of positive
+numbers should always be possible, but raising a \emph{negative} number
+to an inverse power generates an error even when, mathematically,
+it should not:
+\begin{centred}
+\verb`\eval{$ (-125)^{1/3} $}` $\Longrightarrow$ \eval{$ (-125)^{1/3} $}
+\end{centred}
+This is a feature of floating point arithmetic. When a number is raised
+to a rational power, say $p/q$ where $p$ and $q$ are non-zero integers,
+then the result is the $p$-th power of the $q$-th root of the number.
+Can a $q$-th root be taken? If our floating point system used (for
+ease of illustration) only $4$ significant digits, $p/q=1/3$ would
+be the fraction $3333/10^{4}$, an odd numerator over an even denominator.
+But a negative number does not possess an even ($10^{4}$-th) root.
+
+\subsection{Integer argument errors}
+
+\label{subsec:errorsIntegerArgs}Some functions require integer arguments
+– factorials, the second argument of a binomial coefficient, and (in
+\texttt{numerica}) $n$-th roots using the optional argument of \texttt{\textbackslash sqrt};
+also summation and product variables. If integers are explicitly entered
+for these arguments there is no problem, but if the value of the argument
+is the result of a calculation, rounding errors require thinking about.
+What accumulation of rounding errors is \emph{too} much so that the
+result of the calculation \emph{cannot} be considered an integer?
+\texttt{numerica} is generous: in the default setup, if a calculation
+rounds to an integer at rounding value $14$ the result of the calculation
+is considered an integer (obviously, the integer resulting from the
+rounding). Since \texttt{l3fp} works to $16$ significant figures
+that gives ample room for rounding errors to `get lost in' and be
+ignored, while still ruling out such things as (recall the example
+in §\ref{subsec:evalBoolean-output}),
+\begin{centred}
+\verb`\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]` $\Longrightarrow$
+\eval{\[ \sum_{n=1}^N n \]}[N=1/0.0123456789]
+\end{centred}
+where $N$ differs from $81$ not until the seventh decimal place.
+
+The default rounding value of $14$ for `int-ifying' calculations
+can be changed: see §\ref{subsec:defaultsIntifyingRounding}.
+
+\subsection{Comparison errors}
+
+Should a user try to make a comparison using a combination like \verb`>=`
+rather than \verb`\geq`, \texttt{numerica} admonishes like this:
+\begin{centred}
+\verb`$\eval{ \pi^e >= e^\pi }[?]$` $\Longrightarrow$ $\eval{ \pi^e >= e^\pi }[?]$
+\end{centred}
+(The relation is false by the way.)
+
+\subsection{Invalid base for \texttt{\textbackslash log}}
+
+ISO recommends using \verb`\log` only with a subscripted base specified.
+Otherwise how is one to know whether the base is \verb`e` or $10$
+or $2$ or whatever? Nonetheless \texttt{numerica} assumes that when
+\verb`\log` is used unsubscripted, the base is 10. Suppose you want
+to make $12$ the base, but forget to put braces around the $12$:
+\begin{centred}
+\verb`$\eval{ \log_12 1728 }$` $\Longrightarrow$ $\eval{ \log_12 1728 }$
+\end{centred}
+Here, \texttt{numerica} has taken \texttt{1} as the base (and $21728$
+as the argument) of the logarithm and responds accordingly.
+
+\subsection{\texttt{l3fp} errors}
+
+Some errors arising at the \texttt{l3fp} level are trapped and a message
+displayed.
+\begin{itemize}
+\item Dividing by zero
+\end{itemize}
+\begin{centred}
+\verb`$\eval{1/\sin x}[x=0]$` $\Longrightarrow$ \eval{1/\sin x}[x=0]
+\end{centred}
+Note however that \verb`$\eval{1/\sin x}[x=\pi]$` $\Longrightarrow\,\eval{1/\sin x}[x=\pi]$,
+because of rounding errors in distant decimal places. No doubt this
+is true for other functions as well.
+\begin{itemize}
+\item Invalid operation
+\end{itemize}
+\begin{centred}
+\verb`$\eval{\arccos x}[x=2]$` $\Longrightarrow$ \eval{\arccos x}[x=2]
+\end{centred}
+In this case the inverse cosine has been fed a value of $x$ outside
+its domain of definition, the interval $[-1,1]$. Trying to evaluate
+an expression that resolves to $0/0$ also produces this message:
+\begin{centred}
+\verb`$\eval{\frac{1-y}{x-2}}[x=2,y=1]$`$\Longrightarrow$ \eval{\frac{1-y}{x-2}}[x=2,y=1]
+\end{centred}
+\begin{itemize}
+\item Overflow/underflow
+\end{itemize}
+The factorial (discussed in §\ref{subsec:evalFactorialsBinom}) provides
+an example of overflow:
+\begin{centred}
+\verb`$\eval{3249!}$`\texttt{ }$\Longrightarrow$ \eval{3249!}
+\end{centred}
+This is hardly surprising since
+\begin{centred}
+\verb`$\eval{3248!}[x]$` $\Longrightarrow$ $\eval{3248!}[x]$.
+\end{centred}
+There is a limit on the size of exponents that \texttt{l3fp} can handle.
+A number in the form $a\times10^{b}$ must have $-10001\le b<10000$.
+If this is not the case an overflow or underflow condition occurs.
+As the examples show, an overflow condition generates a \texttt{numerica}
+error. For underflow, where the number is closer to $0$ than $10^{-10001}$,
+\texttt{l3fp} assigns a zero value to the quantity. \texttt{numerica}
+accepts the zero value.
+
+\chapter{Settings}
+
+\label{chap:Settings}A calculation is effected against a background
+of default values for various quantities. For a particular calculation,
+these values may not be appropriate; or you may have different preferences.
+The way to change settings for a particular calculation is through
+the settings option of \verb`\nmcEvaluate` discussed next. The way
+to change a \emph{default} setting is by creating a configuration
+file \texttt{numerica.cfg} discussed in §\ref{sec:settingsDefaults}.
+
+\section{Settings option}
+
+\label{sec:settingsOption}The second argument of the \verb`\nmcEvaluate`
+command is the settings option, delimited by square brackets. This
+option is a \emph{key=value} list, hence comma-separated. \emph{Key=value
+}lists tend to be wordy. For back-of-envelope calculations one wants
+to be able to `dash off' the calculation, hence the short, cryptic
+nsture of the keys. Most settings are generic, applicable not only
+to \verb`\nmcEvaluate` but also to other commands that are available
+if \texttt{numerica} is loaded with the \texttt{plus} option; see
+the associated document \texttt{numerica-plus.pdf}.
+
+\subsection{\textquoteleft Debug\textquoteright{} facility}
+
+\label{subsec:settingsDebug}It is rather grandiose to call this a
+debug facility, but if a calculation goes wrong or produces a surprising
+result, \texttt{numerica} offers a means of examining some intermediate
+stages on the way to the final result. To use the facility, enter
+\begin{lyxcode}
+dbg~=~<integer>
+\end{lyxcode}
+into the settings option. (White space around the equals sign is optional.)
+\begin{itemize}
+\item \texttt{dbg=0 }turns off the debug function, displays the result or
+error message (this is the default);
+\item \texttt{dbg=1 }equivalent to \texttt{dbg=2{*}3{*}5{*}7};
+\end{itemize}
+The `magic' integers are the following primes and their products:
+
+\begin{table}[t]
+\centering
+\noindent \centering{}\caption{Settings options}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. funcions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}} & \tabularnewline
+{\small\texttt{p}} & token(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+\begin{itemize}
+\item \texttt{dbg=2} displays the vv-list after multi-token variables have
+been converted to their single token form, \texttt{\textbackslash\_nmca},
+\texttt{\textbackslash\_nmcb}, etc.;
+\item \texttt{dbg=3} displays the formula after multi-token variables have
+been converted to their single token form;
+\item \texttt{dbg=5} displays the stored variables and their evaluated values
+(\texttt{dbg=2} lists the values as expressions; here they have been
+evaluated); note that any saved values (Chapter~\ref{chap:Saving-and-retrieving})
+that have been loaded will also feature in this list which might lead
+to a messy display depending on the nature of those values;
+\item \texttt{dbg=7} displays the formula after it has been fp-ified (but
+before it has been fed to \texttt{l3fp} to evaluate);
+\begin{itemize}
+\item should the formula successfully evaluate, the result of the evaluation
+is also displayed.
+\end{itemize}
+\end{itemize}
+\vspace{-1ex}
+
+\begin{wraptable}{o}{0.5\columnwidth}%
+\centering{}\caption{Magic integers}\label{tab:settingsMagic-integers}
+\begin{center}
+\begin{tabular}{rrl}
+\toprule
+integer & & \multicolumn{1}{c}{factors}\tabularnewline
+\midrule
+6 & & 2,3\tabularnewline
+10 & & 2,5\tabularnewline
+14 & & 2,7\tabularnewline
+15 & & 3,5\tabularnewline
+21 & & 3,7\tabularnewline
+30 & & 2,3,5\tabularnewline
+35 & & 5,7\tabularnewline
+42 & & 2,3,7\tabularnewline
+70 & & 2,5,7\tabularnewline
+105 & & 3,5,7\tabularnewline
+210 & & 2,3,5,7\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+To display two or more of these elements simultaneously, use the product
+of their debug numbers for the magic integer. This can be entered
+either as the multiplied-out product, or as the `waiting to be evaluated'
+product with asterisks (stars) between the factors. Thus \texttt{dbg=6}
+or \verb`dbg=2*3` display both the vv-list and formula after multi-token
+variables have been converted to single token form; \texttt{dbg=10}
+or \verb`dbg=2*5` display both the vv-list after multi-token variables
+have been converted to single token form and the recorded variables
+with their evaluated values. And similarly for the other magic integers
+listed. For other integers, if they are divisible by $2$ or $3$
+or $5$ or $7$, they will display the corresponding component. Both
+\texttt{dbg=210} and \verb`dbg=2*3*5*7` display all four elements,
+but rather than remembering this product, it suffices to put \texttt{dbg=1}.
+This is equivalent and displays all elements.
+
+The debug option uses an \texttt{aligned }or \texttt{align{*}} environment
+to display its wares, depending on the presence or absence of math
+delimiters around the \texttt{\textbackslash eval} command. In the
+next example I have used multi-token variables to illustrate the different
+elements in the debug display, and a chain of comparisons to show
+how \texttt{numerica} treats these (§\ref{subsec:evalBoolean-output}).
+\begin{verbatim}
+ \eval[dbg=1]{ a_1<2a_2<3a_3<\pi+e }
+ [a_1=\pi,a_2=\phi,a_3=e\gamma][6?'']
+\end{verbatim}
+$\Longrightarrow$ \eval[dbg=1]{ a_1<2a_2<3a_3<\pi+e }
+ [a_1=\pi,a_2=\phi,a_3=e\gamma][6?'']
+
+Note that the four elements are displayed in temporal order: first
+comes the vv-list after conversion of multi-token to single-token
+variables, then the formula in the single-token variables; these are
+created essentially at the same time. The vv-list is presented in
+left-to-right order because that is the direction of evaluation \emph{internally}
+in \texttt{numerica}. Next the stored values of the variables are
+displayed. These are the values \emph{after }evaluation. The fourth
+element both in the display and chronologically is the fp-ified formula;
+this is often a thicket of parentheses. The final element of the display
+and chronologically is the result of evaluating the formula. This
+is displayed only if $7$ is a factor of the \texttt{dbg} integer,
+and there is no error.
+
+When interpreting the fp-form, one should be aware of differences
+between \texttt{numerica} and \texttt{l3fp}. In particular be aware
+that in \texttt{l3fp} function calls bind most tightly so that, for
+example, \verb`sin 2pi` evaluates not to zero but to $(\sin2)\times\pi$
+and \verb`sin x^2` evaluates to $(\sin x)^{2}$. This should not
+be of any concern to the user except as here in debug mode when interpreting
+fp-forms.
+
+Finally, note that those mathematical operations that have no direct
+representation in \texttt{l3fp} contribute only their value to the
+fp-form. This applies to sums and products, double factorials and
+partly to binomial coefficients as illustrated in the followng (ridiculous)
+example:
+\begin{verbatim}
+ \eval[dbg=1]{\[
+ \sum_{n=1}^5 n + \binom{10}{m}
+ + \prod_{n=2}^5 (1-1/n) + m!! \][m=6]
+\end{verbatim}
+$\Longrightarrow$ \eval[dbg=1]{\[
+ \sum_{n=1}^5 n+\binom{10}{m}
+ +\prod_{n=2}^5(1-1/n) +m!! \]}[m=6]
+
+\noindent The various contributions to the overall result are displayed
+simply as numbers because \texttt{l3fp} does not (at least as yet)
+handle these elements natively.
+
+\subsubsection{Negative \texttt{dbg} values}
+
+Negative \texttt{dbg} values are possible: \texttt{dbg=-2}, \texttt{dbg=-3},
+etc. (and \texttt{dbg=-1} meaning \texttt{dbg=-210}) have exactly
+the same effects as the corresponding positive values except for some
+details of display. The display for positive \texttt{dbg} values is
+the one evident in the examples above. Lines wrap, the left margin
+is not indented and the display occupies the page width. For negative
+\texttt{dbg} values, lines do not wrap, the left margin is indented
+and the display occupies the text width. An example is presented in
+§\ref{subsec:nestDebugging} below where the display for a nested
+\verb`\eval` is significantly improved with a negative \texttt{dbg}
+value.
+
+\subsection{Reuse setting}
+
+This setting determines whether the entire display or only the numerical
+result is saved to file with the \verb`\nmcReuse` command. See below,
+Chapter~\ref{chap:Saving-and-retrieving}, §\ref{subsec:reuse=00005Ceval's-reuse-setting}.
+
+\subsection{Inputting numbers in scientific notation}
+
+\label{subsec:settingsInputtingSciNotation}\emph{Outputting} numbers
+in scientific notation is controlled by the final trailing argument
+of the \texttt{\textbackslash eval} command. That is turned off by
+default and needs to be explicitly ordered. Similarly, \emph{inputting}
+numbers in scientific notation is turned off by default and needs
+to be explicitly ordered. To turn it on, write
+\begin{lyxcode}
+\textasciicircum ~=~<char>
+\end{lyxcode}
+\noindent in the settings option, where \texttt{<char>} is any single
+character, usually \texttt{e} or \texttt{d} or their upper-casings,
+but not restricted to them: \texttt{\textasciicircum =@} for instance
+is perfectly possible, and has the advantage over \texttt{e} or \texttt{d}
+that it doesn't conflict with the use of the character as a variable
+or constant.
+\begin{centred}
+\verb`$ \eval[^=@]{ 1.23@-1 } $` $\Longrightarrow$ $ \eval[^=@]{ 1.23@-1 } $.
+\end{centred}
+With letters for the exponent mark – say \verb`d` or \verb`e` –
+the problem is interpreting forms like \texttt{8d-3} or \texttt{2e-1}.
+Does such a form denote a number in scientific notation or an algebraic
+expression? In \texttt{numerica}, if the settings option shows \texttt{\textasciicircum =d},
+then a form like \texttt{8d-3} is treated as a number in scientific
+notation. Similarly for \texttt{e} or any other letter used as the
+exponent marker for the input of scientific numbers. (But only one
+character can be so used at a time.) Note that the number \emph{must}
+start with a digit: \verb`e-1` for instance does not and will be
+treated as an algebraic expression involving the exponential constant
+(unless \verb`e` is assigned a different value in the vv-list).
+\begin{centred}
+\verb`$ \eval[^=e]{ x+e-1 }[x=2e-1] $` $\Longrightarrow$ $ \eval[^=e]{ x+e-1 }[x=2e-1] $.
+\end{centred}
+The problem here is that \verb`2e-1` is treated as a number in scientific
+notation but displays in the vv-list as if it were an algebraic expression.
+The solution is to put \verb`2e-1` into an \verb`\mbox` or \verb`\text`
+command in the vv-list:
+\begin{centred}
+\verb`$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $` $\Longrightarrow$
+$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $.
+\end{centred}
+If you use a particular character as the exponent marker for inputting
+numbers in scientific notation, it is good practice \emph{not} to
+use that character as a variable, not because it will cause an error
+but because it makes expressions harder to read.
+
+\subsection{Multi-token variables}
+
+\label{subsec:settingsMultitokSwitch}Variables need not consist of
+a single character or token (like $x$ or $\alpha$). Multi-token
+symbols like $x'$ or $t_{i}$ or $var$ are perfectly acceptable.
+For its internal operations, \texttt{numerica} converts such multi-token
+names to single tokens (as discussed in §\ref{subsec:evalVariableNames}).
+This conversion takes time. Even if there are no multi-token variables
+used at all, \texttt{numerica} still needs to check that that is so.
+There is a setting that allows a user to turn off or turn on the check
+for such variables by entering
+\begin{lyxcode}
+xx~=~<integer>
+\end{lyxcode}
+into the settings option. If \texttt{<integer>} is \texttt{0}, the
+check for (and conversion of) multi-token variables is turned off;
+if \texttt{<integer>} is \texttt{1} (or any other\emph{ non-zero}
+integer), the check, and conversion if needed, goes ahead. By default,
+checking for multi-token variables and converting them if found is
+turned \emph{on}. (The name for the key, \texttt{xx}, is chosen because
+\texttt{x} is the most familiar variable of all, introduced in elementary
+algebra, and doubling it like this suggests multi-token-ness.)
+
+If checking is turned off when a multi-token variable is present,
+an error results. We don't need to enter \texttt{xx=1} in the first
+of the following examples because the check for multi-token variables
+is on by default. Explicitly turning it off in the second produces
+an error.
+\begin{centred}
+\verb`\eval{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval{$ x_0^{\,2} $}[x_0=5],\medskip{}
+\verb`\eval[xx=0]{$ x_0^{\,2} $}[x_0=5]` $\Longrightarrow$ \eval[xx=0]{$ x_0^{\,2} $}[x_0=5]
+\end{centred}
+
+\subsection{Parsing arguments of trigonometric functions}
+
+This setting allows a wider range of arguments to trigonometric functions
+to be used (think Fourier series) without needing to insert extra
+parentheses in order for them to be read correctly by \verb`\eval`;
+see §\ref{subsec:parseTrigFns}.
+
+\subsection{Using degrees rather than radians}
+
+\label{subsec:settingsDegrees}You may find it more convenient to
+use degrees rather than radians with trigonometric functions.This
+can be switched on simply by entering a lowercase \texttt{o} in the
+settings option. (The author's fond hope is that the charitable eye
+might see a degree symbol in the \texttt{o}.) Thus
+\begin{centred}
+\verb`\eval[o]{$ \sin 30 $}` $\Longrightarrow$ \eval[o]{$ \sin 30 $},
+
+\verb`\eval[o]{$ \arcsin 0.5 $}` $\Longrightarrow$ \eval[o]{$ \arcsin 0.5 $}.
+\end{centred}
+
+\subsection{Specifying a logarithm base}
+
+\label{subsec:settingsLogBase}If you wish to use \verb`\log` without
+a subscripted base in a particular calculation, then add an entry
+like
+\begin{lyxcode}
+log~=~<positive~number${}\,\mathtt{\ne1}$>
+\end{lyxcode}
+to the settings option of the \verb`\eval` command. The \verb`<positive number>`
+does not need to be an integer. It could be \verb`e` (if you object
+to writing \verb`\ln`) but is more likely to be $2$ or another small
+integer; $10$ is the default. If you want to use this changed base
+not for one but most calculations, then add an entry with your choice
+of base to a configuration file; see §\ref{sec:settingsDefaults}.
+
+\subsection{Calculation mode}
+
+\label{subsec:settingsRecalcMode}A variable may change in the course
+of a calculation. This is certainly true of sums and products. If
+a parameter in the vv-list depends on the variable then that parameter
+will need to be recalculated, perhaps repeatedly, in the course of
+a calculation. By entering \label{subsec:settingsCalculationMode}
+\begin{lyxcode}
+vvmode~=~<integer>
+\end{lyxcode}
+in the settings option it is possible to turn on or off the ability
+to repeatedly evaluate the vv-list; \verb`<integer>` here takes two
+possible values, \texttt{0} or \texttt{1}. \texttt{vvmode=0} means
+the vv-list is evaluated once at the start of the calculation; \texttt{vvmode=1}
+means the vv-list is recalculated every time the relevant variable
+changes.
+
+For example, it may be desirable to place the summand, or some part
+of it, in the vv-list. Since the summation variable obviously changes
+during the course of the calculation, we need to enter \texttt{vvmode=1}
+in the settings option. Repeating an earlier sum (the seting \verb`p=.`
+is discussed in §\ref{subsec:settingsPunctuation}) \medskip{}
+\begin{verbatim}
+ \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
+ [N=100,f(k)=1/k^3,{k}=1][4]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
+ [N=100,f(k)=1/k^3,{k}=1][4]
+
+As you can see, the summand \texttt{f(k)} has been given explicit
+form in the vv-list – equated to \texttt{1/k\textasciicircum 3}.
+That means we need to give a preceding value to \texttt{k} in the
+vv-list; hence the rightmost entry. But we don't want \texttt{k=1}
+appearing in the final display, so we wrap \texttt{k} in braces (see
+§\ref{subsec:evalVvSuppressVars}). Since the value \texttt{k=1} applies
+only to the first term in the sum, to ensure it is not used for all
+terms, we enter \texttt{vvmode=1} in the settings option. This turns
+vv-recalculation mode on and ensures \texttt{k=1} is overwritten by
+\texttt{k=2}, \texttt{k=3} and so on, and the vv-list recalculated
+each time. The final result is the same as before, although recalculating
+the vv-list at each step is a more resource-hungry process. The difference
+may not be marked for this example; with more complicated expressions
+it noticeably takes longer.
+
+Because it is necessary to activate this switch when using \emph{implicit}
+notations – like $f(k)$ in the example – rather than the explicit
+form of the function in the main argumet it seems natural to call
+\texttt{vvmode=1} \emph{implicit }mode and \texttt{vvmode=0} (the
+default) \emph{explicit }mode. Most calculations are explicit mode
+– the vv-list is evaluated only once.\emph{ }
+
+\subsection{Changing the vv-list display format}
+
+\label{subsec:settingsVvDisplayChangeLocal}In previous formulas with
+variables the vv-list has been displayed following the result. It
+is wrapped in parentheses following a comma followed by a space. These
+formatting elements – comma, space, parentheses – can all be changed
+with the settings option.
+
+The default format specification is
+\begin{lyxcode}
+\{,\}\textbackslash mskip~12mu~plus~6mu~minus~9mu(vv)
+\end{lyxcode}
+for a text-style display (an inline formula) and
+\begin{lyxcode}
+\{,\}\textbackslash mskip~36mu~minus~24mu(vv)
+\end{lyxcode}
+in a display-style context. The commas are wrapped in braces because
+these are items in a comma-separated list. Both entries exhibit the
+elements: punctuation (comma), preceding a variable space, preceding
+the parenthesized vv-list (the \texttt{vv} placeholder). No full stop
+is inserted after the closing parentheses because the \texttt{\textbackslash eval}
+command may occur in the middle of a sentence (even in display style).
+For inline use, the elasticity of the space becomes relevant when
+\TeX{} is adjusting individual lines to fit sentences into paragraphs
+and paragraphs into pages. The largest spacing that can be stretched
+to is a quad, $18$~mu (mu~=~math unit), and the smallest that
+can be shrunk to is a thin space, $3$~mu. In display style,\texttt{
+}the largest spacing specified is the double quad, in line with the
+recommendation in \emph{The \TeX{} Book}, Chapter~18, but this can
+shrink to a single quad, for instance if the vv-list is heavily populated
+with variables so that the evaluated result is pushed well to the
+left by the vv-list. (But see below, §\ref{subsec:settings New-line-display}.)
+
+If you want to change these defaults, enter in the settings option
+\begin{lyxcode}
+vvi~=~<new~specification>
+\end{lyxcode}
+to change the inline display and
+\begin{lyxcode}
+vvd~=~<new~specification>
+\end{lyxcode}
+to change the display-style display For example the settings
+\begin{lyxcode}
+vvi~=~\{,\}\textbackslash quad(vv)
+
+vvd~=~\{,\}\textbackslash qquad(vv)
+\end{lyxcode}
+would give a comma (in braces since the settings option is a comma-separated
+list) and a fixed space (of one or two quads) between the result and
+the parenthesized vv-list.
+
+The vv-list itself in the display specification is represented by
+the placeholder \texttt{vv}. If the \texttt{vv} is omitted from the
+specification, then the vv-list will not appear at all:
+\begin{centred}
+\verb`\eval[vvi=?!]{$ \pi $}[\pi=3]` $\Longrightarrow$ \eval[vvi=?!]{$ \pi $}[\pi=3]
+\end{centred}
+More relevantly, it may well be the case that all variables in the
+vv-list are suppressed (wrapped in braces). In that case the display
+would look something like \verb`, ()`. To prevent this enter \verb`vvi=`
+in the vv-list, in the inline case, or \verb`vvd=` in the display-style
+case, i.e. enter an empty value. (Alternatively, use the star option
+of the \verb`\eval` command.)
+
+Another minor wrinkle occurs if you want to change parentheses around
+the vv-list to square brackets. Because the settings option is a square-bracket
+delimited argument, the square brackets in the specification will,
+like commas, need to be hidden in braces, although you can get away
+with braces around the whole spec.:
+\begin{lyxcode}
+vvi=\{,\textbackslash mskip~12mu~plus~6mu~minus~9mu~{[}vv{]}\}
+\end{lyxcode}
+
+\subsection{Displaying the vv-list on a new line }
+
+\label{subsec:settings New-line-display}Display of a long formula
+with many variables, hence a full vv-list, may not fit comfortably
+on a line. In an earlier example I used Brahmagupta's formula to calculate
+the area of a triangle. It squeezed onto a line. I shall now use his
+formula for the area of a cyclic quadrilateral:
+\[
+A=\sqrt{(s-a)(s-b)(s-c)(s-d)}.
+\]
+The extra side (quadrilateral as against triangle) means there is
+a further variable to accommodate, not only in the formula but also
+in the vv-list. In the following example, the cyclic quadrilateral
+is formed by a 45-45-90 triangle of hypotenuse 2 joined along the
+hypotenuse to a 30-60-90 triangle. The sides are therefore $\surd2,\surd2,\surd3,1$.
+Adding the areas of the two triangles, the area of the quadrilateral
+is $A=1+\tfrac{1}{2}\surd3$, or in decimal form, \verb`$\eval{1+\tfrac12\surd3}$`
+$\Longrightarrow$ $\eval{1+\tfrac12\surd3}$. Let's check with Brahmagupta's
+formula:
+\begin{verbatim}
+ \eval[p=.,vvd={,}\\(vv),*]
+ {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
+ [s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.,vvd={,}\\(vv),*]
+ {\[ \sqrt{(s-a)(s-b)(s-c)(s-d)} \]}
+ [s=\tfrac12(a+b+c+d),
+ a=\surd2,b=\surd2,c=\surd3,d=1]
+
+\noindent The values agree. The point to note here is the\texttt{
+vvd=\{,\}\textbackslash\textbackslash (vv)} and the \texttt{{*}}
+in the settings option. The \texttt{\textbackslash\textbackslash}
+in a specification for \texttt{vvd} acts as a trigger for \texttt{numerica}
+to replace whatever math delimiters are enclosed by the \verb`\eval`
+command with a \verb`multline` environment. As you can see, the specification
+inserts a comma after the formula and places the parenthesized vv-list
+on a new line. The star \texttt{{*}} if present suppresses equation
+numbering by turning the \verb`multline` into a \verb`multline*`
+environment.
+
+Things to note in the use of\texttt{ \textbackslash\textbackslash}
+in a \texttt{vvd} specification are that
+\begin{itemize}
+\item it applies only to the \texttt{vvd} specification, not the \texttt{vvi}
+spec.;
+\item it applies only when\emph{ }\verb`\eval`\emph{ wraps around }a math
+environment of some kind;
+\item it has no effect when the \verb`\eval` command is used \emph{within}
+a math environment when the presentation of the result is of the form
+\emph{result, vv-list}. The formula is not displayed and so the pressure
+on space is less and the `ordinary' vv-list specification is used.
+\end{itemize}
+
+\subsection{Punctuation}
+
+\label{subsec:settingsPunctuation}The \verb`\eval` command can be
+used within mathematical delimiters or it can be wrapped around mathematical
+delimiters. The latter gives a \emph{formula=result} style of display
+automatically, which is convenient. One doesn't need to write the
+\emph{formula=} part of the expression, but it causes a problem when
+\verb`\eval` wraps around a display-style or similar environment:
+how to display a following punctuation mark? For an inline display
+we can simply follow the \verb`\eval` command with the appropriate
+punctuation, for instance: \verb`\eval{$ 1+1 $}.` $\Longrightarrow$
+\eval{$ 1+1 $}. But with \verb`\[ \]` delimiters used \emph{within}
+the \verb`\eval` command – \verb`\eval{\[ 1+1 \]}.` – the fullstop
+slides off to the start of the next line, since it is beyond the closing
+delimiter. We want it to display as if it were the last element \emph{before}
+the closing delimiter.
+
+Explicitly putting it there – \verb`\eval{\[ 1+1. \]}` – means the
+punctuation mark becomes part of the formula. Potentially \texttt{numerica}
+then needs to check not just for a fullstop but also other possible
+punctuation marks like comma, semicolon, perhaps even exclamation
+and question marks. All these marks have roles in mathematics or \texttt{l3fp}.
+Including them in the formula means distinguishing their punctuation
+role from their mathematical role and can only cause difficulties
+(and code bloat).
+
+Instead, \texttt{numerica} uses the setting
+\begin{lyxcode}
+p~=~<char(s)>~
+\end{lyxcode}
+to place the \texttt{char(s)} after the result but within the environment
+delimiters. The default punctuation mark is the comma so that simply
+entering \texttt{p} will produce a comma in the appropriate place.
+This saves having to write \texttt{p=\{,\}} as would otherwise be
+required, since the settings option is a \emph{comma}-separated list.
+
+Nor is one limited to a single punctuation mark:
+\begin{centred}
+\verb`\eval[p=\ (but no 8!)]{\[ \frac{1}{81} \]}[9]` $\Longrightarrow$
+\eval[p=\ \text{(but no 8!)}]{\[ \frac{1}{81} \]}[9]
+\end{centred}
+
+\section{Infinite sums and products}
+
+\label{sec:settingsInfiniteSumsProds}There are ways of tweaking various
+default settings to nudge infinite sums and products to a correct
+limit. These tweaks are applied via the settings option of the \verb`\eval`
+command.
+
+\begin{table}
+\centering
+\noindent \centering{}\caption{Settings for infinite sums \& products}\label{tab:settingsSumsProducts}
+\noindent \begin{center}
+{\small{}%
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+key & type & meaning & default\tabularnewline
+\midrule
+\texttt{S+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
+\texttt{S?} & $\text{int}\ge0$ & stopping criterion query terms for sums & \texttt{0}\tabularnewline
+\texttt{P+} & int & extra rounding for stopping criterion & \texttt{2}\tabularnewline
+\texttt{P?} & $\text{int}\ge0$ & stopping criterion query terms for products & \texttt{0}\tabularnewline
+\bottomrule
+\end{tabular}}
+\par\end{center}
+\end{table}
+The normal convergence criterion used by \texttt{numerica} to determine
+when to stop adding/multiplying terms in an infinite sum/product is
+\emph{when the next term added/multiplied leaves the total unaltered
+when rounded to 2 more digits than the specified rounding value.}
+Suppose $T_{k}$ is the sum/product after the inclusion of $k$ terms,
+and $r$ is the rounding value. Denote $T_{k}$ rounded to $r$ figures
+by $\left(T_{k}\right)_{r}$. \emph{The infinite sum or product stops
+at the $(k+1)$-th term (and the value is attained at the $k$-th
+term) when }$\left(T_{k+1}\right)_{r+2}=\left(T_{k}\right)_{r+2}$.
+The hope is that if this is true at rounding value $r+2$ then at
+rounding value $r$ the series or product will have attained a stable
+value at that level of rounding.
+
+For a series of monotonic terms converging quickly to a limit, this
+stopping criterion works well, less so if convergence is slower, as
+seen earlier with the infinite product for $\sinh1$. The criterion
+can fail completely when terms behave in a non-monotonic manner. Distant
+terms of a Fourier series, for example, may take zero values; the
+criterion is necessarily satisfied but the series may still be far
+from its limit. In a product the equivalent would be a distant term
+taking unit value. Such series or products may also have initial `irregular'
+terms including zero/unit terms. A summation/product might stop after
+only one or two additions/multiplications if the criterion were applied
+to them.
+
+To cope with these possibilities, \texttt{numerica} offers two settings
+for sums, two for products, summarized in Table~\ref{tab:settingsSumsProducts}.
+These are entered in the settings option of the \verb`\eval` command.
+\begin{itemize}
+\item \texttt{S+=<integer> }or \texttt{P+=<integer>} additional rounding
+on top of the specified (or default) rounding for the calculation;
+default = $2$
+\begin{itemize}
+\item the larger the additional \texttt{<integer>} is, the more likely that
+sum or product has attained a stable value at the specified rounding
+$r$
+\end{itemize}
+\item \texttt{S?=<integer${}\,\mathtt{\ge0}$> }or \texttt{P?=<integer${}\,\mathtt{\ge0}$>}
+the number of final terms to query after the stopping criterion has
+been achieved to confirm that it is not an `accident' of particular
+values; default = $0$
+\begin{itemize}
+\item a final few terms to be summed/multiplied and the rounded result after
+each such operation to be compared with the rounded result at the
+time the stopping criterion was achieved. Suppose the additional rounding
+(\texttt{S+} or \texttt{P+}) is $n$ on top of the specified rounding
+$r$ and let the number of final checking terms be $m$. Suppose $T_{k_{0}}$
+is the first term at which the stopping criterion is achieved: $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1}\right)_{r+n}$.
+What we require of the final query terms is that $\left(T_{k_{0}}\right)_{r+n}=\left(T_{k_{0}+1+j}\right)_{r+n}$
+for $j=0,1,\ldots,m$.
+\end{itemize}
+\end{itemize}
+Previously we found that the infinite product for $\sinh1$ with the
+default settings gave the wrong value, $0.174$, deficient by $1$
+in the last digit. We now have the means to tweak the stopping criterion
+by increasing the additional rounding:
+\begin{verbatim}
+ \eval[p,P+=3]{\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+ \]}[x=1][3] \nmcInfo{prod}.
+\end{verbatim}
+\noindent $\Longrightarrow$ \noindent \eval[p,P+=3]{\[
+ \prod_{k=1}^{\infty}
+ \biggl(\frac{x^2}{k^2\pi^2} +1\biggr)
+\]}[x=1][3] \nmcInfo{prod}.
+
+\noindent To obtain that last item of information (350 factors), I've
+anticipated a little and used the command \verb`\nmcInfo` with the
+argument \verb`prod`; see Chapter~\ref{sec:info}. The product now
+produces the correct three-figure value, but it takes $350$ factors
+to do so.
+
+Knowing how many terms or factors have been needed helps assess how
+trustworthy the result from an infinite sum or product is. For example,
+for the exponential series,
+\begin{verbatim}
+ \eval[p]{\[
+ \sum_{k=0}^\infty \frac1{k!}
+ \]}[9] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[\sum_{k=0}^\infty \frac1{k!} \]}[9] \nmcInfo{sum}.
+
+To $9$ places of decimals, using the default value \texttt{S+=2},
+the exponential series arrives at the right sum after only $15$ terms.
+Convergence is rapid. We can trust this result (and it is in fact
+the correct nine-figure value). By contrast, if we didn't know the
+value of $\sinh1$ beforehand, noting the number of factors required
+would make us justly cautious about accepting the result of the infinite
+product calculation.
+
+One way to gain confidence in a result is to choose a possibly unrealistic
+rounding value – say, the default $6$ for the infinite product \-\-
+then use \emph{negative} values for the extra rounding, \texttt{S+=-5},
+\texttt{S+=-4}, \ldots{} , so that the stopping criterion applies at
+rounding values $s$ of $6+(-5)=1$, $6+(-4)=2$, and so on, but the
+result is always presented to $6$ decimal places. One can then see
+how the $6$-figure results behave relative to the number of terms
+it takes to meet the stopping criterion. A little experimenting shows
+that for our infinite product for $\sinh1$ the number of factors
+$N_{s}$ at a stopping rounding value $s$ increases in geometric
+proportion with a scale factor of about $3$: $N_{s+1}/N_{s}\approx3$.
+For the exponential series on the other hand $N_{s}=4+s$, the number
+of terms increasing in direct proportion to the stopping rounding
+value.
+
+A similar calculation for the sum of inverse fourth powers of the
+integers $\zeta(4)=\sum_{n=1}^{\infty}\frac{1}{n^{4}}$, inverse third
+powers, $\zeta(3)$, and inverse squares, $\zeta(2)$, using \verb`\nmcInfo`
+to find how many terms are required at each stopping rounding value,
+shows that at least over the rounding value range $1$ to $8$, for
+inverse fourth powers $N_{s+1}/N_{s}\approx1.7$, for inverse third
+powers $N_{s+1}/N_{s}\gtrsim2$ and for inverse squares $N_{s+1}/N_{s}\approx3$.
+All are geometric rather than arithmetic progressions, but for inverse
+fourth powers the scale factor ($\approx1.7$) is sufficiently small
+that for these low values of $s$ the number of terms required does
+not grow too quickly. It is a standard result (Euler) that the series
+sums to $\pi^{4}/90$: \verb`$ \eval{ \pi^4/90 } $` $\Longrightarrow$
+$ \eval{ \pi^4/90 } $ to six places, and indeed, with the default
+\texttt{S+=2},
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}` $\Longrightarrow$
+\eval[p]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}
+\end{centred}
+there is complete agreement.
+
+For inverse third powers, the number of terms required to reach the
+stopping criterion grows rapidly for rounding values from $7$ onwards
+($2^{7}=128$, $2^{8}=256$, \ldots ). This suggests trying for a
+five-figure result (with the default setting \texttt{S+=2} the stopping
+rounding value is $7$). Doing this gives a result $1.20205$ to five
+decimal places. \emph{HMF} Table 23.3 has this quantity tabulated
+to $20$ places and shows our result is too small by $1$ in the final
+figure.
+
+For inverse second powers, the number of terms required to reach the
+stopping criterion increase even more quickly: $3^{4}=81$, $3^{5}=243$,
+and so on. A three figure answer (with the default setting \texttt{S+=2}
+the stopping rounding value is $5$) seems the best we can hope for.
+Doing the evaluation gives $1.642$ whereas we know that $\zeta(2)=\pi^{2}/6$
+(Euler's famous result), evaluating to $1.645$. Even with \texttt{S+=3},
+the sum is still too small, $1.644$ after 1007 terms. Increasing
+the additional rounding to $4$, \texttt{S+=4}, does finally give
+the correct three-figure result, $1.645$, but only after summing
+$3180$ terms.
+
+\subsection{Premature ending of infinite sums}
+
+All the series considered so far have been monotonic. Trigonometric
+series will generally not be so, nor even single-signed.
+
+Trigonometric sums are computationally intensive and so, for the following
+example, I have specified a rounding value of 2. The series
+\[
+\sum_{n=1}^{\infty}\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt
+\]
+is the Fourier series for the triangular wave function /\!\textbackslash\!/\!\textbackslash\!/\!\textbackslash\!/\!\textbackslash{} \ldots{}
+of period 1, symmetric about the origin where it takes its maximum
+value 1, crossing the axis at $t=0.25$ and descending to its minimum
+$-1$ at $t=0.5$, before ascending to a second maximum at $t=1$
+(and so on). In the interval $[0,0.5)$ the series should sum to $1-4t$.
+The problem is that the summand $\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt$
+vanishes both when $n$ is even and when $4nt$ is an odd integer.
+If $t=0.1$ then $4nt$ is never an odd integer so the summand vanishes
+only for $n$ even, every second term. We expect the result to be
+$1-4\times0.1=0.6$.
+\begin{verbatim}
+ \eval[p]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+ \]}[t=0.1][2] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+\]}[t=0.1][2] \info{sum}.
+
+\noindent Only one term? Of course – since the second term $n$ is
+even; the term vanishes and the stopping criterion is satisfied. The
+way around this problem is to query terms \emph{beyond} the one where
+the stopping criterion is achieved, i.e., to set \texttt{S?} to a
+nonzero value. We try \texttt{S?=1}:
+\begin{verbatim}
+ \eval[p,S?=1]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+ \]}[t=0.1][2] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p,S?=1]{\[
+ \sum_{n=1}^{\infty}
+ \frac{4}{n^{2}\pi^{2}}
+ (1-\cos n\pi)\cos2\pi nt
+\]}[t=0.1][2] \info{sum}.
+
+Table~\ref{tab:settingsFinite-sums} lists the results of evaluating
+the \emph{finite }sums from $n=1$ to $N$ for values of $N$ around
+$65$. Since we have specified a rounding value of $2$ for the calculation,
+the stopping criterion\emph{ }applies at a rounding value of $2$
+more than that, $4$. Since $N=64$ is even, the summand for the $64$th
+term is zero and the sum takes the same value as for $N=63$. The
+$65$th term is the query term and the sum differs, so the summation
+continues. The $66$th term vanishes, so the stopping criterion is
+met. This time for the query term, the $67$th, the sum retains the
+same $4$-figure value, and the summation stops. The result was attained
+at the $65$th term.
+
+\begin{wraptable}{o}{0.5\columnwidth}%
+\centering{}\vspace{-4ex}
+\caption{Finite sums}\label{tab:settingsFinite-sums}
+\setlength{\abovetopsep}{1ex}%
+\begin{tabular}{cc}
+\toprule
+$N$ & $\Sigma$\tabularnewline
+\midrule
+$63$ & $0.6001$\tabularnewline
+$64$ & $0.6001$\tabularnewline
+$65$ & $0.5999$\tabularnewline
+$66$ & $0.5999$\tabularnewline
+$67$ & $0.5999$\tabularnewline
+\bottomrule
+\end{tabular}\end{wraptable}%
+Should we be confident in the result? Increase the number of query
+terms to $3$ (there is no point in increasing \texttt{S?} to $2$
+because of the vanishing of the even terms), the sum stops after $113$
+terms, with the same $0.6$ result. Indeed, increasing \texttt{S?}
+to $5,7,\ldots$ makes no difference. It still takes $113$ terms
+to reach the stable two-figure result $0.6$.
+
+For a final example, consider the error function
+\[
+\erf z=\dfrac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}dt
+\]
+which can also be rendered as an infinite sum (\emph{HMF }7.1.5):
+\[
+\erf z=\sum_{n=0}^{\infty}(-1)^{n}\frac{z^{2n+1}}{n!(2n+1)}.
+\]
+(\verb`\erf` expanding to \verb`erf` has been defined in the preamble
+to this document using \verb`\DeclareMathOperator`.) We calculate
+this sum for $z=2$ to $10$ places of decimals. Although this is
+an alternating series, it is obvious that the summand never vanishes
+when $z\ne0$ as here. Hence there seems no need to change the default
+value \texttt{S?=0}.
+\begin{verbatim}
+ \eval[p]{\[
+ \frac2{\sqrt{\pi}}
+ \sum_{n=0}^\infty(-1)^n
+ \frac{z^{2n+1}}{n!(2n+1)}
+ \]}[z=2][10*] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p]{\[
+ \frac2{\sqrt{\pi}}
+ \sum_{n=0}^\infty(-1)^n
+ \frac{z^{2n+1}}{n!(2n+1)}
+\]}[z=2][10*] \nmcInfo{sum}.
+
+According to \emph{HMF }Table 7.1, this calculated value of $\erf2$
+is correct to all $10$ places. But beyond $z=2$ errors will begin
+to interfere with the result. Note that $26$ terms means $n=26$
+was the last value of $n$ for which the summand was evaluated. (The
+sum stops at the $26$th term, $n=25$, but the next term $n=26$
+needs to be calculated for the stopping criterion.) Fortuitously,
+$2^{2\times26+1}=2^{53}$ is the greatest power of $2$ that can be
+\emph{exactly} rendered to the $16$ significant figures that \texttt{l3fp}
+uses. But $n!$ exceeds the $16$-significant figure limit of \texttt{l3fp}
+when $n>21$, so despite the 10-figure result, errors have already
+begun to occur in the denominator of the summand and accrue in the
+sum when $z=2$. For larger $z$ values the errors can only get worse
+and at some point will render the calculated value worthless at any
+meaningful rounding value. For example, when $z=7$ the sum apparently
+`evaluates' to over $929$ whereas we know that
+\[
+\erf z<\dfrac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-t^{2}}dt=1.
+\]
+
+
+\subsection{Double sums or products}
+
+Sums or products can be iterated. For instance, the exponential function
+can be calculated this way:
+\begin{centred}
+\verb`\eval[p]{\[ \sum_{k=0}^{\infty}\prod_{m=1}^{k}\frac{x}{m} \]}[x=2]`
+$\Longrightarrow$ \eval[p]{\[ \sum_{k=0}^{\infty}\prod_{m=1}^{k}\frac{x}{m} \]}[x=2]
+\end{centred}
+\noindent which is \verb`\eval{$ e^2 $}` $\Longrightarrow\eval{\ensuremath{e^{2}}}$.
+
+A second example is afforded by Euler's transformation of series (\emph{HMF}
+3.6.27). To calculate $e^{-1}$ we use
+\begin{verbatim}
+ \eval[p={,}\quad \mbox{\nmcInfo{sum}}.]
+ {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p={,}\quad \mbox{\nmcInfo{sum}}.]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+
+(Note the placement of the information command as the value of the
+punctuation key. This keeps it \emph{within }the \texttt{\textbackslash{[}
+\textbackslash{]}} delimiters.) Following Euler, this series can
+be transformed to the form
+\begin{verbatim}
+ \eval[p,S?=1]{\[
+ \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}
+ \sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!}
+ \]}[3] \nmcInfo{sum}.
+\end{verbatim}
+$\Longrightarrow$ \eval[p,S?=1]{\[ \sum_{k=0}^\infty \frac{(-1)^k}{2^{k+1}}\sum_{n=0}^k(-1)^n\binom kn \frac1{(k-n)!} \]}[3] \nmcInfo{sum}.
+
+\noindent Note the setting \verb`S?=1`. Without it, the summation
+stops after $1$ term, the $k=0$ term, because the $k=1$ term vanishes.
+With \verb`S?=1` it takes $16$ terms of the \emph{outer }sum to
+reach the stopping criterion. Since that sum starts at $0$, that
+means that changing the upper limit from $\infty$ to $15$ should
+give the same result – which it does – but it takes $\tfrac{1}{2}\times16\times17=136$
+terms in total to get there, to be compared with the $9$ terms of
+the earlier simpler sum, and the terms are more complicated. Obviously
+such double sums are computationally intensive.
+
+\section{Changing default values}
+
+\label{sec:settingsDefaults}The settings option enables various settings
+to be changed for an individual calculation. You may find yourself
+wanting to make such changes sufficiently often that a change of default
+value is a better plan than encumbering each calculation with a list
+of settings.
+
+\begin{table}[h]
+\centering
+\noindent \centering{}\caption{Default values, \texttt{\textbackslash eval} command}\label{tab:settingsDefaults}
+\noindent \begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small value}\tabularnewline
+\midrule
+{\small rounding} & {\small\texttt{6}}\tabularnewline
+{\small pad} & {\small\texttt{0}}\tabularnewline
+{\small output-sci-notation } & {\small\texttt{0}}\tabularnewline
+{\small output-exponent-char} & {\small\texttt{e}}\tabularnewline
+{\small\%} & \tabularnewline
+{\small input-sci-notation } & {\small\texttt{0}}\tabularnewline
+{\small input-exponent-char} & {\small\texttt{e}}\tabularnewline
+{\small multitoken-variables } & {\small\texttt{1}}\tabularnewline
+logarithm-base & {\small\texttt{10}}\tabularnewline
+{\small vv-display} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{\small vv-inline} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\%} & \tabularnewline
+{\small intify-rounding} & {\small\texttt{14}}\tabularnewline
+\% & \tabularnewline
+{\small sum-extra-rounding} & {\small\texttt{2}}\tabularnewline
+{\small sum-query-terms} & {\small\texttt{0}}\tabularnewline
+{\small prod-extra-rounding} & {\small\texttt{2}}\tabularnewline
+{\small prod-query-terms} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+The way to do that is to create a \emph{configuration file }with the
+name \texttt{numerica.cfg} in a text editor. Its entries, one per
+line, are of the form \emph{key=value} followed by a comma, and for
+clarity preferably one entry per line (although this is not essential).The
+key names are noticeably more verbose than the corresponding keys
+of the settings option.\emph{ }The possible keys are listed in Table~\ref{tab:settingsDefaults},
+together with their current default values.
+
+Keys taking one of two possible values, \verb`0` (for \verb`false/off`)
+or \verb`1` (for \verb`true/on`), are \verb`pad` (the result with
+zeros), \verb`output-sci-notation`, \verb`input-sci-notation`, and
+(check for) \verb`multitoken-variables`.
+
+The table is divided into four parts.
+\begin{itemize}
+\item The top four rows concern elements that can be changed for individual
+calculations with the trailing optional argument of \verb`\eval`:
+rounding, padding with zeros, and outputting in scientific notation;
+see §\ref{subsec:evalRoundingEtc}.
+\begin{itemize}
+\item Note that to output the result always in scientific notation requires
+two settings, first setting \texttt{output-sci-notation} to \texttt{1},
+and then choosing a character to act as the exponent marker. Because
+\texttt{l3fp} uses \texttt{e} for this character, \texttt{numerica}
+has made \texttt{e} its default. But this option is turned off by
+default (hence the \texttt{0} against this key).
+\end{itemize}
+\item The next block of rows concern general elements that can be changed
+for individual calculations with the settings option of \verb`\eval`;
+see §\ref{sec:settingsOption}. Obviously the key names are more expansive
+in the present context but the effect is the same.
+\begin{itemize}
+\item But note that to input numbers in scientific notation requires two
+settings, first setting \texttt{input-sci-notation} to \texttt{1},
+and then choosing a character to act as the exponent marker. Because
+\texttt{l3fp} uses \texttt{e} for this character, \texttt{numerica}
+has made \texttt{e} its default. The option is turned off by default
+(hence the \texttt{0} against this key).
+\end{itemize}
+\item The third block is a single row specifying at what rounding value
+a floating point should be considered an integer; see §\ref{subsec:defaultsIntifyingRounding}
+below.
+\item The last four rows concern default settings for infinite sums and
+products. These correspond to the keys \texttt{S+}, \texttt{S?} and
+\texttt{P+}, \texttt{P?} of the settings option that can be used to
+tweak the behaviour of the stopping criterion for such sums or products;
+see §\ref{sec:settingsInfiniteSumsProds}.
+\end{itemize}
+If you are dissatisfied with any of the default values listed, then
+in a text editor create a new file called \texttt{numerica.cfg} and
+assign \emph{your} values to the relevant keys. For instance, if you
+find yourself working to $4$ figures, that rounding to $6$ is too
+many, then make the entry \texttt{rounding=}4. If also you want results
+always presented in proper scientific notation,\emph{ $d.d_{1}d_{2}d_{3}d_{4}\times10^{n}$},
+then add a comma after \texttt{4} and enter on a new line (recommended
+but not strictly necessary; the comma is the crucial thing), \texttt{output-sci-notation=1,}
+(note the comma) and on another new line, \texttt{output-exponent-char=x}.
+
+Perhaps you also want a non-zero setting for the final query terms
+for infinite sums and products. This makes sense if you are largely
+dealing with non-monotonic series – like Fourier series. Even the
+Euler transformation of the exponential series for $e^{-1}$ discussed
+above required a non-zero \texttt{S?}. If you wish to make this change
+then add a comma and on a new line add (for instance) \texttt{sum-query-terms
+= 1,} and again on a new line, \texttt{prod-query-terms = }1. If this
+is all you wish to change, then no comma is necessary after this final
+entry. Your newly created file should look something like
+\begin{lyxcode}
+rounding~~~~~~~~~~~~~=~4,
+
+output-sci-notation~~=~1,
+
+output-exponent-char~=~x,
+
+sum-query-terms~~~~~~=~1,
+
+prod-query-terms~~~~~=~1
+\end{lyxcode}
+The white spacing may be different; white space is ignored by \texttt{numerica}
+when reading the file. Using it to align the equals signs helps \emph{us}
+read the file. Note that the last entry, because it is the last entry,
+lacks a comma. Now save the file with the name \texttt{numerica.cfg}.
+This file will be read by \texttt{numerica} near the end of its loading
+process. These settings will be \texttt{numerica}'s defaults for the
+relevant keys.
+
+\subsection{Location of \texttt{numerica.cfg}}
+
+Save, yes, but where to? If the new settings are likely to apply only
+to your current document, then the document's directory is a sensible
+place to put it and \texttt{numerica} will certainly find it there
+since it is part of \LaTeX 3 file handling that file searches are
+not limited to the \TeX{} distribution (including your personal texmf
+tree) but also include the current document directory. But what happens
+when you start working on another document? Will you remember to copy
+\texttt{numerica.cfg} to its new location? That is why your \emph{personal
+texmf tree} is a better place.
+
+\subsubsection{Personal texmf tree? }
+
+This is a directory for `waifs and strays' of the \TeX{} system that
+are not included in the standard distributions like MiK\TeX{} or \TeX Live.
+Here you place personal packages designed for your own particular
+circumstances. These may include your own \TeX{} or \LaTeX{} package,
+say \texttt{mypackage.sty}, achieving some small or singular effect
+that doesn't warrant wider distribution on CTAN. Here you might place
+configuration files for other packages with your preferences (unless
+the package requires some specific location). Here you can put your
+personal bibliography files.
+
+Your personal texmf tree is structured like the standard MiK\TeX{}
+or \TeX Live hierarchy but placed in another location so that there
+is no chance of its being overwritten when packages in MiK\TeX{} or
+\TeX Live are updated. But these distributions need to be alerted
+to its existence.
+
+For example, in the MiK\TeX{} console, click on \textsf{Settings},
+and then on the \textsf{Directories} tab of the resulting dialog.
+Here you get to add your personal texmf hierarchy to the list of paths
+that MiK\TeX{} searches, by clicking on the \textsf{+} button, browsing
+to your texmf folder and selecting it. By using the up and down arrow
+keys that the MiK\TeX{} console provides, ensure that it lies \emph{above
+}the the entry for the main MiK\TeX{} tree. That way, files in your
+personal texmf tree will be found first and loaded. Now go to the
+\textsf{Tasks} menu and click on \textsf{Refresh the filename database}.
+This will let MiK\TeX{} know what is held in your personal texmf tree.
+Files there can then be used like standard \LaTeX{} packages.
+
+\subsection{Rounding in \textquoteleft int-ifying\textquoteright{} calculations}
+
+\label{subsec:defaultsIntifyingRounding}Factorials, binomial coefficients,
+summation and product variables, and (in \texttt{numerica}) $n$-th
+roots from the \texttt{\textbackslash sqrt} command, all require
+integer arguments. These integers may indeed be entered explicitly
+as integers, but they can also be determined as the result of a calculation.
+Rounding errors may mean the result is not an exact integer. How much
+leeway should be allowed before it is clear that the calculation did
+not give an integer result? In the default setup, \texttt{numerica}
+is generous. A number is considered an integer if it rounds to an
+integer when the rounding value is $14$. Since \texttt{l3fp} works
+to $16$ significant figures this provides more than enough `elbowroom'
+for innocuous rounding errors to be accommodated. If a calculation
+does not round to an integer at a rounding value of $14$ then it
+seems reasonable to conclude that it has \emph{really }not given an
+integer answer, not just that rounding errors have accumulated. If
+you want to change this `int-ifying' value for a particular calculation,
+then add a line to \texttt{numerica.cfg} like
+\begin{lyxcode}
+intify-rounding~=~<integer>
+\end{lyxcode}
+Since \texttt{l3fp} works to $16$ significant figures, values of
+\texttt{integer} greater than $16$ are pointless. Generally int-ifying
+rounding values will be less than but close to $16$ (although when
+testing the code I used some ridiculous values like $3$ or $4$).
+If other entries follow this one in the file, then conclude the line
+with a comma.
+
+\section{Parsing mathematical arguments}
+
+\label{sec:Argument-parsing} A main aim of the \texttt{numerica}
+package is to require minimal, preferably no, adjustment to the \LaTeX{}
+form in which an expression is typeset in order to evaluate it. But
+mathematicians do not follow codified rules of the kind programming
+languages insist on when writing formulas – like parenthesizing the
+arguments of functions, or inserting explicit multiplication signs
+({*}) between juxtaposed terms. Hence the question of where the arguments
+of mathematical functions end is acute. For a few functions \LaTeX{}
+delimits the argument: think of \verb`\sqrt`, \verb`\frac`, \verb`\binom`;
+also \verb`^`. But for functions like \verb`\sin` or \verb`\tanh`
+or \verb`\ln`, unary functions, this is not so. Nor is it for sums
+and products, and comparisons.
+
+Before discussing the parsing rules for different groups of functions,
+I discuss the means \texttt{numerica} provides to handle exceptions
+to those rules, when one \emph{does} need to make some adjustment
+to a formula.
+
+\subsection{The cleave commands \texttt{\textbackslash q} and \texttt{\textbackslash Q}}
+
+\label{subsec:parseCleave}The word \emph{cleave} has two opposed
+meanings: to adhere or cling to, and to split or sever. \texttt{numerica}
+defines two commands, \verb`\q` and \verb`\Q` to achieve these opposite
+effects. When a mathematical argument is being parsed, the \verb`\q`
+command joins the next token to the argument (\emph{cleaves to});
+the \verb`\Q` command severs the next token from the argument (\emph{cleaves
+apart}). Neither command is added to the argument nor leaves a visible
+trace in the output.
+
+Thus, without \verb`\q`,
+\begin{centred}
+\verb`\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
+\eval{$ \sin(n+\tfrac12)(x-t) $}[n=3,x=t+\pi,t=1.234],
+\end{centred}
+which is $(\sin\tfrac{7}{2})\times\pi$. With \verb`\q` between the
+bracketed factors,
+\begin{centred}
+\verb`\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234]` $\Longrightarrow$
+\eval{$ \sin(n+\tfrac12)\q(x-t) $}[n=3,x=t+\pi,t=1.234],
+\end{centred}
+which is $\sin(\tfrac{7}{2}\pi)$. Similarly, without \verb`\q`,
+\begin{centred}
+\verb`\eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]`
+$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}n(t+\tfrac12T) \]}[T=2,t=1,n=3]
+\end{centred}
+which is $(\cos\pi)\times3\times(1+\tfrac{1}{2}\times2)$. With \verb`\q`
+used twice, once after the fraction and once before the left parenthesis,
+\begin{centred}
+\verb`\eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]`
+$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]
+\end{centred}
+which is $\cos(\pi\times3\times2)$.
+
+It should be noted that for \emph{trigonometric }functions, because
+of their use in Fourier series especially, there is another way of
+handling arguments with parentheses (and fractions). This is discussed
+in §\ref{subsec:parseTrigFns} below.
+
+For the \verb`\Q` command which splits an argument we have, without
+it,
+\begin{centred}
+\verb`\eval{$ 1/2e $}` $\Longrightarrow$ \eval{$ 1/2e $},
+\end{centred}
+which is the reciprocal of $2e$, whereas with the \verb`\Q` command
+inserted before \verb`e`,
+\begin{centred}
+\verb`\eval{$ 1/2\Q e $}` $\Longrightarrow$ \eval{$ 1/2\Q e $},
+\end{centred}
+which is a half of $e$. Of course, the meaning in this example would
+be clearer if $1/2$ were parenthesized or presented as a \verb`\tfrac`.
+
+\subsubsection{Mnemonic}
+
+As mnemonic, best seen in sans serif for the Latin Modern fonts used
+in this document, think of the letter \textsf{q} as a circle \emph{clinging
+}to a vertical descender; think of the letter \textsf{Q} as a circle
+\emph{cut} by the diagonal stroke.
+
+\subsection{Parsing groups}
+
+The arguments of different groups of functions are handled in different
+ways. The criterion used for deciding when an argument ends for one
+group will not be that used for the others. Table §\ref{subsec:defaultsIntifyingRounding}
+lists the different groups that \texttt{numerica} takes account of.
+At the top are functions or operations that have the smallest reach
+when determining where their arguments end; at the bottom are operations
+that have the greatest reach. The denominator of a slash fraction
+is treated as a unary function and is assigned to group II. By default
+trigonometric functions are treated the same as other unary functions
+but there is a setting which enables the direct (rather than inverse)
+trigonometric functions to accept a wider range of arguments, as occurs
+in Fourier series. Hence they are separated into their own group.
+
+\begin{wraptable}{o}{0.5\columnwidth}%
+\noindent \centering{}\caption{Parsing groups}\label{tab:settingsParsing-groups}
+{\ttfamily\noindent \begin{center}
+{\ttfamily{}%
+\begin{tabular}{ll}
+\toprule
+{\small\textrm{group}} & {\small\textrm{function/operation}}\tabularnewline
+\midrule
+{\small\textrm{I}} & {\small\textrm{surd, logical Not}}\tabularnewline
+{\small\textrm{II}} & {\small\textrm{unary functions, /}}\tabularnewline
+{\small\textrm{III}} & {\small\textrm{direct trig. functions}}\tabularnewline
+{\small\textrm{IV}} & {\small\textrm{sums, products}}\tabularnewline
+{\small\textrm{V}} & {\small\textrm{comparisons}}\tabularnewline
+{\small\textrm{VI}} & {\small\textrm{logical And, logical Or}}\tabularnewline
+\bottomrule
+\end{tabular}}
+\par\end{center}}\end{wraptable}%
+ A formula is a sequence of tokens and brace groups. All parsing occurs
+from the left, \LaTeX{} argument by \LaTeX{} argument, where \emph{argument}
+means either a token (an N-type argument in \verb`expl3`-speak) or
+a brace group (an n-type argument). To distinguish \LaTeX{} arguments
+from mathematical arguments I shall when necessary refer to L-args
+and M-args. A mathematical argument may end \emph{at} an L-arg, meaning
+immediately before the L-arg, or end \emph{with} the L-arg, meaning
+immediately after the L-arg. Ending or not will in general depend
+on whether the argument is in \emph{first position} – the position
+immediately following a function token like \verb`\sin` or \verb`\log`
+– or in \emph{general position} – any later position (although for
+trigonometric functions we will also need to consider \emph{second}
+and even \emph{third }position).
+
+For counting position, we need to allow for formatting elements and
+multi-token numbers – in both decimal and scientific formats. Formatting
+elements do not change the position count. This applies to things
+like thin spaces or phantoms (and their arguments) or modifiers like
+\verb`\left` or \verb`\biggl`. Multi-token numbers (in decimal or
+scientific formats) are treated as single items; they advance the
+position count by exactly one. \LaTeX{} functions – like \verb`\frac`
+– which take \LaTeX{} arguments again advance the position count only
+by one. Mathematically, the fraction is viewed as a single unit.
+
+I shall refer to a token or a token and its \LaTeX{} arguments – like
+\verb`\frac` and its arguments – as an \emph{item}. Similarly, a
+(possibly multi-token) number is an item. Also it will help to distinguish
+tokens within brackets where both brackets lie to the right of a function
+from those that do not. The former I call \emph{clothed}; the latter
+are \emph{naked}. Thus the plus sign in $(\sin x+y)$ is naked relative
+to the sine (one bracket to the left of the function), but is clothed
+in $\sin(x+y)$ (both brackets to the right of the function).
+
+\subsubsection{Parsing group I}
+
+The only functions in this category are the surd and logical Not.
+
+Why distinguish the surd from other unary functions? Surely we all
+agree that \verb`\sin2\pi`, displaying as $\sin2\pi$, vanishes?
+The argument of the sine extends beyond the $2$ to include the $\pi$.
+But \verb`\surd2\pi`, displaying as $\surd2\pi$, is understood to
+be the product $\surd2\times\pi$. The argument of the surd ends with
+the $2$. The surd binds more tightly to its argument than is true
+of unary functions generally.
+
+For parsing group I
+\begin{enumerate}
+\item if a left bracket is in first position, the mathematical argument
+ends with the matching right bracket; otherwise
+\item the argument ends with the item in first position and any L- or M-args
+required by that item.
+\end{enumerate}
+If the factorial sign \verb`!` \emph{preceded} its argument, it too
+would belong to this parsing state, for it also binds tightly like
+the surd. This means that an expression like $\surd4!$ is intrinsically
+ambiguous. Is it the square root of $24$ or the factorial of $2$?
+In \texttt{numerica} it produces the (perhaps rather odd) error
+\begin{centred}
+\verb`\eval{$ \surd 4! $}` $\Longrightarrow$ \eval{$ \surd4! $}
+\end{centred}
+The surd has seized the argument; there is nothing for the factorial
+to operate on. The same error arises if the $4$ is parenthesized,
+but parenthesizing like either \verb`(\surd 4)!` or \verb`\surd(4!)`
+repairs the situation. Because other unary functions (like the sine
+or logarithm) do not bind as tightly, this ambiguity does not arise
+for them.
+
+Exponents cause no problem because taking square roots and raising
+to a power are commutative operations – the result is the same whichever
+is performed first.
+\begin{centred}
+\verb`\eval{$ \surd 3^4 $}` $\Longrightarrow$ \eval{$ \surd 3^4 $}.
+\end{centred}
+
+\subsubsection{Parsing group II: unary functions, slash fractions}
+
+In the default setup this category includes the trigonometric and
+hyperbolic functions, their inverses, the various logarithms and the
+exponential functions, the signum function \verb`\sgn`, and the slash
+fraction \verb`/` where the argument to be determined is its denominator.
+Note however that there is a setting switch which enables trigonometric
+ functions to handle parentheses in arguments more generally; see
+§\ref{subsec:parseTrigFns}.
+\begin{itemize}
+\item In parsing group II we wish to accommodate usages like $\ln z^{n}=n\ln z$
+(\emph{HMF} 4.1.11), or $\gd z=2\arctan e^{z}-\frac{1}{2}\pi$ (\emph{HMF}
+4.3.117), defining the Gudermannian. The exponent is included in the
+argument. Considering $\ln(1+1/n)^{n}$ exponents must also be part
+of parenthesized arguments.
+\item An approximation to Stirling's formula for the factorial is often
+written $\ln N!\approx N\ln N-N$ (widely used in texts on statistical
+mechanics). Hence the factorial sign should also be considered part
+of the argument.
+\item $\ln xy=\ln x+\ln y$ means the argument must reach over a product
+of variables. Identities like $\sin2z=2\sin z\cos z$ mean the argument
+also reaches over numbers, and expressions like $\sin\tfrac{1}{2}\pi x$
+(\emph{HMF} 4.3.104) mean that it further reaches over \verb`\tfrac`-s
+and constants.
+\item Essentially \emph{anything }can be in first position, and without
+parentheses; e.g.
+\begin{itemize}
+\item unary functions: $\ln\ln z$ (\emph{HMF} 4.1.52), $\ln\tan\dfrac{z}{2}$
+(\emph{HMF} 4.3.116),
+\item fractions: $\ln\dfrac{z_{1}}{z_{2}}$ (\emph{HMF} 4.1.9), $\arcsin\dfrac{(2ax+b)}{(b^{2}-4ac)^{1/2}}$
+(\emph{HMF} 3.3.36), $\ln\dfrac{\tan z}{z}$ (\emph{HMF} 4.3.73),
+\item absolute values: $\ln\abs*{\dfrac{a+x}{a-x}}$ (\emph{HMF} 3.3.25),
+\item square roots: $\arctan\sqrt{\dfrac{\nu_{1}}{\nu_{2}}F}$ (\emph{HMF
+}26.6.8)
+\end{itemize}
+\end{itemize}
+With these examples in mind, for parsing group II
+\begin{enumerate}
+\item if a left bracket is in first position, the mathematical argument
+ends with the matching right bracket and any attached exponent, or
+factorial or double factorial sign; otherwise
+\item the mathematical argument includes the item in first position and
+any L- or M-args required by that item;
+\begin{enumerate}
+\item if the item in first position is a number, variable, constant or \verb`\tfrac`
+\begin{enumerate}
+\item the argument appends the next item if it is a number, variable, constant
+or \verb`\tfrac`, and so on recursively; or
+\item the argument appends the next item if it is an exponent, or facorial
+or double factorial sign, and ends there; otherwise
+\item the argument ends.
+\end{enumerate}
+\item if the item in first position is not a number, variable, constant
+or \verb`\tfrac`
+\begin{enumerate}
+\item the argument appends the next item if it is an exponent, or factorial
+or double factorial sign, and ends there; otherwise
+\item the argument ends.
+\end{enumerate}
+\end{enumerate}
+\end{enumerate}
+An argument may extend over (see 2(a)i) numbers, constants, variables
+and \verb`\tfrac`-s: $\sin2\tfrac{p}{q}\pi x$ exhibits all elements.
+
+Illustrating 1, the exponent is included in the argument but not the
+following variable:
+\begin{centred}
+\verb`\eval{$ \log_{10}(1+2+3+4)^3n $}[n=5]` $\Longrightarrow$ \eval{$ \log_{10}(1+2+3+4)^3n $}[n=5].
+\end{centred}
+For the sake of the reader, and as one naturally does in any case
+to avoid ambiguity, the formula should be written with the variable
+$n$ preceding the logarithm: $n\log_{10}(1+2+3+4)^{3}$. The way
+the example is written suggests that the writer wished the $n$ to
+be considered part of the argument. If that is the case, an outer
+set of parentheses would make intentions clear, but it is possible
+to leave the argument as written but insert a \verb`\q` command before
+$n$:
+\begin{centred}
+\verb`\eval{$ \log_{10}(1+2+3+4)^3\q n $}[n=5]` $\Longrightarrow$
+\eval{$ \log_{10}(1+2+3+4)^{3}\q n $}[n=5],
+\end{centred}
+which is $\log_{10}5000.$
+
+Illustrating 2(a)ii, again the exponent is included in the argument
+but not the following variable:
+\begin{centred}
+\verb`\eval{$ \log_{10}m^3n $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}m^3n $}[m=10,n=5].
+\end{centred}
+Again, for the sake of the reader and as one naturally does to avoid
+ambiguity, the variable $n$ should precede the logarithm. If in fact
+one wants the $n$ included in the argument of the logarithm, the
+\verb`\q` command is again available or, better in this case, the
+$n$ can be shifted to precede the $m$, which illustrates 2(a)i:
+\begin{centred}
+\verb`\eval{$ \log_{10}nm^3 $}[m=10,n=5]` $\Longrightarrow$ \eval{$ \log_{10}nm^3 $}[m=10,n=5].
+\end{centred}
+Is \texttt{numerica} being too strict when $nm^{3}$ is included in
+the argument of the logarithm, but $m^{3}n$ is not? Any criterion
+is going to miss some instances where a different outcome might be
+desirable. Where an argument ends is affected by visual appearance.
+It is simple and easy to remember if it is understood that anything
+that breaks the visual appearance of juxtaposed numbers, variables,
+constants and \verb`\tfrac`-s ends the argument. An exponent does
+just that.
+
+Illustrating 2(b)ii, the argument stops with the \verb`\dfrac` and
+its arguments and does not extend to the following constant:
+\begin{centred}
+\verb`\eval{$ \sin\dfrac12\pi $}` $\Longrightarrow$ \eval{$ \sin\dfrac12\pi $}.
+\end{centred}
+Obviously, someone writing an expression like this intends the $\pi$
+to be part of the argument. In that case, a \verb`\tfrac` should
+be used. The \verb`\dfrac` breaks the `visual flow' of an argument.
+\begin{description}
+\item [{Fractions}]~
+
+But why not a plain \verb`\frac`? After all, for an inline expression
+it displays in the same way as a \verb`\tfrac`. I considered making
+the argument-behaviour of \verb`\frac` the same as \verb`\tfrac`
+for text-style contexts, and the same as \verb`\dfrac` for display-style
+contexts, but that would have meant the same expression evaluating
+to different results depending on the context, text-style or display-style,
+which ruled it out. Because \verb`\frac` sometimes displays as \verb`\dfrac`,
+it necessarily is treated like \verb`\dfrac` (but see §\ref{subsec:parseTrigFns},
+specifically \texttt{()=2}).
+\item [{Slash~fractions}]~
+
+It is easy to write ambiguous expressions using the slash $/$ to
+indicate fractions or division. How should $\pi/2n$ be interpreted?
+With from-the-left evaluation and calculator precedence rules which
+give equal precedence to {*} (multiplication) and / (division), this
+would be interpreted as $(\pi/2)\times n$, but most people will instinctively
+interpret it as $\pi/(2n)$. By placing \verb`/` in parsing group
+II, this is what \texttt{numerica} does.
+
+It treats the right-hand argument of the slash \emph{as if it were
+the argument of a named function}. This means that $1/2\sin(\pi/6)$
+is parsed as $(1/2)\sin(\pi/6)$ rather than as $1/(2\sin(\pi/6))$.
+It also means that $1/2\exp(1)$ and $1/2e$ give different results,
+which is acceptable since (in the author's view) they display differently
+and are not instinctively read in the same way.
+\end{description}
+
+\subsubsection{Parsing group III}
+
+\label{subsec:parseTrigFns}By default trigonometric functions are
+set to parsing group II. This accommodates many instances of how arguments
+are used with these functions, but Fourier series in particular require
+more. For them we need to take account of how \emph{parentheses} are
+used in arguments. I find $\tan\tfrac{1}{2}(A+B)$ (\emph{HMF }4.3.148),
+$\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$ (\emph{HMF }19.3.3), $\cos(2m+p)z$
+(\emph{HMF }20.2.3), $\sin(2n+1)v$ (\emph{HMF }16.38.1). Looking
+through various texts discussing Fourier series it is easy to find
+examples like
+\[
+\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),
+\]
+and
+\[
+\cos(N+\tfrac{1}{2})\frac{2\pi\tau}{T},\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
+\]
+In the last of these \verb`\left` and \verb`\right` have been used
+to enlarge the parentheses.
+
+All these usages can be accommodated by adjusting a setting in the
+settings option (§\ref{sec:settingsOption}) of the \verb`\eval`
+command:
+\begin{lyxcode}
+()~=~integer
+\end{lyxcode}
+where \texttt{integer} is one of \texttt{0, 1, 2}. For convenience
+of statement in what follows call parentheses, square brackets or
+braces \emph{brackets}. If preceded by a \verb`\left` or \verb`\right`
+or \verb`\biggl` or \verb`\biggr` etc. modifier, call them \emph{Brackets},
+with an uppercase `B'. Modifiers do not contribute to the position
+count, so that a left Bracket in first position means the modifier
+and left bracket are both considered to be in first position. When
+it is immaterial whether it is a bracket or a Bracket I write b/Bracket.
+The rules that follow do not prescribe what mathematicians \emph{ought}
+to do but are intended to be descriptive of certain patterns of mathematical
+practice as discerned in \emph{HMF} and a number of texts (about half
+a dozen) on Fourier series.
+\begin{description}
+\item [{\texttt{()=0}}] is the \emph{default} setting; b/Brackets are included
+in the argument only if
+\begin{itemize}
+\item the left b/Bracket is in first position;
+\begin{itemize}
+\item if the first item beyond the matching right b/Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, otherwise
+\item the argument ends with the right b/Bracket.
+\end{itemize}
+\end{itemize}
+\item [{\texttt{()=1}}] includes a b/Bracketed expression in the argument,
+provided
+\begin{itemize}
+\item the left Bracket is in first position;
+\begin{itemize}
+\item if the first item beyond the matching right Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, otherwise
+\item the argument ends with the right Bracket.
+\end{itemize}
+\item or the item in first position is a number, variable, constant or \verb`\tfrac`
+and the left bracket is in second position;
+\begin{itemize}
+\item if the first item beyond the matching right bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, or
+\item if the first item beyond the matching right bracket is a number, variable,
+constant, or \verb`\tfrac` it is appended to the argument, and so
+on recursively, until
+\begin{itemize}
+\item an exponent, or factorial or double factorial sign is met, which is
+appended to the argument which ends there, or
+\item an item is met which is \emph{not} an exponent, or factorial or double
+factorial sign, or a number, variable, constant or \verb`\tfrac`,
+at which point the argument ends, or
+\item the end of the formula is reached.
+\end{itemize}
+\end{itemize}
+\end{itemize}
+\item [{\texttt{()=2}}] includes a b/Bracketed expression in the argument
+provided
+\begin{itemize}
+\item the left b/Bracket is in first position, or the item in first position
+is a number, variable, constant, \verb`\dfrac`, \verb`\frac` or
+\verb`\tfrac` and the left b/Bracket is in second position, or the
+items in first and second positions are numbers, variables, constants,
+\verb`\dfrac`-s, \verb`\frac`-s or \verb`\tfrac`-s and the left
+b/Bracket is in third position;
+\begin{itemize}
+\item if the first item beyond the matching right b/Bracket is an exponent,
+or factorial or double factorial sign, it is appended to the argument,
+which ends there, or
+\item if the first item beyond the matching right b/Bracket is a number,
+variable, constant, \verb`\dfrac`, \verb`\frac` or \verb`\tfrac`
+it is appended to the argument, and so on recursively, until
+\begin{itemize}
+\item an exponent, or factorial or double factorial sign is met, which is
+appended to the argument which ends there, or
+\item an item is met which is \emph{not} an exponent, or factorial or double
+factorial sign, or a number, variable, constant, \verb`\dfrac`, \verb`\frac`
+or \verb`\tfrac`, at which point the argument ends, or
+\item the end of the formula is reached.
+\end{itemize}
+\end{itemize}
+\end{itemize}
+\end{description}
+The default setting is \texttt{()=0} which is parsing group II behaviour.
+
+\texttt{()=1} allows $\tan\tfrac{1}{2}(A+B)$ and $\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$,
+and $\cos(2m+p)z$ and $\sin(2n+1)v$, and also items on \emph{both}
+sides of the bracketed part like $\sin\tfrac{1}{2}(m+n)\pi$ provided
+there is only one item between the function and the left bracket:
+\begin{centred}
+\verb`\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2]`. $\Longrightarrow$
+\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2].
+\end{centred}
+Note that \texttt{numerica} does not check what is included between
+the brackets – it could be anything. However inserting \verb`\left`,
+\verb`\right` modifiers before the parentheses restricts the argument
+of the sine in this example to the \verb`\tfrac`:
+\begin{centred}
+\verb`\eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2]`.
+$\Longrightarrow$ \eval[()=1]{$ \sin\tfrac16\left(m+n\right)\pi $}[m=1,n=2].
+\end{centred}
+\texttt{()=2} draws no distinction between brackets and Brackets.
+It allows all \texttt{()=1} possibilities but also \emph{two }items
+(of a suitable kind) before the left b/Bracket; it also allows \verb`\dfrac`-s
+and \verb`\frac`-s in addition to \verb`\tfrac`-s.
+
+The following examples are taken from different texts on Fourier series.
+The first shows a \verb`\frac` being included in the argument, the
+second shows \emph{two} items – including a \verb`\frac` – preceding
+the left parenthesis, the third shows a \verb`\frac` to the right
+of the parentheses, and the fourth shows parentheses using \verb`\left`-\verb`\right`
+modifiers with two items preceding them:
+\[
+\cos\frac{2\pi}{T}nt,\quad\cos\frac{2\pi}{T}n(t+\tfrac{1}{2}T),\quad\text{\ensuremath{\sin(N+\tfrac{1}{2})\frac{2\pi\tau}{T}}\ensuremath{\quad}and}\quad\sin2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right).
+\]
+All these usages are accommodated by the \verb`()=2` setting. For
+instance
+\begin{verbatim}
+ \eval[p,()=2]
+ {
+ \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
+ }[N=1,\tau=2,T=3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p,()=2]
+ {
+ \[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
+ }[N=1,\tau=2,T=3]which is the sine of $(\tfrac{3}{2})\times(\tfrac{4}{3}\pi)=2\pi$
+(and \emph{not }$(\sin\tfrac{3}{2})(\tfrac{4}{3}\pi)$ ), where a
+\verb`\frac` trailing the parentheses has been included in the argument.
+Or consider
+\begin{verbatim}
+ \eval[p,()=2]
+ {\[
+ \sin2\pi\left(\frac{x}{\lambda}
+ -\frac{t}{T}\right)
+ \]}[x=1,\lambda=2,t=3,T=4]
+\end{verbatim}
+$\Longrightarrow$ \eval[p,()=2]
+ {\[
+ \sin2\pi\left(\frac{x}{\lambda}
+ -\frac{t}{T}\right)
+ \]}[x=1,\lambda=2,t=3,T=4] which is the sine of $2\pi\times(-\tfrac{1}{4})=-\tfrac{1}{2}\pi$
+(and \emph{not} $\sin2\pi$ times the parenthesised expression) where
+there are two items before the parentheses which surround two \verb`\frac`-s
+and \verb`\left` and \verb`\right` modifiers have been used with
+the parentheses.
+
+However a usage like $\sin(n+\tfrac{1}{2})(x-t)$, noted in two different
+texts, is not available without explicit use of the \verb`\q` command
+between the parenthesized groups.
+
+\subsubsection{Parsing group IV}
+
+The only members of this group are \verb`\sum` and \verb`\prod`.
+
+For parsing group IV
+\begin{enumerate}
+\item the argument ends
+\begin{enumerate}
+\item at the first naked plus or minus sign encountered, or
+\item at the first comparison sign or comparison command encountered, or
+\item at the first logical And or logical Or sign encountered, or
+\item at the end of the formula.
+\end{enumerate}
+\end{enumerate}
+In practice this means mainly (a) and (d), and seems to be the instinctive
+practice. \emph{HMF} has multiple examples in multiple chapters of
+the argument to a sum ending at a naked plus sign: 7.3.12 \& 7.3.14,
+9.1.11 \& 9.1.77, 9.6.35 \& 9.6.43, 11.1.9, \ldots{} (at that point
+I stopped looking). They were all of the form
+\[
+\sum\text{argument}+\ldots
+\]
+ A minus sign serving the same purpose was harder to find but \emph{HMF}
+10.4.65 \& 10.4.67 are two instances. I considered whether a \verb`\times`
+or slash fraction sign \verb`/` might end the argument of a sum,
+but surely we need to allow things like $\sum1/n^{2}$ which rules
+out the slash and \emph{HMF} 9.9.11 provides two of a number of instances
+in \emph{HMF} of sum arguments continuing past explicit \verb`\times`
+signs (at line breaks when a summand spills onto a second line).
+
+Because they are evaluated using the same code as sums I (unthinkingly)
+placed products with sums but doubts later intruded. In \emph{HMF}
+products occur only occasionally and are almost all of the form
+\[
+\prod\left(\text{argument}\right)
+\]
+where the argument is bracketed (often with \verb`\left \right` modifiers)
+and the multiplicand ends with the right bracket. At least twice (\emph{HMF
+}6.1.25 and 24.2.2.1) an exponent ($-1$) is attached to the right
+bracket and the argument ends there. Looking further afield, a text
+on number theory has examples where the argument of the product extends
+to \emph{three} parenthesised factors, $\prod\left(\text{arg}1\right)\left(\text{arg2}\right)\left(\text{arg3}\right)$
+and a number of others where it extends to two. A text on theory of
+functions has
+\[
+\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right)e^{z/n}
+\]
+although \emph{HMF}, for the same expression, encloses the two factors
+within (large) square brackets, as if some ambiguity existed as to
+how far the reach of the \verb`\prod` extended.
+
+\emph{Tentatively} I retain products here in the same group as sums.
+
+\subsubsection{Parsing group V}
+
+Comparison symbols compose this group: \texttt{=}, \texttt{<}, \texttt{>},
+\verb`\ne`, \verb`\le`, \verb`\ge`, \verb`\leq`, \verb`\geq`,
+and the various comparison commands from the \texttt{amssymb} package
+listed in §\ref{subsec:evalAmssymb-comparisons}. Because of the way
+\texttt{numerica} handles comparisons, it is the argument on the right-hand
+side of the relation that needs determining.
+
+For parsing group V
+\begin{enumerate}
+\item the argument ends at
+\begin{enumerate}
+\item the first logical And or logical Or encountered, or
+\item the first comparison sign or command encountered, or
+\item the end of the formula.
+\end{enumerate}
+\end{enumerate}
+
+\subsubsection{Parsing group VI}
+
+Logical And and logical Or are the sole members of this group. It
+is the right-hand side of the And or Or command that needs determining.
+
+For parsing group VI
+\begin{enumerate}
+\item the argument ends at
+\begin{enumerate}
+\item the first logical And or logical Or encountered, or
+\item the end of the formula.
+\end{enumerate}
+\end{enumerate}
+
+\subsubsection{Disclaimer}
+
+The parsing rules of the different groups are not normative; they
+are not statements of how mathematical formulas should be written.
+Rather they are attempts to discern regularities in how mathematicians
+often do write formulas. It is how things look in the pdf, not \LaTeX ,
+that is the guide. You are always free to parenthesize as you see
+fit and to insert cleave commands (\verb`\q` or \verb`\Q`) to force
+outcomes.
+
+(But note that parenthesizing has its limits. For sums, writing
+\[
+\sum\left(\mathtt{<stuff>}\right)\mathtt{<more-stuff>}
+\]
+ does not necessarily end the summand at the right parenthesis: it
+ends at the first naked $+$ or $-$ sign, or \verb`\Q` command,
+encountered.)
+
+The rule should always be to write expressions that are clear to the
+reader of the pdf. An expression that is ambiguous to the reader,
+even if it fits within the parsing rules, is to be deplored. The \emph{intent}
+is that \verb`\eval` can parse unambiguous expressions correctly.
+
+\chapter{Nesting \texttt{\textbackslash eval} commands}
+
+\label{chap:Nesting}\verb`\eval` commands can be used within \verb`\eval`
+commands, both as part of the formula being evaluated or as part of
+the vv-list or both. Only in special circumstances is this likely
+to be useful – perhaps when a calculation can be divided into two
+or more parts where different settings are appropriate for the different
+parts. One can imagine cases in which trigonometric functions are
+involved and different \texttt{()} settings would be helpful in different
+parts of the formula. Nesting of command within command becomes especially
+significant with the additional commands available when \texttt{numerica}
+is loaded with the \texttt{plus} or \texttt{tables} options; see the
+associated documents \texttt{numerica-plus.pdf} and \texttt{numerica-tables.pdf}.
+Since those additional commands are not available in this document,
+I restrict myself here to some `toy' examples of \verb`\eval` commands
+within \verb`\eval` commands to show how things work.
+
+\section{Star option for inner \texttt{\textbackslash eval}}
+
+The \verb`\eval` command `digests' a \LaTeX{} formula to produce
+an \verb`l3fp`-readable formula. This is then fed to \verb`l3fp`
+to be evaluated. The evaluated output is then formatted in various
+ways to be displayed. If the inner \verb`\eval` command produces
+formatted output, it is \emph{that} that the outer \verb`\eval` command
+will attempt to digest – and fail. Hence \emph{always} use the star
+option for the inner \verb`\eval` command. That means the outer \verb`\eval`
+is feeding on a number only:
+\begin{centred}
+\verb`\eval{$ \sin(\eval*{\sin x}[x=\pi/6]\pi) + 1 $}` $\Longrightarrow$
+\eval{$ \sin(\eval*{\sin x}[x=\pi/6]\pi) + 1$}.
+\end{centred}
+Also no math delimiters are used in the inner command. These are irrelevant
+with the star option in any case, but in the present context would
+cause error if included because they would be treated as part of the
+formula and thereby produce an `unknown token' error message. In
+the presentation of the overall result that the inner \verb`\eval`
+command is evaluated, showing as $0.5$.
+
+\section{Nesting in the vv-list}
+\begin{centred}
+\verb`\eval{$ \sin k\pi + 1 $}[k=\eval*{\sin x},x=\pi/6]` $\Longrightarrow$
+\eval{$ \sin k\pi + 1 $} [k=\eval*{\sin x},x=\pi/6].
+\end{centred}
+When the inner \verb`\eval` command is in the vv-list of the outer
+command and has a vv-list of its own, then the entire inner command
+needs to be placed in braces:
+\begin{centred}
+\verb`\eval{$ \sin k\pi + z $}[k={\eval*{y\sin x}[x=\pi/4,y=1/\surd2]},z=1]`
+$\Longrightarrow$ \eval{$ \sin k\pi + z $} [k={\eval*{y\sin x}[x=\pi/4,y=1/\surd2]},z=1].
+\end{centred}
+The vv-list of the inner \verb`\eval` command contains both a comma
+and square brackets. Both elements need to be hidden from the outer
+\verb`\eval` in order that \emph{its }vv-list be parsed correctly.
+Hence braces surround the inner \verb`\eval` and its arguments. The
+same need arises if the inner \verb`\eval` has a non-empty settings
+option – another comma-separated square-bracketed option.
+
+The values of variables used in an inner \verb`\eval` command are
+restricted to that command; they do not `leak' into the outer calculation.
+But variables and their values in the outer vv-list are available
+for use in the inner \verb`\eval` command (unless a value is explicitly
+changed in the inner vv-list).
+
+\subsection{Debugging}
+
+\label{subsec:nestDebugging}It is worth looking at the debug display
+when \verb`\eval` commands are nested. For the outer \verb`\eval`
+command:
+\begin{centred}
+\verb`\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}` $\Longrightarrow$
+\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}
+\end{centred}
+and when the inner \verb`\eval` is in the vv-list,
+\begin{centred}
+\verb`\eval[dbg=210]{$ \sin k\pi + 1 $}[k=\eval*{\sin x},x=\pi/6]`
+$\Longrightarrow$ \eval[dbg=210]{$ \sin k\pi + 1 $} [k=\eval*{\sin x},x=\pi/6]
+\end{centred}
+For the inner \verb`\eval` command debugging still works but in an
+idiosyncratic way. To clarify exactly what is going on I have added
+a \verb`\left( \right)` pair around the entire inner \verb`\eval`
+command. Note that I have also used a \emph{negative} \texttt{dbg}
+value. With a positive value, the right parenthesis is pressed toward
+the right margin of the page. The negative value limits the display
+to the text width and gives the much neater result shown.
+\begin{verbatim}
+ \eval[()=2]{$
+ \sin\left(
+ \eval*[dbg=-210]{ \sin x }[x=\pi/6]
+ \right)\pi + 1 $}
+\end{verbatim}
+$\Longrightarrow$ \eval[()=2]{$
+ \sin\left(
+ \eval*[dbg=-210]{ \sin x }[x=\pi/6]
+ \right)\pi + 7 $}
+
+\medskip{}
+The debug display from the inner \verb`\eval` command has been inserted
+into the formula of the outer \verb`\eval` in the position occupied
+by the inner \verb`\eval`. I did not deliberately code for this,
+but have decided to leave it as is, since there can be no confusion
+about which \verb`\eval` command is being `debugged', despite the
+potential for some rather odd displays. In this last example, in order
+to both use \verb`\left(...\right)` and have the calculation give
+the previous result I have employed the setting \verb`()=2` in the
+outer \verb`\eval`; see §\ref{subsec:parseTrigFns}.
+
+\chapter{\texttt{\textbackslash nmcInfo (\textbackslash info)}}
+
+\label{sec:info}Used after the evaluation of an `infinite' process,
+the \verb`\nmcInfo` command, or its equivalent short-name form \verb`\info`
+will tell you how many terms or factors or iterations or steps were
+needed to arrive at the result. The syntax of the \verb`\nmcInfo`
+command is
+\begin{lyxcode}
+\textbackslash nmcInfo\{<arg>\}~
+\end{lyxcode}
+where \texttt{<arg>} is restricted to one of two choices at present,
+either \texttt{sum} or \texttt{prod}. If the package \texttt{numerica-plus.def}
+is loaded (see §\ref{subsec:introPackagesOptions} and the associated
+document \texttt{numerica-plus.pdf}) two further arguments are possible:
+\texttt{iter} and \texttt{solve}.
+
+There is a starred form of the command:
+\begin{lyxcode}
+\textbackslash nmcInfo{*}\{<arg>\}
+\end{lyxcode}
+(or \verb`\info*{<arg>}`). As with the \verb`\eval` command the
+star has the effect of suppressing anything other than the numerical
+result from the display.
+
+As an example, let's test a standard identity, $\cosh^{2}x-\sinh^{2}x=1$,
+`the hard way'. We know that $\cosh x=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}$
+and $\sinh x=x\prod_{k=1}^{\infty}\left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)$.
+The difference of their squares should be $1$:
+\begin{verbatim}
+ \eval{\[
+ \left[\sum_{n=0}^{\infty}
+ \frac{x^{2n}}{(2n)!}
+ \right]^2-
+ \left[x\prod_{k=1}^{\infty}
+ \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)
+ \right]^2
+ \]}[x=1][3] \info{sum}\quad \info{prod}
+\end{verbatim}
+$\Longrightarrow$ \eval{\[
+ \left[\sum_{n=0}^{\infty}
+ \frac{x^{2n}}{(2n)!}\right]^2-
+ \left[x\prod_{k=1}^{\infty}
+ \left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)\right]^2
+ \]}[x=1][3] \info{sum}\quad \info{prod}.
+
+Nearly right. Obviously the product converges only slowly which is
+where the error comes from (see the discussion in §\ref{sec:settingsInfiniteSumsProds},
+where we needed the extra rounding setting \texttt{P+=3} and $350$
+factors to get a correct 3-figure value). The point of the example
+is to show the information command being used for both sum and product
+in the one evaluation. One does not exclude the other.
+
+The information command can also be placed in the settings option
+as the value of the punctuation setting. An example of this has already
+been provided earlier which I'll repeat here:
+\begin{verbatim}
+ \eval[p=\mbox{,\quad\nmcInfo{sum}.}]
+ {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=\mbox{,\quad\nmcInfo{sum}.}]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+
+Because of the \texttt{\textbackslash{[} \textbackslash{]}} delimiters,
+if the information command had been placed \emph{after} the \texttt{\textbackslash eval}
+command, it would have slid down to the next line. As it is, it resides
+\emph{inside} the \texttt{\textbackslash{[} \textbackslash{]}}
+delimiters, on the same line as the expression. This may be significant
+for adjusting vertical spacing of subsequent elements of the document.
+
+\section{Errors}
+
+Should the wrong argument be used in the \verb`\nmcInfo` command,
+no harm is done:
+\begin{verbatim}
+ \eval{$
+ \sum_{k=0}^{\infty}\binom \alpha k x^k
+ $}[x=1/2,\alpha=3], \ \info{prod}
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\ \info{prod}.
+
+$119$ \emph{factors}? The information command is remembering a previous
+result, the last time \verb`prod` was used as its argument. Changing
+the argument from \verb`prod` to \verb`sum` reveals the correct
+number of \emph{terms}.
+
+Should a non-existent argument be used, an error message is generated:
+\begin{verbatim}
+ \eval{$
+ \sum_{k=0}^{\infty}\binom \alpha k x^k
+ $}[x=1/2,\alpha=3], \\ \info{Fred}
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\\ \info{Fred}
+
+\chapter{Saving and reusing results }
+
+\label{chap:Saving-and-retrieving}You may want to use at some place
+in a document a result calculated earlier. It would be good to be
+able to do so without having to do the calculation again at the new
+location. \texttt{numerica} offers a command \verb`\nmcReuse` which
+saves a result to a control sequence which can be used elsewhere in
+the document, expanding to the saved result. The control sequence
+and its content are also saved to file for use on other occasions.
+
+The syntax of \verb`\nmcReuse` is simple. The command takes two optional
+arguments, a star (asterisk) and a conventional square-bracket delimited
+argument. If both are used it looks like
+\begin{lyxcode}
+{\small\textbackslash nmcReuse{*}{[}csname{]}}{\small\par}
+\end{lyxcode}
+where \texttt{csname} is the proposed name of what will become the
+control sequence \verb`\csname` containing the latest result from
+the \verb`\eval` command. The name should be composed of letters
+only.
+
+As with \verb`\nmcEvaluate` and \verb`\nmcInfo` there is a short-name
+form, \verb`\reuse`, for \verb`\nmcReuse`. If a conflict with another
+package arises, it should be possible to fall back on \verb`\nmcReuse`.
+
+\section{Use without optional argument: \texttt{\textbackslash nmcReuse}}
+
+Suppose your document is \verb`mydoc.tex` (so that the \LaTeX{} command
+{\small\verb`\jobname`} expands to \verb`mydoc`). If \verb`\nmcReuse`
+is used without optional arguments, then \texttt{numerica} checks
+for the existence of a file \verb`mydoc.nmc` in the current document
+directory\texttt{ }and if found loads and records the contents of
+\verb`mydoc.nmc`. The contents should be a comma separated list of
+control sequences and braced values like \texttt{\textbackslash csname1
+\{value1\},\textbackslash csname2 \{value2\},...} The control sequences
+can then be used elsewhere in the document. In particular, control
+sequences containing numerical values can be used in expressions within
+\verb`\eval` commands and in vv-lists.\footnote{The associated document \texttt{numerica-plus.pdf} describes how other
+quantities like tables and sequences of numerical values (iterates,
+recurrences) can also be saved and reused.}
+
+Note that the control sequences are given \LaTeX{} definitions using
+\verb`xparse`'s \verb`\NewDocumentCommand` (a little more general
+than \LaTeXe 's \verb`\newcommand`) and can be used at any later
+point in the document simply by entering the control sequence (e.g.,
+\verb`\csname1`) there (but see §\ref{subsec:reuseGroup-level}).
+Should there already be a control sequence with the same name, \LaTeX{}
+will generate an error and halt compilation.
+
+\section{Use with optional name: \texttt{\textbackslash nmcReuse{[}csname{]}}}
+
+If a name \emph{is} supplied, say \verb`\nmcReuse[csname]`, then
+not only does \texttt{numerica} first look for \verb`mydoc.nmc` (assuming
+your document is called \verb`mydoc.tex`) and load the values stored
+in that file if they have not already been loaded, but it also defines
+\verb`\csname` to contain the latest result from the \verb`\eval`
+command. Should \verb`\csname` already be present in \verb`mydoc.nmc`
+and so have been loaded with the other values from \verb`mydoc.nmc`,
+the old value is overwritten with the new value using \verb`xparse`'s
+\verb`\RenewDocumentCommand` and the new value is saved to the file
+\verb`mydoc.nmc`.
+
+\subsection{Group level}
+
+\label{subsec:reuseGroup-level}Control sequences like \verb`\csname`
+defined by \verb`\nmcReuse` are defined within the current group
+level. A usage like
+\begin{centred}
+\verb`$ \eval{1+1}\reuse[two] $`
+\end{centred}
+confines the definition of \verb`\two` to the environment delimited
+by the dollar signs. A usage like
+\begin{centred}
+\verb`$ \eval{1+1} $ \reuse[two]`,
+\end{centred}
+where the command has been moved outside the math delimiters, still
+confines the definition of \verb`\two` to whatever larger environment
+the \verb`\reuse` command might lie within. If in fact the definition
+occurs at document level then \verb`\two` (in the present example)
+is available for use throughout the document, otherwise it is available
+only within the confines of the current environment.
+
+This is likely to be \emph{not }what is wanted. The remedy is simple:
+precede the saved control sequence – \verb`\two` in the present instance
+– with a `naked' \verb`\reuse` command. This loads the value stored
+in \verb`mydoc.nmc`. In the following example, \verb`\reuse` is
+used within a math environment which is followed by some text (\verb`blah blah result:`)
+then a `naked' \verb`\reuse` command and the control sequence \verb`\two`
+(between math delimiters, which are necessary for reasons explained
+in the next section).
+\begin{centred}
+\verb`$ \eval{1+1} \reuse[two] $, blah blah result: \reuse $\two$.`
+$\Longrightarrow$ $ \eval{1+1} \reuse[two]$, blah blah result: \reuse $\two$.
+\end{centred}
+
+\section{What is saved?}
+
+\label{sec:reuseWhat-is-saved}In the default configuration \verb`\nmcReuse`
+saves the entirety of the display resulting from the latest \verb`\eval`-uation.
+This may include invisible formatting elements meaning that what one
+expects to be only a number cannot be inserted into text without causing
+a \LaTeX{} error; it requires a math environment to print – see the
+last example.
+
+The simplest way to avoid this awkwardness is to use \verb`\eval`
+with the star option. This produces a numerical result with \emph{no
+}formatting. In the following example, I calculate \verb`11+11` with
+\verb`\eval*` and store the value in the control sequence \verb`\twos`.
+I then enter some text (\verb`Blah blah blah:`) and insert the control
+sequence \verb`\twos` into the text without math delimiters. As you
+can see, \verb`\twos` has expanded to $22$, the result of the calculation.
+\begin{centred}
+\verb`\eval*{$ 11+11 $}. \nmcReuse[twos] Blah blah blah: \twos` $\Longrightarrow$
+ \eval*{$ 11+11 $}. \nmcReuse[twos] Blah blah blah: \twos
+\end{centred}
+
+\subsection{Viewing what is saved: \textbackslash reuse\texttt{{*}}}
+
+To view what is saved in the \verb`.nmc` file append a star (asterisk)
+to the \verb`\reuse` command. (This makes particular sense when using
+\texttt{numerica} in a program like \LyX{} with a preview facility;
+see Chapter~\ref{chap:LyX}.)
+\begin{centred}
+\verb`\reuse*` $\Longrightarrow$ \reuse*
+\end{centred}
+Particularly notable here is the invisible formatting \verb`\mathchoice{}{}{}{}`
+accompanying \verb`2` in the value of \verb`\two`.
+
+The two options of the\verb`\reuse` command can be used together,
+in which case the control sequence resulting from the name supplied
+in the square-bracketed option will appear in the list resulting from
+the star option, i.e. the list contains not just what has been saved
+earlier but also the current control sequence saved.
+
+\subsection{\texttt{\textbackslash eval}'s \texttt{reuse} setting}
+
+\label{subsec:reuse=00005Ceval's-reuse-setting}The star option of
+the \verb`\eval` command allows a purely numerical result to be saved,
+but also only a number is displayed. By using the \verb`reuse` setting
+of the \verb`\eval` command it is possible to have both a full display
+of an evaluation, vv-list and all, and to save only a numerical result.
+
+For the \emph{starred} form of the \verb`\eval` command it is always
+\emph{only the numerical result} that is saved, whatever the value
+of the \texttt{reuse} key in the settings option of the \verb`\eval`
+command.
+
+For the \emph{unstarred} form of the \verb`\eval` command exactly
+what is saved with \verb`\nmcReuse` depends on the \texttt{reuse}
+setting:
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+where \verb`<integer>` can take one of two values,
+\begin{itemize}
+\item \texttt{reuse=0} (the default) saves\emph{ the form that is displayed}
+including a formatting component. If the result is displayed in the
+form \emph{formula=result (vv-list)} then that is what is saved; if
+the display is of the form \emph{result (vv-list)} then that is what
+is saved; if the vv-list is empty, an empty formatting component is
+still present in the saved result;
+\item \texttt{reuse=1} (or, indeed, any non-zero integer) saves only the
+numerical result with no other elements of the display (meaning no
+formatting component).
+\end{itemize}
+Thus, with the default setting (\texttt{reuse=0}) the full content
+of the display is saved:
+\begin{centred}
+\verb`\eval{$ x + y $}[x=1,y=1] \reuse[iandi]` $\Longrightarrow$
+\eval{$ x + y $}[x=1,y=1] \reuse[iandi].
+\end{centred}
+To check that this is the case, \verb`\reuse \iandi` $\Longrightarrow$
+\reuse \iandi .
+
+On the other hand, with \texttt{reuse=1} only the numerical value
+is saved:
+\begin{centred}
+\verb`\eval[reuse=1]{$ x + y $}[x=1,y=1] \reuse[iandi]` $\Longrightarrow$
+\eval[reuse=1]{$ x + y $}[x=1,y=1] \reuse[iandi],
+\end{centred}
+which we can check here: \verb`\reuse\iandi` $\Longrightarrow$ \reuse\iandi .
+
+\subsubsection{\texttt{\textbackslash reuse} in the preamble}
+
+To gain access from the outset to the control sequences stored in
+the file \texttt{mydoc.nmc}, place \verb`\nmcReuse` without an optional
+argument in the preamble (but after \verb`\usepackage{numerica}`).
+
+\chapter{Using \texttt{numerica} with \protect\LyX}
+
+\label{chap:LyX}The document processor \LyX{} has a facility that
+enables snippets from a larger document to be compiled separately
+and the results presented to the user without having to compile the
+entire document. The present document was written in \LyX . The demonstration
+calculations were evaluated using this \emph{instant preview} facility.
+
+To use \texttt{numerica} in \LyX{} go to \textsf{Document \lyxarrow{}
+Settings \lyxarrow{} LaTeX Preamble} and enter
+\begin{lyxcode}
+\textbackslash usepackage\{numerica\}
+\end{lyxcode}
+then click \textsf{OK}. However preview poses problems for the straightforward
+use of the \verb`\nmcReuse` command. If you wish to use this command
+in \LyX{} then \texttt{numerica} should be loaded with the \texttt{lyx}
+package option. Thus in \textsf{Document \lyxarrow{} Settings \lyxarrow{}
+LaTeX Preamble} enter
+\begin{lyxcode}
+\textbackslash usepackage{[}lyx{]}\{numerica\}
+\end{lyxcode}
+then click \textsf{OK}, or you may wish to follow the above line in
+the preamble with \verb`\nmcReuse`,
+\begin{lyxcode}
+\textbackslash usepackage{[}lyx{]}\{numerica\}
+
+\textbackslash nmcReuse
+\end{lyxcode}
+and \emph{then} click \textsf{OK}. The additional line ensures all
+saved values are available in your document from the outset.
+
+\section{Instant~preview}
+
+Preview performs localised mini-\LaTeX{} runs on selected parts of
+a document (for instance, the mathematical parts) and displays the
+results in \LyX{} while the user continues to work on the surrounding
+document.\texttt{ numerica} uses these local \LaTeX{} runs to do its
+evaluations and display their results. That means you get feedback
+on your calculations almost immediately.
+
+To use this facility first ensure that instant preview is turned on.
+This means selecting \textsf{Tools \lyxarrow Preferences \lyxarrow Look
+\& Feel \lyxarrow{} Display} and against \textsf{Instant preview} selecting
+\textsf{On}, then clicking \textsf{OK}.
+
+\subsection{Conflict with hyperref support?}
+
+There may be a conflict in \LyX{} between hyperref support and preview,
+not that the previews do not form but that their formation takes a
+circuitous path that noticeably slows their display.\footnote{At least there is on the author's Windows 10 system, but I'm not sure
+that this is general.} If this occurs on your system, go to \textsf{Document \lyxarrow{}
+Settings \lyxarrow{} PDF Properties} and ensure the check box \textsf{Use
+Hyperref Support} is cleared. By all means reset the check box when
+you come finally to compile your document, but until then it should
+result in a noticeably brisker display of previews if the checkbox
+is cleared.
+
+\section{Mathed}
+
+(Mathed = the \LyX{} mathematics editor.) If you have instant preview
+\emph{on} then one way to use \texttt{numerica} in \LyX{} is to enter
+an \verb`\eval` command in mathed. Clicking the cursor outside the
+editor with the mouse or moving it outside with the arrow keys will
+then trigger formation of a preview of the editor's contents – a snippet
+of what will be shown in the pdf. This will be displayed in mathed's
+place after a generally short `pause for thought' as the mini-\LaTeX{}
+run progresses behind the scenes.
+
+The original expression can be recovered by clicking on the preview.
+The content of mathed is immediately displayed and can be edited.
+
+\subsection{\protect\LaTeX{} braces~\{~~\}}
+
+\LyX{} does not support \texttt{numerica}'s \verb`\eval` command `out
+of the box' as it does, say, \verb`\frac` or \verb`\sqrt`. To use
+the \verb`\eval` command in mathed you will need to supply the braces
+used to delimit its mandatory argument. (For \verb`\frac` and \verb`\sqrt`
+by contrast, \LyX{} supplies these automatically.) Unfortunately the
+\verb`{` key\footnote{\textsf{Shift+{[}} on my keyboard.} does not
+insert a left brace into the document but rather an escaped left brace
+\verb`\{` as you can see by looking at \textsf{View \lyxarrow{} Code
+Preview Pane}. Escaped braces like this are used for grouping terms
+in \emph{mathematics}; they are not the delimiters of a \LaTeX{} argument.
+
+The brace delimiters for \LaTeX{} arguments are entered in mathed by
+typing a backslash \textsf{\textbackslash{} }then\textsf{ \{} (two
+separate key presses rather than a single combined press). This enters
+a balanced pair of (unescaped) braces with the cursor sitting between
+them waiting for input. Alternatively, if you have already written
+an expression that you want to place between braces, select it, then
+type \textsf{\textbackslash{} }then\textsf{ \{}.
+
+\section{Preview insets vs mathed}
+
+There are problems with using mathed for calculations.
+\begin{itemize}
+\item Expressions entered in mathed are necessarily of the form \verb`$ \eval... $`
+or more generally \verb`delimiter` \verb`\eval...` \verb`delimiter`.
+But you may wish to wrap the \verb`\eval` command \emph{around} the
+math delimiters to produce a \emph{formula=result} form of display.
+In mathed the only way to do that is to write the \emph{formula= }part
+yourself – which may involve no more than copy and paste but is still
+additional mouse work/key pressing.
+\item Mathed does not accept carriage returns. If you want to format a complicated
+expression for readability by breaking it into separate lines, you
+can't. The expression is jammed into the one line, along with the
+settings option content and the vv-list.
+\end{itemize}
+For these reasons I have come to prefer \emph{not} using mathed for
+calculations but instead to use preview insets wrapped around \TeX -code
+(ERT) insets. \LyX{} uses the shortcut \textsf{Ctrl+L} to insert an
+ERT inset. Since \LyX{} now does no printing itself, the shortcut \textsf{Ctrl+P}
+that was formerly used for printing is available for other purposes.
+On my keyboard, the \textsf{P} key lies diagonally up and to the right
+but adjacent to the \textsf{L} key. I suggest assigning \textsf{Ctrl+P}
+to inserting a preview inset. Then typing \textsf{Ctrl+P Ctrl+L} –
+which means holding the \textsf{Ctrl} key down and tapping two adjacent
+keys, \textsf{P} followed immediately by \textsf{L} – will insert
+an ERT inset inside a preview inset with the cursor sitting inside
+the ERT inset waiting for input. In the ERT inset you can enter carriage
+returns, and so format complicated expressions. You can place the
+vv-list on a separate line or onto consecutive lines. And when you
+have finished, clicking outside the preview inset will trigger preview
+into doing its thing and present the result `before your eyes'.
+
+To assign the suggested shortcut, go to \textsf{Tools \lyxarrow{} Preferences
+\lyxarrow{} Editing \lyxarrow{} Shortcuts}. Under \textsf{Cursor, Mouse
+and Editing Functions} in the main window on the right, scroll down
+until you come to \textsf{preview-insert}, select it, then click \textsf{Modify}.
+Now press \textsf{Ctrl+P}. The shortcut will magically appear in the
+greyed, depressed key.\textsf{ }Click \textsf{OK} and then \textsf{OK}
+in the \textsf{Preferences} window to close it. (Most of the examples
+in this document have been evaluated in this way, using \textsf{Ctrl+P
+Ctrl+L.)}
+
+\section{Errors }
+
+Instant preview will display \texttt{numerica} error messages in \LyX{}
+just as it does the results of calculations. Clicking on the message
+will show the underlying expression which can then be edited. However
+\LaTeX{} errors will \emph{not} produce a preview; formation of the
+preview will stall. To find precisely what has gone wrong, you will
+need to look at the \LaTeX{} log, but not the log of the overall document;
+rather the \emph{preview} log. Unfortunately this is tucked away in
+a temporary directory and is not immediately accessible in \LyX{} (unlike
+the main \LaTeX{} log from \textsf{Document \lyxarrow{} \LaTeX{} Log}).
+When \LyX{} is started, it sets up a temporary directory in which to
+perform various tasks. On Windows systems this will be located in
+\texttt{C:\textbackslash Users\textbackslash <your name>\textbackslash AppData\textbackslash Local\textbackslash Temp}
+and will have a name like \texttt{lyx\_tmpdir.XOsSGhBc1344}.
+
+One of the tasks \LyX{} uses this temporary directory for is to create
+preview images when a document is opened. If you look inside \LyX 's
+temporary directory when a document is first loaded, you will see
+a subdirectory created, with a name like \texttt{lyx\_tmpbuf0}. There
+may already be such directories there, in which case the number on
+the end will be greater than \texttt{0} – it depends on whether other
+documents are or have been open in the current instance of \LyX .
+Inside the appropriate \texttt{lyx\_tmpbuf}\texttt{\emph{n}} folder
+will be the preview log with a name like \texttt{lyxpreviewZL1344.log}.
+It will usually be accompanied by other files with extensions like
+\texttt{.dvi}, \texttt{.tex}, and perhaps quite a number with the
+extension \texttt{.png}, each one of which is a preview, or part of
+a preview. For a document just loaded there will be only the one preview
+log, but if you have added preview insets or math insets to your document\textsf{
+}in the current editing session there will be a number of such logs
+and you will need to determine the relevant one by the time stamp.
+
+The log files are text files and can be opened in a text editor. The
+relevant part of the log is towards the end (just before the final
+statistical summary) where you will find a list of entries like \texttt{Preview: Snippet
+1 641947 163840 7864588}. If there is an error, it will be noted here
+among these snippets and will generally make clear what needs remedying.
+
+\subsection{CPU usage, \protect\LaTeX{} processes}
+
+It is possible when a preview fails to resolve that the \LaTeX{} process
+associated with the preview will continue to run, using CPU cycles,
+slowing overall computer performance, and perhaps resulting in extra
+fan use giving a different sound to the computer. In Windows 10, the
+\textsf{Task Manager} (\textsf{Ctrl+Shift+esc}) under the \textsf{Details}
+tab shows the current executables running. The \textsf{CPU} column
+will show which processes are preoccupying the CPU. Check whether
+one or more of these processes looks \LaTeX -related (e.g. \texttt{latex.exe}
+or \texttt{pdflatex.exe}, or \texttt{miktex-pdftex.exe} if using MiK\TeX ).
+Click the \textsf{Name} column to sort the processes by name and look
+for the relevant name in the list, select it, and end the process
+(click the \textsf{End Task} button).
+
+I am not familiar with the corresponding situation on Linux or Mac.
+
+\section{Using \texttt{\textbackslash nmcReuse}}
+
+As noted, \LyX{} creates its previews in a temporary directory, not
+the document directory. If you want to save values from your current
+document – say, \texttt{mydoc.lyx} – to \texttt{mydoc.nmc} then you
+can do so without drama, but \texttt{mydoc.nmc} will be located in
+the temporary directory, and when \LyX{} is closed will be deleted
+along with the temporary directory.
+
+Suppose first that at the end of a session you manually copy \texttt{mydoc.nmc}
+back to the document directory. How can you ensure that the values
+saved in this file are available the next time you open \texttt{mydoc.lyx}?
+As noted at the start of this chapter entering
+\begin{lyxcode}
+\textbackslash usepackage{[}lyx{]}\{numerica\}
+
+\textbackslash nmcReuse
+\end{lyxcode}
+in the preamble ensures that these saved values are available for
+use from the outset – available to the mini-\LaTeX{} runs creating
+previews in the temporary directory.
+
+That leaves the problem of saving new values from the current session,
+which are stored in \texttt{mydoc.nmc} in the \emph{temporary} directory,\texttt{
+}back to \texttt{mydoc.nmc} in the \emph{document} directory. When
+\LyX{} is closed the temporary directory with all its contents is deleted.
+As suggested already we could manually copy \texttt{mydoc.nmc} from
+the temporary directory to the document directory but that means remembering
+to do so before closing \LyX . Inevitably we will sometimes forget.
+
+Fortunately \LyX{} has a copying mechanism for getting files out of
+the temporary directory. When a document is exported – say to pdf
+– it is possible to specify a \emph{copier} to automatically copy
+back to the document directory or subdirectory various files in the
+temporary directory. We want the \texttt{.nmc} file containing the
+saved values to be copied back. Go to \textsf{Tools \lyxarrow{} Preferences
+\lyxarrow{} File Handling \lyxarrow{} File Formats} and find \textsf{PDF
+(pdflatex)} (assuming export to \texttt{pdf} by this route) in the
+list of formats. In\textsf{ }the \textsf{Copier} slot of the dialogue
+insert the following line of code:
+\begin{lyxcode}
+{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc,pdf~-d~\$\$i~\$\$o}{\small\par}
+\end{lyxcode}
+\verb`ext_copy.py` is a python script that is supplied with \LyX .
+The \texttt{-e nmc,pdf -d} part of the line tells \texttt{ext\_copy.py}
+that on export to \texttt{pdf} by the \texttt{pdflatex} route\texttt{
+}to copy any files with the extensions \texttt{.nmc} or \texttt{.pdf}
+from the temporary directory where \LyX{} does its work back to the
+document directory – the \verb`-d` option (which became available
+from \LyX{} 2.3.0).
+
+But if you have a complex document, it may take too much time to want
+to export to pdf before closing \LyX , particularly if there are a
+lot of evaluations in the document. Much faster is to export to \emph{plain
+text}, not because you want a plain text version of your document
+but because it too can be used to trigger the copier mechanism. Go
+to \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} File Handling \lyxarrow{}
+File Formats} and find \textsf{Plain text} in the list of formats.
+In the \textsf{Copier} slot enter
+\begin{lyxcode}
+{\small python~-tt~\$\$s/scripts/ext\_copy.py~-e~nmc~-d~\$\$i~\$\$o}{\small\par}
+\end{lyxcode}
+The only difference from the previous copier command is the absence
+of \texttt{pdf}.\footnote{I'm assuming that you don't actually want the plain text version of
+the file copied back. If you do, then change \texttt{-e nmc} to \texttt{-e
+nmc,txt}.} This will copy \texttt{mydoc.nmc} with its saved values from the
+temporary directory back to the document directory. To effect the
+export, go to \textsf{File \lyxarrow{} Export }and find \textsf{Plain
+text} in the list of formats and click on it.
+
+A shortcut would be nice. For that go to \textsf{Tools \lyxarrow{}
+Preferences \lyxarrow{} Editing \lyxarrow{} Shortcuts}, click on \textsf{New},
+enter \texttt{buffer-export text} in the \textsf{Function:} slot,
+click on the blank key against \textsf{Shortcut:} and type your shortcut.
+You may have to try a number before you find one that hasn't already
+been assigned. (I'm using \textsf{Ctrl+}; for no particular reason
+beyond the fact that it fits under the fingers easily and saving values
+to the document directory has a punctuation-like feel to it, a pause
+in the process of writing.) It is now an easy matter to press the
+shortcut at the end of a \LyX{} session to copy all the values saved
+in \texttt{mydoc.nmc} back to a file of the same name in the document
+directory. And it is brisk, not least because plain text export ignores
+ERT insets (and hence preview insets wrapped around ERT insets), nor
+does it evaluate \verb`\eval` commands in math insets.
+
+\subsection{A final tweak?}
+
+But one still needs to \emph{remember} to press the shortcut. The
+thought arises: can \emph{closing} the current document trigger the
+copying process? \LyX{} provides a means of linking two commands and
+assigning a keyboard shortcut to them with its \texttt{command-sequence}
+\LyX{} function. I suggest assigning a shortcut\textsf{ }to
+\begin{lyxcode}
+command-sequence~buffer-export~text;~view-close
+\end{lyxcode}
+Indeed, why not reassign the current shortcut for \texttt{view-close},\texttt{
+}which is \textsf{Ctrl+W} on my system, to this command sequence?
+(I use the \texttt{cua} key bindings – check the \textsf{Bind file:}
+slot in \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} Editing \lyxarrow{}
+Shortcuts}.)
+
+Please note, however, that \emph{this will work as intended only from
+\LyX{} 2.4.0}.\footnote{Due for release in the first half of 2021.}
+For \LyX{} 2.3 and earlier, the command sequence will generally fail
+because of `asynchronous' processing – \texttt{buffer-export }and
+\texttt{view-close} use different threads and the latter may well
+start before the former is complete. From \LyX{} 2.4.0 this defect
+has been fixed. You press your shortcut, the export to plain text
+occurs and the \texttt{.nmc} file is copied back to the document directory,
+then the current view is closed.
+
+\section{Using \protect\LyX{} notes}
+
+The central fact about a \LyX{} note is that it does not contribute
+to the pdf. But instant preview still works there. This suggests a
+possibility: that a calculation be performed within a \LyX{} note and
+the result saved using \verb`\nmcReuse` within the same preview inset.
+The saved value is now available \emph{from file} for use elsewhere
+in the document. In this way, some selected content from a LyX note
+\emph{can} find its way into the pdf when the document is compiled.
+
+\chapter{Reference summary}
+
+\section{Commands defined in \texttt{numerica}}
+\begin{enumerate}
+\item \texttt{\textbackslash nmcEvaluate, \textbackslash eval }
+\item \texttt{\textbackslash nmcInfo, \textbackslash info, }
+\item \texttt{\textbackslash nmcReuse, \textbackslash reuse}
+\item \texttt{\textbackslash q, \textbackslash Q }(`cleave' commands)
+\end{enumerate}
+Provided they have not already been defined when \texttt{numerica}
+is loaded, the following commands are defined in \texttt{numerica}
+using \verb`\DeclareMathOperator` from \texttt{amsmath} :
+\begin{enumerate}
+\item \texttt{\textbackslash arccsc, \textbackslash arcsec, \textbackslash arccot}
+\item \texttt{\textbackslash csch, \textbackslash sech}
+\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
+\textbackslash acsch, \textbackslash asech, \textbackslash acoth}
+\item \texttt{\textbackslash sgn, \textbackslash lb}
+\end{enumerate}
+Provided they have not already been defined, the following commands
+are defined in \texttt{numerica} using \verb`\DeclarePairedDelimiter`
+from \texttt{mathtools}:
+\begin{lyxcode}
+\textbackslash abs,~\textbackslash ceil,~\textbackslash floor
+\end{lyxcode}
+The following commands have been redefined in \texttt{numerica} to
+give more spacing around the underlying \verb`\wedge` and \verb`\vee`
+symbols:
+\begin{lyxcode}
+\textbackslash land,~\textbackslash lor
+\end{lyxcode}
+
+\section{\textquoteleft Digestible\textquoteright{} content}
+
+\texttt{numerica} knows how to deal with the following content, meaning
+that any of these elements occurring within an \verb`\eval` command
+should not of itself cause a \texttt{numerica} error. Not all formatting
+commands affect display of the output.
+\begin{enumerate}
+\item variable names (sequences of tokens given values in the variable~=~value
+list)
+\item digits, decimal point
+\begin{enumerate}
+\item \texttt{1, 2, 3, 4, 5, 6, 7, 8, 9, 0, .}
+\end{enumerate}
+\item constants
+\begin{enumerate}
+\item \texttt{e, \textbackslash pi, \textbackslash gamma, \textbackslash phi,
+\textbackslash deg, \textbackslash infty }(sometimes)
+\end{enumerate}
+\item arithmetic operators
+\begin{enumerate}
+\item \texttt{+, -, {*}, /, \textasciicircum , \textbackslash times, \textbackslash cdot,
+\textbackslash div}
+\end{enumerate}
+\item logical operators
+\begin{enumerate}
+\item \texttt{\textbackslash wedge, \textbackslash land, \textbackslash vee,
+\textbackslash lor, \textbackslash neg, \textbackslash lnot}
+\end{enumerate}
+\item comparisons
+\begin{enumerate}
+\item \texttt{=, <, >, \textbackslash ne, \textbackslash neq, \textbackslash le,
+\textbackslash leq, \textbackslash ge, \textbackslash geq}
+\item (if \texttt{amssymb} loaded) \texttt{\textbackslash nless, \textbackslash ngtr,
+\textbackslash geqq, \textbackslash geqslant, \textbackslash leqq,
+\textbackslash leqslant, \textbackslash ngeq, \textbackslash ngeqq,
+\textbackslash ngeqslant, \textbackslash nleq, \textbackslash nleqq,
+\textbackslash nleqslant}
+\end{enumerate}
+\item brackets, bracket-like elements, modifiers
+\begin{enumerate}
+\item \texttt{( ), {[} {]}, \textbackslash\{ \textbackslash\}}
+\item \texttt{\textbackslash lparen \textbackslash rparen} (from \texttt{mathtools})\texttt{,
+\textbackslash lbrack \textbackslash rbrack, \textbackslash lbrace
+\textbackslash rbrace}
+\item \texttt{\textbackslash lvert \textbackslash rvert, \textbackslash lfloor
+\textbackslash rfloor, \textbackslash lceil \textbackslash rceil}
+\item \texttt{| |} (no nesting, deprecated)
+\item \texttt{\textbackslash left \textbackslash right, \textbackslash bigl
+\textbackslash bigr, \textbackslash Bigl \textbackslash Bigr, \textbackslash biggl
+\textbackslash biggr, \textbackslash Biggl \textbackslash Biggr}
+\item \texttt{.} \texttt{/ |} (used with a modifier)
+\item \texttt{\textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
+\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
+\textbackslash ceil{*}\{\}}
+\end{enumerate}
+\item unary functions (in the mathematical sense)
+\begin{enumerate}
+\item \texttt{\textbackslash sin, \textbackslash cos, \textbackslash tan,
+\textbackslash csc, \textbackslash sec, \textbackslash cot}
+\item \texttt{\textbackslash arcsin, \textbackslash arccos, \textbackslash arctan,
+arccsc, \textbackslash arcsec, \textbackslash arccot }
+\item \texttt{\textbackslash sin\textasciicircum\{-1\}, \textbackslash cos\textasciicircum\{-1\},
+\textbackslash tan\textasciicircum\{-1\}, \textbackslash csc\textasciicircum\{-1\},
+\textbackslash sec\textasciicircum\{-1\}, \textbackslash cot\textasciicircum\{-1\}}
+\item \texttt{\textbackslash sinh, \textbackslash cosh, \textbackslash tanh,
+\textbackslash csch, \textbackslash sech, \textbackslash coth }
+\item \texttt{\textbackslash asinh, \textbackslash acosh, \textbackslash atanh,
+\textbackslash csch, \textbackslash sech, \textbackslash acoth}
+\item \texttt{\textbackslash sinh\textasciicircum\{-1\}, \textbackslash cosh\textasciicircum\{-1\},
+\textbackslash tanh\textasciicircum\{-1\}, \textbackslash csch\textasciicircum\{-1\},
+\textbackslash sech\textasciicircum\{-1\}, \textbackslash acoth\textasciicircum\{-1\}}
+\item \texttt{\textbackslash exp, \textbackslash lb, \textbackslash lg,
+\textbackslash ln, \textbackslash log, \textbackslash log\_\{\},
+\textbackslash sgn, \textbackslash surd}
+\item \texttt{\textbackslash sqrt\{\}, \textbackslash abs{[}{]}\{\}, \textbackslash abs{*}\{\},
+\textbackslash floor{[}{]}\{\}, \textbackslash floor{*}\{\}, \textbackslash ceil{[}{]}\{\},
+\textbackslash ceil{*}\{\}}
+\item \texttt{!, !! }(prepended argument)
+\end{enumerate}
+\item binary functions
+\begin{enumerate}
+\item \texttt{\textbackslash tfrac\{\}\{\}, \textbackslash frac\{\}\{\},
+\textbackslash dfrac\{\}\{\}}
+\item \texttt{\textbackslash tbinom\{\}\{\}, \textbackslash binom\{\}\{\},
+\textbackslash dbinom\{\}\{\}}
+\item \texttt{\textbackslash sqrt{[}{]}\{\}}
+\end{enumerate}
+\item $n$-ary functions
+\begin{enumerate}
+\item \texttt{\textbackslash min, \textbackslash max, \textbackslash gcd}
+\end{enumerate}
+\item sum, prod
+\begin{enumerate}
+\item \texttt{\textbackslash sum\_\{\}\textasciicircum , \textbackslash prod\_\{\}\textasciicircum{} }
+\end{enumerate}
+\item formatting commands
+\begin{enumerate}
+\item \texttt{,} (comma, in $n$-ary functions)
+\item \texttt{\{\}, \textbackslash\textbackslash , \&, \textbackslash to}
+\item \texttt{\textbackslash dots, \textbackslash ldots, \textbackslash cdots,}
+\item \texttt{\textbackslash{} , \textbackslash ,{}, \textbackslash ;,
+\textbackslash :, \textbackslash !, \textbackslash >}
+\item \texttt{\textbackslash thinspace, \textbackslash quad, \textbackslash qquad
+, \textbackslash hfill, \textbackslash hfil}
+\item \texttt{\textbackslash phantom\{\}, \textbackslash vphantom\{\},
+\textbackslash hphantom\{\}}
+\item \texttt{\textbackslash xmathstrut{[}{]}\{\}} \texttt{, \textbackslash splitfrac\{\}\{\},
+\textbackslash splitdfrac\{\}\{\} }(from \texttt{mathtools}), \texttt{\textbackslash mathstrut}
+\item \texttt{\textbackslash displaystyle, \textbackslash textstyle, \textbackslash scriptstyle,
+\textbackslash scriptscriptstyle}
+\item \texttt{\textbackslash label\{\}, \textbackslash ensuremath\{\},
+\textbackslash text\{\}, \textbackslash mbox\{\}}
+\item \texttt{\textbackslash begin\{\}, \textbackslash end\{\}}
+\end{enumerate}
+\item font commands
+\begin{enumerate}
+\item \texttt{\textbackslash mathrm\{\}, \textbackslash mathit\{\}, \textbackslash mathcal\{\},
+\textbackslash mathtt\{\}, \textbackslash mathbf\{\}, \textbackslash mathbb\{\},
+\textbackslash mathsf\{\}, \textbackslash mathfrak\{\}, \textbackslash mathscr\{\},
+\textbackslash mathnormal\{\}, \textbackslash boldsymbol\{\}}
+\end{enumerate}
+\end{enumerate}
+
+\section{Settings}
+
+\subsection{Available \texttt{\textbackslash nmcEvaluate} settings}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. funcions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small token(s)} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small switch to suppress equation numbering (if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}}{\small )} & \tabularnewline
+{\small\texttt{p}} & char(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{,}}\tabularnewline
+{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion, sums} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, sums} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion, products} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small query stopping with these final terms, products} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{reuse}} & {\small int} & {\small form of result saved with }{\small{\small\verb`\nmcReuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+\subsection{Available configuration file settings}
+\begin{center}
+\bigskip{}
+\begin{tabular}{ll}
+\toprule
+key & default\tabularnewline
+\midrule
+{\small\texttt{rounding}} & {\small\texttt{6}}\tabularnewline
+{\small\texttt{pad}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{output-sci-notation}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{output-exponent-char}} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{input-sci-notation}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{input-exponent-char}} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{multitoken-variables}} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{logarithm-base}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{intify-rounding}} & {\small\texttt{14}}\tabularnewline
+{\small\texttt{vv-display}} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{\small\texttt{vv-inline}} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{sum-extra-rounding}} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{sum-query-terms}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{prod-extra-rounding}} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{prod-query-terms}} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{eval-reuse}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{document}
diff --git a/macros/latex/contrib/numerica/numerica-lyx.def b/macros/latex/contrib/numerica/numerica-lyx.def
new file mode 100644
index 0000000000..96b0f7b193
--- /dev/null
+++ b/macros/latex/contrib/numerica/numerica-lyx.def
@@ -0,0 +1,44 @@
+% This is file `numerica-lyx.def',
+% part of the `numerica' package
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe@gmail.com)
+%
+\ProvidesExplFile
+ {numerica-lyx.def}
+ {2021/02/15}
+ {1.0.0}
+ { Enable reuse command to preview in LyX }
+%----------------------------------------------------------
+% copy (only once) .nmc file from doc to temp
+% directory; use nmcDummy.txt to mark success
+\tl_set:NV \l_tmpb_tl \jobname
+\tl_put_right:Nn \l_tmpb_tl {.nmc}
+\file_if_exist:nTF { nmcDummy.txt }
+ { \tl_gset_eq:NN \g__nmc_reuse_filename_tl \l_tmpb_tl }
+ {
+ \iow_open:Nn \g__nmc_iow { nmcDummy.txt }
+ \iow_close:N \g__nmc_iow
+ \makeatletter
+ \tl_set:Nx \l_tmpa_tl \input@path
+ \makeatother
+ \exp_last_unbraced:NNV
+ \tl_gset:Nn \g__nmc_reuse_filename_tl \l_tmpa_tl
+ \tl_gput_right:NV \g__nmc_reuse_filename_tl \l_tmpb_tl
+ \file_get:VnNTF \g__nmc_reuse_filename_tl {} \l__nmc_reuse_retrieved_tl
+ {
+ \tl_gset_eq:NN \g__nmc_reuse_filename_tl \l_tmpb_tl
+ \iow_open:NV \g__nmc_iow \g__nmc_reuse_filename_tl
+ \bool_lazy_or:nnF
+ { \tl_if_blank_p:V \l__nmc_reuse_retrieved_tl }
+ { \tl_if_head_eq_meaning_p:VN \l__nmc_reuse_retrieved_tl \par }
+ { \iow_now:NV \g__nmc_iow \l__nmc_reuse_retrieved_tl }
+ \iow_close:N \g__nmc_iow
+ }
+ { \tl_gset_eq:NN \g__nmc_reuse_filename_tl \l_tmpb_tl }
+ }
+% end of `numerica-lyx.def' \ No newline at end of file
diff --git a/macros/latex/contrib/numerica/numerica-plus.def b/macros/latex/contrib/numerica/numerica-plus.def
new file mode 100644
index 0000000000..6301dc9643
--- /dev/null
+++ b/macros/latex/contrib/numerica/numerica-plus.def
@@ -0,0 +1,876 @@
+% This is file `numerica-plus.def',
+% part of the `numerica' package.
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe@gmail.com)
+%
+\ProvidesExplFile
+ {numerica-plus.def}
+ {2021/02/15}
+ {1.0.0}
+ { Iterate functions, find fixed points, zeros, extrema and recurrence terms }
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_plus_reuse:nNN #1#2#3
+ {
+ \int_case:nnF { #1 }
+ {
+ { 1 }
+ {
+ \seq_pop:NN #2 \l_tmpa_tl
+ \seq_pop:NN #3 \l_tmpb_tl
+ \tl_gset:Nx \g__nmc_reuse_tl { { \l_tmpa_tl , \l_tmpb_tl } }
+ \seq_mapthread_function:NNN #2 #3 \__nmc_plus_reuse_display:nn
+ }
+ { 2 }
+ {
+ \tl_gset:Nx \g__nmc_reuse_tl
+ { \seq_use:Nn #3 { , } }
+ }
+ }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+\cs_new_protected:Npn \__nmc_plus_reuse_display:nn #1#2
+ { \tl_gput_right:Nx \g__nmc_reuse_tl { , { #1 , #2 } } }
+%--------------------------------------
+% \nmcIterate
+\int_new:N \l__nmc_iter_total_int
+\int_new:N \l__nmc_iter_view_int
+\int_new:N \l__nmc_iter_round_int
+\tl_new:N \l__nmc_iter_var_tl
+\fp_new:N \l__nmc_iter_first_fp
+\seq_new:N \l__nmc_iter_result_seq
+\seq_new:N \l__nmc_iter_index_seq
+
+\fp_new:N \l__nmc_iter_fixedpti_fp
+\fp_new:N \l__nmc_iter_fixedptii_fp
+
+\tl_new:N \l__nmc_iter_begin_tl
+\tl_new:N \l__nmc_iter_end_tl
+\tl_new:N \g__nmc_iter_info_tl
+
+\nmc_define:nnN { \nmcIterate } { iter } \iter
+
+\cs_new_protected:Npn \__nmc_iter_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \int_set:Nn \l__nmc_iter_total_int
+ { \int_max:nn { \l__nmc_iter_total_int } { 1 } }
+ \int_set:Nn \l__nmc_iter_view_int
+ { \int_min:nn { \int_max:nn { \l__nmc_iter_view_int }
+ { 1 } } { \l__nmc_iter_total_int } }
+ }
+\cs_new_protected:Npn \__nmc_iter_vv_digest:N #1
+ {
+ \tl_if_empty:NTF \l__nmc_iter_var_tl
+ { \__nmc_vv_digest:N #1 }
+ {
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_iter_var_tl
+ \__nmc_vv_digest:N #1
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_iter_var_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_process:
+ {
+ \int_set:Nn \l__nmc_iter_round_int
+ { \l__nmc_round_int + \l__nmc_iter_extra_int }
+ % first iterate
+ \__nmc_error_where:n { formula }
+ \tl_clear:N \l__nmc_fp_expr_tl
+ \__nmc_fpify:VN \l__nmc_formula_tl \l__nmc_fp_expr_tl
+ \fp_set:Nn \l__nmc_result_fp { \l__nmc_fp_expr_tl }
+ \__nmc_error_fpflag:
+
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \fp_set_eq:NN \l__nmc_iter_first_fp \l__nmc_result_fp
+ \tl_if_empty:NT \l__nmc_iter_var_tl
+ { % get iter. var if unspecified
+ \seq_get:NN \l__nmc_vv_all_seq \l_tmpa_tl
+ \__nmc_vv_split_item:V \l_tmpa_tl
+ \tl_set_eq:NN \l__nmc_iter_var_tl \l__nmc_eq_var_tl
+ }
+ \__nmc_iter_do:
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ }
+ \bool_if:nF { \g__nmc_error_bool || \l__nmc_num_only_bool }
+ {
+ \__nmc_iter_environ:
+ \__nmc_iter_write:
+ }
+ }
+%--------------------------------------
+\cs_new_protected:Npn \__nmc_iter_do:
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ {
+ \__nmc_iter_fixed_pt:
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_iter_fixedpti_fp }
+ \l_nmc_result_tl { \l__nmc_round_int }
+ \l__nmc_sci_num_out_bool
+ \int_decr:N \l__nmc_iter_total_int
+ \tl_gset:Nx \g__nmc_iter_info_tl
+ { \int_use:N \l__nmc_iter_total_int }
+ }
+ }
+ { % don't print initial iterations
+ \int_step_function:nnnN { 1 } { 1 }
+ { \l__nmc_iter_total_int - \l__nmc_iter_view_int - 1 }
+ \__nmc_iter_current:n
+ \bool_if:NF \g__nmc_error_bool
+ { % store then print these ones
+ \seq_clear:N \l__nmc_iter_result_seq
+ \int_step_function:nnnN { 1 } { 1 }
+ { \int_min:nn { \l__nmc_iter_view_int }
+ { \l__nmc_iter_total_int - 1 } } \__nmc_iter_current_store:n
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_current:n #1
+ { % stepping function
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_iter_var_tl \l__nmc_formula_tl
+ { \l__nmc_result_fp } \l__nmc_result_fp
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_current_store:n #1
+ { % stepping function
+ \__nmc_calc_fn_val:VNnN \l__nmc_iter_var_tl \l__nmc_formula_tl
+ { \l__nmc_result_fp } \l__nmc_result_fp
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_result_fp } \l_nmc_result_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:NV \l__nmc_iter_result_seq \l_nmc_result_tl
+ \int_set:Nn \l_tmpa_int
+ { \l__nmc_iter_total_int - \l__nmc_iter_view_int + #1 }
+ \seq_put_right:Nx \l__nmc_iter_index_seq { \int_use:N \l_tmpa_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_fixed_pt:
+ { % already 1 iteration
+ \int_set:Nn \l__nmc_iter_total_int { 1 }
+ \fp_set_eq:NN \l__nmc_iter_fixedpti_fp \l__nmc_iter_first_fp
+ \fp_set:Nn \l__nmc_iter_fixedptii_fp { \l__nmc_iter_fixedpti_fp + 1 }
+ \bool_until_do:nn
+ {
+ \fp_compare_p:nNn { 0 } =
+ { round( \l__nmc_iter_fixedpti_fp - \l__nmc_iter_fixedptii_fp,
+ \l__nmc_iter_round_int ) }
+ || \g__nmc_error_bool
+ }
+ {
+ \int_incr:N \l__nmc_iter_total_int
+ \fp_set_eq:NN \l__nmc_iter_fixedptii_fp \l__nmc_iter_fixedpti_fp
+ \__nmc_calc_fn_val:VNnN \l__nmc_iter_var_tl \l__nmc_formula_tl
+ { \l__nmc_result_fp } \l__nmc_result_fp
+ \fp_set_eq:NN \l__nmc_iter_fixedpti_fp \l__nmc_result_fp
+ \int_compare:nNnF { \l__nmc_iter_total_int } <
+ { \l__nmc_iter_max_int }
+ {
+ \bool_gset_true:N \g__nmc_error_bool
+ \__nmc_error_what:n
+ {
+ No~fixed~point~attained~after~$\int_use:N
+ \l__nmc_iter_total_int$~iterations~of
+ }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_environ:
+ {
+ \tl_set:Nx \l__nmc_iter_begin_tl
+ {
+ \mode_if_math:F { \exp_not:o \l__nmc_math_delimi_tl }
+ \exp_not:N \begin{array}{r@{}l}
+ \bool_if:NTF \l__nmc_wraps_math_bool
+ { \exp_not:o \l__nmc_formula_dup_tl &{}= }
+ { &{} }
+ }
+ \tl_set:Nx \l__nmc_iter_end_tl
+ {
+ \exp_not:N \end{array}
+ \mode_if_math:F { \exp_not:o \l__nmc_math_delimii_tl }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_write:
+ {
+ \__nmc_num_format:nNnN { \l__nmc_iter_first_fp } \l_tmpa_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_gset:Nx \g__nmc_iter_info_tl { \int_use:N \l__nmc_iter_total_int }
+ \__nmc_if_mod_zero:nnTF { \l__nmc_debug_int } { 7 }
+ { \seq_get_right:NN \l__nmc_iter_result_seq \l_nmc_result_tl }
+ {
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_iter_begin_tl
+ \l_tmpa_tl \l__nmc_vv_display_tl
+ \int_compare:nNnT
+ { \l__nmc_iter_total_int } > { \l__nmc_iter_view_int + 1 }
+ {
+ \\ & \exp_not:N \ldots
+ \ \exp_not:N\mbox{final\
+ \int_use:N \l__nmc_iter_view_int\ of\ \g__nmc_iter_info_tl :}
+ }
+ \int_compare:nNnT { \l__nmc_iter_total_int } > { 1 }
+ { \\ & \exp_not:N \hookrightarrow }
+ \seq_use:Nn \l__nmc_iter_result_seq { \\ & \hookrightarrow }
+ \exp_not:o \l__nmc_iter_end_tl
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_iter_display:
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ {
+ \__nmc_plus_reuse:nNN { \l__nmc_iter_reuse_int }
+ \l__nmc_iter_index_seq \l__nmc_iter_result_seq
+ }
+ \l_nmc_result_tl
+ }
+%--------------------------------------
+%\nmcSolve
+\int_new:N \l__nmc_solve_round_int
+\int_new:N \l__nmc_solve_steps_int
+\int_new:N \l__nmc_solve_slope_int
+\int_new:N \l__nmc_solve_slopei_int
+\int_new:N \l__nmc_solve_signs_int
+\fp_new:N \l__nmc_solvea_fp
+\fp_new:N \l__nmc_solveb_fp
+\fp_new:N \l__nmc_solvec_fp
+\fp_new:N \l__nmc_solved_fp
+
+\fp_new:N \l__nmc_solvefa_fp
+\fp_new:N \l__nmc_solvefb_fp
+\fp_new:N \l__nmc_solvefc_fp
+\bool_new:N \l__nmc_solve_stop_bool
+\tl_new:N \g__nmc_sove_info_tl
+
+\nmc_define:nnN { \nmcSolve } { solve } \solve
+
+\cs_new_protected:Npn \__nmc_solve_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \tl_if_empty:NF \l__nmc_solve_step_tl
+ {
+ \__nmc_fpify_set:NN \l__nmc_solve_step_fp \l__nmc_solve_step_tl
+ \fp_compare:nNnT \l__nmc_solve_step_fp = { 0 }
+ { \__nmc_error_what:n { Non-zero~initial~step~required~in } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_vv_digest:N #1
+ {
+ \tl_if_empty:NTF \l__nmc_solve_var_tl
+ { \__nmc_vv_digest:N #1 }
+ {
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_solve_var_tl
+ \__nmc_vv_digest:N #1
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_solve_var_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_process:
+ {
+ \int_set:Nn \l__nmc_solve_round_int
+ { \l__nmc_round_int + \l__nmc_solve_extra_int }
+ % get equation var if unspecified
+ \tl_if_empty:NT \l__nmc_solve_var_tl
+ {
+ \seq_get:NN \l__nmc_vv_all_seq \l_tmpa_tl
+ \__nmc_vv_split_item:V \l_tmpa_tl
+ \tl_set_eq:NN \l__nmc_solve_var_tl \l__nmc_eq_var_tl
+ }
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_solve_get_trial_vals: }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_error_where:n { function }
+ \__nmc_solve_do:
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_solvea_fp } \l_nmc_result_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_gset:Nx \g__nmc_solve_info_tl
+ { \clist_use:Nn \g__nmc_solve_info_tl { + } }
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_get_trial_vals:
+ {
+ \prop_get:NVN \g__nmc_subst_var_prop
+ \l__nmc_solve_var_tl \l__nmc_subst_tl
+ % ensure a < b
+ \int_case:nn { \fp_sign:n { \l__nmc_solve_step_fp } }
+ {
+ { 1 }
+ {
+ \fp_set:Nn \l__nmc_solvea_fp { \l__nmc_subst_tl }
+ \fp_set:Nn \l__nmc_solveb_fp
+ { \l__nmc_solvea_fp + \l__nmc_solve_step_fp }
+ }
+ { -1 }
+ {
+ \fp_set:Nn \l__nmc_solveb_fp { \l__nmc_subst_tl }
+ \fp_set:Nn \l__nmc_solvea_fp
+ { \l__nmc_solveb_fp + \l__nmc_solve_step_fp }
+ }
+ }
+ }
+% find opp. signs, zero, or fn min.
+% a b = var vals; fa fb = fn vals
+\cs_new_protected:Npn \__nmc_solve_do:
+ {
+ \__nmc_solve_calc_values:
+ \int_zero:N \l__nmc_solve_steps_int
+ \tl_gclear:N \g__nmc_solve_info_tl
+ \bool_do_until:nn
+ {
+ \g__nmc_error_bool || \l__nmc_solve_stop_bool
+ || \int_compare_p:nNn
+ { \l__nmc_solve_steps_int } > { \l__nmc_solve_max_int }
+ || \fp_compare_p:nNn { 0 } =
+ { round(\l__nmc_solveb_fp - \l__nmc_solvea_fp,
+ \l__nmc_solve_round_int) }
+ }
+ {
+ \int_incr:N \l__nmc_solve_steps_int
+ \int_case:nn { \l__nmc_solve_signs_int }
+ {
+ { 0 } { \__nmc_solve_do_bingo: }
+ { -1 } { \__nmc_solve_do_bisect: }
+ { 1 } { \__nmc_solve_do_slope: }
+ }
+ }
+ \bool_if:nF { \g__nmc_error_bool || \l__nmc_solve_stop_bool }
+ {
+ \__nmc_error_where:n { $\l__nmc_formula_tl$ }
+ \__nmc_error_what:n
+ {
+ No~zero/extremum~found~after~$\int_use:N
+ \l__nmc_solve_max_int$~steps~for~function
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_do_bingo:
+ { % fn = 0 to 16 figures
+ \fp_compare:nNnTF { \l__nmc_solvefb_fp } = { 0 }
+ {
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solveb_fp
+ \fp_set_eq:NN \l__nmc_solvefa_fp \l__nmc_solvefb_fp
+ }
+ {
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvea_fp
+ \fp_set_eq:NN \l__nmc_solvefb_fp \l__nmc_solvefa_fp
+ }
+ \bool_set_true:N \l__nmc_solve_stop_bool
+ }
+\cs_new_protected:Npn \__nmc_solve_do_bisect:
+ {
+ \tl_gset:Nx \g__nmc_solve_info_tl { \int_use:N \l__nmc_solve_steps_int }
+ \int_zero:N \l__nmc_solve_steps_int
+ \fp_set:Nn \l__nmc_solvec_fp { ( \l__nmc_solvea_fp + \l__nmc_solveb_fp ) / 2 }
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl \l__nmc_formula_tl
+ { \l__nmc_solvec_fp } \l__nmc_solvefc_fp
+ \fp_set:Nn \l__nmc_solved_fp { \l__nmc_solvec_fp + 1 }
+ \bool_until_do:nn
+ {
+ \g__nmc_error_bool ||
+ \fp_compare_p:nNn { 0 } =
+ { round( \l__nmc_solvec_fp - \l__nmc_solved_fp ,
+ \l__nmc_solve_round_int ) }
+ }
+ {
+ \int_incr:N \l__nmc_solve_steps_int
+ \fp_set_eq:NN \l__nmc_solved_fp \l__nmc_solvec_fp
+ \fp_compare:nNnTF { 0 } = { \l__nmc_solvefc_fp }
+ {
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp
+ \fp_set_eq:NN \l__nmc_solvefa_fp \l__nmc_solvefc_fp
+ }
+ {
+ \fp_compare:nNnTF
+ { sign(\l__nmc_solvefa_fp)sign(\l__nmc_solvefc_fp) } = { 1 }
+ {
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp
+ \fp_set_eq:NN \l__nmc_solvefa_fp \l__nmc_solvefc_fp
+ }
+ {
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvec_fp
+ \fp_set_eq:NN \l__nmc_solvefb_fp \l__nmc_solvefc_fp
+ }
+ \fp_set:Nn \l__nmc_solvec_fp
+ { ( \l__nmc_solvea_fp + \l__nmc_solveb_fp ) / 2 }
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl
+ \l__nmc_formula_tl { \l__nmc_solvec_fp } \l__nmc_solvefc_fp
+ }
+ }
+ \bool_set_true:N \l__nmc_solve_stop_bool
+ \clist_gput_right:Nx \g__nmc_solve_info_tl { \int_use:N \l__nmc_solve_steps_int }
+ }
+\cs_new_protected:Npn \__nmc_solve_do_slope:
+ {
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_if_zero:nTF { \l__nmc_solve_slope_int }
+ { % contract
+ \fp_add:Nn \l__nmc_solvea_fp
+ { ( \l__nmc_solveb_fp - \l__nmc_solvea_fp ) / 4 }
+ \fp_sub:Nn \l__nmc_solveb_fp
+ { ( \l__nmc_solveb_fp - \l__nmc_solvea_fp ) / 4 }
+ }
+ { % always towards x-axis
+ \fp_compare:nNnTF { 0 } <
+ { \l__nmc_solvefa_fp * \l__nmc_solve_slope_int }
+ { \__nmc_solve_do_slope_left: }
+ { \__nmc_solve_do_slope_right: }
+ }
+ \fp_set_eq:NN \l__nmc_solved_fp \l__nmc_solvea_fp
+ \int_set_eq:NN \l__nmc_solve_slopei_int \l__nmc_solve_slope_int
+ \__nmc_solve_calc_values:
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_compare:nNnF { \l__nmc_solve_slope_int } =
+ { \l__nmc_solve_slopei_int }
+ {
+ \fp_set:Nn \l__nmc_solvec_fp { ( \l__nmc_solvea_fp + \l__nmc_solveb_fp ) / 2 }
+ \int_case:nn { \l__nmc_solve_slopei_int }
+ {
+ { 1 } { \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp }
+ { -1 } { \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvec_fp }
+ }
+ \__nmc_solve_calc_values:
+ }
+ }
+ \fp_compare:nNnT { 0 } =
+ { round(\l__nmc_solveb_fp - \l__nmc_solvea_fp,
+ \l__nmc_solve_round_int) }
+ { \bool_set_true:N \l__nmc_solve_stop_bool }
+ }
+\cs_new_protected:Npn \__nmc_solve_do_slope_left:
+ {
+ \fp_set:Nn \l__nmc_solvec_fp { 2 \l__nmc_solvea_fp - \l__nmc_solveb_fp }
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvea_fp
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solvec_fp
+ }
+\cs_new_protected:Npn \__nmc_solve_do_slope_right:
+ {
+ \fp_set:Nn \l__nmc_solvec_fp { 2 \l__nmc_solveb_fp - \l__nmc_solvea_fp }
+ \fp_set_eq:NN \l__nmc_solvea_fp \l__nmc_solveb_fp
+ \fp_set_eq:NN \l__nmc_solveb_fp \l__nmc_solvec_fp
+ }
+\cs_new_protected:Npn \__nmc_solve_calc_values:
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl \l__nmc_formula_tl
+ { \l__nmc_solvea_fp } \l__nmc_solvefa_fp
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_solve_var_tl \l__nmc_formula_tl
+ { \l__nmc_solveb_fp } \l__nmc_solvefb_fp
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_set:Nn \l__nmc_solve_slope_int
+ { \fp_eval:n { sign(\l__nmc_solvefb_fp - \l__nmc_solvefa_fp) } }
+ \int_set:Nn \l__nmc_solve_signs_int
+ { \fp_eval:n { sign(\l__nmc_solvefa_fp) sign(\l__nmc_solvefb_fp) } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_solve_display:
+ {
+ \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \__nmc_num_format:nNnN { \l__nmc_solvefa_fp } \l_tmpa_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_math_delimi_tl
+ \bool_if:NTF \l__nmc_wraps_math_bool
+ {
+ \exp_not:o \l__nmc_formula_tl = \l_tmpa_tl
+ \bool_if:NTF \l__nmc_vv_multline_bool
+ { \exp_not:o \l__nmc_vv_display_tl }
+ {
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \rightarrow \exp_not:N \quad
+ \l__nmc_solve_var_tl = \l_nmc_result_tl
+ }
+ {
+ \l__nmc_solve_var_tl = \l_nmc_result_tl
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \l__nmc_punc_tl
+ \exp_not:o \l__nmc_math_delimii_tl
+ }
+ \int_if_zero:nT { \l__nmc_solve_reuse_int }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+ \l_nmc_result_tl
+ }
+%--------------------------------------
+% \nmcRecur
+\bool_new:N \l__nmc_recur_ellipsis_bool
+\int_new:N \l__nmc_recur_last_int
+\fp_new:N \l__nmc_recur_result_fp
+
+\int_new:N \l__nmc_recur_subscr_ini_int
+\int_new:N \l__nmc_recur_subscr_val_int
+\int_new:N \l__nmc_recur_order_int
+\int_new:N \l__nmc_recur_var_int
+
+\tl_new:N \l__nmc_recurrence_tl
+\tl_new:N \l__nmc_recur_base_var_tl
+\tl_new:N \l__nmc_recur_subscr_var_tl
+
+\seq_new:N \l__nmc_recur_result_seq
+\seq_new:N \l__nmc_recur_index_seq
+\seq_new:N \l__nmc_recur_vars_seq
+
+\nmc_define:nnN { \nmcRecur } { recur } \recur
+
+\cs_new_protected:Npn \__nmc_recur_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \int_set:Nn \l__nmc_recur_total_int
+ { \int_max:nn { \l__nmc_recur_total_int } { 1 } }
+ \int_set:Nn \l__nmc_recur_last_int
+ { \int_max:nn { 0 } { \int_min:nn
+ { \l__nmc_recur_last_int } { \l__nmc_recur_total_int } } }
+ \int_set:Nn \l__nmc_recur_first_int
+ { \int_max:nn { 0 } { \int_min:nn { \l__nmc_recur_first_int }
+ { \l__nmc_recur_total_int - \l__nmc_recur_last_int } } }
+ \int_if_zero:nT { \l__nmc_recur_first_int }
+ { \int_decr:N \l__nmc_recur_first_int }
+ }
+\cs_new_protected:Npn \__nmc_recur_vv_digest:N #1
+ { % #1 = reversed vv clist
+ \bool_set_true:N \l__nmc_multitok_bool
+ \__nmc_recur_elements:
+ \__nmc_recur_vars_change:N #1
+ \__nmc_vv_digest:N #1
+ \__nmc_recur_vv_post:
+ \tl_set_eq:NN \l__nmc_formula_tl \l__nmc_recurrence_tl
+ }
+% \l__nmc_recurrence_tl, \l__nmc_recur_base_var_tl,
+% \l__nmc_recur_subscr_var_tl, \l__nmc_recur_subscr_val_int
+\cs_new_protected:Npn \__nmc_recur_elements:
+ {
+ \tl_clear:N \l_tmpa_tl
+ \bool_set_false:N \l_tmpa_bool
+ \tl_map_inline:Nn \l__nmc_formula_tl
+ {
+ \bool_if:NTF \l_tmpa_bool
+ {
+ \tl_set:Nn \l_tmpa_tl { ##1 }
+ \tl_map_break:
+ }
+ {
+ \token_if_math_subscript:NTF ##1
+ {
+ \tl_set:NV \l__nmc_recur_base_var_tl \l_tmpa_tl
+ \bool_set_true:N \l_tmpa_bool
+ }
+ { \tl_put_right:Nn \l_tmpa_tl { ##1 } }
+ }
+ }
+ \__nmc_recur_parse_subscr:N \l_tmpa_tl
+ \exp_last_unbraced:NV\__nmc_split_eq:w \l__nmc_formula_tl \q_stop
+ \tl_set:NV \l__nmc_recurrence_tl \l__nmc_eq_val_tl
+ \tl_set_rescan:Nno \l__nmc_recurrence_tl { \ExplSyntaxOn } \l__nmc_recurrence_tl
+ }
+\cs_new_protected:Npn \__nmc_recur_parse_subscr:N #1
+ {
+ \tl_clear:N \l__nmc_recur_subscr_var_tl
+ \tl_set:Nn \l_tmpb_tl { 0 }
+ \int_zero:N \l__nmc_recur_subscr_val_int
+ \bool_set_false:N \l_tmpa_bool
+ \tl_map_inline:Nn #1
+ {
+ \bool_if:NTF \l_tmpa_bool
+ { \tl_put_right:Nn \l_tmpb_tl { ##1 } }
+ {
+ \tl_if_in:nnTF { +- } { ##1 }
+ {
+ \tl_put_right:Nn \l_tmpb_tl { ##1 }
+ \bool_set_true:N \l_tmpa_bool
+ }
+ { \tl_put_right:Nn \l__nmc_recur_subscr_var_tl { ##1 } }
+ }
+ }
+ \int_set:Nn \l__nmc_recur_subscr_val_int { \l_tmpb_tl }
+ }
+\cs_new_protected:Npn \__nmc_recur_vars_change:N #1
+ { % f_{1} etc ==> f_{n-1} etc in #1 (reverse order vv-list)
+ \clist_reverse:N #1
+ \int_zero:N \l__nmc_recur_order_int
+ \clist_clear:N \l_tmpa_clist % --> \l__nmc_recur_vars_seq
+ \clist_clear:N \l_tmpb_clist % --> #1
+ \int_set:Nn \l_tmpb_int { \l__nmc_recur_subscr_val_int - 1 }
+ \tl_set_rescan:Nno #1 { \ExplSyntaxOn } #1
+ \clist_map_inline:Nn #1
+ {
+ \seq_set_split:Nnn \l_tmpa_seq {_} { ##1 }
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_if_empty:NTF \l_tmpa_seq
+ { \clist_put_left:NV \l_tmpb_clist \l_tmpa_tl }
+ {
+ \tl_if_eq:NNTF \l_tmpa_tl \l__nmc_recur_base_var_tl
+ { % change e.g. f_{1}(x) to f_{n-1}(x)
+ \int_incr:N \l__nmc_recur_order_int
+ \tl_put_right:Nn \l_tmpa_tl { _ }
+ \tl_set_eq:NN \l_tmpb_tl \l__nmc_recur_subscr_var_tl
+ \int_case:nn { \int_sign:n { \l_tmpb_int } }
+ {
+ { -1 }
+ {
+ \tl_put_right:Nx \l_tmpb_tl
+ { \int_use:N \l_tmpb_int }
+ }
+ { 0 } { \prg_do_nothing: }
+ { 1 }
+ {
+ \tl_put_right:Nn \l_tmpb_tl { + }
+ \tl_put_right:Nx \l_tmpb_tl
+ { \int_use:N \l_tmpb_int }
+ }
+ }
+ \tl_put_right:Nx \l_tmpa_tl { { \l_tmpb_tl } }
+ \int_decr:N \l_tmpb_int
+ \seq_pop:NN \l_tmpa_seq \l_tmpb_tl
+ \int_set:Nn \l__nmc_recur_subscr_ini_int
+ { \tl_head:N \l_tmpb_tl }
+ \tl_put_right:Nx \l_tmpa_tl
+ { \tl_range:Nnn \l_tmpb_tl { 2 } { -1 } }
+ \clist_put_left:NV \l_tmpb_clist \l_tmpa_tl
+ \seq_set_split:NnV \l_tmpb_seq { = } \l_tmpa_tl
+ \seq_pop:NN \l_tmpb_seq \l_tmpa_tl
+ \clist_put_left:NV \l_tmpa_clist \l_tmpa_tl
+ }
+ { \clist_put_left:Nn \l_tmpb_clist { ##1 } }
+ }
+ }
+ \int_set:Nn \l__nmc_recur_var_int
+ { \l__nmc_recur_subscr_ini_int + \l__nmc_recur_order_int
+ - \l__nmc_recur_subscr_val_int }
+ \clist_set_eq:NN #1 \l_tmpb_clist
+ \clist_put_left:NV \l_tmpa_clist \l__nmc_recur_subscr_var_tl
+ \clist_concat:NNN \l__nmc_formula_tl \l__nmc_recurrence_tl \l_tmpa_clist
+ }
+\cs_new_protected:Npn \__nmc_recur_vv_post:
+ {
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_recurrence_tl
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_recur_subscr_var_tl
+ \tl_set:Nx \l_tmpa_tl { \int_use:N \l__nmc_recur_var_int }
+ \__nmc_vv_record:NVN \l__nmc_recur_subscr_var_tl \l_tmpa_tl \c_empty_prop
+ \tl_set_eq:NN \l_tmpa_tl \l__nmc_recur_subscr_var_tl
+ \tl_put_right:Nn \l_tmpa_tl { =0 } % formal value
+ \seq_put_left:NV \l__nmc_calc_fn_seq \l_tmpa_tl
+ \seq_set_from_clist:NN \l__nmc_recur_vars_seq \l__nmc_formula_tl
+ }
+%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_recur_process:
+ { % store initial vals; generate later vals
+ \__nmc_recur_store_ini:
+ \__nmc_error_where:n { recurrence~formula }
+ \__nmc_recur_generate:
+ \seq_get_right:NN \l__nmc_recur_result_seq \l_nmc_result_tl
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ }
+\cs_new_protected:Npn \__nmc_recur_store_ini:
+ {
+ \seq_set_eq:NN \l_tmpa_seq \l__nmc_recur_vars_seq
+ \int_step_inline:nnnn { 1 } { 1 } { \l__nmc_recur_order_int }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \prop_get:NVN \g__nmc_subst_var_prop \l_tmpa_tl \l__nmc_subst_tl
+ \__nmc_num_format:nNnN { \l__nmc_subst_tl } \l_tmpa_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:NV \l__nmc_recur_result_seq \l_tmpa_tl
+ \tl_set:Nx \l_tmpb_tl { \int_eval:n
+ { \l__nmc_recur_subscr_ini_int + ##1 -1 } }
+ \seq_put_right:NV \l__nmc_recur_index_seq \l_tmpb_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_recur_generate:
+ {
+ \prop_get:NVN \g__nmc_subst_var_prop
+ \l__nmc_recur_subscr_var_tl \l__nmc_subst_tl
+ \int_set:Nn \l__nmc_recur_var_int { \l__nmc_subst_tl }
+ \int_set:Nn \l_tmpa_int { \l__nmc_recur_var_int +
+ \l__nmc_recur_total_int - \l__nmc_recur_order_int - 1 }
+ \__nmc_error_where:n { recurrence~relation }
+ \__nmc_if_mod_zero:nnT { \l__nmc_debug_int } { 7 }
+ { \__nmc_fpify:VN \l__nmc_recurrence_tl \l__nmc_fp_expr_tl }
+ \int_step_function:nnnN { \l__nmc_recur_var_int } { 1 }
+ {\l_tmpa_int } \__nmc_recur_generate_loop:n
+ }
+\cs_new_protected:Npn \__nmc_recur_generate_loop:n #1
+ {
+ \bool_if:NF \g__nmc_error_bool
+ { % calc. the next term
+ \fp_set:Nn \l_tmpa_fp { #1 }
+ \__nmc_calc_fn_val:VNnN \l__nmc_recur_subscr_var_tl
+ \l__nmc_recurrence_tl { \l_tmpa_fp } \l__nmc_recur_result_fp
+ }
+ \bool_if:NF \g__nmc_error_bool
+ { % store the result
+ \__nmc_num_format:nNnN { \l__nmc_recur_result_fp }
+ \l_tmpa_tl { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:NV \l__nmc_recur_result_seq \l_tmpa_tl
+ \seq_put_right:Nn \l__nmc_recur_index_seq { #1 }
+ % shift vals "down variable"; tmpa above, tmpb below
+ \seq_set_eq:NN \l_tmpa_seq \l__nmc_recur_vars_seq
+ \seq_pop:NN \l_tmpa_seq \l_tmpb_tl % low var
+ \int_step_inline:nnnn {2} { 1 } { \l__nmc_recur_order_int }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl % hi var
+ \prop_get:NVN \g__nmc_subst_var_prop \l_tmpa_tl \l__nmc_subst_tl
+ \prop_put:NVV \g__nmc_subst_var_prop \l_tmpb_tl \l__nmc_subst_tl
+ \prop_put:NVV \l__nmc_vv_change_prop \l_tmpb_tl \l__nmc_subst_tl
+ \tl_set_eq:NN \l_tmpb_tl \l_tmpa_tl
+ }
+ % use tmpb, not tmpa, in case order = 1
+ \prop_put:NVx \g__nmc_subst_var_prop \l_tmpb_tl
+ { \fp_use:N \l__nmc_recur_result_fp }
+ \prop_put:NVx \l__nmc_vv_change_prop \l_tmpb_tl
+ { \fp_use:N \l__nmc_recur_result_fp }
+ }
+ }
+\cs_new_protected:Npn \__nmc_recur_display:
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ {
+ \seq_clear:N \l_tmpa_seq
+ \seq_clear:N \l_tmpb_seq
+ \seq_clear:N \l_tmpc_seq
+ \__nmc_recur_result:NN \l_tmpa_seq \l_tmpb_seq
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_math_delimi_tl
+ \bool_if:NT \l__nmc_wraps_math_bool
+ {
+ \exp_not:o \l__nmc_formula_dup_tl
+ \bool_if:NTF \l__nmc_vv_multline_bool
+ { \exp_not:o \l__nmc_vv_display_tl }
+ {
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \rightarrow \exp_not:N \quad
+ }
+ \seq_use:Nn \l_tmpa_seq { ,\ }
+ \l__nmc_punc_tl
+ \exp_not:o \l__nmc_math_delimii_tl
+ }
+ \__nmc_plus_reuse:nNN { \l__nmc_recur_reuse_int } \l_tmpc_seq \l_tmpb_seq
+ }
+ \l_nmc_result_tl
+ }
+\cs_new_protected:Npn \__nmc_recur_result:NN #1#2
+ {
+ \int_zero:N \l_tmpa_int
+ \seq_map_inline:Nn \l__nmc_recur_result_seq
+ {
+ \seq_pop:NN \l__nmc_recur_index_seq \l__nmc_toss_tl
+ \int_compare:nNnTF { \l_tmpa_int } < { \l__nmc_recur_first_int }
+ { \seq_put_right:Nn #1 { ##1 } }
+ {
+ \int_compare:nTF { \l_tmpa_int = \l__nmc_recur_first_int
+ < \l__nmc_recur_total_int - \l__nmc_recur_last_int }
+ { \seq_put_right:Nn #1 { \ldots } }
+ {
+ \int_compare:nNnT { 1 + \l_tmpa_int } >
+ { \l__nmc_recur_total_int - \l__nmc_recur_last_int }
+ {
+ \seq_put_right:Nn #1 { ##1 }
+ \seq_put_right:Nn #2 { ##1 }
+ \seq_put_right:Nx \l_tmpc_seq { \int_eval:n { \l_tmpa_int + \l__nmc_recur_subscr_val_int } }
+ }
+ }
+ }
+ \int_incr:N \l_tmpa_int
+ }
+ \bool_if:NT \l__nmc_recur_ellipsis_bool
+ { \seq_put_right:Nn #1{ \ldots } }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\keys_define:nn { numerica }
+ {
+ iter-do .int_set:N = \l__nmc_iter_total_int,
+ iter-see-last .int_set:N = \l__nmc_iter_view_int,
+ iter-max-iterations .int_set:N = \l__nmc_iter_max_int,
+ iter-extra-rounding .int_set:N = \l__nmc_iter_extra_int,
+ iter-reuse .int_set:N = \l__nmc_iter_reuse_int,
+
+ iter-do .default:n = 5,
+ iter-see-last .default:n = 4,
+ iter-max-iterations .default:n = 100,
+ iter-extra-rounding .default:n = 0,
+ iter-reuse .default:n = 0,
+%
+ solve-max-steps .int_set:N = \l__nmc_solve_max_int,
+ solve-extra-rounding .int_set:N = \l__nmc_solve_extra_int,
+ solve-first-step .tl_set:N = \l__nmc_solve_step_tl,
+ solve-reuse .int_set:N = \l__nmc_solve_reuse_int,
+
+ solve-max-steps .default:n = 100,
+ solve-extra-rounding .default:n = 0,
+ solve-first-step .default:n = 1,
+ solve-reuse .default:n = 0,
+%
+ recur-do .int_set:N = \l__nmc_recur_total_int,
+ recur-see-first .int_set:N = \l__nmc_recur_first_int,
+ recur-see-last .int_set:N = \l__nmc_recur_last_int,
+ recur-reuse .int_set:N = \l__nmc_recur_reuse_int,
+
+ recur-do .default:n = 7,
+ recur-see-first .default:n = 3,
+ recur-see-last .default:n = 2,
+ recur-reuse .default:n = 0
+ }
+\keys_set_known:nn { numerica }
+ {
+ iter-do,iter-see-last,iter-max-iterations,
+ iter-extra-rounding,iter-reuse,
+ solve-variable,solve-first-step,solve-max-steps,
+ solve-extra-rounding,solve-reuse,
+ recur-do,recur-see-first,recur-see-last,recur-reuse
+ }
+\keys_define:nn { numerica/iter }
+ {
+ var .tl_set:N = \l__nmc_iter_var_tl,
+ + .int_set:N = \l__nmc_iter_extra_int,
+ max .int_set:N = \l__nmc_iter_max_int,
+ do .int_set:N = \l__nmc_iter_total_int,
+ see .int_set:N = \l__nmc_iter_view_int,
+ reuse .int_set:N = \l__nmc_iter_reuse_int
+ }
+\keys_define:nn { numerica/solve }
+ {
+ var .tl_set:N = \l__nmc_solve_var_tl,
+ + .int_set:N = \l__nmc_solve_extra_int,
+ max .int_set:N = \l__nmc_solve_max_int,
+ dvar .tl_set:N = \l__nmc_solve_step_tl,
+ reuse .int_set:N = \l__nmc_solve_reuse_int
+ }
+\keys_define:nn { numerica/recur }
+ {
+ do .int_set:N = \l__nmc_recur_total_int,
+ see1 .int_set:N = \l__nmc_recur_first_int,
+ see2 .int_set:N = \l__nmc_recur_last_int,
+ ... .code:n = \bool_set_true:N \l__nmc_recur_ellipsis_bool,
+ reuse .int_set:N = \l__nmc_recur_reuse_int
+ }
+% end of `numerica-plus.def' \ No newline at end of file
diff --git a/macros/latex/contrib/numerica/numerica-plus.pdf b/macros/latex/contrib/numerica/numerica-plus.pdf
new file mode 100644
index 0000000000..7115e44e67
--- /dev/null
+++ b/macros/latex/contrib/numerica/numerica-plus.pdf
@@ -0,0 +1,6722 @@
+%PDF-1.5
+%ÐÔÅØ
+1 0 obj
+<< /S /GoTo /D (chapter.1) >>
+endobj
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+(\376\377\0001\000\040\000I\000n\000t\000r\000o\000d\000u\000c\000t\000i\000o\000n)
+endobj
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+<< /S /GoTo /D (section.1.1) >>
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+endobj
+9 0 obj
+<< /S /GoTo /D (subsection.1.1.1) >>
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+<< /S /GoTo /D (section.1.2) >>
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+<< /Size 587
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+startxref
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+%%EOF
diff --git a/macros/latex/contrib/numerica/numerica-plus.tex b/macros/latex/contrib/numerica/numerica-plus.tex
new file mode 100644
index 0000000000..049418f1a1
--- /dev/null
+++ b/macros/latex/contrib/numerica/numerica-plus.tex
@@ -0,0 +1,1973 @@
+%% LyX 2.4.0-alpha1 created this file. For more info, see https://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english,tableposition=top]{report}
+\usepackage{lmodern}
+\renewcommand{\sfdefault}{lmss}
+\renewcommand{\ttdefault}{lmtt}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage[latin9]{inputenc}
+\setcounter{secnumdepth}{3}
+\usepackage{color}
+\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{wrapfig}
+\usepackage{booktabs}
+\usepackage{framed}
+\usepackage{url}
+\usepackage{amsmath}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=2,
+ breaklinks=true,pdfborder={0 0 1},backref=section,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+\newenvironment{centred}%
+ {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}}
+\newenvironment{lyxcode}
+ {\par\begin{list}{}{
+ \setlength{\rightmargin}{\leftmargin}
+ \setlength{\listparindent}{0pt}% needed for AMS classes
+ \raggedright
+ \setlength{\itemsep}{0pt}
+ \setlength{\parsep}{0pt}
+ \normalfont\ttfamily}%
+ \item[]}
+ {\end{list}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+\usepackage[plus]{numerica}
+
+\newcommand\rel{\,\varrho\;}
+\DeclareMathOperator{\erf}{erf}
+\DeclareMathOperator{\gd}{gd}
+
+\reuse
+
+\usepackage{upquote}
+
+\makeatother
+
+\begin{document}
+\title{\texttt{numerica-plus}~\\
+}
+\author{Andrew Parsloe\\
+(\url{ajparsloe@gmail.com})}
+\maketitle
+\begin{abstract}
+In this module of the \verb`numerica` package, commands are defined
+which iterate and find fixed points of functions of a single variable,
+which find the zeros or extrema of such functions, and which calculate
+the terms of recurrence relations.\\
+\\
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\paragraph*{Note:}
+\begin{itemize}
+\item {\normalsize This document applies to version 1.0.0 of }{\normalsize\texttt{numerica-plus}}{\normalsize .def.}{\small\par}
+\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required.}{\small\par}
+\item {\normalsize I refer a number of times in this document to }{\normalsize\emph{Handbook
+of Mathematical Functions}}{\normalsize , edited by Milton Abramowitz
+and Irene A. Segun, Dover, 1965. This is abbreviated to }{\normalsize\emph{HMF}}{\normalsize .}{\small\par}
+\end{itemize}
+\end{shaded}%
+\end{minipage}
+\end{abstract}
+\tableofcontents{}
+
+\chapter{Introduction}
+
+By calling \texttt{numerica} with the \texttt{plus} package option
+in the preamble,
+\begin{lyxcode}
+\textbackslash usepackage{[}plus{]}\{numerica\}
+\end{lyxcode}
+a number of additional commands are loaded:
+\begin{itemize}
+\item \verb`\nmcIterate`, a command to iterate a function (apply it repeatedly
+to itself), including finding fixed points (values $x$ where $f(x)=x$);
+\item \verb`\nmcSolve`, a command to find the zeros of functions of a single
+variable (values $x$ for which $f(x)=0$) or, failing that, local
+maxima or minima of such functions;
+\item \verb`\nmcRecur`, a command to calculate the values of terms in recurrence
+relations in a single (recurrence) variable (like the terms of the
+Fibonacci sequence or Legendre polynomials).
+\end{itemize}
+These commands are defined in the package \texttt{numerica-plus.def}
+which is loaded with \texttt{numerica.sty} when the \texttt{plus}
+option is used. The new commands all share the syntax of the \verb`\nmcEvaluate`
+command. I will discuss them individually in later chapters but turn
+first to an example that illustrates their use and gives a sense of
+`what they are about'.
+
+\section{Example of use: the rotating disk}
+
+\label{sec:introExampleOfUse}Consider a disk rotating uniformly with
+angular velocity $\omega$ in an anticlockwise sense in an inertial
+system in which the disk's centre \textbf{0} is at rest. Three distinct
+points \textbf{1}, \textbf{2}, \textbf{3} are fixed in the disk and,
+in a co-rotating polar coordinate system centred at \textbf{0}, have
+polar coordinates $(r_{i},\theta_{i})$ ($i,j=1,2,3$). Choose \textbf{01}
+as initial line so that $\theta_{1}=0$.
+
+The cosine rule for solving triangles tells us that the time $t_{ij}$
+in the underlying inertial system for a signal to pass from \textbf{i}
+to \textbf{j} satisfies the equation
+\[
+t_{ij}=c^{-1}\sqrt{r_{i}^{2}+r_{j}^{2}-2r_{i}r_{j}\cos(\theta_{j}-\theta_{i}+\omega t_{ij})}\equiv f(t_{ij}),
+\]
+where $c$ is the speed of light. (Equally, we could be describing
+an acoustic signal between points on a disk rotating uniformly in
+a still, uniform atmosphere \textendash{} in which case $c$ would
+be the speed of sound.) Although the equation doesn't solve algebraically
+for the time $t_{ij},$ it does tell us that $t=t_{ij}$ is a \emph{fixed
+point} of the function $f(t)$. To calculate fixed points we use the
+command \verb`\nmcIterate`, or its short-name form \verb`\iter`,
+with the star option, \verb`\iter*`. For \verb`\iter` the star option
+means: continue iterating until a fixed point has been reached and,
+as with the \verb`\eval` command, suppress all elements from the
+display save for the numerical result.
+
+First, though, values need to be assigned to the various parameters.
+Suppose we use units in which $c=30,$ $r_{1}=10$ and $r_{3}=20$;
+also $\omega=0.2$ radians per second and $\theta_{13}\equiv\theta_{3}-\theta_{1}=0.2$
+radians. We shall give $t$ an initial value $1$ (plucked from the
+air). Its position as the rightmost item in the vv-list tells \verb`\iter`
+that $t$ is the iteration variable:
+\begin{verbatim}
+ \iter*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ], \quad\info{iter}.
+\end{verbatim}
+$\Longrightarrow$ \iter*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ],
+\quad\info{iter}. The \verb`\nmcInfo` command or, more briefly \verb`\info`, can
+be used to display the number of iterations required to attain the
+fixed-point value.
+
+To six figures, only five iterations are needed, which seems rapid
+but we can check this by substituting $t=0.356899$ back into the
+formula and \verb`\eval`-uating it:
+\begin{verbatim}
+ \eval*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0.356899 ]
+\end{verbatim}
+$\Longrightarrow$ \eval*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0.356899 ], confirming that we have indeed calculated a fixed point. That it
+took only $5$ iterations can be checked by omitting the asterisk
+from the \verb`\iter` command and specifying the total number of
+iterations to perform. I choose \texttt{do=}7 to show not just the
+$5$th iteration but also the next two just to confirm that the result
+is stable. We shall view all $7$: \texttt{see=7}. I have also suppressed
+display of the vv-list by giving the key \texttt{vvd}\emph{ }an empty
+value,\footnote{See the associated document \texttt{numerica-basics.pdf}, the chapter
+on settings.} since there are too many variables to display on a line:
+\begin{verbatim}
+ \iter[do=7,see=7,vvd=]
+ {\[ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ]
+\end{verbatim}
+$\Longrightarrow$ \iter[do=7,see=7,vvd=]
+ {\[ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=1 ]
+\noindent \begin{flushleft}
+This display makes clear that on the $5$th iteration, the $6$-figure
+value has been attained.
+\par\end{flushleft}
+
+Alternatively, we could use the \verb`\nmcRecur` command, or its
+short-name form \verb`\recur`, to view the successive iterations
+since an iteration is a first-order recurrence: $f_{n+1}=f(f_{n})$:
+\begin{verbatim}
+ \recur[do=8,see1=0,see2=5,vvd={,\\(vv)\\},*]
+ {\[ f_{n+1}=c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega f_{n})} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2, f_{0}=1 ]
+\end{verbatim}
+$\Longrightarrow$ \recur[do=8,see1=0,see2=5,vvd={\\(vv)\\},*]
+ {\[ f_{n+1}=c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega f_{n})} \]}
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2, f_{0}=1 ]
+
+\noindent I have specified \texttt{do=8} terms rather than $7$ since
+the zero-th term ($f_{0}=1$) is included in the count. I've chosen
+to view the last $5$ of them but none prior to those by writing \texttt{see1=0,see2=5}.
+The vv-list has been suppressed from the display (\texttt{vvd=} )
+and the numerical results forced onto a new line (without an equation
+number \textendash{} the \texttt{{*}} in the vv-list).
+
+Another and perhaps more obvious way to find the value of $t_{13}$,
+is to look for a zero of the function $f(t)-t$. That means using
+the command \verb`\nmcSolve` or its short-name form \verb`\solve`.
+I shall do so with the star option \verb`\solve*` which suppresses
+display of all but the numerical result. A trial value for $t$ is
+required. I have chosen \texttt{t=0}:
+\begin{verbatim}
+ \solve*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)} - t }
+ [ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0 ], \quad\nmcInfo{solve}.
+\end{verbatim}
+$\Longrightarrow$ \solve*{ c^{-1}\sqrt{r_1^2+r_3^2-2r_1 r_3
+ \cos(\theta_{13}+\omega t)}-t
+ }[ c=30,r_1=10,r_3=20,\theta_{13}=0.2,
+ \omega=0.2,t=0],
+\quad\nmcInfo{solve}.
+
+Nearly the same answer as before is attained but this time many more
+steps have been required. This is to be expected. The \verb`\solve`
+command uses the bisection method. Since $1/2^{10}\approx1/10^{3}$
+or $10$ bisections for $3$ decimal places, we can expect about $20$
+bisections being needed for a $6$-decimal-place answer. The particular
+form of the \verb`\nmcInfo` command display, `$1+20$ steps', indicates
+that it took $1$ search step to find an interval in which the function
+vanished and, having found that interval, $20$ bisections to narrow
+the position of the zero to $6$-figures.
+
+\subsection{Circuits}
+
+Okay, so we can calculate the time taken in the underlying inertial
+system for a signal to pass from one point of the rotating disk to
+another. How long does it take to traverse the circuit \textbf{1}
+to \textbf{2} to \textbf{3} back to \textbf{1}? That means forming
+the sum $t_{12}+t_{23}+t_{31}$, hence calculating the separate $t_{ij}$
+and then using \verb`\eval` to calculate their sum.
+
+To make things neater and easier to read, I assume a little symmetry.
+Let the coordinates of \textbf{1} be $(a,0),$ of \textbf{2} be $(r,-\theta)$,
+and of \textbf{3} be $(r,\theta)$: \textbf{2} and \textbf{3} are
+at the same radial distance from the centre \textbf{0} and at the
+same angular distance but on opposite sides of the line \textbf{01},
+\textbf{3} an angle $\theta$ ahead of \textbf{1}, \textbf{2} an angle
+$\theta$ behind \textbf{1}. The rotation is in the direction of positive
+$\theta$. Rather than just calculate $t_{12}+t_{23}+t_{31}$ for
+the circuit \textbf{1231}, I also calculate the time $t_{13}+t_{32}+t_{21}$
+for a signal to traverse the opposite circuit \textbf{1321} and compare
+them (form the difference).
+
+Note that with \textbf{2} and \textbf{3} positioned as they are relative
+to \textbf{1},
+\[
+t_{31}=c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta-\omega t_{31})}
+\]
+and
+\[
+t_{12}=c^{-1}\sqrt{a^{2}+r^{2}-2ar\cos(\theta-\omega t_{12})}.
+\]
+Both $t_{12}$ and $t_{31}$ are fixed points of the same function
+\[
+F_{-1}(t)=c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta-\omega t).}
+\]
+Similarly, both $t_{13}$ and $t_{21}$ are fixed points of
+\[
+F_{+1}(t)=c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta+\omega t)}.
+\]
+ Geometrically one can show that $t_{31}=t_{12}$ and $t_{13}=t_{21}$.
+Analytically, with the values $c=30,a=10,r=20$,
+\[
+0<F_{\pm1}(t)\le c^{-1}\sqrt{r^{2}+a^{2}+2ar}=c^{-1}(r+a)=1,
+\]
+so that we need look for fixed points only in the interval $[0,1]$.
+Differentiating $F_{-1}(t)$ shows the derivative is negative on $[0,1]$,
+hence that $F_{-1}$ is strictly decreasing there and can intersect
+the identity mapping at most once on the interval. Similarly, differentiating
+$F_{+1}(t)$ shows the derivative
+\[
+\frac{ar\omega\sin(\theta+\omega t)}{c\sqrt{r^{2}+a^{2}-2ar\cos(\theta+\omega t)}}
+\]
+is positive on $[0,1]$, but with the given values for $c,a$ and
+$r$, and with $\theta=0.2,\omega=0.2$, $F'_{+1}(t)<0.1$ on $[0,1]$:
+hence $F_{+1}$ increases sufficiently slowly on the interval that
+the identity mapping can intersect it at most once there. In both
+cases, there can be at most one fixed point.
+
+It follows that $t_{12}=t_{31}$ and $t_{13}=t_{21}$ and the round
+trip times are $2t_{12}+t_{23}$ and $2t_{13}+t_{32}$.
+
+\subsubsection{Nesting commands}
+
+The calculation to compare the round trip times can be done `in one
+go' by nesting the \verb`\iter*` commands to calculate the $t_{ij}$
+inside an \verb`\eval*` command to form the sum of their results
+and then take the difference.
+\begin{verbatim}
+ $
+ \eval*{ % circuit 1231
+ 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ }[ c=30,a=10,r=20,\theta=0.2,\omega=0.2, t=1 ]
+ $
+\end{verbatim}
+$\Longrightarrow$ $
+ \eval*{ % circuit 1231
+ 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\iter*{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ }[ c=30,a=10,r=20,\theta=0.2,\omega=0.2, t=1 ]
+ $.
+
+By itself this result is of little interest beyond seeing that \texttt{numerica}
+can handle it. What \emph{is} interesting is to find values of $\theta$,
+say, given the other parameters and especially the value of $r$,
+such that the time difference vanishes. Is there a circuit, despite
+the rotation of the disk, such that it takes a signal the same time
+to travel in opposite directions around the circuit? That is a job
+for \verb`\nmcSolve` (or \verb`\solve`):
+\begin{verbatim}
+ \solve[p=.,var=\theta,vvi=]{$ % circuit 1231
+ 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ $}[ c=30,a=10,r=20,\theta=0.1,\omega=0.2, {t}=1 ]
+\end{verbatim}
+$\Longrightarrow$ \solve[p=.,var=\theta,vvi=]{$ % circuit 1231
+ 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta-\omega t)} }
+ + \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta+\omega t)} }
+ % circuit 1321
+ - 2\times\iter*[var=t]{ c^{-1}\sqrt{a^2+r^2-2ar
+ \cos(\theta+\omega t)} }
+ - \iter*[var=t]{ c^{-1}\sqrt{2r^2-2r^2
+ \cos(2\theta-\omega t)} }
+ $}[ c=30,a=10,r=20,\theta=0.1,\omega=0.2, {t}=1 ]
+
+One point to note here is the use of \verb`\times` (in \verb`2\times\iter*`)
+as against the previous example with \verb`\eval*` where \verb`\times`
+was not used. In the present case the formula is displayed (\verb`\solve`
+wraps around math delimiters); the star on the \verb`\eval` command
+in the previous example suppressed display of the formula. Without
+the \verb`\times` the `$2$'s would be juxtaposed against the following
+decimals, making it look as if the signal travel times were $20.537778$
+and $20.614419$ and no doubt causing perplexity. Also note the \texttt{vvi=}
+to suppress display of the vv-list.
+
+But this is not a research paper on the rotating disk. I wished to
+show how the different commands of \texttt{numerica-plus} can be used
+to explore a meaningful problem. And although it looks as if a lot
+of typing is involved, once $c^{-1}\sqrt{r^{2}+a^{2}-2ra\cos(\theta-\omega t)}$
+has been formed in \LaTeX{} and values specified in the vv-list, much
+of the rest is copy-and-paste followed by minor editing.
+
+\section{Shared syntax of the new commands}
+
+\texttt{numerica-plus} offers three new commands for three processes:
+\verb`\nmcIterate` (short-name form \verb`\iter`) for iterating
+functions, \verb`\nmcSolve` (short-name form \verb`\solve`) for
+finding the zeros or (local) extrema of functions, and \verb`\nmcRecur`
+(short-name form \verb`\recur`) for calculating terms of recurrence
+relations.
+
+All three commands share the syntax of the \verb`\nmcEvaluate` (or
+\verb`\eval`) command detailed in the associated document \texttt{numerica-basics.pdf}.
+When all options are used then the command looks like
+\begin{centred}
+\noindent \verb`\nmcIterate*[settings]{expr.}[vv-list][num. format]`
+\end{centred}
+You can substitute \verb`\nmcSolve`, or \verb`\nmcRecur` for \verb`\nmcIterate`
+here. The arguments are similar to those for \verb`\nmcEvaluate`.
+\begin{enumerate}
+\item \verb`*` optional switch; if present ensures a single number output
+with no formatting, or an appropriate error message if the single
+number cannot be produced;
+\item \verb`[settings]` optional comma-separated list of \emph{key=value
+}settings for this particular command and calculation;
+\item \verb`{expr.}` the only mandatory argument; the mathematical expression
+in \LaTeX{} form that is the object of interest;
+\item \verb`[vv-list]` optional comma-separated list of \emph{variable=value
+}items; for \verb`\iter` and \verb`\solve` the \emph{rightmost}
+(or innermost) variable in the vv-list may have special significance;
+\item \verb`[num. format]` optional format specification for presentation
+of the numerical result (rounding, padding with zeros, scientific
+notation); boolean output is suppressed for these commands.
+\end{enumerate}
+Boolean output makes little sense for these three commands and is
+turned \emph{off}. Entering a question mark in the trailing number-format
+option of each command has no effect \textendash{} see §.
+
+All commands also share with \verb`\nmcEvaluate` changes in the way
+the result is displayed depending on whether they wrap around math
+delimiters, or are wrapped within math delimiters, or are employed
+in the absence of math delimiters. These distinctions are relevant\emph{
+only if the optional star {*} is absent.}
+\begin{itemize}
+\item When the star option is used, the \emph{result} is a number only without
+any formatting or vv-list display, or an error message is displayed.
+\item When the star option is not used and one of the following is the case
+\begin{itemize}
+\item the command wraps around math delimiters, e.g. \verb`\iter{$ expr. $}`,
+then the result is displayed in the form \emph{expression~=~result}
+or the form \emph{expression~$\rightarrow$~result} as appropriate,
+where \emph{result} may be multi-line and may include the vv-list,
+\item the command is used within math delimiters, e.g. \verb`\[\iter...\]`,
+then only the \emph{result} is presented, where \emph{result} may
+be multi-line and may include the vv-list,
+\item the command is used in the absence of delimiters, then the result
+is presented as if it had been used between \verb`\[` and \verb`\]`.
+\end{itemize}
+\end{itemize}
+Looking at the various examples in the preceding section on the rotating
+disk you will see illustrations of all these situations.
+
+\subsection{Settings}
+
+\begin{table}
+\noindent \centering{}\caption{\protect\label{tab:introSettingsInherited}Inherited settings options}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (0/1)} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (0/1/2)} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. funcions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small tokens} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small token(s)} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}} & \tabularnewline
+{\small\texttt{p}} & char(s) & {\small punctuation (esp. in display-style)} & {\small\texttt{, (comma)}}\tabularnewline
+{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion for sums} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for sums} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion for products} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for products} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+Nearly all the settings available to the \verb`\eval` command are
+available to these other commands. To save switching between documents
+I reproduce in Table~\ref{tab:introSettingsInherited} the options
+found in \texttt{numerica-basics.pdf}, although for discussion of
+the options you will need to refer to that document. In addition,
+each of the present commands also has settings of its own, discussed
+at the relevant parts of the following chapters.
+
+\subsection{Nesting}
+
+Provided the starred form of a command actually does produce a numerical
+result and not an error message then it can be nested within the main
+argument of any one of the commands, including itself. The example
+of use, §\ref{sec:introExampleOfUse} above, shows several examples
+of this. The starred form can also be used in the vv-list of any one
+of the commands, including itself. The associated document \texttt{numerica-basics.pdf}
+shows examples of an \verb`\eval*` command being used in the vv-list
+of an \verb`\eval` command.
+
+\chapter{Iterating functions: \texttt{\textbackslash nmcIterate}}
+
+\label{chap:Iterating-functions}Only in desperation would one try
+to evaluate a continued fraction by stacking fraction upon fraction
+upon fraction like so:
+\[
+1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1}}}}}}}}}}}}
+\]
+\texttt{numerica-plus} provides a command for tackling problems like
+this sensibly. In such problems a function is repeatedly applied to
+itself (\emph{iterated}). This can be achieved in \texttt{numerica}
+through the command \verb`\nmcIterate` or (short-name form) \verb`\iter`.
+Thus to evaluate this continued fraction we write
+\begin{centred}
+\verb`\iter[do=15,see=5]{\[ 1+1/x \]}[x=1]` $\Longrightarrow$ \iter[do=15,see=5]{\[ 1+1/x \]}[x=1]
+\end{centred}
+The \verb`\iter` command evaluates $1+1/x$ when $x=1$ and then
+uses this value as a new $x$-value to substitue into $1+1/x$, to
+once again evaluate and use as a new $x$-value, and so on. It looks
+as if the repeated iterations are approaching \verb`\eval{$ \tfrac{\sqrt{5}+1}2 $}`
+$\Longrightarrow$ \eval{$ \tfrac{\sqrt{5}+1}2 $}.
+
+Increasing the number of iterations in the example from \texttt{do=15}
+to, say, \texttt{do=18}, shows that this is indeed the case. I've
+taken the opportunity to also put a question mark in the number-format
+option to get boolean output in order to show that it has \emph{no}
+effect:
+\begin{centred}
+\verb`\iter[do=18,see=5]{\[ 1+1/x \]}[x=1][?]` $\Longrightarrow$
+\iter[do=18,see=5]{\[ 1+1/x \]}[x=1][?]
+\end{centred}
+But iteration of functions is not limited to continued fractions.
+Particularly since the emergence of chaos theory, iteration has become
+an important study in its own right. Any function whose range lies
+within its domain can be iterated \textendash{} repeatedly applied
+to itself. The cosine, for instance,
+\begin{centred}
+\verb`\iter[do=20,see=4]{\[ \cos x \]}[x=\pi/2]` $\Longrightarrow$
+\iter[do=20,see=4]{\[ \cos x \]}[x=\pi/2]
+\end{centred}
+which displays the first one and last four of 20 iterations of $\cos x$
+when $x=\tfrac{\pi}{2}$. It looks as if the cosine is `cautiously'
+approaching a limit, perhaps around $0.738$ or $0.739$. You need
+to nearly double the number of iterations (\texttt{do=40}) to confirm
+that this is so.
+
+The logistic function $kx(1-x)$ exhibits a variety of behaviours
+depending on the value of $k$. For instance, with $k=3.5$ we get
+a period-4 cycle:
+\begin{centred}
+\verb`\iter[do=12,see=8]{\[ kx(1-x) \]}[k=3.5,x=0.5]` $\Longrightarrow$
+ \iter[do=12,see=8]{\[ kx(1-x) \]}[k=3.5,x=0.5]
+\end{centred}
+and with $k=3.1$ we get a period-2 cycle, although it takes many
+more iterations to stabilize there:
+\begin{centred}
+\verb`\iter[do=44,see=8]{\[ kx(1-x) \]}[k=3.1,x=0.5]` $\Longrightarrow$
+\iter[do=44,see=8]{\[ kx(1-x) \]}[k=3.1,x=0.5]
+\end{centred}
+
+\section{Star (\texttt{{*}}) option: fixed points}
+
+In the first two of these examples, iteration eventually ended at
+a \emph{fixed point}. This is a point $x$ where $f(x)=x$ and, by
+induction, $f^{n}(x)=x$. Appending a star (asterisk) to the \verb`\iter`
+command is the signal for iteration to continue until a fixed point
+has been reached at the specified rounding value:
+\begin{centred}
+\verb`\iter*{ 1+a/x }[a=n(n+1),n=1,x=1]` $\Longrightarrow$ \iter*{ 1+a/x }[a=k(k+1),k=1,x=1]
+\end{centred}
+(with the default rounding value $6$).\footnote{For your own interest try also putting $n=2,3,4,\ldots$ .}
+The star overrides any value for the number of iterations to perform
+(the \texttt{do} key) that may have been entered in the settings option.
+It also overrides any elements of the display other than the numerical
+result. With the star option math delimiters are irrelevant \textendash{}
+other than displaying minus signs correctly when \verb`\iter*` is
+between them.
+
+A function may not approach a fixed point when iterated \textendash{}
+see the examples with the logistic function above. To prevent an infinite
+loop \verb`\iter*` counts the number of iterations performed and
+when that number reaches a certain cut-off value \textendash{} the
+default is $100$ \textendash{} the loop terminates and a message
+is displayed:
+\begin{centred}
+\verb`\iter*{kx(1-x)}[k=3.5,x=0.5]` $\Longrightarrow$ \iter*{kx(1-x)}[k=3.5,x=0.5]
+\end{centred}
+In this case we \emph{know} that a fixed point does not exist, but
+that may not always be the case. One response is to change parameter
+values or starting value of the iteration variable. For instance,
+changing the parameter value to $k=1.5$,
+\begin{centred}
+\verb`\iter*{kx(1-x)}[k=1.5,x=0.5]` $\Longrightarrow$ \iter*{kx(1-x)}[k=1.5,x=0.5],
+\end{centred}
+and a fixed point is now attained.
+
+But should a fixed point still not eventuate after `fiddling' with
+parameter and start values, there are two general adjustments one
+might try: either
+\begin{enumerate}
+\item reduce the rounding value, from the default $6$ (or the one specified),
+to a smaller value, or
+\item increase the cut-off figure from the default 100 to some higher value.
+\end{enumerate}
+The former is done via the trailing \verb`[num format]` optional
+argument; the latter is done via the settings option, see §\ref{sec:iterSettings-option}.
+
+\subsection{Use with \texttt{\textbackslash nmcInfo}}
+
+It is of interest to know how many iterations are required to reach
+a fixed point at a particular rounding value. That knowledge allows
+a good guess as to whether a fixed point will be attained at a greater
+rounding value. Thus when iterating the function
+\[
+f(t_{ij})=c^{-1}\sqrt{r_{i}^{2}+r_{j}^{2}-2r_{i}r_{j}\cos(\theta_{j}-\theta_{i}+\omega t_{ij})}
+\]
+in §\ref{sec:introExampleOfUse} only $5$ iterations were required
+to attain $6$-figure accuracy for the fixed point. That information
+came by following the \verb`\iter*` command with \verb`\nmcInfo`
+(or \verb`\info`) with the argument \verb`iter`. And generally,
+for any `infinite' process, follow the command with an \verb`\info`
+command if you want to know how many `steps' \textendash{} in the
+present case iterations \textendash{} are required to achieve the
+result. So, if $5$ iterations achieve $6$-figure accuracy, presumably
+something like $10$ iterations will achieve $12$-figure accuracy:
+\begin{verbatim}
+ \iter*{ c^{-1}\sqrt{r_i^2+r_j^2-2r_i r_j
+ \cos(\theta_{ij}+\omega t)}
+ }[ c=30,r_i=10,r_j=20,\theta_{ij}=0.2,
+ \omega=0.2,t=1 ][12],
+ \quad\info{iter}.
+\end{verbatim}
+$\Longrightarrow$ \iter*{ c^{-1}\sqrt{r_i^2+r_j^2-2r_i r_j
+ \cos(\theta_{ij}+\omega t)}
+ }[ c=30,r_i=10,r_j=20,\theta_{ij}=0.2,
+ \omega=0.2,t=1 ][12],
+\quad\info{iter}. Only $9$ iterations suffice to achieve $12$-figure accuracy. Or
+again, with another earlier example,
+\begin{centred}
+\verb`$ \iter*{\cos x}[x=\pi/2] $,\ \info{iter}.` $\Longrightarrow$
+$\iter*{\cos x}[x=\pi/2] $,\ \info{iter}.
+\end{centred}
+That suggests that around $74$ iterations will give $12$-figure
+accuracy, well within the cut-off figure of $100$:
+\begin{centred}
+\verb`$ \iter*{\cos x}[x=\pi/2][12] $,\ \info{iter}.` $\Longrightarrow$
+$\iter*{\cos x}[x=\pi/2][12] $,\ \info{iter}.
+\end{centred}
+
+\section{Settings option}
+
+\label{sec:iterSettings-option}The settings option is a comma-separated
+list of items of the form \emph{key~=~value}.
+
+\subsection{Inherited settings}
+
+Nearly all of the keys discussed in the settings\emph{ }option for
+\verb`\nmcEvaluate` are available for \verb`\nmcIterate`. Table~\ref{tab:introSettingsInherited}
+above lists these, repeating a table from \texttt{numerica-basics.pdf}.
+Thus should a quantity in the vv-list depend on the iteration variable,
+forcing an implicit mode calculation, simply enter (as with \verb`\eval`)
+\texttt{vvmode=1} in the settings option:
+\begin{centred}
+\verb`\iter*[vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1]` $\Longrightarrow$
+\iter*[vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1].
+\end{centred}
+Implicit in this example is the default multi-token setting \texttt{xx=1}
+inherited from \verb`\eval` and ensuring that the multi-token variable
+$f(x)$ is treated correctly.
+
+Let's add \verb`dbg=1` to the example:
+\begin{centred}
+\verb`\iter*[dbg=1,vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1]` $\Longrightarrow$
+\iter*[dbg=1,vvmode=1]{$ 1+f(x) $}[f(x)=a/x,a=12,x=1]
+\end{centred}
+The multi-token variable \verb`f(x)` has been changed to the single-token
+\verb`\_nmcu`. The values of \verb`x` and \verb`\_nmcu` shown are
+those of the \emph{final} iteration.
+
+\subsection{\texttt{\textbackslash iter}-specific settings}
+
+In addition to the inherited settings there are some specific to \verb`\nmcIterate`.
+These are listed in Table~\ref{tab:iterSettings}.
+
+\subsubsection{Iteration variable}
+
+In nearly all of the examples so far, the iteration variable has been
+the rightmost variable in the vv-list and has not needed to be otherwise
+specified. However it is sometimes not feasible to indicate the variable
+in this way. In that case, entering
+\begin{lyxcode}
+var~=~<variable~name>
+\end{lyxcode}
+in the settings option enables the variable to be specified, irrespective
+of what the rightmost variable in the vv-list is. Here, \texttt{<variable
+name>} will generally be a character like \texttt{x} or \texttt{t}
+or a token like \verb`\alpha`, but it could also be a multi-token
+name like {\ttfamily\verb`x'`}\texttt{ }or \texttt{\textbackslash beta\_\{ij\}}
+(or even \texttt{Fred} if you so chose). Although the iteration variable
+can be independently specified like this, it must still be given an
+initial \emph{value} in the vv-list \textendash{} only it need not
+be the rightmost variable.
+
+In the following example the rightmost variable is $a$ which is clearly
+\emph{not} the iteration variable:
+\begin{centred}
+\verb`\iter[var=x'',do=40,see=5]{$ 1+a/x'' $}[x''=a/6,a=6][*]` $\Longrightarrow$
+\iter[var=x'',do=40,see=5]{$ 1+a/x'' $}[x''=a/6,a=6][*]
+\end{centred}
+
+\subsubsection{Extra rounding for fixed-point calculations}
+
+\begin{table}
+\centering{}\caption{\protect\label{tab:iterSettings}Settings for \texttt{\textbackslash nmcIterate}}
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small iteration variable} & \tabularnewline
+{\small\texttt{+}} & {\small int} & {\small fixed point extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. iteration count (fixed points)} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{do}} & {\small int > 0} & {\small number of iterations to perform} & {\small\texttt{5}}\tabularnewline
+{\small\texttt{see}} & {\small int > 0} & {\small number of final iterations to view} & {\small\texttt{4}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+\texttt{numerica} determines that a fixed point has been reached when
+the difference between successive iterations vanishes when rounded
+to the current rounding value. One might want reassurance that this
+really is the correct value by seeking a fixed point at a higher rounding
+value than that displayed. This extra rounding is achieved by entering
+\begin{lyxcode}
++~=~<integer>
+\end{lyxcode}
+in the settings option. By default this extra rounding is set to zero.
+
+We have seen before that $\cos x$ starting at $x=\tfrac{1}{2}\pi$
+takes $37$ iterations to reach a $6$-figure fixed point $0.739085$,
+about $6$ iterations per decimal place. By entering \texttt{+=1}
+in the settings option the number of iterations is increased to $43$,
+$6$ more than $37$ but, reassuringly, the $6$-figure result that
+is displayed remains unchanged:
+\begin{centred}
+\verb`$ \iter*[+=1]{\cos x}[x=\pi/2] $,\ \info{iter}.` $\Longrightarrow$
+$ \iter*[+=1]{\cos x}[x=\pi/2] $,\ \info{iter}.
+\end{centred}
+
+\subsubsection{Maximum {\small iteration count for fixed point searches}}
+
+To prevent a fixed-point search from continuing indefinitely when
+no fixed point exists, there needs to be a maximum number of iterations
+specified after which point the search is called off. By default this
+number is $100$. To change it enter
+\begin{lyxcode}
+max~=~<positive~integer>
+\end{lyxcode}
+in the settings option.
+
+\subsubsection{Number of iterations to perform}
+
+To specify the number of iterations to perform enter
+\begin{lyxcode}
+do~=~<positive~integer>
+\end{lyxcode}
+in the settings option. Note that if the \verb`*` option is present
+this value will be ignored and iteration will continue until either
+a fixed point or the maximum iteration count is reached. By default
+\texttt{do} is set to $5$. (Note that \texttt{do} can be set to a
+greater number than \texttt{max}; \texttt{max} applies only to \verb`\iter*`.)
+
+\subsubsection{Number of iterations to view}
+
+To specify the number of final iteations to view enter
+\begin{lyxcode}
+see~=~<positive~integer>
+\end{lyxcode}
+in the settings option. By default \texttt{see} is set to $4$. Always
+it is the \emph{last} \texttt{see} iterations that are displayed.
+If \texttt{see} is set to a greater value than \texttt{do}, all iterations
+are shown. If the star option is used the \texttt{see} value is ignored.
+
+\subsubsection{Form of result saved by \texttt{\textbackslash reuse}}
+
+By entering
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+in the settings option of the \verb`iter` command it is possible
+to specify the form of result that is saved when using \verb`\nmcReuse`.
+(This setting has no effect when the star option is used with \verb`\nmcIterate`.
+In that case only the numerical result of the fixed point calculation
+\textendash{} if successful \textendash{} is saved.) The possibilities
+are:
+\begin{itemize}
+\item \texttt{int=0} (or any integer $\mathtt{\ne1,2}$) saves the display
+resulting from the \verb`\iter` command (the default);
+\item \texttt{int=1} saves a comma-separated list of pairs of the form:\texttt{ }~\\
+\texttt{\{$k$, value-of-$k$-th-iterate\}};
+\item \texttt{int=2} saves a comma-separated list of iterate values.
+\end{itemize}
+Note that the number and content of the items in the lists are those
+resulting from the \texttt{see} setting (the number of iterations
+to view).
+\begin{verbatim}
+ \iter[reuse=1,do=12,see=4]
+ {\[ kx(1-x) \]}[k=3.5,x=0.5]
+ \reuse[logistic]
+\end{verbatim}
+$\Longrightarrow$ \iter[reuse=1,do=12,see=4]{\[ kx(1-x) \]}[k=3.5,x=0.5] \reuse[logistic]
+
+\noindent whence \verb`\logistic` $\Longrightarrow$ \logistic.
+As you can see the control sequence \verb`\logistic` displays as
+a comma-separated list of numbers, alternating between the ordinal
+of the iterate, and the iterate value. In fact these values are grouped
+as braced pairs separated by commas in \verb`\logistic`, as can be
+seen by using \TeX 's \verb`\meaning` command:
+\begin{centred}
+\verb`\meaning\logistic` $\Longrightarrow$ \meaning\logistic
+\end{centred}
+
+\subsection{Changing default values}
+
+\begin{wraptable}[12]{o}{0.5\columnwidth}%
+\centering{}\caption{\protect\label{tab:iterSettingsDefaults}Defaults for \texttt{\textbackslash nmcIterate}}
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small iter-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small iter-max-iterations} & {\small\texttt{100}}\tabularnewline
+{\small iter-do} & {\small\texttt{5}}\tabularnewline
+{\small iter-see-last} & {\small\texttt{4}}\tabularnewline
+{\small iter-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+If you wish to change the default values of the various settings for
+\verb`\nmcIterate` this can be done by entering new values in a configuration
+file \texttt{numerica.cfg} as described in the chapter on settings
+in the associated document \texttt{numerica-basics.pdf}. The relevant
+keys are listed in Table~\ref{tab:iterSettingsDefaults}, corresponding
+to the \verb`+`, \verb`max`, \verb`do`, \verb`see` and \verb`reuse`
+settings of the \verb`\iter` command. (Obviously it makes no sense
+to have a default setting for the iteration variable. That will change
+from case to case.)
+
+\section{Errors}
+
+By errors I refer to \texttt{numerica} errors rather than \LaTeX{}
+errors. We have already met one in the discussion of fixed points:
+\begin{centred}
+\verb`\iter*{kx(1-x)}[k=3.5,x=0.5]` $\Longrightarrow$ \iter*{kx(1-x)}[k=3.5,x=0.5]
+\end{centred}
+For a function to be iterated indefinitely, its range must lie within
+or be equal to its domain. If even part of the range of a function
+lies outside its domain, then on repeated iteration there is a chance
+that a value will eventually be calculated which lies in this `outside'
+region. Iteration cannot continue beyond this point and an error message
+is generated. As an example consider the inverse cosine, \verb`\arccos`.
+This can be iterated only so far as the iterated values lie between
+$\pm1$ inclusive. If we try to iterate \verb`\arccos` at 0 for example,
+since $\cos\frac{1}{2}\pi=0$, $\arccos0=\eval{0.5\pi}[4]$ (which
+is $\tfrac{1}{2}\pi$) so only a first iterate is possible. But we
+could choose an initial value more carefully; $37$ iterations of
+the cosine at $\tfrac{1}{2}\pi$ led to a fixed point $0.739085$,
+so let's choose $0.739085$ as initial point and perform $37$ iterations:
+\begin{centred}
+\verb`\iter[do=37,see=4]{\[ \arccos x \]}[x=0.739085]` $\Longrightarrow$
+\iter[do=37,see=4]{\[ \arccos x \]}[x=0.739085]
+\end{centred}
+The result of the $37$th iteration is greater than $1$. Thus increasing
+the number of iterations to 38 should generate an error message:
+\begin{centred}
+\verb`\iter[do=38,see=4]{\[ \arccos x \]}[x=0.739085]` $\Longrightarrow$\iter[do=38,see=4]{\[ \arccos x \]} [x=0.739085]
+\end{centred}
+\verb`l3fp` objects when asked to find the inverse cosine of a number
+greater than $1$.
+
+\chapter{Finding zeros and extrema: \texttt{\textbackslash nmcSolve}}
+
+\texttt{numerica} provides a command\textbf{ }\verb`\nmcSolve` (short-name
+form \verb`\solve`) for finding a zero of a function, should it have
+one. In the following example,
+\begin{centred}
+\verb`\solve[p]{\[ e^{ax}-bx^2 \]}[a=2,b=3,{x}=0]` $\Longrightarrow$
+\solve[p]{\[ e^{ax}-bx^2 \]}[a=2,b=3,{x}=0]
+\end{centred}
+I have sought and found a solution $x$ to the equation $e^{ax/2}-bx^{2}=0$
+when $a=2$ and $b=3$, starting with a trial value $x=0$, entered
+as the \emph{rightmost} variable in the vv-list (and em-braced since
+I don't want this trial value displaying in the presentation of the
+result). Although $x$ has been found to the default six-figure accuracy,
+it is evident that the function vanishes only to five figures. Let's
+check:
+\begin{centred}
+\verb`\eval{$ bx^2 $}[b=3,x=x=-0.390647]` $\Longrightarrow$ \eval{$ bx^2 $}[b=3,x=-0.390647],
+
+\verb`\eval{$ e^{ax} $}[a=2,x=-0.390646]` $\Longrightarrow$ \eval{$ e^{ax} $}[a=2,x=-0.390647];
+\end{centred}
+the values agree save in the final digit.
+
+This discrepancy in the final decimal place or places is a general
+feature of solutions found by \verb`\solve`. It is the value of $x$,
+not the value of $f(x)$, that is being found (in this case) to six
+figures. If the graph of a function crosses the $x$-axis steeply
+then the $x$ value (the zero) may be located to a higher precision
+than the function value. Conversely, if the graph of a function crosses
+the $x$-axis gently (at a shallow angle) then the function value
+will vanish to a greater number of decimal places than the zero (the
+$x$ value) is located.
+
+A second example, which we can check with values tabulated in \emph{HMF},
+is
+\begin{centred}
+\verb`\solve{$ \tan x - \lambda x $}[\lambda=-1/0.8,{x}=1][5]` $\Longrightarrow$
+\solve{$ \tan x - \lambda x $}[\lambda=-1/0.8,{x}=1][5].
+\end{centred}
+Table 4.19 of \emph{HMF }lists values of $x$ against $\lambda$ and
+this is the value tabulated there.
+
+\section{Extrema}
+
+A function may not have a zero or for the given initial trial value
+and initial step in the search for a zero there may be a local extremum
+in the way. In that case \texttt{numerica} may well locate the local
+extremum (maximum or minimum but not a saddle point). For example
+for the quadratic $(2x-1)^{2}+3x+1$ the \verb`\solve` command gives
+the result
+\begin{centred}
+\verb`\solve[vvi=]{$ (2x-1)^2+3x+1 $}[x=2]` $\Longrightarrow$\solve[vvi=]{$ (2x-1)^2+3x+1 $}[{x}=2].
+\end{centred}
+Since $(2x-1)^{2}+3x+1\ne0$ for any (real number) $x$, we deduce
+that the quadratic takes a minimum value $1.9375$ at $x=0.125$ \textendash{}
+easily confirmed analytically. This particular minimum is a global
+minimum but in general any extremum found is only \emph{local}. The
+function may well take larger or smaller values (or vanish for that
+matter) further afield.
+
+It is also worth noting in this example the \verb`vvi=` in the settings
+option which suppresses display of the vv-list. (The only member of
+the vv-list is the trial value \verb`x=2` which we do not want to
+display.)
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+Note that the function for which a zero is being sought is \emph{not}
+equated to zero when entered in the \verb`\solve` command. It is
+\verb`\solve{ f(x) }`, not \verb`\solve{ f(x)=0 }`. This is precisely
+because it may be an extremum that is found rather than a zero (if
+extremum or zero is found at all \textendash{} think $e^{x}$). The
+display of the result makes clear which is which, equating $f(x)$
+to its value, zero or extremum depending on what has been found, as
+you can see in the preceding examples.\end{shaded}%
+\end{minipage}
+
+\subsection{The search strategy}
+
+\label{subsec:solveSearch-strategy}If you have some sense of where
+a function has a zero, then choose a trial value in that vicinity.
+\verb`\solve` uses a bisection method to home in on the zero. It
+therefore needs \emph{two} initial values. For the first it uses the
+trial value you specify, call it $a$ and for the second, by default,
+it uses $a+1$. (The default value $1$ for the initial step from
+the trial value can be changed in the settings option; see §\ref{sec:solveSettings-option}.)
+If $f(a)$ and $f(a+1)$ have opposite signs then that is good. Bisection
+of the interval $[a,a+1]$ can begin immediately in order to home
+in on the precise point where $f$ vanishes. Write $b=a+1$.
+\begin{itemize}
+\item Let $c=\tfrac{1}{2}(a+b)$; if $f(c)=0$ the zero is found; otherwise
+either $f(a),f(c)$ are of opposite signs or $f(c),f(b)$ are of opposite
+signs. In the former case write $a_{1}=a,$ $b_{1}=c$; in the latter
+case write $a_{1}=c$, $b_{1}=b$ and then redefine $c=\tfrac{1}{2}(a_{1}+b_{1})$.
+Continue the bisection process, either until an exact zero $c$ of
+$f$ is reached ($f(c)=0$) or a value $c$ is reached where the difference
+between $a_{n+1}$ and $b_{n+1}$ is zero at the specified rounding
+value. (But note, $f(c)$ may not vanish at that rounding value \textendash{}
+the zero might be elsewhere in the interval and $f$ might cross the
+axis at a steep slope.)
+\end{itemize}
+However $f(a)$ and $f(b)=f(a+1)$ may not have opposite signs. If
+we graph the function $y=f(x)$ and suppose $f(a),f(b)$ are distinct
+but of the same sign, then the line through the points $(a,f(a))$,
+$(b,f(b))$ will intersect the $x$-axis to the left of $a$ or the
+right of $b$ depending on its slope. We search always \emph{towards
+the $x$-axis} in steps of $b-a$ ($=1$ with default values).
+\begin{itemize}
+\item If the line intersects the axis to the left of $a$ then $c=a-(b-a)$
+and we set $a_{1}=c,b_{1}=a$; if the line intersects the axis to
+the right of $b$ then $c=b+(b-a)$ and we set $b_{1}=c,a_{1}=b$.
+The hope is that by always taking steps in the direction towards the
+$x$-axis that eventually $f(c)$ will be found to lie on the \emph{opposite}
+side of the axis from $f(a_{n})$ or $f(b_{n})$, at which point the
+bisection process begins.
+\item Of course this may not happen. At some point $c$ may lie to the left
+of $a_{n}$ but $\left|f(c)\right|>\left|f(a_{n})\right|$, or $c$
+may lie to the right of $b_{n}$ but $\left|f(c)\right|>\left|f(b_{n})\right|$.
+The slope has reversed. In that case we halve the step value to $\tfrac{1}{2}(b-a)$
+and try again in the same direction as before from the same point
+as before ($a_{n}$ or $b_{n}$ as the case may be).
+\item Should we find at some point that $f(a_{n})=f(b_{n})$ then the previous
+strategy does not apply. In this case we choose $a_{n+1}$ and \textbf{$b_{n+1}$}
+at the quarter and three-quarter marks between $a_{n}$ and $b_{n}$.
+Either $f(a_{n+1})$ and $f(b_{n+1})$ will differ and the previous
+search strategy can start again or we are on the way to finding an
+extremum of $f$.
+\end{itemize}
+As already noted it is also possible that our function has neither
+zeros nor extrema. To prevent the search continuing indefinitely,
+\texttt{numerica} uses a cut-off value for the maximum number of steps
+pursued \textendash{} by default set at 100.
+
+\subsubsection{Elusive extrema}
+
+The strategy `search always towards the $x$-axis' has a consequence:
+it means that a local maximum above the $x$-axis will almost certainly
+not be found, since `towards the $x$-axis' pulls the search away
+from the maximum. Similarly a local minimum below the $x$-axis will
+also not be found since `towards the $x$-axis' pulls the search
+away from the minimum.
+
+One way of countering this elusiveness is to add a constant value
+(possibly negative) to the function whose zeros and extrema are being
+sought. The zeros of the function will change but the abscissae ($x$
+values) of the extrema remain unchanged. If the constant is big enough
+it will push a local minimum above the axis where it can be found
+or, for a negative constant, push a local maximum below the axis where
+it can be found.
+
+For example $f(x)=x^{3}-x$ has roots at $-1,0,1$, a local maximum
+at $-\tfrac{1}{\surd3}$ and a local minimum at $\tfrac{1}{\surd3}$.
+To locate the minimum, I have added an unnecessarily large constant
+$k$ to $f(x)$. ($k=1$ would have sufficed, but note, $k=0$ fails.)
+\begin{centred}
+\verb`\solve{$ x^3-x+k $}[k=5,{x}=0.5]` $\Longrightarrow$ \solve{$ x^3-x+k $}[k=5,{x}=0.5].
+\end{centred}
+Checking, \verb`\eval{$\tfrac1{\surd 3}$}` $\Longrightarrow$ \eval{$\tfrac1{\surd 3}$}.
+There is a discrepancy in the $6$th decimal place which can be eliminated
+by using the extra rounding setting; see §\ref{subsec:solveExtraRounding}.
+
+\section{Star (\texttt{{*}}) option}
+
+A starred form of the\textbf{ }\verb`\nmcSolve` command suppresses
+all elements of display of the result apart from the numerical value.
+When nesting a \verb`\solve` command within another command this
+is the form to use. Thus with the `elusive' extremum example above,
+we can find the actual value of the minimum by nesting the starred
+\verb`\solve` command within the vv-list of an \verb`\eval` command:
+\begin{centred}
+\verb`\eval{$ x^3-x $}[x={\solve*{y^3-y+k}[k=5,y=0.5]}]` $\Longrightarrow$
+\eval{$ x^3-x $}[x={\solve*{y^3-y+k}[k=5,y=0.5]}].
+\end{centred}
+(Note the braces around the \verb`\solve*` and arguments to hide
+\emph{its} square-bracketed vv-list from the parsing of the vv-list
+of the \verb`\eval` command.) The result is to be compared with \verb`$\eval*{x^3-x}[x=\tfrac1{\surd3}]$`
+$\Longrightarrow$ $\eval*{x^3-x}[x=\tfrac1{\surd3}]$.
+
+\section{Settings option}
+
+\label{sec:solveSettings-option}The settings option is a comma-separated
+list of items of the form \emph{key~=~value}.
+
+\subsection{Inherited settings}
+
+The keys discussed in the settings\emph{ }option for \verb`\nmcEvaluate`
+are also available for \verb`\nmcSolve`. The very first example in
+this chapter used the punctuation option \texttt{p} (\verb`\solve[p]{\[... `)
+inherited from the \verb`\eval` command to ensure a comma after the
+display-style presentation of the result. We also saw in the quadratic
+example illustrating extrema the use of \texttt{vvi} with no value
+to suppress display of the vv-list: \verb`\solve[vvi=]{$ ...`.
+
+Putting \verb`dbg=1` produces a familiar kind of display. Using the
+function
+\[
+ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+\]
+from the rotating disk problem,
+\begin{verbatim}
+ \solve[dbg=1,var=t,p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\end{verbatim}
+$\Longrightarrow$ \solve[dbg=1,var=t,p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=1][4]
+
+\subsubsection{Multi-line display of the result}
+
+\label{subsec:solveMulti-line-display}By default the result is presented
+on a single line. Unless the star option is being used, this can be
+of the form \emph{function = function value, (vv-list) $\rightarrow$
+result}. It takes only a slightly complicated formula and only a few
+variables in the vv-list before this becomes a crowded line, likely
+to exceed the line width and extend into the margin. To split the
+display over two lines choose a \texttt{vvd} specification in the
+vv-list like, for instance, \texttt{vvd=\{,\}\textbackslash\textbackslash (vv)}.
+The \texttt{\textbackslash\textbackslash} is a trigger for \texttt{numerica}
+to replace whatever environment the \texttt{\textbackslash eval}
+command is wrapped around with a \texttt{multline} environment. An
+asterisk in the vv-list replaces \texttt{multline} with \texttt{multline{*}}
+so that no equation number is used:
+\begin{verbatim}
+ \solve[p=.,vvd={,}\\(vv),*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\end{verbatim}
+$\Longrightarrow$ \solve[p=.,vvd={,}\\(vv),*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\begin{verbatim}
+
+\end{verbatim}
+You could introduce a third line if you wished to display the result
+on a line of its own by using the spec. \texttt{vvd=\{,\}\textbackslash\textbackslash (vv)\textbackslash\textbackslash}:
+\begin{verbatim}
+ \solve[p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+\end{verbatim}
+$\Longrightarrow$ \solve[p=.,vvd={,}\\(vv)\\,*]
+ {$ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4]
+
+The function evaluates to $-0.0015$. Is this a zero that has been
+found or an extremum? To find out, the calculation needs to be carried
+out to a higher rounding value which is the reason why \verb`\nmcSolve`
+has an extra rounding setting; see §\ref{subsec:solveExtraRounding}
+below.
+
+\subsection{\texttt{\textbackslash solve}-specific settings}
+
+In addition there are some settings peculiar to \verb`\nmcSolve`.
+These are listed in Table~\ref{tab:solveSettings}.
+
+\begin{table}
+\centering{}\caption{\protect\label{tab:solveSettings}Settings for \texttt{\textbackslash nmcSolve}}
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small equation variable} & \tabularnewline
+{\small\texttt{dvar}} & {\small real $\ne0$ } & {\small initial step size} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{+}} & {\small int} & {\small extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. number of steps before cut off} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+
+\subsubsection{Equation variable}
+
+By default the equation variable is the \emph{rightmost} variable
+in the vv-list. This may not always be convenient. A different equation
+variable can be specified by entereing
+\begin{lyxcode}
+var~=~<variable~name>
+\end{lyxcode}
+in the vv-list. \texttt{<variable name>} will generally be a single
+character or token \textendash{} $x$, $t$, $\alpha$, $\omega$
+\textendash{} but is not necessarily of this kind. Multi-token names
+are perfectly acceptable (with the default \texttt{xx=1} multi-token
+setting).
+
+\subsubsection{Initial step size}
+
+The vv-list must contain the equation variable set to some trial value.
+But \verb`\solve` needs \emph{two} initial values to begin its search
+for a zero or extremum; see §\ref{subsec:solveSearch-strategy}. If
+the equation variable is set to a trial value $a$ then by default
+the second value is $a+1$. The `$+1$' here can be changed by entering
+in the settings option
+\begin{lyxcode}
+dvar~=~<non-zero~real~number>
+\end{lyxcode}
+For instance, \texttt{dvar=-1}, or \texttt{dvar=\textbackslash pi}
+are two valid specifications of initial step size.
+
+The notation is prompted by the use of expressions like $x+dx$ in
+calculus.
+
+An example where the default step value is too big and a smaller one
+needs to be specified is provided by Planck's radiation function (\emph{HMF
+}Table 27.2),
+\[
+f(x)=\frac{1}{x^{5}(e^{1/x}-1)}.
+\]
+From the (somewhat coarse-grained) table in \emph{HMF }it is clear
+that there is a maximum of about 21.2 when $x$ is a little more than
+$0.2$. This is a maximum above the $x$-axis and hence `elusive'.
+To find it, substract $100$ (say) from the formula and again use
+the ability to nest commands to display the result. Note the \verb`dvar=0.1`
+in the settings option of the \verb`\solve*` command:
+\begin{verbatim}
+ \eval[p=.]{\[ \frac1{x^5(e^{1/x}-1)} \]}
+ [ x={ \solve*[dvar=0.1]
+ { \frac1{y^5(e^{1/y}-1)}-100 }[y=0.1]
+ } ]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{\[ \frac1{x^5(e^{1/x}-1)} \]}
+ [ x={ \solve*[dvar=0.1]
+ { \frac1{y^5(e^{1/y}-1)}-100 }[y=0.1]
+ } ]
+
+\noindent The maximum is indeed a little over $21.2$ and the $x$
+value a little more than $0.2$.
+
+The default \verb`dvar=1` is too big for this problem. From the table
+in \emph{HMF},\emph{ }$f(0.1)=4.540$ and $f(1.1)=0.419$. By subtracting
+$100$ from $f(x)$ the `towards the $x$-axis' search strategy
+leads to negative values of $x$, an artifact of the subtraction.
+
+\subsubsection{Extra rounding}
+
+\label{subsec:solveExtraRounding}\verb`\solve` determines that a
+zero or an extremum has been reached when the difference between two
+successive bisection values or step values vanishes at the specified
+rounding value (the value in the final trailing optional argument
+of the \verb`\solve` command; $6$ by default). If our function is
+$f(x)$ then $\abs{x_{n+1}-x_{n}}=0$ to the specified rounding value
+and (assuming $x_{n+1}>x_{n}$) there is a critical value $x_{c}\in[x_{n},x_{n+1}]$
+such that $f(x_{c})=0$ (to all 16 places of decimals that \texttt{l3fp}
+works to). But in general the critical value $x_{c}$ will not coincide
+with $x_{n}$ or $x_{n+1}$. If $f(x)$ crosses the $x$-axis at a
+steep angle it may well be that although $f(x_{c})$ vanishes to all
+$16$ places of decimals, $f(x_{n}),f(x_{n+1})$ may well not vanish
+at the (generally smaller) specified rounding value. For instance,
+suppose $f(x)=1000x-3000$.
+\begin{centred}
+\verb`\solve[vvi=]{$ 1000x-3000 $}[x=e][4*]` $\Longrightarrow$ \solve[vvi=]{$ 1000x-3000 $}[x=e][4*].
+\end{centred}
+In this example, although the difference between successive $x$ values
+vanishes to $4$ places of decimals, $f(x)$ does not, not even to
+$2$ places.
+
+This suggests it would be helpful to be able to use two rounding values,
+one to determine when a zero has been found and one for the visible
+display. This is done through the extra rounding key in the settings
+option. Enter
+\begin{lyxcode}
++~=~<integer>
+\end{lyxcode}
+in the settings option of the \verb`\solve` command to add \texttt{<integer>}
+to the rounding value determining the conclusion of the calculation.
+By default the setting is \texttt{+=}0. The extra rounding setting
+leaves the display rounding unaffected.
+
+With this option available it is easy to check that \verb`+=3` suffices
+to ensure that both $x$ and $f(x)$ vanish to $4$ places of decimals,
+\begin{centred}
+\verb`\solve[+=3]{$ 1000x-3000 $}[x=e][4*]` $\Longrightarrow$ \solve[+=3]{$ 1000x-3000 $}[x=e][4*],
+\end{centred}
+and that \texttt{+=2} does not, i.e., we need to locate the zero to
+$4+3=7$ figures to ensure the function vanishes to $4$ figures.
+
+There is no need for the \texttt{<integer>} to be positive. In fact
+negative values can illuminate what is going on. In the first of the
+following, the display is to $10$ places but the calculation is only
+to $10-4=6$ places. In the second, the display is again to $10$
+places, but the calculation is to $10-3=7$ places.
+\begin{centred}
+\verb`\solve[+=-4]{$ 1000x-3000 $}[x=e][10*]` $\Longrightarrow$
+\solve[+=-4]{$ 1000x-3000 $}[x=e][10*],
+
+\verb`\solve[+=-3]{$ 1000x-3000 $}[x=e][10*]` $\Longrightarrow$
+\solve[+=-3]{$ 1000x-3000 $}[x=e][10*].
+\end{centred}
+Only in the second does $f(x)=1000x-3000$ vanish when rounded to
+$4$ figures.
+
+Returning to an earlier example (§\ref{subsec:solveMulti-line-display})
+in which it was not entirely clear whether a zero or an extremum had
+been found, we can now resolve the confusion. Use the extra rounding
+setting (and pad with zeros to emphasize the $4$-figure display by
+adding an asterisk in the trailing optional argument):
+\begin{verbatim}
+ \solve[+=2,vvd={,}\\(vv),*]
+ {$
+ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4*]
+\end{verbatim}
+$\Longrightarrow$ \solve[+=2,vvd={,}\\(vv),*]
+ {$
+ ct-\sqrt{a^{2}+b^{2}-2ab\cos(\beta+\omega t)}
+ $}[c=30,a=10,b=20,\beta=1,\omega=0.1,{t}=0][4*]
+
+\subsubsection{Maximum number of steps before cut-off}
+
+Once two function values have been found of opposite sign, bisection
+is guaranteed to arrive at a result. The problem is the \emph{search}
+for two such values. This may not terminate \textendash{} think of
+functions like $e^{x}$or $1/x$ which lack both zeros and extrema.
+To prevent an infinite loop, \verb`\solve` cuts off the search after
+$100$ steps. This cut-off value can be changed for a calculation
+by entering
+\begin{lyxcode}
+max~=~<positive~integer>
+\end{lyxcode}
+in the settings option.
+
+To illustrate, we know that $1/x$ has neither zero nor extremum,
+but we do not get an infinite loop; we get an error message if we
+attempt to `solve' $1/x$:
+\begin{centred}
+\verb`\solve{ 1/x }[x=1]` $\Longrightarrow$ \solve{ 1/x }[x=1]
+\end{centred}
+
+\subsubsection{Form of result saved by \texttt{\textbackslash reuse}}
+
+As wth \verb`\eval` and \verb`\iter` it is possible to specify to
+some extent what is saved to file when using \verb`\reuse` after
+a \verb`\solve` command. The form of entry in the settings option
+is
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+If the star option is used with the \verb`\solve` command the numerical
+result is the only thing saved, but in the absence of the star option,
+\begin{itemize}
+\item \texttt{reuse=0} saves\emph{ the form that is displayed}. For example,
+if the display is of the form \emph{function = function value, (vv-list)
+$\rightarrow$ result} then that is what is saved; this is the default
+behaviour;
+\item \texttt{reuse=1} (or any non-zero integer) saves only the numerical
+result.
+\end{itemize}
+
+\subsection{Changing default values}
+
+\begin{wraptable}[12]{o}{0.5\columnwidth}%
+\centering{}\caption{\protect\label{tab:solveSettingsDefaults}Defaults for \texttt{\textbackslash nmcSolve}}
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small solve-first-step} & 1\tabularnewline
+{\small solve-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small solve-max-steps} & {\small\texttt{100}}\tabularnewline
+{\small solve-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+If you wish to change the default values of the various settings for
+\verb`\nmcSolve` this can be done by entering new values in a configuration
+file \texttt{numerica.cfg} as described in the chapter on settings
+in the associated document \texttt{numeric-basics.pdf}. The relevant
+keys are listed in Table~\ref{tab:solveSettingsDefaults}, corresponding
+to the \verb`dvar`, \verb`+`, \verb`max` and \verb`reuse` settings
+of the \verb`\solve` command. (Obviously it makes no sense to have
+a default setting for the solution variable. That will change from
+case to case.)
+
+\chapter{Recurrence relations: \texttt{\textbackslash nmcRecur}}
+
+One of the simplest recurrence relations is that determining the Fibonacci
+numbers, $f_{n+2}=f_{n+1}+f_{n}$, with initial values $f_{0}=f_{1}=1$.
+The command \verb`\nmcRecur`, short-name form \verb`\recur`, allows
+calculation of the terms of this sequence:
+\begin{verbatim}
+ $ \nmcRecur[do=8,see1=8,...]
+ { f_{n+2}=f_{n+1}+f_{n} }
+ [f_{1}=1,f_{0}=1] $
+\end{verbatim}
+$\Longrightarrow$ $\nmcRecur[do=8,see1=8,...]
+ { f_{n+2}=f_{n+1}+f_{n} }
+ [f_{1}=1,f_{0}=1]$
+
+The recurrence relation is entered in the main argument (between braces),
+the initial values in the vv-list trailing the main argument, and
+the display specification is placed in the settings option: \texttt{do=8}
+terms to be calculated, all $8$ to be viewed (\texttt{see1=8}), and
+the display to be concluded by an ellipsis to indicate that the sequence
+continues (but those are three dots/periods/full stops in the settings
+option).
+
+A more complicated recurrence relation determines the Legendre polynomials:
+\[
+(n+2)P_{n+2}(x)-(2n+3)xP_{n+1}(x)+(n+1)P_{n}(x)=0.
+\]
+For the purposes of \verb`\recur` we need $P_{n+2}$ expressed in
+terms of the lower order terms:
+\[
+P_{n+2}(x)=\frac{1}{n+2}\left((2n+3)xP_{n+1}(x)-(n+1)P_{n}(x)\right).
+\]
+ It is this standard form \textendash{} the term to be calculated
+on the left, equated to an expression involving a fixed number of
+lower-order terms on the right \textendash{} that \texttt{numerica}
+works with. For $P_{0}(x)=1,~P_{1}(x)=x$ and $x=0.5$, the terms
+are calculated thus:
+\begin{verbatim}
+ \recur[p,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl((2n+3)xP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl((2n+3)xP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+
+\noindent where $P_{9}(0.5)$ and $P_{10}(0.5)$ are the last two
+displayed values (and to $6$-figures are the values listed in \emph{HMF
+}Table 8.1).
+
+These examples also illustrate a common behaviour of the \texttt{numerica}
+commands: when wrapped around math delimiters: the display is of the
+\emph{expression=result} form, and when placed between math delimiters
+the display is simply of the \emph{result}. When used without math
+delimiters, \texttt{numerica} treats the command as if it had been
+placed between \texttt{\textbackslash{[} \textbackslash}{]}.
+
+\section{Notational niceties}
+
+More than the other commands in \texttt{numerica}, \verb`\nmcRecur`
+depends on getting the notation into a standard form.
+\begin{itemize}
+\item The terms of the recurrence must be \emph{subscripted}: $f_{n}$,
+$P_{n}(x)$ are examples.
+\item The recurrence relation is placed in the main (mandatory) argument
+of \verb`\nmcRecur` in the form: \emph{high-order term=function of
+lower-order terms}.
+\item The initial-value terms in the vv-list must occur left-to-right in
+the order \emph{high }to \emph{low} order.
+\item The recurrence variable changes by $1$ between successive terms.
+\end{itemize}
+The example for Legendre polynomials in particular shows what is required.
+The Fibonacci example is simpler, since the recurrence variable does
+not occur independently in the recurrence relation as it does with
+the Legendre polynomials. In both cases though the recurrence variable
+is absent from the vv-list.
+
+\subsection{Vv-list and recurrence variable}
+
+The recurrence variable is required in the vv-list only when an implicit
+mode calculation is undertaken. Suppose we write $A$ and $B$ for
+the coefficients $2n+3$ and $n+1$ respectively in the Legendre recurrence.
+$A$ and $B$ will now need entries in the vv-list which means the
+recurrence variable will need a value assigned to it there too, and
+we will need to add \texttt{vvmode=1} to the settings option.
+\begin{verbatim}
+ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5,A=2n+3,B=n+1,n=0]
+\end{verbatim}
+$\Longrightarrow$ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[P_{1}(x)=x,P_{0}(x)=1,x=0.5,A=2n+3,B=n+1,n=0]
+
+Since the vv-list is evaluated from the right, the left-to-right high-to-low
+ordering of the initial-value terms means the value of the lowest
+order term is read first. Although \texttt{numerica} depends on this
+order of occurrence of the terms, they do not need to be \emph{consecutive}
+as in the examples so far (although it is natural to enter them in
+this way). \texttt{numerica} reads the value of the subscript of only
+the right-most term (the lowest order term), increments it by $1$
+when reading the next recurrence term to the left, and so on. The
+reading of the subscript of the lowest order term in the vv-list provides
+the initial value of the recurrence variable.
+
+In the following example I have placed other items between $P_{1}(x)$
+and $P_{0}(x)$ in the vv-list (but maintained their left-to-right
+order) and given the recurrence variable $n$ a ridiculous initial
+value $\pi^{2}/12$. (Because of the order in which things get done
+`behind the scenes', \emph{some} value is necessary so that the
+$n$ in `$B=n+1$' does not generate an `unknown token' message.)
+The result is unchanged.
+\begin{verbatim}
+ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[A=2n+3,P_{1}(x)=x,B=n+1,n=\pi^2/12,P_{0}(x)=1,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p,vvmode=1,do=11,see1=4,see2=2,vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(AxP_{n+1}(x)-BP_{n}(x)\Bigr)
+ \]}[A=2n+3,P_{1}(x)=x,B=n+1,n=\pi^2/12,P_{0}(x)=1,x=0.5]
+
+\subsection{Form of the recurrence relation}
+
+As noted earler, the form of the recurrence must be entered in the
+main argument in the form: \emph{highest order term = function of
+consecutive lower order terms}. The number of lower\emph{ }order terms
+is the order of the recurrence. The Fibonacci and Legendre polynomial
+recurrences are both second order and presented in the form: \emph{$n+2$-th
+term = function of $n+1$-th term and $n$-th term}. We could equally
+have done
+\begin{verbatim}
+ \nmcRecur[p,do=8,see1=8,...]
+ {$ f_{n}=f_{n-1}+f_{n-2} $}
+ [f_{1}=1,f_{0}=1]
+\end{verbatim}
+$\Longrightarrow$ \nmcRecur[p,do=8,see1=8,...]
+ {$ f_{n}=f_{n-1}+f_{n-2} $}
+ [f_{1}=1,f_{0}=1] where now the recurrence is of the form $n$\emph{-th term = function
+of $n-1$-th term and $n-2$-th term}, or (adjusting the coefficients
+as well as the recurrence terms),
+\begin{verbatim}
+ \recur[p=.,do=10,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p=.,do=10,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+
+\noindent The recurrence here is of the form $n+1$\emph{-th term
+= function of $n$-th term and $n-1$-th term}. This last example
+has one further `wrinkle'. I've made $P_{1}(x)$ the lowest order
+term and decreased the number of terms to calculate by $1$ accordingly.
+
+\subsection{First order recurrences (iteration)}
+
+The recurrence relations for both the Fibonacci sequence and Legendre
+polynomials are second order. There is no reason why the recurrence
+should not be of third or higher order or, indeed, lower. A first
+order recurrence provides an alternative means of iterating functions.
+\verb`\recur` therefore provides a means to display the results of
+an iteration in a different form from \verb`\iter`.
+
+Iterating $1+a/x$ in this way, $16$ terms gives the sequence
+\begin{verbatim}
+ \recur[do=16,see1=0,see2=3,...]{$
+ x_{n+1}=1+a/x_{n}
+ $}[x_{0}=1,a=1]
+\end{verbatim}
+$\Longrightarrow$ \recur[do=16,see1=0,see2=3,...]{$
+ x_{n+1}=1+a/x_{n}
+ $}[x_{0}=1,a=1]
+
+\noindent to be compared with the example near the start of Chapter~\ref{chap:Iterating-functions}.
+(\emph{That} effected $15$ iterations; \emph{this} uses $16$ terms
+because of the extra $x_{0}=1$ term.)
+
+\section{Star (\texttt{{*}}) option}
+
+When the star option is used with the \verb`\nmcRecur` command, only
+a single term, the \emph{last}, is presented as the result. Repeating
+the last calculation, but with the star option produces
+\begin{verbatim}
+ \recur*[p=.,do=10]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur*[p=.,do=10]{\[
+ P_{n+1}(x)=\frac{1}{n+1}
+ \Bigl((2n+1)xP_{n}(x)-nP_{n-1}(x)\Bigr)
+ \]}[P_{2}(x)=-0.125,P_{1}(x)=x,x=0.5]
+
+Although punctuation (a full stop) was specified in the settings,
+it has been ignored in the display of the result. Other settings would
+also have been ignored with the exception of the \verb`do` key which
+is required to know exactly which term to calculate. The star option
+produces a purely numerical answer without any trimmings.
+
+\section{Settings}
+
+The settings option is a comma-separated list of items of the form
+\emph{key~=~value}.
+
+\subsection{Inherited settings}
+
+Because recurrence terms are necessarily multi-token, the multi-token
+key is hard-coded in \verb`\recur` to \texttt{xx=1}.
+
+\subsubsection{Multi-line formatting of result}
+
+When the \verb`\recur` command wraps around math delimiters, the
+\texttt{vvd} setting is available to split display of the result over
+two or more lines. For example, \texttt{vvd=\{,\}\textbackslash\textbackslash (vv)}pushes
+the vv-list and sequence of calculated values to a second line; or,
+\texttt{vvd=\{,\}\textbackslash qquad(vv)\textbackslash\textbackslash}
+pushes only the sequence of calculated values to a second line; or
+\texttt{vvd=\{,\}\textbackslash\textbackslash (vv)\textbackslash\textbackslash}
+pushes the vv-list, centred, to a second line and the sequence of
+values, right aligned, to a third line. The \texttt{{*}} setting is
+available to suppress equation numbering (by substituting \verb`multline*`
+for \verb`multline`).
+\begin{verbatim}
+ \nmcRecur[do=8,see1=8,...,vvd={,}\qquad(vv)\\,*]
+ {$ f_{n+2}=f_{n+1}+f_{n} $}
+ [f_{1}=1,f_{0}=1]
+\end{verbatim}
+$\Longrightarrow$ \nmcRecur[do=8,see1=8,...,vvd={,}\qquad(vv)\\,*]
+ {$ f_{n+2}=f_{n+1}+f_{n} $}
+ [f_{1}=1,f_{0}=1]
+
+\subsection{\texttt{\textbackslash recur}-specific settings}
+
+\label{subsec:recurSpecific-settings}
+
+\subsubsection{Number of terms to calculate}
+
+By entering
+\begin{lyxcode}
+do~=~<integer>
+\end{lyxcode}
+in the settings option you can specify how many terms of a recurrence
+to calculate. The default is set to $7$ (largely to show a sufficient
+number of terms of the Fibonacci series to begin to be interesting).
+Note that \texttt{<integer>} will generally not correspond to the
+subscript on the last term calculated since that also depends on the
+value of the subscript of the lowest order term in the vv-list.
+
+\subsubsection{Number of terms to display}
+
+\begin{table}
+\centering{}\caption{\protect\label{tab:solveSettings-1}Settings for \texttt{\textbackslash nmcRecur}}
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{do}} & {\small int$\ge0$} & {\small number of terms to calculate} & {\small\texttt{7}}\tabularnewline
+{\small\texttt{see1}} & {\small int$\ge0$} & {\small number of initial terms to display} & {\small\texttt{3}}\tabularnewline
+{\small\texttt{see2}} & {\small int$\ge0$} & {\small number of final terms to display} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{...}} & {\small chars} & {\small follow display of values with an ellipsis} & \tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+By entering
+\begin{lyxcode}
+see1~=~<integer1>,~see2=<integer2>
+\end{lyxcode}
+in the settings option, you can specify how many initial terms of
+the recurrence and how many of the final terms calculated you want
+to view. If the sum of these settings is less than the \texttt{do}
+setting, then the terms are displayed with an intervening ellipsis.
+If the sum is greater than the \texttt{do} setting, then the values
+are adjusted so that their sum equals the \texttt{do} setting and
+all terms are displayed.
+
+The adjustment is preferentially to \texttt{see1}. Suppose \texttt{do=}7,
+\texttt{see1=5}, \texttt{see2=4}. Then \texttt{see2} is left unchanged
+but \texttt{see1} is reduced to \texttt{7-4=3}. If, say, \texttt{do=}7,
+\texttt{see1=5}, \texttt{see2=8}, then \texttt{see2} is reduced to
+\texttt{7} and \texttt{see1} to \texttt{-1} (rather than zero, for
+technical reasons). The reason for preserving \texttt{see2} over \texttt{see1}
+is for the functioning of the \texttt{reuse} setting (see above).
+
+The default value for \texttt{see1} is $3$; the default value for
+\texttt{see2} is $2$.
+
+\subsubsection{Ellipsis}
+
+Including three dots in the settings option
+\begin{lyxcode}
+...
+\end{lyxcode}
+ensures that a (proper) ellipsis is inserted after the final term
+is displayed. An example is provided by the display of the Fibonacci
+sequence at the start of this chapter. By default this option is turned
+off.
+
+\subsubsection{Form of result saved by \texttt{\textbackslash reuse}}
+
+By entering
+\begin{lyxcode}
+reuse~=~<integer>
+\end{lyxcode}
+it is possible to specify the form of result that is saved when using
+\verb`\nmcReuse`. (This setting has no effect when the star option
+is used with \verb`\nmcRecur`. In that case only the numerical result
+of the final term calculated is saved.) There are three different
+outputs possible:
+\begin{itemize}
+\item \texttt{int=0} (or any integer $\mathtt{\ne1,2}$) saves the full
+display (the default);
+\item \texttt{int=1} saves a comma-separated list of braced pairs of the
+form: \texttt{\{$k$, value-of-term-$k$\}} for the last \texttt{see2}
+terms calculated;
+\item \texttt{int=2} saves a comma-separated list of the values of the last
+\texttt{see2} terms calculated.
+\end{itemize}
+As an example, using \texttt{reuse=1},
+\begin{verbatim}
+ \recur[reuse=1,p=.,vvmode=1,do=11,see1=4,see2=2,
+ vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(kxP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[k=2n+3,n=123,P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+ \reuse[legendre]
+\end{verbatim}
+$\Longrightarrow$ \recur[reuse=1,p=.,vvmode=1,do=11,see1=4,see2=2,
+ vvd={,}\\(vv)\\,*]
+ {\[ P_{n+2}(x)=\frac{1}{n+2}
+ \Bigl(kxP_{n+1}(x)-(n+1)P_{n}(x)\Bigr)
+ \]}[k=2n+3,n=123,P_{1}(x)=x,P_{0}(x)=1,x=0.5]
+ \reuse[legendre]
+
+\noindent Now check to see what has been saved:
+\begin{centred}
+\verb`$\legendre$` $\Longrightarrow$ $ \legendre$.
+\end{centred}
+As you can see, the final two (because of \texttt{see2=2}) of the
+$12$ Legendre polynomials calculated have been saved, each value
+preceded by its index value. If the setting had been \texttt{reuse=2},
+only the two values would have been saved. The \verb`\legendre` control
+sequence contains the values as comma-separated braced pairs, as can
+be seen by using \TeX 's \verb`\meaning` command:
+\begin{centred}
+\verb`\meaning\legendre` $\Longrightarrow$ \meaning\legendre
+\end{centred}
+
+\subsection{Changing default values}
+
+\begin{wraptable}[12]{o}{0.5\columnwidth}%
+\centering{}\caption{\protect\label{tab:recurSettingsDefaults}Defaults for \texttt{\textbackslash nmcRecur}}
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small recur-do} & {\small\texttt{7}}\tabularnewline
+{\small recur-see-first} & {\small\texttt{3}}\tabularnewline
+{\small recur-see-last} & {\small\texttt{2}}\tabularnewline
+{\small recur-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\end{wraptable}%
+If you wish to change the default values of the various settings for
+\verb`\nmcRecur` this can be done by entering new values in a configuration
+file \texttt{numerica.cfg} as described in the chapter on settings
+in the associated document \texttt{numerica-basics.pdf}. The relevant
+keys are listed in Table~\ref{tab:recurSettingsDefaults}, corresponding
+to the \verb`do`, \verb`see1`, \verb`see2` and \verb`reuse` settings
+of the \verb`\recur` command.
+
+\subsection{Orthogonal polynomials}
+
+I've used Legendre polynomials in examples above, but orthogonal polynomials
+generally lend themselves to the \verb`\recur` treatment. Quoting
+from \emph{HMF} 22.7, orthogonal polynomials $f_{n}$ satisfy recurrence
+relations of the form
+\[
+a_{1n}f_{n+1}(x)=(a_{2n}+a_{3n}x)f_{n}(x)-a_{4n}f_{n-1}(x),
+\]
+or in the standard form required by \verb`\recur`,
+\[
+f_{n+1}(x)=\frac{a_{2n}+a_{3n}x}{a_{1n}}f_{n}(x)-\frac{a_{4n}}{a_{1n}}f_{n-1}(x).
+\]
+\emph{HMF} 22.7 provides a listing of the coefficients $a_{in}$ for
+the polynomials of Jacobi, Chebyshev, Legendre, Laguerre, Hermite
+and others, and tables for these polynomials.
+
+For example, Laguerre polynomials satisfy the recurrence
+\[
+L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-\frac{n}{n+1}L_{n-1}(x).
+\]
+with initial values $L_{0}(x)=1$ and $L_{1}(x)=1-x$. So let's calculate
+the first $13$ Laguerre polynomials for, say, $x=0.5$:
+\begin{verbatim}
+ \recur[do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=0.5]
+\end{verbatim}
+$\Longrightarrow$ \recur[do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=0.5]
+
+\noindent and for $x=5$:
+\begin{verbatim}
+ \recur[p=.,do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+\end{verbatim}
+$\Longrightarrow$ \recur[p=.,do=13,see1=4,see2=2,vvd={,}\\(vv)\\,*]{\[
+ L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)
+ \]}[L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+
+\noindent The results (reassuringly) coincide with those provided
+in \emph{HMF }Table 22.11.
+
+\subsection{Nesting}
+
+It is possible to use the \verb`\recur*` command (but only the starred
+form) in the \verb`\eval`, \verb`\iter`, and \verb`\solve` commands,
+and indeed in \verb`\recur` itself, but with this caveat: if \verb`\recur*`
+is nested within another command, the initial terms of the recurrence
+\textendash{} e.g., $f_{1}=1,f_{0}=1$, for the Fibonacci series,
+or $L_{1}(x)=1-x,L_{0}(x)=1$ for the Laguerre polynomials \textendash{}
+\emph{must be located in the vv-list of that inner }\verb`\recur*`\emph{
+command}. Other shared variables can often be shifted to the vv-list
+of the outer command, but not these initial terms.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+The terms of a recurrence relation are multi-token variables but \texttt{numerica}
+requires single tokens for its calculations. The problem for \verb`\recur`
+is that the terms in the recurrence relation in the main (mandatory)
+argument differ from the terms in the vv-list: for instance $f_{n}$
+in the main argument, $f_{0}$ in the vv-list. If left like that,
+when \texttt{numerica} does its conversion from multi-token to single
+token variables, $f_{n}$ would not be found since it differs from
+$f_{0}$. Hence a crucial first step for \verb`\recur` is to reconcile
+the different forms, which it does by converting the forms in the
+vv-list to the forms in the recurrence in the main argument. To be
+available for this form change, they must reside in the \emph{inner}
+vv-list. In the outer vv-list they would be inaccessible to the inner
+command.
+
+{*}{*}{*}
+
+This suggests an alternative way of proceeding: write the inital values
+of the recurrence terms in the \emph{same} form in which they occur
+in the recurrence relation, together with an initial value for the
+recurrence variable: $f_{n+1}=1,f_{n}=1,n=0$, say. This is not how
+mathematicians write the initial values in recurrence relations, which
+is why I did not pursue it, but it neatly sidesteps what is otherwise
+an initial awkwardness. \end{shaded}%
+\end{minipage}
+
+In the following example I multiply together (rather futilely) the
+third and fourth members of the sequence of Laguerre polynomials for
+$x=5$ (the answer expected is \verb`$ \eval{3.5\times2.666667} $`
+$\Longrightarrow$ $ \eval{3.5\times2.666667} $). Note that although
+it is tempting to shift the shared vv-lists of the inner \verb`\recur*`
+commands to the vv-list of the outer \verb`\eval` command, in fact
+only the \verb`x=5` entry has been transferred:
+\begin{verbatim}
+ \eval[p=.]{$
+ \recur*[do=3]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1]
+ \times
+ \recur*[do=4]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1]
+ $}[x=5]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=.]{$
+ \recur*[do=3]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+ \times
+ \recur*[do=4]
+ { L_{n+1}(x)=\frac{2n+1-x}{n+1}L_{n}(x)-
+ \frac{n}{n+1}L_{n-1}(x)}
+ [L_{1}(x)=1-x,L_{0}(x)=1,x=5]
+ $}
+
+\chapter{Reference summary}
+
+\section{Commands defined in \texttt{numerica-plus}}
+\begin{enumerate}
+\item \texttt{\textbackslash nmcIterate, \textbackslash iter}
+\item \texttt{\textbackslash nmcSolve, \textbackslash solve}
+\item \textbackslash\texttt{nmcRecur, \textbackslash recur}
+\end{enumerate}
+
+\section{Settings for the three commands}
+
+\subsection{Settings for \texttt{\textbackslash nmcIterate}}
+
+Settings option of \verb`\nmcIterate`:
+
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small iteration variable} & \tabularnewline
+{\small\texttt{+}} & {\small int} & {\small fixed point extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. iteration count (fixed points)} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{do}} & {\small int > 0} & {\small number of iterations to perform} & {\small\texttt{5}}\tabularnewline
+{\small\texttt{see}} & {\small int > 0} & {\small number of final iterations to view} & {\small\texttt{4}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+Configuration settings for \verb`\nmcIterate`:
+
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small iter-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small iter-max-iterations} & {\small\texttt{100}}\tabularnewline
+{\small iter-do} & {\small\texttt{5}}\tabularnewline
+{\small iter-see-last} & {\small\texttt{4}}\tabularnewline
+{\small iter-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+\subsection{Settings for \texttt{\textbackslash nmcSolve}}
+
+Settings option of \verb`\nmcSolve`:
+
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{var}} & {\small token(s)} & {\small equation variable} & \tabularnewline
+{\small\texttt{dvar}} & {\small real $\ne0$} & {\small initial step size} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{+}} & {\small int} & {\small extra rounding} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{max}} & {\small int > 0} & {\small max. number of steps before cut off} & {\small\texttt{100}}\tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+Configuration settings for \verb`\nmcSolve`:
+
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small solve-first-step} & 1\tabularnewline
+{\small solve-extra-rounding} & {\small\texttt{0}}\tabularnewline
+{\small solve-max-steps} & {\small\texttt{100}}\tabularnewline
+{\small solve-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+\subsection{Settings for \texttt{\textbackslash nmcRecur}}
+
+Settings option of \verb`\nmcRecur`:
+
+\begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{do}} & {\small int$\ge0$} & {\small number of terms to calculate} & {\small\texttt{7}}\tabularnewline
+{\small\texttt{see1}} & {\small int$\ge0$} & {\small number of initial terms to display} & {\small\texttt{3}}\tabularnewline
+{\small\texttt{see2}} & {\small int$\ge0$} & {\small number of final terms to display} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{...}} & {\small chars} & {\small follow display of values with an ellipsis} & \tabularnewline
+{\small\texttt{reuse}} & {\small int ($\mathtt{0}/\mathtt{1}/\mathtt{2}$)} & {\small form of result saved with }{\small{\small\verb`\reuse`}} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+
+Configuration settings for \verb`\nmcRecur`:
+
+\begin{center}
+\begin{tabular}{ll}
+\toprule
+{\small key} & {\small default}\tabularnewline
+\midrule
+{\small recur-do} & {\small\texttt{7}}\tabularnewline
+{\small recur-see-first} & {\small\texttt{3}}\tabularnewline
+{\small recur-see-last} & {\small\texttt{2}}\tabularnewline
+{\small recur-reuse} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{document}
diff --git a/macros/latex/contrib/numerica/numerica-tables.def b/macros/latex/contrib/numerica/numerica-tables.def
new file mode 100644
index 0000000000..5fb2efd5b4
--- /dev/null
+++ b/macros/latex/contrib/numerica/numerica-tables.def
@@ -0,0 +1,1138 @@
+% This is file `numerica-tables.def',
+% part of the `numerica' package.
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe@gmail.com)
+%
+\RequirePackage{booktabs}
+%
+\ProvidesExplFile
+ {numerica-tables.def}
+ {2021/02/15}
+ {1.0.0}
+ {Create tables of function values}
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_table_prepad:Nnn #1#2#3
+ {
+ \tl_put_left:Nx #1 { \prg_replicate:nn
+ { \int_max:nn { #2 - \tl_count:N #1 } { 0 } } { #3 } }
+ }
+\cs_new:Npn \__nmc_sci_num_table_plus:nn #1#2
+ { % #1 significand, #2 exponent
+ \fp_compare:nTF { #1 >= 0 }
+ { ( #2 )\,{ \__nmc_sci_num_table:n { #1 } } }
+ { ( #2 )\,{ #1 } }
+ }
+\cs_new:Npn \__nmc_sci_num_table:n #1
+ {
+ \int_compare:nNnF { \l__nmc_num_sgn_int } < { 0 }
+ {
+ \int_case:nn { \l__nmc_table_signs_int }
+ {
+ {-2 } { \exp_not:N \hphantom{-} }
+ {-1 } { \bool_if:NTF \l__nmc_table_signs_bool
+ { + }{ \exp_not:N \hphantom{-} } }
+ { 1 } { \bool_if:NT \l__nmc_table_signs_bool { + } }
+ { 2 } { + }
+ }
+ }
+ #1
+ }
+%--------------------------------------
+\tl_new:N \l__nmc_x_tl
+\seq_new:N \l_tmpd_seq
+\tl_const:Nn \c__nmc_r_tl { r }
+\tl_const:Nn \c__nmc_l_tl { l }
+\bool_new:N \l__nmc_table_clist_bool
+\clist_new:N \l__nmc_formula_clist
+\tl_new:N \l__nmc_table_valign_tl
+\seq_new:N \l_nmc_table_reuse_seq
+
+\tl_new:N \l__nmc_table_rvar_tl
+\tl_new:N \l__nmc_table_rvari_tl
+\tl_new:N \l__nmc_table_rstart_tl
+\tl_new:N \l__nmc_table_rstop_tl
+\tl_new:N \l__nmc_table_rstep_tl
+\fp_new:N \l__nmc_table_rstart_fp
+\fp_new:N \l__nmc_table_rstop_fp
+\fp_new:N \l__nmc_table_rstep_fp
+\fp_new:N \l__nmc_table_r_fp
+\fp_new:N \l__nmc_table_c_fp
+\int_new:N \l__nmc_table_rnum_int
+\clist_new:N \l__nmc_table_rspec_clist
+\bool_new:N \l__nmc_table_rspec_bool
+
+\tl_new:N \l__nmc_table_cvar_tl
+\tl_new:N \l__nmc_table_cstart_tl
+\tl_new:N \l__nmc_table_cstop_tl
+\tl_new:N \l__nmc_table_cstep_tl
+\fp_new:N \l__nmc_table_cstart_fp
+\fp_new:N \l__nmc_table_cstop_fp
+\fp_new:N \l__nmc_table_cstep_fp
+\int_new:N \l__nmc_table_cnum_int
+\clist_new:N \l__nmc_table_cspec_clist
+\bool_new:N \l__nmc_table_cspec_bool
+
+\seq_new:N \l__nmc_table_rvraw_seq
+\seq_new:N \l__nmc_table_rvcol_seq
+\seq_new:N \l__nmc_table_rvcoli_seq
+\seq_new:N \l__nmc_table_this_col_seq
+\seq_new:N \l__nmc_table_body_cols_seq
+\seq_new:N \l__nmc_table_body_rows_seq
+
+\seq_new:N \l__nmc_table_header_seq
+\tl_new:N \l__nmc_table_header_tl
+\tl_new:N \l__nmc_table_rfont_tl
+\tl_new:N \l__nmc_table_rhead_tl
+\tl_new:N \l__nmc_table_rheadi_tl
+\tl_new:N \l__nmc_table_rhnudge_tl
+\tl_new:N \l__nmc_table_rhnudgei_tl
+\int_new:N \l__nmc_table_chround_int
+\int_new:N \l__nmc_table_chstyle_int
+\tl_new:N \l__nmc_table_chnudge_tl
+
+\tl_new:N \l__nmc_table_ctitle_tl
+\bool_new:N \l__nmc_table_ctitle_bool
+\bool_new:N \l__nmc_table_ctitle_vv_bool
+\int_new:N \l__nmc_table_rules_int
+\tl_new:N \l__nmc_table_footer_tl
+\bool_new:N \l__nmc_table_foot_bool
+\bool_new:N \l__nmc_table_footops_bool
+\tl_new:N \l__nmc_table_csum_tl
+\tl_new:N \l__nmc_table_cave_tl
+\tl_new:N \l__nmc_table_cmax_tl
+\tl_new:N \l__nmc_table_cmin_tl
+\tl_new:N \l__nmc_table_cdel_tl
+\tl_new:N \l__nmc_table_rbloc_tl
+\tl_new:N \l__nmc_table_rblock_tl
+\int_new:N \l__nmc_table_rbloc_int
+\clist_new:N \l__nmc_table_rbloc_clist
+
+\int_new:N \l__nmc_table_signs_int
+\int_new:N \l__nmc_table_tsigns_int
+\int_new:N \l__nmc_table_c_int
+\int_new:N \l__nmc_table_b_int
+\int_new:N \l__nmc_table_a_int
+\bool_new:N \l__nmc_table_signs_bool
+\bool_new:N \l__nmc_table_diffs_bool
+\tl_new:N \l__nmc_table_diff_tl
+\tl_new:N \l__nmc_table_diff_null_tl
+
+\tl_new:N \l__nmc_table_Q_tl
+\tl_new:N \l__nmc_table_A_tl
+\fp_new:N \l__nmc_table_max_fp
+\fp_new:N \l__nmc_table_min_fp
+\bool_new:N \l__nmc_table_QA_max_bool
+\bool_new:N \l__nmc_table_QA_min_bool
+\bool_new:N \l__nmc_table_QA_bool
+\bool_new:N \l__nmc_table_num_only_bool
+\tl_new:N \l__nmc_table_num_only_tl
+\tl_new:N \l__nmc_table_query_tl
+%------------------------------------------------
+\nmc_define:nnN { \nmcTabulate } { table } \tabulate
+
+\cs_new_protected:Npn \__nmc_table_settings_digest:
+ {
+ \bool_set_false:N \l__nmc_allow_TF_out_bool
+ \int_compare:nTF { 0 <= \l__nmc_table_rpos_int <= 4 }
+ {
+ \int_compare:nNnT { \l__nmc_table_rpos_int } = { 4 }
+ { \int_incr:N \l__nmc_table_rpos_int }
+ }
+ { \int_set:Nn \l__nmc_table_rpos_int { 1 } }
+ \clist_if_empty:NF \l__nmc_table_rspec_clist
+ { \__nmc_table_settings_spec:N r }
+ \clist_if_empty:NF \l__nmc_table_cspec_clist
+ { \__nmc_table_settings_spec:N c }
+ \tl_if_empty:NT \l__nmc_table_rhead_tl
+ { \tl_set_eq:NN \l__nmc_table_rhead_tl \l__nmc_table_rvar_tl }
+ \tl_if_empty:NT \l__nmc_table_rheadi_tl
+ {
+ \tl_if_empty:NTF \l__nmc_table_rvari_tl
+ { \tl_set_eq:NN \l__nmc_table_rheadi_tl \l__nmc_table_rhead_tl }
+ { \tl_set_eq:NN \l__nmc_table_rheadi_tl \l__nmc_table_rvari_tl }
+ }
+ \tl_if_empty:NT \l__nmc_table_rhnudgei_tl
+ { \tl_set_eq:NN \l__nmc_table_rhnudgei_tl \l__nmc_table_rhnudge_tl }
+
+ \tl_if_empty:NF \l__nmc_table_ctitle_tl
+ { \l__nmc_table_settings_ctitle: }
+ \tl_if_empty:NTF \l__nmc_table_Q_tl
+ { \tl_set:Nn \l__nmc_table_query_tl { *~option~ } }
+ {
+ \tl_set:Nn \l__nmc_table_query_tl { table~value~satisfies~ }
+ \__nmc_table_settings_QA:
+ }
+ \clist_if_empty:NTF \l__nmc_table_rbloc_clist
+ { \tl_set:Nn \l__nmc_table_rbloc_tl { 10000 } }
+ { \clist_pop:NN \l__nmc_table_rbloc_clist \l__nmc_table_rbloc_tl }
+ \tl_set_eq:NN \l__nmc_table_rblock_tl \l__nmc_table_rbloc_tl
+ \int_set:Nn \l__nmc_table_rbloc_int { \l__nmc_table_rbloc_tl }
+
+ \__nmc_table_settings_rules:
+ \int_if_zero:nTF \l__nmc_table_diffs_int
+ { \bool_set_false:N \l__nmc_table_diffs_bool }
+ { \bool_set_true:N \l__nmc_table_diffs_bool }
+ \tl_if_empty:NF \l__nmc_table_chead_tl
+ { \int_set:Nn \l__nmc_table_chstyle_int { 4 } }
+ \tl_if_empty:NF \l__nmc_table_footer_tl
+ {
+ \tl_if_eq:VnTF \l__nmc_table_footer_tl { * }
+ { \bool_set_true:N \l__nmc_table_foot_bool }
+ { \__nmc_table_footer:N \l__nmc_table_footer_tl }
+ }
+ }
+\cs_new_protected:Npn \l__nmc_table_settings_ctitle:
+ {
+ \bool_set_true:N \l__nmc_table_ctitle_bool
+ \tl_if_head_eq_charcode:VNT \l__nmc_table_ctitle_tl *
+ {
+ \tl_if_eq:VnT \l__nmc_table_ctitle_tl { ** }
+ { \bool_set_true:N \l__nmc_table_ctitle_vv_bool }
+ \tl_if_head_eq_charcode:VNT \l__nmc_table_ctitle_tl *
+ {
+ \tl_set:Nn \l__nmc_table_ctitle_tl
+ { \exp_not:o \l__nmc_formula_dup_tl }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_settings_rules:
+ {
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { T } { *2 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { t } { *3 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { h } { *5 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { f } { *7 }
+ \tl_replace_once:Nnn \l__nmc_table_rules_tl { B } { *11 }
+ \int_set:Nn \l__nmc_table_rules_int { 1 \l__nmc_table_rules_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_settings_QA:
+ {
+ \bool_set_true:N \l__nmc_table_QA_bool
+ \regex_replace_all:nnN { (\@) } { \c{l_tmpb_fp} } \l__nmc_table_Q_tl
+ \tl_set_rescan:Nno \l__nmc_table_Q_tl { \ExplSyntaxOn } \l__nmc_table_Q_tl
+ \regex_replace_all:nnN { (\cC.) } { \c{exp_not:N}\1 } \l__nmc_table_A_tl
+ \regex_replace_all:nnN { (\@) } { \c{l_tmpb_tl} } \l__nmc_table_A_tl
+ \tl_if_in:NnT \l__nmc_table_Q_tl { MAX }
+ {
+ \bool_set_true:N \l__nmc_table_QA_max_bool
+ \regex_replace_all:nnN { MAX } { \c{l__nmc_table_max_fp} }
+ \l__nmc_table_Q_tl
+ \fp_set:Nn \l__nmc_table_max_fp { -inf }
+ }
+ \tl_if_in:NnT \l__nmc_table_Q_tl { MIN }
+ {
+ \bool_set_true:N \l__nmc_table_QA_min_bool
+ \regex_replace_all:nnN { MIN } { \c{l__nmc_table_min_fp} }
+ \l__nmc_table_Q_tl
+ \fp_set:Nn \l__nmc_table_min_fp { inf }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_footer:N #1
+ {
+ \tl_set_eq:NN \l_tmpa_tl #1
+ \regex_replace_all:nnN { SUM } { \c{l__nmc_table_csum_tl} } #1
+ \regex_replace_all:nnN { AVE } { \c{l__nmc_table_cave_tl} } #1
+ \regex_replace_all:nnN { MAX } { \c{l__nmc_table_cmax_tl} } #1
+ \regex_replace_all:nnN { MIN } { \c{l__nmc_table_cmin_tl} } #1
+ \regex_replace_all:nnN { DEL } { \c{l__nmc_table_cdel_tl} } #1
+ \tl_if_eq:NNF \l_tmpa_tl #1
+ { \bool_set_true:N \l__nmc_table_footops_bool }
+ }
+\cs_new_protected:Npn \__nmc_table_settings_spec:N #1
+ {
+ \bool_set_true:c { l__nmc_table_ #1 spec_bool }
+ \clist_pop:cc { l__nmc_table_#1spec_clist } { l__nmc_table_ #1 var_tl }
+ \clist_pop:cc { l__nmc_table_ #1 spec_clist } { l__nmc_table_ #1 step_tl }
+ \clist_pop:cN { l__nmc_table_ #1 spec_clist } \l_tmpa_tl
+ \tl_if_head_eq_charcode:VNTF \l_tmpa_tl (
+ {
+ \int_set:cn { l__nmc_table_ #1 num_int }
+ { \exp_last_unbraced:NV \__nmc_paren_arg:w \l_tmpa_tl }
+ }
+ { \tl_set_eq:cN { l__nmc_table_ #1 stop_tl } \l_tmpa_tl }
+ }
+\cs_new:Npn \__nmc_paren_arg:w (#1) { #1 }
+
+\cs_new_protected:Npn \__nmc_table_vv_digest:N #1
+ {
+ \bool_if:NTF \l__nmc_multitok_bool
+ {
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_table_cvar_tl
+ \clist_push:NV \l__nmc_formula_tl \l__nmc_table_rvar_tl
+ \__nmc_vv_digest:N #1
+ \clist_pop:NN \l__nmc_formula_tl \l__nmc_table_rvar_tl
+ \tl_if_empty:NF \l__nmc_table_cvar_tl
+ { \clist_pop:NN \l__nmc_formula_tl \l__nmc_table_cvar_tl }
+ }
+ { \__nmc_vv_digest:N #1 }
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_table_get_ini_vals:
+ \bool_if:nT
+ {
+ \int_if_zero_p:n { \l__nmc_table_rpos_int } &&
+ \int_if_zero_p:n { \l__nmc_table_cnum_int }
+ }
+ { \__nmc_error_what:n { Null~table~specified~in } }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \bool_if:NT \l__nmc_table_ctitle_vv_bool
+ {
+ \exp_args:NNnx\tl_replace_once:Nnn \l__nmc_vv_inline_tl { vv }
+ { \seq_use:Nn \l__nmc_vv_visible_seq { , } }
+ \tl_put_right:Nn \l__nmc_table_ctitle_tl
+ { \exp_not:o \l__nmc_vv_inline_tl }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_get_ini_vals:
+ {
+ \__nmc_error_where:n { settings }
+ \__nmc_table_get_ini_vals_aux:N r
+ \tl_if_empty:NTF \l__nmc_table_cvar_tl
+ {
+ \int_set:Nn \l__nmc_table_cnum_int
+ {
+ \int_compare:nNnTF { \l__nmc_table_cnum_int } = { -1 }
+ { 1 } { \int_min:nn { 1 }
+ { \int_abs:n { \l__nmc_table_cnum_int } } }
+ }
+ }
+ { \__nmc_table_get_ini_vals_aux:N c }
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ }
+\cs_new_protected:Npn \__nmc_table_get_ini_vals_aux:N #1
+ {
+ \prop_get:NvNTF \g__nmc_subst_var_prop { l__nmc_table_ #1 var_tl } \l_tmpa_tl
+ {
+ \fp_set:cn { l__nmc_table_ #1 start_fp } { \l_tmpa_tl }
+ \bool_if:cTF { l__nmc_table_ #1 spec_bool }
+ {
+ \__nmc_fpify_set:cc { l__nmc_table_ #1 step_fp }
+ { l__nmc_table_ #1 step_tl }
+ \int_compare:vNnT { l__nmc_table_ #1 num_int } = { -1 }
+ {
+ \__nmc_fpify_set:cc { l__nmc_table_ #1 stop_fp }
+ { l__nmc_table_ #1 stop_tl }
+ \l__nmc_table_calc_num:cnnn { l__nmc_table_ #1 num_int }
+ { l__nmc_table_ #1 stop_fp } { l__nmc_table_ #1 start_fp }
+ { l__nmc_table_ #1 step_fp }
+ }
+ }
+ { \__nmc_table_get_individual_ini_vals:N #1 }
+ }
+ { \__nmc_error_what:n { No~ \__nmc_table_rc:N #1 ~variable~specified~in } }
+ }
+\cs_new:Npn \__nmc_table_rc:N #1
+ { \tl_if_eq:NNTF #1 \c__nmc_r_tl { row } { column } }
+
+\cs_new_protected:Npn \l__nmc_table_calc_num:Nnnn #1#2#3#4
+ {
+ \int_set:Nn #1 { \fp_to_int:n { ( \exp_not:v { #2 } - \exp_not:v { #3 } )
+ / \exp_not:v { #4 } } + 1 }
+ }
+\cs_generate_variant:Nn \l__nmc_table_calc_num:Nnnn { c }
+
+\cs_new_protected:Npn \__nmc_table_get_individual_ini_vals:N #1
+ {
+ \int_case:vnTF { l__nmc_table_ #1 num_int }
+ {
+ { -1 }
+ {
+ \__nmc_fpify_set:cc { l__nmc_table_ #1 step_fp }
+ { l__nmc_table_ #1 step_tl }
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_fpify_set:cc { l__nmc_table_ #1 stop_fp }
+ { l__nmc_table_ #1 stop_tl } }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \l__nmc_table_calc_num:cnnn { l__nmc_table_ #1 num_int }
+ { l__nmc_table_ #1 stop_fp } { l__nmc_table_ #1 start_fp }
+ { l__nmc_table_ #1 step_fp }
+ }
+ }
+ { 0 }
+ {
+ \fp_set:cn { l__nmc_table_ #1 stop_fp }
+ { \exp_not:v { l__nmc_table_ #1 start_fp } - 1 }
+ \fp_set:cn { l__nmc_table_ #1 step_fp } { 1 }
+ }
+ { 1 }
+ {
+ \fp_set_eq:cc { l__nmc_table_ #1 stop_fp }
+ { l__nmc_table_ #1 start_fp }
+ \fp_set:cn { l__nmc_table_ #1 step_fp } { 1 }
+ }
+ }
+ {
+ \bool_if:NT \g__nmc_error_bool
+ { \__nmc_error_what:n { Missing~
+ \__nmc_table_rc:N #1-related~variable~in } }
+ }
+ {
+ \int_compare:vNnTF { l__nmc_table_ #1 num_int } > { 1 }
+ { \__nmc_fpify_set:cc { l__nmc_table_ #1 step_fp }
+ { l__nmc_table_ #1 step_tl } }
+ { \__nmc_error_what:n { Check~number~of~
+ \__nmc_table_rc:N #1s~specified~in } }
+ }
+ }
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_table_process:
+ {
+ \__nmc_error_where:n { formula }
+ \__nmc_table_calc_rvar_col:
+ \seq_clear:N \l__nmc_table_header_seq
+ \tl_if_head_eq_charcode:VNT \l__nmc_formula_tl ,
+ { \__nmc_table_multifn: }
+ \__nmc_table_calc_cvar_cols:
+ \tl_clear:N \l_nmc_result_tl
+ \__nmc_table_form_body:N \l_nmc_result_tl
+ \tl_set_eq:NN \l__nmc_fp_expr_tl \l__nmc_fp_exprn_tl
+ \bool_if:NT \l__nmc_num_only_bool
+ {
+ \bool_if:NTF \l__nmc_table_num_only_bool
+ {
+ \tl_set_eq:NN \l_nmc_result_tl \l__nmc_table_num_only_tl
+ \tl_gset_eq:NN \g__nmc_reuse_tl \l__nmc_table_num_only_tl
+ }
+ {
+ \__nmc_error_where:n { settings }
+ \__nmc_error_what:n
+ { No~ \l__nmc_table_query_tl query~\__nmc_verb:n {Q?}~in }
+ \tl_clear:N \l_nmc_result_tl
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_multifn:
+ {
+ \bool_set_true:N \l__nmc_table_clist_bool
+ \clist_set:NV \l__nmc_formula_clist \l__nmc_formula_tl
+ \seq_set_from_clist:NN \l__nmc_table_header_seq \l__nmc_formula_clist
+ \int_set:Nn \l__nmc_table_cnum_int
+ { \clist_count:N \l__nmc_formula_clist }
+ \fp_set:Nn \l__nmc_table_cstep_fp { 0 }
+ \fp_set:Nn \l__nmc_table_cstart_fp { 0 }
+ \tl_set:Nn \l__nmc_table_cvar_tl { \l__nmc_x_tl }
+ \prop_put:Nnn \g__nmc_subst_var_prop { \l__nmc_x_tl } { 1 }
+ \int_set:Nn \l__nmc_table_chstyle_int { 4 }
+ }
+\cs_new_protected:Npn \__nmc_table_calc_rvar_col:
+ {
+ \fp_set_eq:NN \l__nmc_table_r_fp \l__nmc_table_rstart_fp
+ \int_step_inline:nnnn { 1 } { 1 } { \l__nmc_table_rnum_int }
+ {
+ \seq_put_right:NV \l__nmc_table_rvraw_seq \l__nmc_table_r_fp
+ \fp_add:Nn \l__nmc_table_r_fp { \l__nmc_table_rstep_fp }
+ }
+ }
+\cs_new:Npn \__nmc_table_calc_cvar_cols:
+ {
+ \fp_set_eq:NN \l__nmc_table_c_fp \l__nmc_table_cstart_fp
+ \int_step_inline:nn { \l__nmc_table_cnum_int }
+ {
+ \bool_if:NTF \l__nmc_table_clist_bool
+ { \clist_pop:NN \l__nmc_formula_clist \l__nmc_formula_tl }
+ { \seq_put_right:NV \l__nmc_table_header_seq { \l__nmc_table_c_fp } }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_clear:N \l__nmc_table_this_col_seq
+ \fp_set_eq:NN \l__nmc_table_r_fp \l__nmc_table_rstart_fp
+ \int_step_function:nN { \l__nmc_table_rnum_int }
+ \__nmc_table_rvar_step_fn:n
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_put_right:NV \l__nmc_table_body_cols_seq
+ \l__nmc_table_this_col_seq
+ \fp_add:Nn \l__nmc_table_c_fp \l__nmc_table_cstep_fp
+ \__nmc_table_prop_mode:n {\l__nmc_mode_int}
+ \l__nmc_table_rvar_tl \l__nmc_table_rstart_fp
+ \__nmc_calc_fn_val:VNnN \l__nmc_table_cvar_tl \l__nmc_formula_tl
+ { \l__nmc_table_c_fp } \l__nmc_table_fn_val_fp
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_prop_mode:n #1
+ {
+ \int_if_zero:nTF { #1 }
+ { \prop_put:NVV \g__nmc_subst_var_prop }
+ { \prop_put:NVV \l__nmc_vv_change_prop }
+ }
+\cs_new_protected:Npn \__nmc_table_rvar_step_fn:n #1
+ {
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_table_rvar_tl
+ \l__nmc_formula_tl { \l__nmc_table_r_fp } \l_tmpa_fp
+ \seq_put_right:NV \l__nmc_table_this_col_seq \l_tmpa_fp
+ \bool_if:NT \l__nmc_table_QA_bool
+ { \__nmc_table_maxmin_do:n { \l_tmpa_fp } }
+ \fp_add:Nn \l__nmc_table_r_fp { \l__nmc_table_rstep_fp }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_maxmin_do:n #1
+ {
+ \bool_if:NT \l__nmc_table_QA_max_bool
+ { \fp_set:Nn \l__nmc_table_max_fp
+ { \fp_max:nn { \l__nmc_table_max_fp } { #1} } }
+ \bool_if:NT \l__nmc_table_QA_min_bool
+ { \fp_set:Nn \l__nmc_table_min_fp
+ { \fp_min:nn { \l__nmc_table_min_fp } { #1 } } }
+ }
+%----------------------------------------------------------
+\cs_new_protected:Npn \__nmc_table_display:
+ {
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \__nmc_table_headercnum:
+ \tl_set:Nx \l__nmc_table_header_tl { \__nmc_table_header: }
+ \bool_if:NT \l__nmc_table_foot_bool
+ {
+ \seq_set_split:NnV \l_tmpa_seq { & }\l__nmc_table_header_tl
+ \seq_reverse:N \l_tmpa_seq
+ \tl_set:Nx \l__nmc_table_footer_tl { \seq_use:Nn \l_tmpa_seq { & } }
+ }
+ \tl_put_left:Nx \l_nmc_result_tl
+ {
+ \__nmc_table_upper:
+ \exp_not:o \l__nmc_table_header_tl
+ \tabularnewline
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 5 }
+ { \exp_not:N \midrule }
+ }
+ \tl_put_right:Nx \l_nmc_result_tl { \__nmc_table_form_lower: }
+ \__nmc_table_reuse:
+ }
+ \l_nmc_result_tl
+ }
+\cs_new:Npn \__nmc_table_upper:
+ {
+ \exp_not:n { \begin{tabular} }
+ \exp_not:o \l__nmc_table_valign_tl
+ {
+ \int_if_odd:nT { \l__nmc_table_rpos_int }
+ { \l__nmc_table_rcalign_tl }
+ \prg_replicate:nn
+ { \l__nmc_table_cnum_int } \l__nmc_table_ccalign_tl
+ \int_compare:nNnT { \l__nmc_table_rpos_int } > { 1 }
+ { \l__nmc_table_rcalign_tl }
+ }
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 2 }
+ { \exp_not:N \toprule }
+ \bool_if:NT \l__nmc_table_ctitle_bool
+ { \__nmc_table_form_ctitle: }
+ }
+\cs_new:Npn \__nmc_table_form_ctitle:
+ {
+ \int_compare:nNnTF { \l__nmc_table_rpos_int } = { 0 }
+ {
+ \int_compare:nNnF { \l__nmc_table_cnum_int } = { 1 }
+ {
+ \__nmc_table_form_ctitle:nnnn {} {} { 1 }
+ { \l__nmc_table_cnum_int }
+ }
+ }
+ {
+ \int_compare:nNnT { \l__nmc_table_cnum_int } > { 1 }
+ {
+ \int_case:nnF { \l__nmc_table_rpos_int }
+ {
+ { 1 } { \__nmc_table_form_ctitle:nnnn { & } {} { 2 }
+ { \l__nmc_table_cnum_int + 1 } }
+ { 2 } { \__nmc_table_form_ctitle:nnnn {} { & } { 1 }
+ { \l__nmc_table_cnum_int } }
+ }
+ { % 3,4/5
+ \__nmc_table_form_ctitle:nnnn { & } { & } { 2 }
+ { \l__nmc_table_cnum_int + 1 }
+ }
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_form_ctitle:nnnn #1#2#3#4
+ {
+ #1 \exp_not:N \multicolumn
+ { \int_eval:n { \l__nmc_table_cnum_int } } { c }
+ { $ \l__nmc_table_ctitle_tl $ }
+ #2 \tabularnewline
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 3 }
+ {
+ \exp_not:N \cmidrule(lr)
+ { #3-\int_eval:n { #4 } }
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%
+\cs_new:Npn \__nmc_table_header:
+ {
+ \int_if_zero:nTF { \l__nmc_table_cnum_int }
+ {
+ \__nmc_table_header_rvar:nn { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ }
+ {
+ \int_case:nnF { \l__nmc_table_rpos_int }
+ {
+ { 0 } { \exp_not:o \l__nmc_table_chead_tl }
+ { 1 }
+ {
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ & \exp_not:o \l__nmc_table_chead_tl
+ }
+ { 2 }
+ {
+ \exp_not:o \l__nmc_table_chead_tl &
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ }
+ }
+ { % 3,4/5
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rhead_tl }
+ \l__nmc_table_rhnudge_tl
+ & \exp_not:o \l__nmc_table_chead_tl &
+ \__nmc_table_header_rvar:nn
+ { \exp_not:o \l__nmc_table_rheadi_tl }
+ \l__nmc_table_rhnudgei_tl
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_headercnum:
+ {
+ \int_compare:nNnTF { \l__nmc_table_cnum_int } = { 1 }
+ { \__nmc_table_headeri: }
+ { \__nmc_table_headern: }
+ }
+\cs_new:Npn \__nmc_table_header_rvar:nn #1#2
+ { \__nmc_table_header_nudge:Nnn r { #1 } { #2 } }
+
+\cs_new:Npn \__nmc_table_headeri:
+ {
+ \tl_if_empty:NT \l__nmc_table_chead_tl
+ {
+ \tl_set:Nx \l__nmc_table_chead_tl
+ {
+ \__nmc_table_header_nudge:Nnn c
+ {
+ \bool_if:NTF \l__nmc_table_ctitle_bool
+ { \exp_not:o \l__nmc_table_ctitle_tl }
+ { \exp_not:o \l__nmc_formula_dup_tl }
+ }
+ \l__nmc_table_chnudge_tl
+ }
+ }
+ }
+\cs_new:Npn \__nmc_table_headern:
+ {
+ \tl_if_empty:NT \l__nmc_table_chead_tl
+ {
+ \seq_clear:N \l_tmpa_seq
+ \bool_set_true:N \l__nmc_table_coli_only_bool
+ \bool_if:NTF \l__nmc_table_clist_bool
+ {
+ \seq_map_inline:Nn \l__nmc_table_header_seq
+ { \__nmc_table_header_auxi:Nn \l_tmpa_seq { ##1 } }
+ }
+ {
+ \seq_map_inline:Nn \l__nmc_table_header_seq
+ {
+ \__nmc_num_format:nNnN { ##1 } \l_tmpb_tl
+ { \l__nmc_table_chround_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_chstyler:NN \l_tmpa_seq \l_tmpb_tl
+ }
+ }
+ \tl_set:Nx \l__nmc_table_chead_tl { \seq_use:Nn \l_tmpa_seq { & } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_chstyler:NN #1 #2
+ {
+ \int_case:nnF { \l__nmc_table_chstyle_int }
+ {
+ { 0 } { \__nmc_table_header_auxi:NV #1 #2 }
+ { 1 }
+ {
+ \bool_if:NTF \l__nmc_table_coli_only_bool
+ {
+ \__nmc_table_header_auxi:Nx #1 { \l__nmc_table_cvar_tl = #2 }
+ \bool_set_false:N \l__nmc_table_coli_only_bool
+ }
+ { \seq_put_right:NV #1 #2 }
+ }
+ { 2 } { \__nmc_table_header_auxi:Nx #1 { \l__nmc_table_cvar_tl = #2 } }
+ { 3 } { \__nmc_table_header_auxiii:NV #1 #2 }
+ { 4 } { \seq_set_split:NnV #1 { & } \l__nmc_table_chead_tl }
+ }
+ { \__nmc_table_header_auxi:NV #1 #2 }
+ }
+\cs_new_protected:Npn \__nmc_table_header_auxi:Nn #1#2
+ {
+ \seq_put_right:Nx #1 { \__nmc_table_header_nudge:Nnn c
+ { \exp_not:n { #2 } } \l__nmc_table_chnudge_tl }
+ }
+\cs_generate_variant:Nn \__nmc_table_header_auxi:Nn { NV,Nx }
+
+\cs_new_protected:Npn \__nmc_table_header_auxiii:Nn #1#2
+ {
+ \tl_set_eq:NN \l_tmpa_tl \l__nmc_formula_dup_tl
+ \regex_replace_all:nnNT { \u{l__nmc_table_cvar_tl} } { #2 } \l_tmpa_tl
+ { \__nmc_table_header_auxi:NV #1 \l_tmpa_tl }
+ }
+\cs_generate_variant:Nn \__nmc_table_header_auxiii:Nn { NV }
+
+\cs_new:Npn \__nmc_table_header_nudge:Nnn #1#2#3
+ { % #1=r(ow)/c(ol), #2=var, #3=nudge
+ \tl_if_eq:cNTF { l__nmc_table_ #1 calign_tl } \c__nmc_r_tl
+ { $ #2 #3 $ }
+ {
+ \tl_if_eq:cNTF { l__nmc_table_ #1 calign_tl } \c__nmc_l_tl
+ { $ #3 #2 $ } { $ #2 $ }
+ }
+ }
+%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_table_form_body:N #1
+ { % #1 \l_nmc_result_tl
+ \int_if_zero:nF { \l__nmc_table_rpos_int }
+ {
+ \__nmc_table_form_rvcol:NN
+ \l__nmc_table_rvraw_seq \l__nmc_table_rvcol_seq
+ }
+ \int_zero:N \l__nmc_table_rbloc_int
+ \int_if_zero:nTF { \l__nmc_table_cnum_int }
+ { \__nmc_table_form_body_rvcol_only:N #1 }
+ { \__nmc_table_form_body_cvcols:N #1 }
+ }
+\cs_new_protected:Npn \__nmc_table_form_rvcol:NN #1#2
+ {
+ \seq_clear:N \l_tmpb_seq
+ \seq_map_variable:NNn #1 \l_tmpa_tl
+ {
+ \__nmc_num_format:nNnN \l_tmpa_tl \l_tmpb_tl
+ { \l__nmc_table_rround_int } \c_false_bool
+ \seq_put_right:Nx \l_tmpb_seq
+ { \exp_not:o \l__nmc_table_rfont_tl { \l_tmpb_tl } }
+ }
+ \seq_set_eq:NN #2 \l_tmpb_seq
+ \int_compare:nNnTF { \l__nmc_table_rpos_int } > { 3 }
+ { \__nmc_table_form_rvcoli:NN #1 #2 } % reverse/pivot
+ { \seq_set_eq:NN \l__nmc_table_rvcoli_seq #2 }
+ }
+\cs_new_protected:Npn \__nmc_table_form_rvcoli:NN #1 #2
+ {
+ \seq_clear:N \l__nmc_table_rvcoli_seq
+ \seq_map_variable:NNn #1 \l_tmpa_tl
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_table_rvar_tl
+ \l__nmc_table_rvari_tl \l_tmpa_tl \l_tmpa_fp
+ \__nmc_num_format:nNnN { \l_tmpa_fp } \l_tmpb_tl
+ { \l__nmc_table_rround_int } \c_false_bool
+ \seq_put_right:Nx \l__nmc_table_rvcoli_seq
+ { \exp_not:o \l__nmc_table_rfont_tl { \l_tmpb_tl } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_form_body_rvcol_only:N #1
+ { % #1 = \l_nmc_result_tl
+ \seq_clear:N \l_tmpb_seq
+ \seq_map_variable:NNn \l__nmc_table_rvcol_seq \l_tmpa_tl
+ {
+ \tl_put_right:Nx #1
+ { \exp_not:o \l_tmpa_tl \tabularnewline }
+ \__nmc_table_rbloc_spc:NN \l__nmc_table_rbloc_int \l_tmpa_bool
+ \bool_if:NT \l_tmpa_bool
+ {
+ \tl_put_right:Nx #1
+ { \exp_not:o \addlinespace[\l__nmc_table_blocsep_tl] }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_form_body_cvcols:N #1
+ { % #1 \l_nmc_result_tl
+ \seq_clear:N \l_tmpd_seq
+ \seq_map_variable:NNn \l__nmc_table_body_cols_seq \l_tmpa_seq
+ {
+ \seq_clear:N \l_tmpb_seq
+ \__nmc_table_form_cols:NN \l_tmpa_seq \l_tmpb_seq
+ \seq_put_right:NV \l_tmpd_seq \l_tmpb_seq
+ }
+ \bool_if:NT \l__nmc_table_footops_bool
+ { \__nmc_table_footops:N \l_tmpd_seq }
+ \__nmc_table_transpose:NN \l_tmpd_seq \l__nmc_table_body_rows_seq
+
+ \seq_map_variable:NNn \l__nmc_table_body_rows_seq \l_tmpa_seq
+ {
+ \__nmc_table_rbloc_spc:NN \l__nmc_table_rbloc_int \l_tmpa_bool
+ \tl_put_right:Nx #1
+ {
+ \seq_use:Nn \l_tmpa_seq { & }
+ \tabularnewline
+ \bool_if:NT \l_tmpa_bool
+ { \exp_not:o \addlinespace[\l__nmc_table_blocsep_tl] }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_footops:N #1
+ {
+ \group_begin:
+ \seq_clear:N \l_tmpb_seq
+ \seq_set_split:NnV \l_tmpa_seq { & } \l__nmc_table_footer_tl
+ \int_if_odd:nT { \l__nmc_table_rpos_int }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_put_left:NV \l_tmpb_seq \l_tmpa_tl
+ }
+ \seq_mapthread_function:NNN #1 \l_tmpa_seq \__nmc_table_footops:nn
+ \int_compare:nNnT { \l__nmc_table_rpos_int } > { 1 }
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_put_right:NV \l_tmpb_seq \l_tmpa_tl
+ }
+ \tl_set:Nx \l__nmc_table_footer_tl { \seq_use:Nn \l_tmpb_seq { & } }
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_table_footer_tl \l__nmc_table_footer_tl
+ }
+\cs_new_protected:Npn \__nmc_table_footops:nn #1#2
+ {
+ \tl_set:Nn \l_tmpc_seq { #1 }
+ \tl_set:Nx \l_tmpa_tl { \seq_use:Nn \l_tmpc_seq { + } }
+ \fp_set:Nn \l_tmpa_fp { \l_tmpa_tl }
+ \fp_set:Nn \l_tmpb_fp { \l_tmpa_fp / \l__nmc_table_rnum_int }
+ \__nmc_num_format:nNnN \l_tmpa_fp \l__nmc_table_csum_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_num_format:nNnN \l_tmpb_fp \l__nmc_table_cave_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_set:Nx \l_tmpa_tl { max(\seq_use:Nn \l_tmpc_seq { , }) }
+ \fp_set:Nn \l_tmpa_fp { \l_tmpa_tl }
+ \__nmc_num_format:nNnN \l_tmpa_fp \l__nmc_table_cmax_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \tl_set:Nx \l_tmpa_tl { min(\seq_use:Nn \l_tmpc_seq { , }) }
+ \fp_set:Nn \l_tmpb_fp { \l_tmpa_tl }
+ \__nmc_num_format:nNnN \l_tmpb_fp \l__nmc_table_cmin_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_num_format:nNnN { \l_tmpa_fp - \l_tmpb_fp } \l__nmc_table_cdel_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \seq_put_right:Nx \l_tmpb_seq { $ #2 $ }
+ }
+\cs_new_protected:Npn \__nmc_table_transpose:NN #1#2
+ { % #1 seq of seq in #2 seq of seq out
+ \int_if_zero:nF { \l__nmc_table_rpos_int }
+ {
+ \int_compare:nNnF { \l__nmc_table_rpos_int } = { 2 }
+ { \seq_put_left:NV #1 \l__nmc_table_rvcol_seq }
+ \int_compare:nNnT { \l__nmc_table_rpos_int } > { 1 }
+ { \seq_put_right:NV #1 \l__nmc_table_rvcoli_seq }
+ }
+ \int_step_inline:nn { \l__nmc_table_rnum_int }
+ {
+ \seq_clear:N \l_tmpb_seq
+ \seq_clear:N \l_tmpc_seq
+ \seq_map_variable:NNn #1 \l_tmpa_seq
+ {
+ \seq_pop:NN \l_tmpa_seq \l_tmpa_tl
+ \seq_put_right:Nx \l_tmpb_seq { $\exp_not:o\l_tmpa_tl $ }
+ \seq_put_right:NV \l_tmpc_seq \l_tmpa_seq
+ }
+ \seq_put_right:NV \l__nmc_table_body_rows_seq \l_tmpb_seq
+ \seq_set_eq:NN #1 \l_tmpc_seq
+ }
+ }
+\cs_new:Npn \__nmc_table_rbloc_spc:NN #1#2
+ { %
+ \int_incr:N #1
+ \bool_set_false:N #2
+ \int_compare:nNnT { #1 } = { \l__nmc_table_rblock_tl }
+ {
+ \int_compare:nNnF { #1 } = { \l__nmc_table_rnum_int }
+ { \bool_set_true:N #2 }
+ \clist_if_empty:NF \l__nmc_table_rbloc_clist
+ { \clist_pop:NN \l__nmc_table_rbloc_clist \l__nmc_table_rbloc_tl }
+ \tl_set:Nx \l__nmc_table_rblock_tl
+ { \int_eval:n { \l__nmc_table_rblock_tl + \l__nmc_table_rbloc_tl } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols:NN #1#2
+ { % #1 raw cseq, #2 <--formatted cseq
+ \int_case:nnF { \l__nmc_table_rnum_int }
+ {
+ { 0 } { \prg_do_nothing: }
+ { 1 } { \__nmc_table_form_cols_ri:NN #1#2 }
+ { 2 } { \__nmc_table_form_cols_rii:NN #1#2 }
+ }
+ { \__nmc_table_form_cols_riii:NN #1#2 }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_ri:NN #1#2
+ {
+ \seq_pop:NN #1 \l_tmpb_fp
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_table_bool
+ \bool_if:NT \l__nmc_table_QA_bool
+ { \__nmc_table_QA:N \l_tmpb_tl }
+ \seq_put_right:Nx #2 { \exp_not:o \l_tmpb_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_rii:NN #1#2
+ {
+ \__nmc_table_form_cols_rii_aux:NN #1#2
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_rii_aux:NN #1#2
+ {
+ \seq_pop:NN #1 \l_tmpb_fp
+ \int_if_zero:nF \l__nmc_table_diffs_int
+ { \__nmc_table_form_diffs_null:n { \l__nmc_table_diffs_int } }
+ \seq_pop:NN #1 \l_tmpa_fp
+ \int_set:Nn \l__nmc_table_c_int { 1 }
+ \int_set:Nn \l__nmc_table_b_int { \fp_sign:n { \l_tmpb_fp } }
+ \int_set:Nn \l__nmc_table_a_int { \fp_sign:n { \l_tmpa_fp } }
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ \int_if_zero:nF \l__nmc_table_diffs_int
+ { \__nmc_table_calc_diffs:nn
+ { \l_tmpb_fp } { \l_tmpa_fp } }
+ \fp_set_eq:NN \l_tmpb_fp \l_tmpa_fp
+ }
+\cs_new_protected:Npn \__nmc_table_form_cols_riii:NN #1#2
+ {
+ \__nmc_table_form_cols_rii_aux:NN #1#2
+ \int_set_eq:NN \l__nmc_table_c_int \l__nmc_table_b_int
+ \int_set_eq:NN \l__nmc_table_b_int \l__nmc_table_a_int
+ \seq_map_variable:NNn #1 \l_tmpa_fp
+ {
+ \int_set:Nn \l__nmc_table_a_int { \fp_sign:n { \l_tmpa_fp } }
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ \int_if_zero:nF \l__nmc_table_diffs_int
+ { \__nmc_table_calc_diffs:nn { \l_tmpb_fp } { \l_tmpa_fp } }
+ \fp_set_eq:NN \l_tmpb_fp \l_tmpa_fp
+ \int_set_eq:NN \l__nmc_table_c_int \l__nmc_table_b_int
+ \int_set_eq:NN \l__nmc_table_b_int \l__nmc_table_a_int
+ }
+ \int_set:Nn \l__nmc_table_a_int { 1 }
+ \__nmc_table_signs:
+ \__nmc_num_format:nNnN \l_tmpb_fp \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \__nmc_table_num_format:N \l_tmpb_tl
+ \seq_put_right:Nx #2
+ { \exp_not:o \l_tmpb_tl \exp_not:o \l__nmc_table_diff_tl }
+ }
+\cs_new_protected:Npn \__nmc_table_num_format:N #1
+ {
+ \bool_if:NTF \l__nmc_sci_num_table_bool
+ {
+ \tl_if_head_eq_charcode:VNTF #1 (
+ { % for nums in [1,10)
+ \exp_last_unbraced:NV
+ \__nmc_table_t_format:wN #1\q_stop \l_tmpa_tl
+ \tl_set_eq:NN #1 \l_tmpa_tl
+ }
+ { \tl_set:Nx #1 { \__nmc_sci_num_table:n { #1 } } }
+ }
+ { \tl_set:Nx #1 { \__nmc_sci_num_table:n { #1 } } }
+ \bool_if:NT \l__nmc_table_QA_bool
+ { \__nmc_table_QA:N #1 }
+ }
+\cs_new_protected:Npn \__nmc_table_QA:N #1
+ {
+ \fp_compare:nNnT { \l__nmc_table_Q_tl } = { 1 }
+ {
+ \bool_lazy_and:nnT
+ { \l__nmc_num_only_bool } { !\l__nmc_table_num_only_bool }
+ {
+ \tl_set_eq:NN \l__nmc_table_num_only_tl \l_tmpb_tl
+ \bool_set_true:N \l__nmc_table_num_only_bool
+ }
+ \tl_set:Nx #1 { \l__nmc_table_A_tl }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_t_format:wN (#1)#2#3\q_stop #4
+ {
+ \tl_set:Nn \l__nmc_toss_tl { #1 }
+ \__nmc_table_prepad:Nnn \l__nmc_toss_tl
+ { \l__nmc_table_tsigns_int } { \exp_not:o \hphantom{-} }
+ \tl_set:Nx #4 { \__nmc_sci_num_table_plus:nn
+ { #3 } { \exp_not:o\l__nmc_toss_tl } }
+ }
+\cs_new_protected:Npn \__nmc_table_signs:
+ { % A prev, B current, C next
+ \int_compare:nNnTF \l__nmc_table_b_int > { -1 }
+ {
+ \bool_lazy_or:nnTF
+ { \int_compare_p:nNn \l__nmc_table_a_int = { -1 } }
+ { \int_compare_p:nNn \l__nmc_table_c_int = { -1 } }
+ { \bool_set_true:N \l__nmc_table_signs_bool }
+ { \bool_set_false:N \l__nmc_table_signs_bool }
+ }
+ { \bool_set_false:N \l__nmc_table_signs_bool }
+ }
+%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_table_calc_diffs:nn #1#2
+ {
+ \__nmc_num_format:nNnN
+ { \fp_abs:n {round( #1,\l__nmc_round_int )
+ - round( #2, \l__nmc_round_int) } } \l_tmpb_tl
+ { \l__nmc_round_int } \c_false_bool
+ \tl_set:Nx \l_tmpb_tl { \fp_eval:n { round( 10^{ \l__nmc_round_int }
+ * \l_tmpb_tl, \l__nmc_round_int + 1) } }
+ \__nmc_table_prepad:Nnn \l_tmpb_tl { \l__nmc_table_diffs_int } { 0 }
+ \tl_set:Nx \l__nmc_table_diff_tl { ^{\,^{ \l_tmpb_tl } } }
+ }
+\cs_new_protected:Npn \__nmc_table_form_diffs_null:n #1
+ {
+ \tl_set:Nn \l_tmpb_tl { 0 }
+ \__nmc_table_prepad:Nnn \l_tmpb_tl { #1 } { 0 }
+ \tl_set:Nx \l__nmc_table_diff_tl
+ { ^{\,^{ \exp_not:o \hphantom{ \l_tmpb_tl } } } }
+ }
+%%%%%%%%%%%%%%%%%%%
+\cs_new:Npn \__nmc_table_form_lower:
+ {
+ \tl_if_empty:NF \l__nmc_table_footer_tl
+ {
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 7 }
+ { \exp_not:N \midrule[\cmidrulewidth] }
+ \exp_not:o \l__nmc_table_footer_tl \tabularnewline
+ }
+ \__nmc_if_mod_zero:nnT { \l__nmc_table_rules_int } { 11 }
+ { \exp_not:N \bottomrule }
+ \exp_not:n { \end{tabular} }
+ }
+%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_table_reuse:
+ {
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \int_case:nnF { \int_sign:n { \l__nmc_table_reuse_int } }
+ {
+ { -1 } { \__nmc_table_reuse_row:n { \l__nmc_table_reuse_int } }
+
+ { 1 } { \__nmc_table_reuse_col:n { \l__nmc_table_reuse_int } }
+ }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_reuse_row:n #1
+ {
+ \tl_set:Nx \l_tmpa_seq
+ { \seq_item:Nn \l__nmc_table_body_rows_seq { -#1 } }
+ \tl_gset:Nx \g__nmc_reuse_tl { \seq_use:Nn \l_tmpa_seq { , } }
+ }
+\cs_new_protected:Npn \__nmc_table_reuse_col:n #1
+ {
+ \int_zero:N \l_tmpa_int
+ \seq_map_variable:NNn \l__nmc_table_body_cols_seq \l_tmpa_seq
+ {
+ \int_incr:N \l_tmpa_int
+ \int_compare:nNnT { \l_tmpa_int } = { #1 }
+ {
+ \seq_mapthread_function:NNN \l__nmc_table_rvraw_seq
+ \l_tmpa_seq \__nmc_table_reuse:nn
+ \seq_map_break:
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_table_reuse:nn #1#2
+ {
+ \__nmc_num_format:nNnN { #1 } \l_tmpa_tl
+ { \l__nmc_table_rround_int } \c_false_bool
+ \__nmc_num_format:nNnN { #2 } \l_tmpb_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool
+ \int_if_zero:nTF \l__nmc_table_rpos_int
+ { \seq_put_right:NV \l_nmc_table_reuse_seq \l_tmpb_tl }
+ {
+ \seq_put_right:Nx \l_nmc_table_reuse_seq
+ { { \l_tmpa_tl , \l_tmpb_tl } }
+ }
+ \tl_gset:Nx \g__nmc_reuse_tl { \seq_use:Nn \l_nmc_table_reuse_seq {,} }
+ }
+%------------------------------------------------
+\keys_define:nn { numerica }
+ {
+ table-rvar-rounding .int_set:N = \l__nmc_table_rround_int,
+ table-rvar-col-pos .int_set:N = \l__nmc_table_rpos_int,
+ table-rvar-align .tl_set:N = \l__nmc_table_rcalign_tl,
+ table-cvar-align .tl_set:N = \l__nmc_table_ccalign_tl,
+ table-block-sep .tl_set:N = \l__nmc_table_blocsep_tl,
+ table-rules .tl_set:N = \l__nmc_table_rules_tl,
+ table-form-diffs .int_set:N = \l__nmc_table_diffs_int,
+ table-reuse .int_set:N = \l__nmc_table_reuse_int,
+ table-rvar-rounding .default:n = 1,
+ table-rvar-align .default:n = r,
+ table-rvar-col-pos .default:n = 1,
+ table-cvar-align .default:n = r,
+ table-block-sep .default:n = 1ex,
+ table-rules .default:n = ThB,
+ table-form-diffs .default:n = 0,
+ table-reuse .default:n = 0
+ }
+\keys_set_known:nn { numerica }
+ {
+ table-rvar-rounding,table-rvar-align,table-rvar-col-pos,
+ table-cvar-align,table-block-sep,table-rules,table-fontsize,
+ table-form-diffs,table-reuse
+ }
+\keys_define:nn { numerica/table }
+ {
+ rvar .tl_set:N = \l__nmc_table_rvar_tl,
+ rstop .tl_set:N = \l__nmc_table_rstop_tl,
+ rstep .tl_set:N = \l__nmc_table_rstep_tl,
+ rows .int_set:N = \l__nmc_table_rnum_int,
+ rows .initial:n = -1,
+ rspec .clist_set:N = \l__nmc_table_rspec_clist,
+ rround .int_set:N = \l__nmc_table_rround_int,
+ rfont .code:n = \tl_set:Nn \l__nmc_table_rfont_tl
+ { \use:c { math#1 } },
+ ralign .tl_set:N = \l__nmc_table_rcalign_tl,
+ rhead .tl_set:N = \l__nmc_table_rhead_tl,
+ rhnudge .code:n = \tl_set:Nn \l__nmc_table_rhnudge_tl
+ { \mkern #1 mu },
+ rpos .int_set:N = \l__nmc_table_rpos_int,
+ rvar' .tl_set:N = \l__nmc_table_rvari_tl,
+ rhead' .tl_set:N = \l__nmc_table_rheadi_tl,
+ rhnudge' .code:n = \tl_set:Nn \l__nmc_table_rhnudgei_tl
+ { \mkern #1 mu },
+%
+ cvar .tl_set:N = \l__nmc_table_cvar_tl,
+ cstop .tl_set:N = \l__nmc_table_cstop_tl,
+ cstep .tl_set:N = \l__nmc_table_cstep_tl,
+ cols .int_set:N = \l__nmc_table_cnum_int,
+ cols .initial:n = -1,
+ cspec .clist_set:N = \l__nmc_table_cspec_clist,
+ calign .tl_set:N = \l__nmc_table_ccalign_tl,
+ chstyle .int_set:N = \l__nmc_table_chstyle_int,
+ chnudge .code:n = \tl_set:Nn \l__nmc_table_chnudge_tl
+ { \mkern #1 mu },
+ chead .tl_set:N = \l__nmc_table_chead_tl,
+ chround .int_set:N = \l__nmc_table_chround_int,
+%
+ ctitle .tl_set:N = \l__nmc_table_ctitle_tl,
+ rules .tl_set:N = \l__nmc_table_rules_tl,
+ foot .tl_set:N = \l__nmc_table_footer_tl,
+ rbloc .clist_set:N = \l__nmc_table_rbloc_clist,
+ rblocsep .tl_set:N = \l__nmc_table_blocsep_tl,
+ signs .int_set:N = \l__nmc_table_signs_int,
+ (pad) .int_set:N = \l__nmc_table_tsigns_int,
+ diffs .int_set:N = \l__nmc_table_diffs_int,
+%
+ Q? .tl_set:N = \l__nmc_table_Q_tl,
+ A! .tl_set:N = \l__nmc_table_A_tl,
+ reuse .int_set:N = \l__nmc_table_reuse_int,
+ valign .code:n = \tl_set:Nn \l__nmc_table_valign_tl { [#1] },
+ }
+% end of `numerica-tables.def' \ No newline at end of file
diff --git a/macros/latex/contrib/numerica/numerica-tables.pdf b/macros/latex/contrib/numerica/numerica-tables.pdf
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diff --git a/macros/latex/contrib/numerica/numerica-tables.tex b/macros/latex/contrib/numerica/numerica-tables.tex
new file mode 100644
index 0000000000..b16f90e3b7
--- /dev/null
+++ b/macros/latex/contrib/numerica/numerica-tables.tex
@@ -0,0 +1,1801 @@
+%% LyX 2.4.0-alpha1 created this file. For more info, see https://www.lyx.org/.
+%% Do not edit unless you really know what you are doing.
+\documentclass[english,tableposition=top]{report}
+\usepackage{lmodern}
+\renewcommand{\sfdefault}{lmss}
+\renewcommand{\ttdefault}{lmtt}
+\usepackage[T1]{fontenc}
+\usepackage{textcomp}
+\usepackage[latin9]{inputenc}
+\setcounter{secnumdepth}{3}
+\usepackage{color}
+\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
+\usepackage{babel}
+\usepackage{array}
+\usepackage{float}
+\usepackage{booktabs}
+\usepackage{framed}
+\usepackage{url}
+\usepackage{multirow}
+\usepackage{amsmath}
+\usepackage[unicode=true,pdfusetitle,
+ bookmarks=true,bookmarksnumbered=true,bookmarksopen=true,bookmarksopenlevel=2,
+ breaklinks=false,pdfborder={0 0 1},backref=section,colorlinks=true]
+ {hyperref}
+
+\makeatletter
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
+%% Because html converters don't know tabularnewline
+\providecommand{\tabularnewline}{\\}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
+\newenvironment{centred}%
+ {\begin{center}\baselineskip=13pt\parskip=1pt}{\end{center}}
+\newenvironment{lyxcode}
+ {\par\begin{list}{}{
+ \setlength{\rightmargin}{\leftmargin}
+ \setlength{\listparindent}{0pt}% needed for AMS classes
+ \raggedright
+ \setlength{\itemsep}{0pt}
+ \setlength{\parsep}{0pt}
+ \normalfont\ttfamily}%
+ \item[]}
+ {\end{list}}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
+\usepackage[tables]{numerica}
+
+\newcommand\rel{\,\varrho\;}
+\DeclareMathOperator{\erf}{erf}
+\DeclareMathOperator{\gd}{gd}
+
+\reuse
+
+\usepackage{upquote}
+
+\makeatother
+
+\begin{document}
+\title{\texttt{numerica-tables}~\\
+}
+\author{Andrew Parsloe\\
+(\url{ajparsloe@gmail.com})}
+\maketitle
+\begin{abstract}
+In this module of the \verb`numerica` package a command is defined
+which enables the creation of multi-column tables of function values
+in a wide variety of table styles. \\
+\\
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\paragraph*{Note:}
+\begin{itemize}
+\item {\normalsize This document applies to version 1.0.0 of }{\normalsize\texttt{numerica-tables.def}}{\normalsize .}{\small\par}
+\item {\normalsize Reasonably recent versions of the \LaTeX 3 bundles }{\normalsize\texttt{l3kernel}}{\normalsize{}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required.}{\small\par}
+\item {\normalsize The }{\normalsize\texttt{booktabs}}{\normalsize{} package
+is required.}{\small\par}
+\item {\normalsize I refer many times in this document to }{\normalsize\emph{Handbook
+of Mathematical Functions}}{\normalsize , edited by Milton Abramowitz
+and Irene A. Segun, Dover, 1965. This is abbreviated to }{\normalsize\emph{HMF}}{\normalsize ,
+often followed by a reference to a specific table like Table 1.2.}{\small\par}
+\end{itemize}
+\end{shaded}%
+\end{minipage}
+
+\tableofcontents{}
+\end{abstract}
+
+\chapter{Introduction}
+
+Calling \texttt{numerica} with the \texttt{tables} package option
+in the preamble,
+\begin{lyxcode}
+~\textbackslash usepackage{[}tables{]}\{numerica\}
+\end{lyxcode}
+\noindent gives access to a command \verb`\nmcTabulate` for creating
+tables of function values. This command is defined in the package
+\texttt{numerica-tables.def} which is loaded with \texttt{numerica.sty}
+when the \texttt{tables} option is used. Note that \verb`\nmcTabulate`
+uses the \texttt{booktabs} package for the construction of its tables.
+The \texttt{booktabs} package is loaded automatically (provided it
+is available in your \TeX{} system) when \texttt{numerica} is loaded
+with the \texttt{tables} option.
+
+\section{Shared syntax}
+
+The \verb`\nmcTabulate` command (short-name form \verb`\tabulate`)
+shares the syntax of \verb`\nmcEvaluate` (see \texttt{numerica-basics.pdf})
+and of \verb`\nmcIterate`, \verb`\nmcSolve` and \verb`\nmcRecur`
+(see \texttt{numerica-plus.pdf}). When all options are used the command
+looks like
+\begin{lyxcode}
+\noindent \textbackslash nmcTabulate{*}{[}settings{]}\{expr.\}{[}vv-list{]}{[}num.~format{]}
+\end{lyxcode}
+\begin{enumerate}
+\item \verb`*` optional switch; if present ensures a single number output
+with no formatting or an appropriate error message if the single number
+cannot be produced;
+\item \verb`[settings]` comma-separated list of \emph{key=value }settings;
+\item \verb`{expr.}` mandatory argument specifying the mathematical expression
+in \LaTeX{} form to be tabulated;
+\item \verb`[vv-list]` comma-separated list of \emph{variable=value }items;
+\item \verb`[num. format]` optional format specification for presentation
+of the numerical result (rounding, padding with zeros, scientific
+notation)
+\end{enumerate}
+Unlike \verb`\nmcEvaluate` and the other commands, for \verb`\nmcTabulate`
+\begin{itemize}
+\item it makes no difference to the display of the result whether the command
+wraps around math delimiters, is wrapped within math delimters, or
+if there are no math delimiters involved whatever;
+\item the two apparently optional arguments straddling the main argument
+(\verb`settings` and \verb`vv-list`) are \emph{essential}; although
+neither is mandatory in the \LaTeX{} sense, each contains items necessary
+for the construction of any table of function values.
+\end{itemize}
+
+\subsection{Inherited settings}
+
+\label{subsec:Inherited-settings}
+\begin{table}
+\noindent \centering{}\caption{\protect\label{tab:introSettingsInherited}Settings options inherited
+from \texttt{\textbackslash nmcEvaluate}}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{dbg}} & {\small int} & {\small debug `magic' integer} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{\textasciicircum}} & {\small char} & {\small exponent mark for sci. notation input} & {\small\texttt{e}}\tabularnewline
+{\small\texttt{xx}} & {\small int (}{\small\texttt{0}}{\small /}{\small\texttt{1}}{\small )} & {\small multi-token variable switch} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{()}} & {\small int (}{\small\texttt{0}}{\small /}{\small\texttt{1}}{\small /}{\small\texttt{2}}{\small )} & {\small trig. function arg. parsing} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{o}} & & {\small degree switch for trig. functions} & \tabularnewline
+{\small\texttt{log}} & {\small num} & {\small base of logarithms for }{\small{\small\verb`\log`}} & {\small\texttt{10}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvd}} & {\small tokens} & {\small vv-list display-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 12mu plus 6mu minus 9mu(vv)}}\tabularnewline
+{\small\texttt{vvi}} & {\small tokens} & {\small vv-list text-style spec.} & {\small\texttt{\{,\}\textbackslash mskip 36mu minus 24mu(vv)}}\tabularnewline
+{*} & & {\small suppress equation numbering if }{\small\texttt{\textbackslash\textbackslash}}{\small{}
+in }{\small\texttt{vvd}} & \tabularnewline
+{\small\texttt{S+}} & {\small int} & {\small extra rounding for stopping criterion for sums} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{S?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for sums} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{P+}} & {\small int} & {\small extra rounding for stopping criterion for products} & {\small\texttt{2}}\tabularnewline
+{\small\texttt{P?}} & {\small$\text{int}\ge0$} & {\small stopping criterion query terms for products} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+To create a table we need to specify a function to tabulate \textendash{}
+this fills the main (mandatory) argument. We also need to decide how
+the function values are to be displayed \textendash{} to how many
+decimal places and whether padded with zeros. These matters are determined
+by what is entered in the trailing optional argument, just as with
+\verb`\eval`, \verb`\iter`, \verb`\solve` and \verb`\recur`, although
+there are some addtional complications peculiar to tables which are
+more appropriately treated in the next chapter.
+
+Many of the settings available to the \verb`\eval` command are also
+available to \verb`\nmcTabulate`. To save switching between documents
+I reproduce the table of options found in \texttt{numerica-basics.pdf}
+(with only the punctuation \texttt{p} setting missing), although for
+discussion of the options you will need to refer to that document.
+But note that the \texttt{dbg} key is of limited usefulness for tables:
+\texttt{dbg=5} and \texttt{dbg=7} display floating point values in
+their `internal' format, and for \texttt{dbg=7}, no result is displayed.
+
+\chapter{\texttt{\textbackslash nmcTabulate}-specific settings}
+
+In addition to the shared settings, \verb`\nmcTabulate` has many
+settings specific to it. They are discussed in groups in subsequent
+sections, some in more than one place.
+
+\section{Row variable settings}
+
+\label{sec:Row-variable-settings}Deciding on a function to tabulate
+(entered in the main or mandatory argument of \verb`\nmcTablate`)
+will inevitably also mean deciding on the tabulation variable (the
+\emph{row} variable \verb`rvar`), and then what value to start tabulating
+from (specified in the vv-list), what value to tabulate to (\verb`rstop`),
+and how fine-grained the tabulation is to be, the step size (\verb`rstep`).
+
+\begin{table}[b]
+\centering{}\caption{Row variable specification}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline
+\midrule
+{\small\texttt{rvar}} & {\small token(s)} & {\small row variable} & \tabularnewline
+{\small\texttt{rstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{rstop}} & {\small real num.} & {\small stop value} & \multirow{2}{3cm}{either {\small\texttt{rstop}}{\small{} or }{\small\texttt{rows}}}\tabularnewline
+{\small\texttt{rows}} & {\small int} & {\small number of rows} & \tabularnewline
+{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small , }{\small\texttt{stop\}}}{\small{}
+or }{\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small ,
+}{\small\texttt{(rows)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+The two tables in the first example below tabulate $\sin x$ and $\cos x$
+between $0$ and $1$ in increments of $0.2$. Note the start value
+of the tabulation variable in the vv-list. The reason for this placement
+is that for more complicated functions other parameters in the function
+and therefore in the vv-list may depend on the row variable. Therefore
+it is always placed in the vv-list.
+
+The difference in appearance of the tables results from padding with
+zeros in the second (the asterisk in the trailing optional argument).
+As you can see, padding applies not only to the values of the function
+but also to the values of the row variable \textendash{} and makes
+an obvious improvement to the table's appearance.
+\begin{verbatim}
+ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \sin x }[x=0]\qquad
+ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \cos x }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \sin x }[x=0] \qquad
+ \tabulate[rvar=x,rstep=0.2,rstop=1]
+ { \cos x }[x=0][*]\medskip{}
+
+Sometimes (perhaps often) it may prove more convenient to specify
+the number of rows (\texttt{rows}) rather than a stop value. Only
+one of \texttt{rows} and \texttt{rstop} should be given, but if both
+(inadvertently) are present, it is the value of \texttt{rows} that
+prevails.
+
+The second and third tables in the next example use an abbreviated
+form of the row variable specification (\texttt{rspec}). This is a
+$3$-element comma list of the form \verb`{rvar,rstep,rstop}` or
+\verb`{rvar,rstep,(rows)}`. Parentheses around the third item signify
+that it is \verb`rows` rather than \verb`rstop` that is being specified.
+In the example below, the second table specifies \texttt{rstop} (value
+\texttt{1}), the third \texttt{rows} (value \texttt{6}).
+
+It is worth noting that \verb`rstep` and \verb`rstop` can be \LaTeX{}
+expressions \textendash{} for instance \texttt{rstop=\textbackslash sin
+\textbackslash pi/2} (just as the expression for the initial value
+in the vv-list can be). However \verb`rows` can only be a simple
+integer expression \textendash{} an integer or integers linked by
+some or all of \verb`+ - * / ( )`.
+\begin{verbatim}
+ \tabulate[rvar=x,rstep=0.2,rows=6]
+ { \sin x/\cos x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rvar=x,rstep=0.2,rows=6]
+ { \sin x/\cos x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ \tabulate[rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]
+
+\subsection{Row-variable column formatting}
+
+\label{subsec:Row-var-col-formatting}Various settings are available
+to format the row variable column in addition to the padding option
+({*}) of the trailing optional argument.
+\begin{table}[H]
+\centering{}\caption{Row-variable column formatting}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline
+{\small\texttt{rhead}} & {\small tokens} & {\small header} & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhnudge}} & int & {\small nudge header }{\small{\small\verb`<int>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small column position(s) } & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rvar'}} & {\small tokens} & {\small 2nd row variable col. spec. } & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhead'}} & {\small tokens} & {\small header of 2nd rv col. (if it exists)} & {\small\texttt{rvar'}}\tabularnewline
+{\small\texttt{rhnudge'}} & int & {\small nudge 2nd rv col. header }{\small{\small\verb`<int>`}}{\small{}
+mu} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+
+\subsubsection{Rounding: \texttt{rround}}
+
+After studying the previous tables, we might decide to adjust the
+step size, say from $0.2$ to $0.25$. But changing \texttt{rstep}
+to the new value gives a disconcerting and \emph{prima facie} false
+result (the first table below). \texttt{numerica} uses a default rounding
+value of $1$ for the row variable and has rounded $0.25$ down to
+$0.2$ and $0.75$ up to $0.8$ accordingly. The second table corrects
+matters by adjusting the row variable rounding (\texttt{rround}) to
+\texttt{2}.
+\begin{verbatim}
+ \tabulate[rvar=x,rstep=0.25,rstop=1]
+ { \sin x }[x=0][*]\qquad
+ \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2]
+ { \sin x }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rvar=x,rstep=0.25,rstop=1]
+ { \sin x }[x=0][*] \qquad
+ \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2]
+ { \sin x }[x=0][*]
+
+\subsubsection{Alignment: \texttt{ralign}}
+
+By default, the alignment of all columns is to the right, as in the
+previous examples. This lends itself to neat output when padding with
+zeros is activated (the \verb`*` in the trailing argument) and when
+some values are negative \textendash{} the minus signs can interfere
+with neat output in left or centred alignments. But in a case like
+the second table in the last example, you might prefer to centre the
+headers for both the row and function value columns. These alignments
+are independently set. For the row variable column the default alignment
+is to the right \texttt{ralign=r}; \texttt{ralign=l} (lowercase L)
+aligns entries in the row variable column to the left, and \texttt{ralign=c}
+centres entries in the row variable column. The tables of the next
+example use a \texttt{c} alignment to centre the row variable column
+header. The third of those tables shows how minus signs spoil the
+effect.
+
+\subsubsection{Font: \texttt{rfont}}
+
+In the second table bolding (\texttt{rfont=bf}) has been applied to
+emphasize the distinction between the row variable values and the
+function values. Possible values for this key are those characters
+that can be adjoined to \verb`\math` to give a meaningful result.
+Thus other valid values are \verb`it` (italic), \verb`sf` (sans
+serif), \verb`tt` (typewriter); \verb`frak` (Fraktur); also \verb`rm`
+(roman) is available, but that is the default.
+
+\subsubsection{Header: \texttt{rhead}}
+
+In the second and third tables, the header for the row variable column
+has also been bolded. The default header is the row variable. That
+can be replaced by giving a value to the key \texttt{rhead}. I have
+used \texttt{rhead=\textbackslash boldsymbol\{x\}} (rather than \verb`\mathbf{x}`)
+in order to get an italicized bold symbol.
+\begin{verbatim}
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,
+ rround=2,ralign=c]
+ { \sin x }[x=0][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,rround=2,
+ ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=0][*]
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=-0.5][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,
+ rround=2,ralign=c]
+ { \sin x }[x=0][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=1,
+ rround=2,ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=0][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ ralign=c,rfont=bf,rhead=\boldsymbol{x}]
+ { \sin x }[x=-0.5][*]\medskip{}
+
+In these tables the row variable column has been given a centred alignment.
+The third table shows what goes wrong when \emph{some} values are
+negative. Better then is to use padding, a right alignment (the default),
+and to use a phantom in the header. The first table below does this.
+The second table incorporates kerning into the header to achieve the
+same effect:
+\begin{verbatim}
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\hphantom{0}]
+ { \sin x }[x=-0.5][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\mkern 9 mu]
+ { \sin x }[x=-0.5][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\hphantom{0}]
+ { \sin x }[x=-0.5][*]\qquad
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x}\mkern 9 mu]
+ { \sin x }[x=-0.5][*]\medskip{}
+
+(To my eye, aligning the $\boldsymbol{x}$ above the first column
+of digits after the decimal point gives a better result than truly
+centring it in the column; compare these examples with the first two
+tables of the previous example.)
+
+\subsubsection{Nudging~the~header: \texttt{rhnudge}}
+
+However, you might prefer to avoid inserting positioning commands
+into the actual row variable header, obscuring its true content. You
+can avoid doing this by using \texttt{numerica}'s nudge setting \texttt{rhnudge}.
+
+The first table below reverts to the default right alignment, avoids
+any positioning commands in the row variable header, but instead nudges
+it into position with the setting \texttt{rhnudge=9}. For positive
+nudge values, nudging works in the opposite sense to the alignment.
+The units for nudging are mu (math units, 18 to a quad), but only
+a number \textendash{} generally an integer \textendash{} should be
+specified; the mu is supplied by \texttt{numerica}.
+
+In the second table below the row variable takes single digit integer
+values, while the row variable name now occupies more than one character.
+With a right alignment the header would protrude out to the left.
+Giving \texttt{rhnudge} a \emph{negative} value (\texttt{rhnudge=-12}
+in the example) brings it back to a centred position in the row variable
+column.
+\begin{verbatim}
+ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x},rhnudge=9]
+ { \sin x }[x=-0.5][4*]\qquad
+ \tabulate
+ [rvar=x_{\text{int}},rstep=1,rstop=4,
+ rround=0,rfont=bf,rhnudge=-12,
+ rhead=\boldsymbol{x_{\text{int}}}]
+ { \sin x_{\text{int}} }[x_{\text{int}}=0][4*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rvar=x,rstep=0.25,rstop=0.5,rround=2,
+ rfont=bf,rhead=\boldsymbol{x},rhnudge=9]
+ { \sin x }[x=-0.5][4*]\qquad
+ \tabulate
+ [rvar=x_{\text{int}},rstep=1,rstop=4,
+ rround=0,rfont=bf,rhnudge=-12,
+ rhead=\boldsymbol{x_{\text{int}}}]
+ { \sin x_{\text{int}} }[x_{\text{int}}=0][4*]
+
+\subsubsection{Position in the table: \texttt{rpos}}
+
+\label{subsec:Row-var-col-pos}By default, the row variable column
+is the \emph{first} column of the table. Its position is determined
+by the value of the key \texttt{rpos}:
+\begin{itemize}
+\item \texttt{rpos=0}, suppressed (no row variable column);
+\item \texttt{rpos=1}, first column (the default);
+\item \texttt{rpos=2}, last column;
+\item \texttt{rpos=3}, first and last columns;\texttt{ }
+\item \texttt{rpos=4}, first and last columns, with the values in the last
+column a user-defined function of the first; see §\ref{subsec:Second-row-var-col};
+\item Any other integer acts like \texttt{rpos=1}.
+\end{itemize}
+An example with \texttt{rpos=3} is given shortly below, §\ref{subsec:Multiple-function-tables}.
+
+\subsubsection{\texttt{rvar'}, \texttt{rhead'}, \texttt{rhnudge'}}
+
+These settings become relevant only when \texttt{rpos=4}; see §\ref{subsec:Second-row-var-col}.
+
+\subsection{Multiple function tables}
+
+How might one tabulate multiple functions simultaneously? \emph{HMF}
+has many, many examples where multiple functions (like the trigonometric
+or the hyperbolic functions) are tabulated in separate columns of
+the same table.
+
+\subsubsection{By adjoining tables}
+
+\label{subsec:Adjoining-tables} With the settings described so far,
+one way is to adjoin single column tables. In the tables below, which
+display as a single multi-columned table, I have used three different
+\texttt{rpos} settings (\texttt{rpos=1} is implicit in the first).
+This is one way to build a table that displays as multi-column. If
+you use this method, note that the \texttt{\%} comment characters
+are essential at the end of the last argument of the \verb`\tabulate`
+commands if you want the tables to abut exactly. Omitting them results
+in a space between the tables.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rhnudge=9]
+ { \sin x }[x=0][*]%
+ \tabulate
+ [rpos=0,rspec={x,0.2,(6)},rround=2]
+ { \cos x }[x=0][*]%
+ \tabulate
+ [rpos=2,rspec={x,0.2,(6)},rhnudge=9]
+ { \tan x }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rhnudge=9]
+ { \sin x }[x=0][*]%
+ \tabulate
+ [rpos=0,rspec={x,0.2,(6)},rround=2]
+ { \cos x }[x=0][*]%
+ \tabulate
+ [rpos=2,rspec={x,0.2,(6)},rhnudge=9]
+ { \tan x }[x=0][*]
+
+\subsubsection{Multiple functions in a single table}
+
+\label{subsec:Multiple-function-tables}However, tabulating more than
+one function at a time is too common a need to have to resort to adjoining
+tables. Instead, it suffices to enter the functions in the main argument
+separated by commas. The critical thing to do is to \emph{precede
+the first function with a comma}. That initial comma is the signal
+\texttt{numerica} needs to make the internal adjustments for a multi-function
+table (and note the \verb`o` setting, to indicate the arguments of
+$\sin$ and $\cos$ are in degrees):
+\begin{verbatim}
+ \tabulate[o,rpos=3.rvar=\theta,rstep=10,rstop=90,
+ rround=0,rules=ThB]
+ { ,\sin \theta,\cos \theta }[\theta=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[o,rpos=3,rvar=\theta,rstep=10,rstop=90,
+ rround=0,rules=ThB]
+ { ,\sin \theta,\cos \theta }[\theta=0][*]\medskip{}
+
+This table also suggests a space saving possibility: since $\sin$
+and $\cos$ are complementary functions ($\cos\theta=\sin(90-\theta)$),
+the values in the bottom half of the table duplicate values in the
+top half, only with the columns reversed. This is the reason for the
+space saving \texttt{rpos=4} setting (§\ref{subsec:Second-row-var-col})
+which enables complementary functions to be tabulated in `half tables';
+see \emph{HMF} Tables 4.10\textendash 4.12 for the trigonometric functions.
+
+\section{Column variable settings}
+
+\label{sec:Column-variable-settings}When a function of \emph{two}
+variables is being tabulated, we generally think of one variable as
+the primary variable and the other as a parameter. To tabulate such
+a function, one way to proceed, would be to create and adjoin separate
+tables, one per parameter value. But this is clumsy. A more systematic
+procedure is to specify, in addition to the row variable, a \emph{column}
+variable and its start, step and stop values.
+
+\begin{table}[b]
+\caption{Column variable specification}
+
+\centering{}\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{cvar}} & {\small token(s)} & {\small column variable} & \tabularnewline
+{\small\texttt{cstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{cstop}} & {\small real num.} & {\small stop value} & {\small either }{\small\texttt{cstop}}\tabularnewline
+{\small\texttt{cols}} & {\small int} & {\small number of columns} & {\small or }{\small\texttt{cols}}\tabularnewline
+{\small\texttt{cspec}} & {\small comma list} & {\small\texttt{\{cvar,cstep}}{\small , }{\small\texttt{cstop\}}}{\small{}
+or}{\small\texttt{ \{cvar,cstep}}{\small ,}{\small\texttt{(cols)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+In the following example \verb`cvar=k` is the column variable. I
+have chosen a step size (\texttt{cstep}) of \texttt{2} and a stop
+value (\texttt{cstop}) of \texttt{9}. As with the row variable, the
+start value (\texttt{k=3}) of the column variable is specified in
+the vv-list. Although in the example these values are numbers, all
+three values could be \LaTeX{} expressions that evaluate to numbers.
+Note also the setting for \texttt{rhead} which shows the reader of
+the table that the numerical values displayed in the column headers
+are values of \verb`k`. This usage occurs throughout \emph{HMF}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ rhead=x\backslash k,
+ cvar=k,cstep=2,cstop=9]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cvar=k,cstep=2,cstop=9]
+ { \sin kx }[k=3,x=0][*] \medskip{}
+
+Again, as with the row variable, rather than using an explicit stop
+value (\texttt{cstop}), you might prefer to specify the number of
+columns (\texttt{cols}) explicitly. I could have replaced \texttt{cstop=9}
+with \texttt{cols=4} to get the same result. Note that the number
+of columns specified here is the number of \emph{function value} columns;
+the row variable column is ignored for this count.
+
+And again, as with the row variable, it is possible to condense the
+specification into a comma list (\texttt{cspec}). This is a $3$-element
+comma list of the form \verb`{cvar,cstep,cstop}` or \verb`{cvar,cstep,(cols)}`
+where the parentheses distinguish \texttt{cols} from \texttt{cstop}
+in the third item. Thus, for the preceding table I could have written
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cvar=k,cstep=2,cols=4]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+or
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cspec={k,2,(4)}]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+or
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x\backslash k,
+ cspec={k,2,9}]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+and produced the same table.
+
+Although \verb`cstep` and \verb`cstop` can be \LaTeX{} expressions
+\textendash{} for instance \texttt{cstop=\textbackslash pi/2} \textendash{}
+\verb`cols` can only be a simple integer expression \textendash{}
+an integer, or integers linked by some or all of \verb`+ - * / ( )`.
+
+\subsection{Column header formatting}
+
+\label{subsec:Column-header-formatting}
+\begin{table}
+\begin{centering}
+\caption{Column variable header formatting}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{chstyle}} & {\small int (0/1/2/3)} & {\small header style} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{ctitle}} & {\small tokens} & {\small single col. alternative header} & \tabularnewline
+{\small\texttt{chead}} & {\small tokens} & {\small user-defined header} & \tabularnewline
+{\small\texttt{calign}} & {\small char (r/c/l)} & {\small column alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{chnudge}} & {\small int} & {\small nudge header }{\small{\small\verb`int`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{chround}} & {\small int} & {\small column header rounding} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{centering}
+\end{table}
+There are four built-in style settings for the header to the column
+variable (or function value) columns (the `ch' prefix evoking `column
+header'). If these don't meet your needs or otherwise satisfy, then
+it is possible to define your own header to the function value columns
+using the key \texttt{chead}. First I discuss the built-in styles.
+
+\subsubsection{Header style: single-column case}
+
+When there is only one column of function values, the function being
+tabulated is by default set as the header to the column. This corresponds
+to setting \texttt{ctitle={*}} (see §\ref{subsec:Title:-ctitle-setting}
+below). You may want some other header. Then give \texttt{ctitle}
+some other value (although note that giving it the value \texttt{{*}{*}}
+will set both the function and the vv-list as the header; again see
+§\ref{subsec:Title:-ctitle-setting}). Whatever value you set, it
+will be typeset between math delimiters (\verb`$` signs) and can
+be nudged (see §\ref{subsec:Nudgingtheheaders:-chnudge}) left or
+right to fine-tune its position in the column. (If you want an asterisk
+as the header, you will need to place it between two pairs of braces,
+\texttt{ctitle=\{\{{*}\}\}} to prevent it being misinterpreted as
+the default setting.)
+
+If you want some more complicated header, perhaps not constrained
+by the \verb`$` delimiters, then give \texttt{chead} a value. This
+key I discuss below in §\ref{subsec:chead}. \texttt{chead} is entirely
+up to the user to specify, including any math environment and positioning.
+
+If both \texttt{ctitle} and \texttt{chead} are given, the \texttt{chead}
+value prevails.
+
+\subsubsection{Header~style: multi-column case}
+
+\texttt{chstyle=0} which is the default gives a header of the form
+displayed in the last example, with only the column-variable value
+at the head of each column. This style generally requires the row-variable
+header to indicate what the values denote, as in \texttt{rhead=x\textbackslash backslash
+k} where the backslash separates row from column variable. \emph{HMF}
+contains a multitude of instances; see Tables~9.7, 17.5, 21.1, 24.3,
+27.4, etc. for examples.
+
+\texttt{chstyle=1} changes the header of the \emph{first} function
+value column to the form \emph{variable=value} \textendash{} in the
+example below, to $k=3$. This may be an apt choice when the columns
+are narrow, perhaps from using a small rounding value. I can find
+only one real instance in \emph{HMF}, Table~26.7. Note that the row
+variable setting \texttt{rhead} no longer needs the `\texttt{\textbackslash backslash
+k}' part since the column variable is now explicitly indicated.
+
+\texttt{chstyle=2} changes the header of all function value columns
+to the form \emph{variable=value}.\emph{ }In \emph{HMF} examples are
+Tables~7.4, 7.9, 10.10, 16.6, etc. Note that the row variable setting
+\texttt{rhead} no longer needs the `\texttt{\textbackslash backslash
+k}' part since the column variable is now explicitly indicated (the
+right-hand table).
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(2)},chstyle=1]
+ { \sin kx }[k=3,x=0][3*]\quad
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=2]
+ { \sin kx }[k=3,x=0][3*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=1]
+ { \sin kx }[k=3,x=0][3*]\quad
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=2]
+ { \sin kx }[k=3,x=0][3*]\medskip{}
+
+Finally, \texttt{chstyle=3} fills each column-variable header with
+the expression being tabulated but with the column variable replaced
+by its respective values. See \emph{HMF} Tables~5.4, 8.1, 9.1, 19.1,
+etc. for examples.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=3]
+ { \sin kx }[k=3,x=0][4*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,
+ cspec={k,2,(3)},chstyle=3]
+ { \sin kx }[k=3,x=0][4*]
+
+\subsubsection{User-defined header: \texttt{chead}}
+
+\label{subsec:chead}Perhaps none of the built-in styles appeal? By
+assigning content to the key \texttt{chead}, users can create their
+own header to the function value columns. \texttt{chead} must contain
+the correct number of tab characters (\verb`&`), allowing for any
+\verb`\multicolumn`-s used, but ignoring the row variable column
+or columns. It is a header only to the function value columns. The
+user will need to insert \verb`$` signs as appropriate.
+
+Non-empty content for the \texttt{chead} key overrides any \texttt{chstyle}
+setting and in the case of a single function value column, overrides
+any \texttt{ctitle} setting.
+
+\subsubsection{Alignment: \texttt{calign}}
+
+The function value columns are aligned right (\texttt{calign=r}) by
+default. Also available are \texttt{calign=c} for centred alignment
+and \texttt{calign=l} (lowercase L) for left alignment. Using centred
+alignment with {\ttfamily\verb`chstyle=2`} in the now familiar
+example table gives
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,ralign=c,
+ cspec={k,2,(3)},chstyle=2,calign=c]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x,ralign=c,
+ cspec={k,2,(3)},chstyle=2,calign=c]
+ { \sin kx }[k=3,x=0][*]\medskip{}
+
+\noindent The first column of function values looks better, but the
+minus signs spoil the effect in the others. Handling signs in tables
+is discussed below; see §\ref{subsec:Signs}.
+
+\subsubsection{Nudging~the~headers: \texttt{chnudge}}
+
+\label{subsec:Nudgingtheheaders:-chnudge}In left or right alignment
+it is possible to nudge the headers in the opposite direction by giving
+a numerical value to the the key \texttt{chnudge}. The header is moved
+by the specified number of mu (math units; 18 to a quad). Note that
+the `mu' does not need to be written. \texttt{numerica} provides
+that. In the example I have chosen \texttt{chnudge=12} to nudge the
+column headers to the left to give a centred effect to the header
+but leaving the function values with their (potentially) awkward minus
+signs right aligned.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=12]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=12]
+ { \sin kx }[k=3,x=0][*]\medskip{}
+
+The \texttt{chnudge} value does not need to be positive. Negative
+nudges can be useful when a column header is \emph{longer} than the
+rounded function values. In the second example below, I've reduced
+the rounding value for function values to $3$, and chosen an initial
+$k$ value of $100$ to ensure this circumstance. To centre the column
+headers I have used \texttt{chnudge=-9}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=-9]
+ { \sin kx }[k=100,x=0][3*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,2,(3)},chstyle=2,chnudge=-9]
+ { \sin kx }[k=100,x=0][3*]
+
+\subsubsection{Rounding: \texttt{chround}}
+
+In the examples so far, the column variable has incremented in integer
+steps. The default rounding value for the column variable is $0$
+(for the row variable it is $1$), so if it increments by some non-integer
+amount, the result will be confusing \textendash{} if $k$ incremented
+by, say, $0.25$, starting from $k=3$, then the next column would
+also have a header $k=3$ (since $3.25$ with a rounding value $0$
+rounds to $3$). The appropriate key to remedy this state of affairs
+is \texttt{chround}. For a step size of $0.25$ the appropriate setting
+is \texttt{chround=2}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.2,(6)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2]
+ { \sin kx }[k=3,x=0][*]
+
+\section{Whole-of-table formatting}
+
+\label{sec:Whole-of-table-formatting}There are a number of settings
+pertaining to the appearance of the table as a whole, things like
+the position of the row variable column, division of the function
+values into blocks to aid readability, the presence of horizontal
+rules or of a collective column title or of a footer row. I discuss
+these here.
+
+\begin{table}[H]
+\noindent \centering{}\caption{\protect\label{tab:Table-formatting-settings}Table formatting}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{ctitle}} & {\small tokens} & {\small collective title for function-value columns} & \tabularnewline
+{\small\texttt{rules}} & {\small chars} & {\small horizontal rules template} & {\small\texttt{ThB}}\tabularnewline
+{\small\texttt{foot}} & {\small tokens} & {\small content of footer line} & \tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row-variable column position(s)} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rbloc}} & {\small comma list} & {\small division of rows into blocks} & \tabularnewline
+{\small\texttt{rblocsep}} & {\small length} & {\small extra spacing between blocks of rows} & {\small\texttt{1 ex}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+
+
+\subsection{Title for function-value columns: \texttt{ctitle}}
+
+\label{subsec:Title:-ctitle-setting}The function value columns have
+individual headers, formatted in the various ways provided by the
+settings discussed above, but it can also be helpful to have a collective
+title for these columns. This is provided by the \texttt{ctitle} key.
+This can be set to whatever you like (e.g. \texttt{ctitle=\textbackslash text\{Fred\}}),
+but for more relevant titles there are two in-built settings: \texttt{ctitle={*}},
+which makes the formula the title, and \texttt{ctitle={*}{*}}, which
+makes a title of the formula and vv-list. In the example below, it
+makes no sense to display the vv-list since it contains only the initial
+values of the row and column variables, already obvious in the table;
+hence \texttt{ctitle={*}}.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,ctitle=*]
+ { \sin kx }[k=3,x=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,ctitle=*]
+ { \sin kx }[k=3,x=0][*]\medskip{}
+
+For an example of \texttt{ctitle={*}{*}} see §\ref{subsec:Rules:-rules-setting}.
+
+\subsection{Rules: \texttt{rules} setting}
+
+\label{subsec:Rules:-rules-setting}The \texttt{booktabs} package
+which \texttt{numerica} uses is most emphatic that one should `1.
+Never, ever use vertical rules. 2. Never use double rules.' Most
+of the tables proper in \emph{HMF} lack rules of any kind although
+closer inspection shows smaller tables within the text generally \emph{are}
+delimited by horizontal rules (often also with vertical rules).\footnote{The tables in \emph{HMF} are often inelegantly typeset, and sometimes
+ugly. For all that, I have used it as a source of table types, of
+the variety of structures that the editors found necessary or at least
+useful for presenting a multitude of different kinds of numerical
+data.} I have used horizontal rules in the various examples in the present
+document because these too are tables within text. Some form of delineation
+seems necessary. Also, to my eye, the table title, $\sin kx$, in
+the last example seems `naked' without a rule to emphasize its domain.
+
+The \texttt{rules} key enables precisely which rules are used to be
+specified. The value of the key is a `word' \textendash{} a sequence
+of letters \textendash{} where the characters have the significance
+and default thicknesses (from \texttt{booktabs}) shown in Table~\ref{tab:Rules}.
+The default setting is \texttt{rules=ThB}. To insert a rule beneath
+the title, for example, change this to \texttt{rules=TthB}. If in
+addition you are using a footer row and want a rule above it, then
+the specification is \texttt{rules=TthfB}.
+
+\begin{table}[H]
+\noindent \centering{}\caption{\protect\label{tab:Rules}Rules}
+\noindent \begin{center}
+\begin{tabular}{llll}
+\toprule
+{\small char} & {\small rule} & {\small position} & {\small default rule thickness}\tabularnewline
+\midrule
+{\small\texttt{T}} & {\small top rule} & {\small above table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline
+{\small\texttt{t}} & {\small title rule} & {\small below title} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline
+{\small\texttt{h}} & {\small header rule} & {\small below header} & {\small\texttt{\textbackslash lightrulewidth=.05em}}\tabularnewline
+{\small\texttt{f}} & {\small footer rule} & {\small above footer} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline
+{\small\texttt{B}} & {\small bottom rule} & {\small below table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\end{table}
+If you wish to change the thickness of a rule from its default, then
+enter new values for any or all of {\small\texttt{\textbackslash heavyrulewidth}},
+{\small\texttt{\textbackslash lightrulewidth}}, {\small\texttt{\textbackslash cmidrulewidth}}
+in the preamble. The values listed in Table~\ref{tab:Rules} are
+the default values in the \texttt{booktabs} package (except for the
+footer rule, which \texttt{booktabs} does not cover; in \texttt{numerica}
+the footer rule is assigned a thickness of {\small\texttt{\textbackslash cmidrulewidth}}).
+
+In the example table below, a rule for the column title has been specified
+(the \verb`t` in the setting \verb`rules=TthB`). Also note the use
+of \texttt{ctitle={*}{*}}. The fornula contains an extra parameter
+$a$, assigned a value in the vv-list. It now makes sense to display
+the vv-list in the column title (but note the braces around \texttt{k}
+and \texttt{x} so that they don't show in the vv-list).
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,
+ ctitle=**,rules=TthB]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,chround=2,
+ ctitle=**,rules=TthB]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*]
+
+\subsection{Footer row: \texttt{foot} setting}
+
+Some tables have a footer row and \texttt{numerica} allows such a
+row to be inserted, but its entire content, with one exception, is
+the responsibility of the user, including insertion of the necessary
+number of tab characters \verb`&`. This will be $1$ less than the
+total number of columns (including row variable columns) in the table
+\textendash{} or some adjustment thereof if you use \verb`\multicol`.
+You can put into the footer what you wish.
+\begin{lyxcode}
+foot=<tokens>
+\end{lyxcode}
+\emph{HMF} uses the footer mainly for cryptic descriptions of the
+accuracy and needs of interpolation methods.
+
+The one exception is when \texttt{foot={*}}. This will fill the footer
+with the header, but \emph{reversed}. This is useful for tabulating
+complementary functions like the sine and cosine or, more generally,
+$f(x)$ and $g(x)$ where $g(x)=f(k-x)$ for some constant $k$. Values
+for the complementary function are read from the bottom up and require
+a reversed row variable column on the right of the table; see §\ref{subsec:Second-row-var-col}.
+
+\subsubsection{Footer functions}
+
+It is also possible to use the footer for displaying the values of
+certain column functions. \texttt{numerica} provides five of these.
+They can be used in the footer (and only in the footer): \verb`SUM`,
+\verb`AVE` (average), \verb`MAX`, \verb`MIN` and \verb`DEL` (for
+$\Delta$=\verb`MAX-MIN`). These functions act on the function values
+of the column they are in. They \emph{do not} combine mathematically:
+entering \verb`MAX-MIN` in the footer of a given column will produce
+a footer entry containing two values (those of \verb`MAX` and \verb`MIN`)
+separated by a minus sign, not the value of \verb`DEL`. The numerical
+output from each function is automatically wrapped in math delimiters
+(\verb`$`) so that minus signs display correctly.
+
+In the following example, I have chosen a column variable step size
+of zero. This is possible because in the column spec., I have also
+specified the exact number of columns. Zeroing the step size means
+the same set of figures can be used for the five footer functions
+to act on.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0,(5)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthfB,
+ foot={\small Func:} & SUM & AVE & MAX & MIN & DEL ]
+ { a\sin kx }[a=2/\pi,{k}=3.5,{x}=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0,(5)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthfB,
+ foot={\small Func:}&SUM&AVE&MAX&MIN&DEL]
+ { a\sin kx }[a=2/\pi,{k}=3.5,{x}=0][*]
+
+\subsection{Second row variable column}
+
+\label{subsec:Second-row-var-col}In §\ref{subsec:Row-var-col-pos}
+I discussed \texttt{rpos=0,1,2,3}. There is another value available
+for this key, \texttt{rpos=}4. Like \texttt{rpos=3} this adds the
+row variable column to both left and right sides of the table, but
+for the right column the values are functions of those in the left
+column.
+
+For example, the sine and cosine are complementary functions; when
+working in degrees $\cos\theta=\sin(90-\theta)$. We can exploit this
+fact to halve the table size needed to tabulate the two functions.
+\begin{verbatim}
+ \tabulate[o,rpos=4,rspec={\theta,5,45},rround=0,
+ chnudge=14,rvar'=90-\theta,rhead'=\theta',
+ rules=ThfB,foot=*]
+ { ,\sin\theta,\cos\theta }[\theta=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[o,rspec={\theta,5,45},rround=0,rpos=4,
+ chnudge=14,rvar'=90-\theta,rhead'=\theta',
+ rules=ThfB,foot=*]
+ { ,\sin\theta,\cos\theta }[\theta=0][*]\ \medskip{}
+
+\noindent where $\theta'=90-\theta$. The first half of the tabulation
+is read normally, down and from the left. The second half of the tabulation
+is read up and from the right. Note the degree setting \verb`o` in
+the settings option and
+\begin{itemize}
+\item the use of \verb`rvar'` to specify the values tabulated on the right
+(\verb`rvar'` defaults to \verb`rvar`);
+\item the use of \verb`rhead'` to specify the content of the header for
+the right-hand row variable column (\verb`rhead'` defaults to \verb`rvar'`);
+\item the footer setting \texttt{foot={*}} to obtain the header reversed
+in the footer;
+\item a rule \emph{above} the footer row specified by the \verb`f` added
+to the \verb`rules` setting, \verb`rules=ThfB`.
+\end{itemize}
+Although there is a significant space saving with tables of this kind
+(see \emph{HMF} Tables 4.10, 4.11, 4.12), they are not `kind to the
+reader'. They require a certain concentration to read and in my view
+should be avoided unless space is seriously constrained.
+
+\emph{HMF} Tables 6.1 and 6.2 are tables of the gamma function and
+its relatives where $y=x-1$ is used (stemming from $y!=\Gamma(x-1)$)
+in the row variable column on the right; Table 6.5 in effect uses
+$\langle1/x\rangle$ (the nearest integer to $1/x$) for the row variable
+on the right.
+
+\subsection{Separating blocks of rows: \texttt{rbloc}}
+
+Readability of long columns of figures can be aided by breaking the
+columns into blocks with extra white space between blocks of rows.
+This is achieved with the \texttt{rbloc} key:
+\begin{lyxcode}
+rbloc~=~<comma~list~of~positive~integers>
+\end{lyxcode}
+specifies how many rows belong to each block. For example, \texttt{rbloc=\{5,5,6\}}
+breaks the table into blocks of $5$ rows, $5$ rows, then $6$ rows.
+If the number of rows in the table is greater than the sum of the
+entries in the comma list, then division into blocks continues as
+specified by the last entry in the comma list. Thus \texttt{rbloc=5}
+(strictly \texttt{rbloc=\{5\}} but the braces can be omitted in this
+case since no comma is enclosed) divides a table into blocks of $5$
+rows; \texttt{rbloc=\{1,5\}} divides a table into $1$ row followed
+by blocks of $5$ rows. A division of this kind may be appropriate
+when, say, the row variable runs from $0$ to $1$ in increments of
+$0.1$ \textendash{} there are $11$ rows of which the first (when
+the row variable is zero) may have distinctive values.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection*{The pull of the nice round number}
+
+However, this is not how \emph{HMF} sets out its tables.\emph{ }The
+dominant practice in \emph{HMF} is division into blocks of (generally)
+$5$ rows, many of which start with a zero value for the row variable.
+Rather than isolate this initial value, they include it in the first
+block of $5$, then continue with blocks of $5$ until a single isolated
+row is left at the bottom of the page or the table. There seems to
+be a psychological need to finish a page or table with the row variable
+set to a nice round number. Thus: tabulate from $0$ to $10$ rather
+than $0$ to $9$, from $0$ to $1$ rather than $0$ to $0.9$, and
+even from $0$ to $30$ or $0$ to $2$ rather than $0$ to $29$
+or $0$ to $1.9$. Using blocks of $5$ the consequence is that there
+is always an isolated line at the end \textendash{} a kind of punctuation,
+marking the end of the page or the table.\end{shaded}%
+\end{minipage}
+
+\medskip{}
+In the next example I have divided the columns into blocks of $5$
+rows by means of the setting \texttt{rbloc=5}.
+\begin{verbatim}
+ \tabulate[o,rspec={\theta,10,90},rround=0,rbloc=5]
+ { ,\sin \theta, \cos \theta}[\theta=0][*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[o,rspec={\theta,10,90},rround=0,rbloc=5]
+ { ,\sin \theta, \cos \theta}[\theta=0][*]
+
+\subsubsection{Adjusting the extra space\texttt{ rblocsep} }
+
+By default \texttt{numerica} sets the extra space between blocks of
+rows at \verb`1 ex`. This value can easily by changed with the setting
+\texttt{rblocsep=<length>}. The units need to be included in the specification.
+The default is $1$ \texttt{ex.}
+
+\section{Function value formatting}
+
+\label{sec:Function-value-formatting}
+\begin{table}
+
+\noindent \begin{centering}
+\caption{\protect\label{tab:Function-value-formatting}Function value formatting}
+\noindent \begin{center}
+\begin{tabular}{cc>{\raggedright}p{4cm}c}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{(pad)}} & {\small int} & {\small (t-notation) phantom padding} & \tabularnewline
+{\small\texttt{signs}} & {\small int} & {\small sign handling for function values} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{diffs}} & {\small int } & {\small insert differences \& pre-pad with zeros} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{Q?}} & {\small tokens} & {\small special cell conditional} & \tabularnewline
+{\small\texttt{A!}} & {\small tokens} & {\small special cell formatting } & \tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{centering}
+\end{table}
+In the examples used so far, function values have been limited to
+a narrow range, generally $[-1,1]$. What happens when function values
+span orders of magnitude?
+
+\subsection{Trailing optional argument}
+
+The primary tool for function value formatting is the trailing optional
+argument of the \verb`\tabulate` command where the rounding value
+is specified, padding with zeros is set or not (generally \emph{set}),
+and scientific notation is set or not. Elegant scientific notation,
+set with an \texttt{x} in the trailing optional argument, is generally
+not appropriate for use in tables; see the first table below. Adding
+a prime to the \texttt{x} in the trailing optional argument (the second
+table) so that scientific notation extends to numbers in the range
+$[1,10)$ helps, particularly with the \emph{left} alignment chosen
+for the function value column, but the result is wasteful of space
+and the repetition of the `$\times10$' is borderline distracting
+and would certainly be so for a larger table. The \texttt{x} specification
+should be used in tables, if at all, only for small tables \textendash{}
+a few function values at most.
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0]
+ { e^x}[x=-5][*x]\qquad
+ \tabulate[rspec={x,1,3},rround=0,calign=l]
+ { e^x}[x=-3][*x']\qquad
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0]
+ { e^x }[x=-3][*x]\qquad
+ \tabulate[rspec={x,1,3},rround=0,calign=l]
+ { e^x}[x=-3][*x']\qquad
+
+\subsubsection{The \texttt{t} option}
+
+\emph{HMF} uses a special notation for coping with function values
+spanning orders of magnitude. This notation can be invoked by inserting
+\texttt{t} in the trailing optional argument. Repeating the previous
+two tables, and adding a \texttt{chnudge} value, gives a more compact
+and visually appealing result:
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t']
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24]
+ { e^x}[x=-3][*t']
+
+\subsection{Padding the exponent: \texttt{(pad)}}
+
+In the second table of the last example some might quibble at the
+lack of alignment of the left parentheses. \emph{HMF} tends to align
+these and \texttt{numerica} offers the setting
+\begin{lyxcode}
+(pad)~=~<integer>
+\end{lyxcode}
+to achieve the effect. \texttt{<integer>} is the number of digits/characters
+to pad to. In the last example, the setting is \texttt{(pad)=2}. Here
+it is repeated with this setting:
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t']
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t]\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=24,(pad)=2]
+ { e^x}[x=-3][*t']\medskip{}
+
+Note that this setting is relevant only when the \texttt{t} option
+is used in the trailing number-formatting argument of the \verb`\tabulate`
+command. Examples in \emph{HMF} of the style exemplified by the first
+table are, among others, Tables 8.6, 9.2, 20.1, and of the style exemplified
+by the second table, among many, Tables 9.9, 10.5, 13.1, 14.1, 19.1.
+
+\subsection{Accommodating signs: \texttt{signs}}
+
+\label{subsec:Signs}Instead of $e^{x}$ as the test function, use
+$e^{x}-1.$ Now there are positive, zero and negative function values
+to contend with. Recall that in the `t-notation' the \emph{exponent}
+is the parenthesized integer part of a number, the \emph{significand}
+the following decimal figures. \texttt{numerica} offers the \texttt{signs}
+key to align (or not) the exponents. The setting is
+\begin{lyxcode}
+signs~=~<integer>
+\end{lyxcode}
+There are four effective values for \texttt{<integer>} and the do-nothing
+default (\texttt{signs=0}):
+\begin{itemize}
+\item \texttt{signs=2 }inserts a $+$ sign between exponent and significand
+of every non-negative number;
+\item \texttt{signs=1 }inserts a $+$ sign between exponent and significand
+of every non-negative number immediately preceding or following a
+negative number;
+\item \texttt{signs=-1 }inserts a $+$ sign between exponent and significand
+of any non-negative number immediately preceding or following a negative
+number, and a phantom $-$ sign between exponent and significand of
+every other non-negative number;
+\item \texttt{signs=-2 }inserts a phantom $-$ sign between exponent and
+significand of every non-negative number;
+\end{itemize}
+With the `t-notation', the negative values and \verb`signs=2` seem
+the appropriate ones to use :
+\begin{verbatim}
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-2]
+ { e^x-1}[x=-3][4*t']\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-1]
+ { e^x-1}[x=-3][4*t']\qquad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=2]
+ { e^x-1}[x=-3][4*t']
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-2]
+ { e^x-1}[x=-3][4*t']\quad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=-1]
+ { e^x-1}[x=-3][4*t']\quad
+ \tabulate[rspec={x,1,3},rround=0,chnudge=9,
+ (pad)=2,signs=2]
+ { e^x-1}[x=-3][4*t']\medskip{}
+
+\noindent In \emph{HMF} Table 23.2 illustrates \verb`signs=-2`; Tables
+10.1, 13.1, 14.1, 19.1 among many others illustrate \verb`signs=-1`;
+and Tables 9.4, 10.6, 20.2, 22.11 among others illustrate \verb`signs=2`.
+
+However the \texttt{signs} key is not limited to the `t-notation'.
+In the following tables where the notation is not used, positive values
+for the key give appropriate results (I've included also the default
+setting):
+\begin{verbatim}
+ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=2]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=1]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2]
+ { 10\sin 5x}[x=-0.4][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=2]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2,signs=1]
+ { 10\sin 5x}[x=-0.4][*4]\qquad
+ \tabulate[rspec={x,0.1,0.4},(pad)=2]
+ { 10\sin 5x}[x=-0.4][*4]\medskip{}
+
+\emph{HMF} seems to use \verb`signs=2` when the sign of function
+values changes every few entries and \verb`signs=1` when there are
+runs of entries of the same sign. They would use the middle table
+of the three here.
+
+\subsection{Differences: \texttt{diffs}}
+
+In fine-grained tables where function values change only slowly from
+entry to entry it can be helpful to include a difference entry between
+function value entries as an aid to interpolation (and a test of eyesight).
+By entering
+\begin{lyxcode}
+diffs~=~<non-negative~integer>
+\end{lyxcode}
+the \verb`tabulate` command will include differences in a table.
+The \texttt{<non-negative integer>} is the maximum number of digits
+in a difference.
+\begin{verbatim}
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=3]
+ { \sinh x }[x=1][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=3]
+ { \sinh x }[x=1][*4]\medskip{}
+
+I have deliberately chosen the function and settings here \textendash{}
+particularly \texttt{diffs=3} \textendash{} to give a good result.
+It is easy to get this wrong. The evidence will be in the misalignment
+of the first row of function values or unnecessary padding of differences
+with leading zeros. It is a good idea to create your table first,
+see how function values change between successive rows and judge how
+many digits there will be in a difference. In the following examples
+I have deliberately put \texttt{diffs=2} and \texttt{diffs=4} to show
+the effect of a misjudgement. In the second table the function is
+\emph{decreasing}, $-\sinh x$, to show how it is the absolute value
+of the difference between successive function values that is tabulated.
+A difference is always a non-negative value.
+\begin{verbatim}
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=2]
+ { \sinh x }[x=1][*4]\qquad
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=4]
+ { -\sinh x }[x=1][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=2]
+ { \sinh x }[x=1][*4]\qquad
+ \tabulate[rspec={x,0.01,1.05},rround=2,
+ rhnudge=9,chnudge=21,diffs=4]
+ { -\sinh x }[x=1][*4]\medskip{}
+
+\noindent When the \texttt{diffs} setting is too small, function values
+in the first row are misaligned, the amount depending on how much
+too small. When the \texttt{diffs} setting is too big, alignment is
+fine but differences are padded with unnecessary leading zeros, meaning
+the column header will need a bigger nudge to bring \emph{it} into
+alignment.
+
+\subsection{Formatting special values: \texttt{Q?} and \texttt{A!}}
+
+You may wish to highlight or display in some special way a particular
+function value or values. \verb`\nmcTabulate` has two related settings
+that enable this: \texttt{Q?=<tokens>} and \texttt{A!=<tokens>}. As
+the names suggest: Question? and Answer!
+
+The question should be an expression that \texttt{l3fp} can digest
+and produce a boolean answer to (1 for `true' or 0 for `false').
+\texttt{numerica} uses \texttt{@} to denote the current function value,
+so queries like \texttt{Q?=@<0} (Is the current function value negative?)
+or \texttt{Q?=\{}@\texttt{>=pi\}} (Is the current function value greater
+than or equal to $\pi$?) are valid questions. (Note the braces in
+the second question, used to hide the equality sign.) Other possible
+useful components of such questions are \texttt{exp(1)} for the number
+$e$, || for logical Or, \texttt{\&\&} for logical And, and \texttt{!}
+for logical Not,\texttt{ }as well as the familiar arithmetic symbols,
+\texttt{+ - {*} /} and \texttt{\textasciicircum}, relation symbols
+\texttt{< > =} and their combinations like \texttt{!= >= <=} etc.,
+and parentheses. For everything \texttt{l3fp} makes available see
+the relevant chapter in \texttt{Interface3.pdf}, part of the documentation
+that comes with the \LaTeX 3 package \texttt{l3kernel}. In addition
+to these components, \texttt{numerica} offers \texttt{MAX} and \texttt{MIN}
+which are the maximum and minimum function values tabulated, so that,
+e.g., \texttt{Q?=\{@=MIN\}} (note the braces) is the question: Is
+the current function value equal to the minimum function value for
+the whole table?
+
+The answer must be in the form of a \LaTeXe{} formatting statement,
+again using \texttt{@} to denote the current function value. Thus
+\texttt{A!=\textbackslash mathbf\{@\}} is a valid answer; so is \texttt{A!=\textbackslash color\{red\}\{@\}}
+(provided you have \texttt{\textbackslash usepackage\{color\}} in
+the preamble); and so is \texttt{A!=(@)}. Another valid answer is
+\texttt{A!=} , meaning that function values satisfying the \texttt{Q?}
+question are omitted from the output.
+
+For example, suppose we wish to focus on the values of $\cos(m\pi/n)$
+lying between $0$ and $\tfrac{1}{2}$ inclusive for certain values
+of $m$ and $n$. Rather than cluttering the table with values outside
+that interval, we suppress them (the two occurrences of `\texttt{1e-14}'
+in the query are there to prevent rounding errors confusing the result):
+\begin{verbatim}
+ \tabulate
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n)}[n=4,m=2][*4]
+
+\section{Table placement}
+
+\label{sec:Table-placement}There is only one setting in this category
+that is part of \texttt{numerica} as such, the tabular vertical alignment
+option; see §\ref{subsec:Verticalalignment}. But \LaTeX{} allows one
+to insert vertical space with its \verb`\bigskip`, \verb`\medskip`,
+\verb`\smallskip`, usually about one line space, a half line space,
+and a quarter line space (with stretch and shrink), and \verb`booktabs`
+provides \verb`\abovetopsep` and \verb`\belowbottomsep`, both set
+by default to \verb`0ex` (or any other unit you care to use) and
+easily changed by writing, e.g., \verb`\abovetopsep=1.25ex` if you
+want to insert \verb`1.25ex` of space above the table (perhaps to
+fit captions).
+
+\subsection{Vertical~alignment}
+
+\label{subsec:Verticalalignment}By writing\texttt{ }
+\begin{lyxcode}
+valign~=~<char>~
+\end{lyxcode}
+where \texttt{<char>} is one of \texttt{t} or \texttt{b} the vertical
+alignment of the table can be set relative to the text baseline.\texttt{ valign=t
+}aligns the top of the table with the text baseline, \texttt{valign=b}
+the bottom of the table with the text baseline. By default (no \texttt{valign}
+setting or \texttt{valign} equated to some character other than \texttt{t}
+or \texttt{b}) the middle of the table is aligned with the text baseline.
+Repeating an example from earlier (§\ref{sec:Row-variable-settings})
+I have added letters A, B, C to show where the baseline is. In the
+first table the top of the table aligns with the baseline; in the
+second table (default case) the middle of the table aligns with the
+baseline; in the third table, the bottom of the table aligns with
+the baseline.
+\begin{verbatim}
+ A \tabulate[valign=t,rvar=x,rstep=0.2,rows=6]
+ { \sin x/\cos x }[x=0][*] \qquad
+ B \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ C \tabulate[valign=b,rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]
+\end{verbatim}
+$\Longrightarrow$ A \tabulate[valign=t,rvar=x,rstep=0.2,rows=6,rstop=2]
+ { \sin x/\cos x }[x=0][*] \qquad
+ B \tabulate[rspec={x,0.2,1}]
+ { \tan x }[x=0][*] \qquad
+ C \tabulate[valign=b,rspec={x,0.2,(6)}]
+ { \sqrt{\sec^2 x - 1} }[x=0][*]\medskip{}
+
+As explained in §\ref{subsec:Adjoining-tables}, tables can be adjoined
+to give the appearance of a single larger table. If tables with different
+numbers of rows are adjoined in this manner, then a middle alignment
+fails and either a top or bottom alignment is necessary.
+
+\section{Star option}
+
+If the \texttt{Q?} question is satisfied by at least one function
+value then adding a star (asterisk) to the \verb`\tabulate` command
+will display the first such instance. I.e., like the other starred
+commands, \verb`\eval*`, \verb`\iter*`, \verb`\solve*` and \verb`\recur*`,
+ \verb`\tabulate*` outputs a single number:
+\begin{verbatim}
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]. Ineed, if you omit the \texttt{Q?} and \texttt{A!} settings from
+the previous table then this is the value that follows $0.5000$ in
+the \texttt{m=2} column \textendash{} the first function value encountered
+satisfying the query \texttt{Q?}.
+
+If \emph{no} function value satisfies the query then a message is
+generated:
+\begin{verbatim}
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?=|@>1]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?=@>1]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+
+And if there is no query at all when the star option is chosen, another
+message is shown:
+\begin{verbatim}
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+\end{verbatim}
+$\Longrightarrow$ \tabulate*
+ [ rspec={n,1,15}, cspec={m,1,5}]
+ { \cos(m\pi/n) }[n=4,m=2][*4]
+
+\subsection{Scientific notation}
+
+If you want the number output in scientific notation when the star
+option is chosen, then enter the exponent mark in the trailing number-format
+option. This is straightforward for a letter like the commonly used
+\texttt{e}, but remember that if it is the \texttt{x} option that
+you enter, then you will need to place the \verb`\tabulate*` command
+between math delimiters, otherwise the \verb`\times` symbol resulting
+from the \texttt{x} option will generate a \LaTeX{} error (`Missing
+\$ inserted'):
+\begin{verbatim}
+ $
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n) }[n=4,m=2][*4x]
+ $
+\end{verbatim}
+$\Longrightarrow$ $
+ \tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@<-1e-14||@>0.5+1e-14},A!=]
+ { \cos(m\pi/n) }[n=4,m=2][*4x]
+ $.
+
+\section{The \texttt{reuse} setting}
+
+By entering
+\begin{lyxcode}
+reuse~=~<non-negative~integer>
+\end{lyxcode}
+it is possible to specify what is saved when the \verb`\tabulate`
+command is followed by a \verb`\reuse` command.
+\begin{itemize}
+\item \texttt{reuse=0} saves the table as displayed (the default);
+\item \texttt{reuse=n} saves the $n$-th \emph{column} of function values
+\begin{itemize}
+\item if \texttt{rpos} is non-zero: in a comma-separated list of braced
+pairs \texttt{\{row-variable value,function value}\};
+\item if \texttt{rpos=0}: in a comma-separated list of function values.
+\end{itemize}
+\item \texttt{reuse=-n} saves the $n$-th \emph{row} of function values
+in a comma-separated list that includes the row variable value in
+its appropriate position(s) if \texttt{rpos} is non-zero.
+\end{itemize}
+In the following example the third row is saved to the control sequence
+\verb`\rowiii` by using the setting \verb`reuse=-3`.
+\begin{verbatim}
+ \tabulate
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,
+ rules=TthB,reuse=-3]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[rowiii]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,
+ rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthB,reuse=-3]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[rowiii]\medskip{}
+
+\noindent Now test the content of the control sequence
+\begin{lyxcode}
+\noindent \verb`\rowiii`~$\Longrightarrow$~\rowiii ~
+\end{lyxcode}
+\noindent You can see that indeed the third row has been `captured
+for posterity'.
+
+Alternatively, we could save the second column in the control sequence
+\verb`\colii`:
+\begin{verbatim}
+ [rspec={x,0.25,(5)},rround=2,rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,
+ rules=TthB,reuse=2]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[colii]
+\end{verbatim}
+$\Longrightarrow$ \tabulate
+ [rspec={x,0.25,(5)},rround=2,
+ rhead=x,ralign=r,rhnudge=9,
+ cspec={k,0.25,(3)},chstyle=2,
+ chround=2,calign=r,ctitle=**,rules=TthB,reuse=2]
+ { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] \reuse[colii]\medskip{}
+
+To see what is saved in \verb`\colii` I use \TeX 's \verb`\meaning`
+command to show that it is indeed braced pairs:
+\begin{verbatim}
+ \meaning \colii
+\end{verbatim}
+$\Longrightarrow$ \meaning\colii
+
+\chapter{Nesting}
+
+\label{sec:tableNesting}Provided the starred form of any one of \verb`\eval`,
+\verb`\iter`, \verb`\solve`, \verb`recur` actually does produce
+a numerical result and not an error message then it can be nested
+within the main argument of any one of the other commands, including
+itself. The associated document \texttt{numerica-plus.pdf} has a number
+of illustrations of this. The starred form can also be used in the
+vv-list of any one of the commands, including itself. \texttt{numerica-plus.pdf}
+has an example of \verb`\solve*` being used in the vv-list of an
+\verb`\eval` command, and the document \texttt{numerica-basics.pdf}
+shows examples of \verb`\eval*` being used in the vv-list of an \verb`\eval`
+command.
+
+These nesting properties are also true of \verb`\tabulate`. It's
+starred form can be used in other commands. Other commands' starred
+forms can be used in \verb`\tabulate`.
+
+For the first, \verb`\tabulate*` inside \verb`\eval`, I have also
+used the example to show how formatting settings (shown for the first
+\verb`\tabulate*` in the example) can in fact be dispensed with (as
+shown for the second \verb`\tabulate*`). Only \emph{functional} settings
+are required. Hence there is no \verb`A!` setting in either. It is
+the \emph{question}, \verb`Q?`, not the answer, that isolates the
+single-value that is the result of \verb`\tabulate*`.
+\begin{verbatim}
+ \eval{$
+ (\tabulate*
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@=MAX}]
+ { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t +
+ (\tabulate*
+ [rspec={n,1,15},cspec={m,1,5},
+ Q?={@=MIN}]
+ { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t
+ $}[t=2][4]
+\end{verbatim}
+$\Longrightarrow$ \eval{$
+ (\tabulate*
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@=MAX}]
+ { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t +
+ (\tabulate*
+ [rspec={n,1,15},rround=0,rpos=2,rules=Tth,
+ cspec={m,1,5},ctitle=*,chstyle=2,
+ Q?={@=MIN}]
+ { \sin(m\pi/n) }[n=4,m=2][*4])\cosh t
+ $}[t=2][4].
+
+For using the starred forms of other commands inside \verb`\tabulate`,
+it makes more sense to use one or more of \verb`\iter*`, \verb`\solve*`
+or \verb`\recur*` inside \verb`\tabulate` than it does \verb`\eval*`,
+but that assumes the reader is familiar with those com$\frac{}{}$mands
+and that \texttt{numerica-plus.def} is loaded for this document, which
+it is not. So, here is a rather pointless illustration of nesting
+\verb`\eval*` in \verb`\tabulate`. (At least it is reassuring that
+the values of $\Delta(N)$ are identical in the two columns.)
+\begin{verbatim}
+ \tabulate[rspec={N,100,500},rround=0,ctitle=\Delta(N),
+ chead=eval*&direct]
+ { ,\eval*{ \sum_{n=1}^N 1/n-\ln N-\gamma },
+ \sum_{n=1}^N 1/n-\ln N-\gamma }[N=100][4*]
+\end{verbatim}
+$\Longrightarrow$ \tabulate[rspec={N,100,500},rround=0,ctitle=\Delta(N),
+ chead=eval*&direct]
+ { ,\eval*{ \sum_{n=1}^N 1/n-\ln N-\gamma },
+ \sum_{n=1}^N 1/n-\ln N-\gamma }[N=100][4*]
+
+\chapter{Reference summary}
+
+\section{Commands defined in \texttt{numerica-tables}}
+
+\texttt{\textbackslash nmcTabulate, \textbackslash tabulate}
+
+\section{Settings for \texttt{\textbackslash nmcTabulate}}
+
+\subsubsection*{Row variable specification §\ref{sec:Row-variable-settings}}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{3cm}}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small comment}\tabularnewline
+\midrule
+{\small\texttt{rvar}} & {\small token(s)} & {\small row variable} & \tabularnewline
+{\small\texttt{rstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{rstop}} & {\small real num.} & {\small stop value} & \multirow{2}{3cm}{either {\small\texttt{rstop}}{\small{} or }{\small\texttt{rows}}{\small ,
+not both}}\tabularnewline
+{\small\texttt{rows}} & {\small int} & {\small number of rows} & \tabularnewline
+{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small , }{\small\texttt{stop\}}}{\small{}
+or }{\small\texttt{\{start}}{\small , }{\small\texttt{step}}{\small ,
+}{\small\texttt{(rows)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Row variable column formatting §\ref{subsec:Row-var-col-formatting}}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline
+{\small\texttt{rhead}} & {\small tokens} & {\small header} & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhnudge}} & int & {\small nudge header }{\small{\small\verb`<int>`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small column position(s)} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rvar'}} & {\small tokens} & {\small 2nd row variable col. spec.} & {\small\texttt{rvar}}\tabularnewline
+{\small\texttt{rhead'}} & {\small tokens} & {\small header of 2nd rv col. (if it exists)} & {\small\texttt{rvar'}}\tabularnewline
+{\small\texttt{rhnudge'}} & int & {\small nudge 2nd rv col. header }{\small{\small\verb`<int>`}}{\small{}
+mu} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Column variable specification §\ref{sec:Column-variable-settings}.}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{cvar}} & {\small token(s)} & {\small column variable} & \tabularnewline
+{\small\texttt{cstep}} & {\small real num.} & {\small step size} & \tabularnewline
+{\small\texttt{cstop}} & {\small real num.} & {\small stop value} & {\small either }{\small\texttt{cstop}}\tabularnewline
+{\small\texttt{cols}} & {\small int} & {\small number of columns} & {\small or }{\small\texttt{cols}}{\small , not both}\tabularnewline
+{\small\texttt{cspec}} & {\small comma list} & {\small\texttt{\{cvar,cstep}}{\small , }{\small\texttt{cstop\}}}{\small{}
+or}{\small\texttt{ \{cvar,cstep}}{\small ,}{\small\texttt{(cols)\}}} & {\small short form spec.}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Column variable header formatting §\ref{subsec:Column-header-formatting}.}
+\begin{center}
+\begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{chstyle}} & {\small int (0/1/2/3)} & {\small header style} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{ctitle}} & {\small tokens} & {\small single col. alternative header} & \tabularnewline
+{\small\texttt{chead}} & {\small tokens} & {\small user-defined header} & \tabularnewline
+{\small\texttt{calign}} & {\small char (r/c/l)} & {\small column alignment} & {\small\texttt{r}}\tabularnewline
+{\small\texttt{chnudge}} & {\small int} & {\small nudge header }{\small{\small\verb`int`}}{\small{} mu} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{chround}} & {\small int} & {\small column header rounding} & {\small\texttt{0}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}\newpage{}
+\par\end{center}
+
+\subsubsection*{Function value formatting §\ref{sec:Function-value-formatting}.}
+\noindent \begin{center}
+\noindent \begin{center}
+\begin{tabular}{cc>{\raggedright}p{4cm}c}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{(pad)}} & {\small int} & {\small (t-notation) phantom padding} & \tabularnewline
+{\small\texttt{signs}} & {\small int} & {\small sign handling for function values} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{diffs}} & {\small int} & {\small insert differences \& pre-pad with zeros} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{Q?}} & {\small tokens} & {\small special cell conditional} & \tabularnewline
+{\small\texttt{A!}} & {\small tokens} & {\small special cell formatting} & \tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Whole-of-table formatting §\ref{sec:Whole-of-table-formatting}.}
+\noindent \begin{center}
+\noindent \begin{center}
+\begin{tabular}{ll>{\raggedright}p{4cm}l}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{ctitle}} & {\small tokens} & {\small collective title for function-value columns} & \tabularnewline
+{\small\texttt{rules}} & {\small chars} & {\small horizontal rules template} & {\small\texttt{ThB}}\tabularnewline
+{\small\texttt{foot}} & {\small tokens} & {\small content of footer line} & \tabularnewline
+{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row-variable column position(s)} & {\small\texttt{1}}\tabularnewline
+{\small\texttt{rbloc}} & {\small comma list} & {\small division of rows into blocks} & \tabularnewline
+{\small\texttt{rblocsep}} & {\small length} & {\small extra spacing between blocks of rows} & {\small\texttt{1 ex}}\tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+
+\subsubsection*{Table placement §\ref{subsec:Verticalalignment}.}
+\noindent \begin{center}
+\noindent \begin{center}
+\begin{tabular}{cc>{\raggedright}p{4cm}c}
+\toprule
+{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
+\midrule
+{\small\texttt{valign}} & {\small char (t/b)} & {\small vertical alignment} & \tabularnewline
+\bottomrule
+\end{tabular}
+\par\end{center}
+\par\end{center}
+\end{document}
diff --git a/macros/latex/contrib/numerica/numerica.sty b/macros/latex/contrib/numerica/numerica.sty
new file mode 100644
index 0000000000..1f1be43342
--- /dev/null
+++ b/macros/latex/contrib/numerica/numerica.sty
@@ -0,0 +1,2692 @@
+% This is file `numerica.sty',
+% the base of the `numerica' package.
+%
+% This work may be distributed and/or modified under the conditions
+% of the LaTeX Project Public License, either version 1.3c of this
+% license or any later version; see
+% http://www.latex-project.org/lppl.txt
+%
+% Andrew Parsloe (ajparsloe@gmail.com)
+%
+\RequirePackage{expl3}[2017/07/15]
+\RequirePackage{xparse,l3keys2e}
+\RequirePackage{amsmath,mathtools}
+
+\ProvidesExplPackage
+ {numerica}
+ {2021/02/15}
+ {1.0.0}
+ {Evaluate math expressions in the LaTeX form they are typeset}
+%------------------------------------------------
+\cs_if_free:NT \arccsc { \DeclareMathOperator{\arccsc}{arccsc} }
+\cs_if_free:NT \arcsec { \DeclareMathOperator{\arcsec}{arcsec} }
+\cs_if_free:NT \arccot { \DeclareMathOperator{\arccot}{arccot} }
+\cs_if_free:NT \csch { \DeclareMathOperator{\csch}{csch} }
+\cs_if_free:NT \sech { \DeclareMathOperator{\sech}{sech} }
+\cs_if_free:NT \asinh { \DeclareMathOperator{\asinh}{asinh} }
+\cs_if_free:NT \acosh { \DeclareMathOperator{\acosh}{acosh} }
+\cs_if_free:NT \atanh { \DeclareMathOperator{\atanh}{atanh} }
+\cs_if_free:NT \acsch { \DeclareMathOperator{\acsch}{acsch} }
+\cs_if_free:NT \asech { \DeclareMathOperator{\asech}{asech} }
+\cs_if_free:NT \acoth { \DeclareMathOperator{\acoth}{acoth} }
+\cs_if_free:NT \sgn { \DeclareMathOperator{\sgn}{sgn} }
+\cs_if_free:NT \lb { \DeclareMathOperator{\lb}{lb} }
+% mathtools
+\cs_if_free:NT \abs {\DeclarePairedDelimiter{\abs}{\lvert}{\rvert} }
+\cs_if_free:NT \ceil { \DeclarePairedDelimiter{\ceil}{\lceil}{\rceil} }
+\cs_if_free:NT \floor { \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} }
+\RenewDocumentCommand \land {} { \,\wedge\, }
+\RenewDocumentCommand \lor {} { \,\vee\, }
+% Qleave apart; qleave to
+\NewDocumentCommand \Q {} { \prg_do_nothing: }
+\NewDocumentCommand \q {} { \prg_do_nothing: }
+\prg_new_conditional:Npnn \int_if_zero:n #1 { p,T,F,TF }
+ {
+ \int_compare:nNnTF { #1 } = { 0 }
+ { \prg_return_true: }
+ { \prg_return_false: }
+ }
+\prg_new_conditional:Npnn \__nmc_if_mod_zero:nn #1#2 { p,T,F,TF }
+ {
+ \bool_if:nTF
+ {
+ !\int_compare_p:nNn { #1 } = { 0 } &&
+ \int_compare_p:nNn { \int_mod:nn { #1 } { #2 } } = { 0 }
+ }
+ { \prg_return_true: }
+ { \prg_return_false: }
+ }
+% constants
+\tl_const:Nn \c__nmc_digits_tl { 123456789 }
+\tl_const:Nn \c__nmc_decimals_tl { 1234567890. }
+\tl_const:Nn \c__nmc_sdigits_tl { -1234567890 }
+\tl_const:Nn \c__nmc_comparisons_tl { <=> }
+\tl_const:Nn \c__nmc_modds_tl { |./ }
+\int_const:Nn \c__nmc_and_int { 0 }
+\int_const:Nn \c__nmc_cmp_int { 1 }
+\int_const:Nn \c__nmc_sum_int { 2 }
+\int_const:Nn \c__nmc_trg_int { 3 }
+\int_const:Nn \c__nmc_uny_int { 4 }
+\int_const:Nn \c__nmc_prn_int { 5 }
+\int_const:Nn \c__nmc_srd_int { 6 }
+\seq_const_from_clist:Nn \c__nmc_T_out_seq { 1,T,\texttt{T} }
+\seq_const_from_clist:Nn \c__nmc_F_out_seq { 0,F,\texttt{F} }
+\int_new:N \l__nmc_trg_int
+\int_new:N \l__nmc_num_sgn_int
+\tl_new:N \l__nmc_toss_tl
+\seq_new:N \l_tmpc_seq
+%-----------------------------------------------------------
+% processing and substitution property lists
+% #1 clist of keys; #2 tl of values
+\cs_new_protected:Npn \__nmc_fill_class:nn #1#2
+ {
+ \clist_map_inline:nn { #1 }
+ { \prop_gput:Nnn \g__nmc_class_prop { ##1 } { #2 } }
+ }
+% math token classes
+\prop_new:N \g__nmc_class_prop
+% decimals
+\__nmc_fill_class:nn { 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, . }
+ { \__nmc_fpify_dec: \__nmc_delim_dec: }
+% constants
+\__nmc_fill_class:nn { e, \pi, \gamma, \phi, \deg, \infty }
+ { \__nmc_fpify_const: \__nmc_delim_const: }
+% plus minus
+\__nmc_fill_class:nn { +, - }
+ { \__nmc_fpify_arith: \__nmc_delim_pm: }
+% arith.
+\prop_put:Nnn \g__nmc_class_prop { * }
+ { \__nmc_fpify_arith: \__nmc_delim_arith: }
+% slash
+\prop_put:Nnn \g__nmc_class_prop { / }
+ { \__nmc_fpify_slash: \__nmc_delim_arith: }
+% arith. alternatives
+\__nmc_fill_class:nn { \times, \cdot, \div }
+ { \__nmc_fpify_arith_alt: \__nmc_delim_arith: }
+% comparisons
+\__nmc_fill_class:nn
+ {
+ =, <, >,
+ \ne, \neq, \nless, \ngtr,
+ \ge, \geq, \geqq, \geqslant,
+ \le, \leq, \leqq, \leqslant,
+ \ngeq, \ngeqq, \ngeqslant,
+ \nleq, \nleqq, \nleqslant,
+ }
+ { \__nmc_fpify_comparison: \__nmc_delim_comparison: }
+% and or
+\__nmc_fill_class:nn { \wedge, \land, \vee, \lor }
+ { \__nmc_fpify_andor: \__nmc_delim_andor: }
+% lparen
+\__nmc_fill_class:nn { (, [, \{, \lparen, \lbrack, \lbrace }
+ { \__nmc_fpify_lparen: \__nmc_delim_lparen: }
+% lvert etc.
+\__nmc_fill_class:nn { |, \lvert, \lfloor, \lceil }
+ { \__nmc_fpify_lvert: \__nmc_delim_lvert: }
+% lmodifiers
+\__nmc_fill_class:nn { \left, \bigl, \Bigl, \biggl, \Biggl }
+ { \__nmc_fpify_lmod: \__nmc_delim_lmod: }
+% rmodifiers
+\__nmc_fill_class:nn { \right, \bigr, \Bigr, \biggr, \Biggr }
+ { \prg_do_nothing: \__nmc_delim_rparen: }
+% trig, hyper, other unary fns
+\__nmc_fill_class:nn { \sin, \cos, \tan, \csc, \sec, \cot }
+ {
+ { \__nmc_fpify_unary:n { \l__nmc_trg_int } }
+ { \__nmc_delim_unary:n { \l__nmc_trg_int } }
+ }
+\__nmc_fill_class:nn
+ {
+ \arcsin, \arccos, \arctan, \arccsc, \arcsec, \arccot,
+ \sinh, \cosh, \tanh, \csch, \sech, \coth,
+ \asinh, \acosh, \atanh, \acsch, \asech, \acoth,
+ \exp, \ln, \lg, \lb, \sgn
+ }
+ {
+ { \__nmc_fpify_unary:n { \c__nmc_uny_int } }
+ { \__nmc_delim_unary:n { \c__nmc_uny_int } }
+ }
+% power
+\prop_put:Nnn \g__nmc_class_prop { ^ }
+ { \__nmc_fpify_power: \__nmc_delim_power: }
+% (d)frac
+\__nmc_fill_class:nn { \frac, \dfrac }
+ { \__nmc_fpify_frac: \__nmc_delim_frac: }
+% tfrac
+\prop_put:Nnn \g__nmc_class_prop { \tfrac }
+ { \__nmc_fpify_frac: \__nmc_delim_tfrac: }
+% binom
+\__nmc_fill_class:nn { \tbinom, \binom, \dbinom }
+ { \__nmc_fpify_binom: \__nmc_delim_frac: }
+% unary with braced arg
+\__nmc_fill_class:nn { \sqrt, \abs, \floor, \ceil }
+ { \__nmc_fpify_unarybrace: \__nmc_delim_unarybrace: }
+% unary subscripted
+\prop_put:Nnn \g__nmc_class_prop { \log }
+ { \__nmc_fpify_unarysub: \__nmc_delim_log: }
+% surd, not
+\__nmc_fill_class:nn { \surd, \neg, \lnot }
+ { \__nmc_fpify_surd: \__nmc_delim_surd: }
+% factorial
+\prop_put:Nnn \g__nmc_class_prop { ! }
+ { \__nmc_fpify_fact: \__nmc_delim_fact: }
+% cleave
+\__nmc_fill_class:nn { \q, \Q }
+ { \prg_do_nothing: \__nmc_delim_qleave: }
+% n-ary
+\__nmc_fill_class:nn { \min, \max, \gcd }
+ { \__nmc_fpify_nary: \__nmc_delim_nary: }
+% comma (n-ary fns)
+\prop_put:Nnn \g__nmc_class_prop { , }
+ { \__nmc_fpify_comma: \__nmc_delim_comma: }
+% absorb
+\__nmc_fill_class:nn
+ {
+ {{}}, \\, &, \to, \q_nil,
+ \dots, \ldots, \cdots,
+ \ , \,, \;, \:, \!, \>,
+ \thinspace, \quad, \qquad , \hfill, \hfil,
+ \mathstrut, \displaystyle, \textstyle,
+ \scriptstyle, \scriptscriptstyle
+ }
+ { \prg_do_nothing: \__nmc_delim_absorb: }
+% absorbm
+\__nmc_fill_class:nn { \vphantom, \hphantom, \phantom, \label }
+ { \__nmc_fpify_absorbm: \__nmc_delim_absorbm: }
+% absorbom
+\__nmc_fill_class:nn { \xmathstrut }
+ { \__nmc_fpify_absorbom: \__nmc_delim_absorbom: }
+% sum, prod
+\__nmc_fill_class:nn { \sum, \prod }
+ { \__nmc_fpify_sum: \__nmc_delim_sum: }
+% begin end
+\__nmc_fill_class:nn { \begin, \end }
+ { \__nmc_fpify_BE: \__nmc_delim_BE: }
+% font
+\__nmc_fill_class:nn
+ {
+ \mathrm, \mathit, \mathcal, \mathtt, \mathbf, \mathbb,
+ \mathsf, \mathfrak, \mathscr, \mathnormal, \boldsymbol
+ }
+ { \__nmc_fpify_font: \__nmc_delim_font: }
+% meta
+\__nmc_fill_class:nn { \ensuremath, \text, \mbox }
+ { \__nmc_fpify_meta: \__nmc_delim_stop: }
+% meta2
+\__nmc_fill_class:nn { \splitfrac, \splitdfrac }
+ { \__nmc_fpify_metamm: \__nmc_delim_metamm: }
+% nmcFn
+\__nmc_fill_class:nn
+ {
+ \eval, \iter, \solve, \recur, \tabulate,
+ \nmcEvaluate, \nmcIterate, \nmcSolve,
+ \nmcRecur, \nmcTabulate
+ }
+ { \__nmc_fpify_cmd: \__nmc_delim_cmd: }
+
+\__nmc_fill_class:nn { \nmcInfo } { \__nmc_fpify_info: \__nmc_delim_info: }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% math token replacements for l3fp evaluation
+% #1 prop; #2 seq (keys); #3 seq (values)
+\cs_new_protected:Npn \__nmc_subst:NNN #1#2#3
+ {
+ \cs_set_protected:Npn \__nmc_substitutions:nn ##1##2
+ { \prop_gput:Nnn #1 { ##1 } { ##2 } }
+ \seq_mapthread_function:NNN #2 #3 \__nmc_substitutions:nn
+ }
+\prop_new:N \g__nmc_subst_fpfn_prop
+\seq_set_from_clist:Nn \l_tmpa_seq
+ {
+ e, \pi, \gamma, \phi, \deg, \infty,
+ \sin, \cos, \tan, \csc, \sec, \cot,
+ \arcsin, \arccos, \arctan, \arccsc, \arcsec, \arccot,
+ \sinh, \cosh, \tanh, \csch, \sech, \coth,
+ \asinh, \acosh, \atanh, \acsch, \asech, \acoth,
+ \exp, \ln, \lg, \lb, \sgn,
+ \max, \min, \surd, \sqrt,
+ \abs, \floor, \ceil,
+ \lvert, \lfloor, \lceil,
+ \times, \cdot, \div,
+ \neg, \lnot, \wedge, \land, \vee, \lor,
+ \le, \leq, \leqq, \leqslant,
+ \ge, \geq, \geqq, \geqslant,
+ \ne, \neq, \nless, \ngtr,
+ \nleq, \nleqq,\nleqslant, \ngeq, \ngeqq, \ngeqslant
+ }
+\seq_set_from_clist:Nn \l_tmpb_seq
+ {
+ exp(1), (pi), (0.5772156649015329),
+ (1.618033988749895), (0.0174532925199433), inf,
+ sin\__nmc_deg:, cos\__nmc_deg:, tan\__nmc_deg:,
+ csc\__nmc_deg:, sec\__nmc_deg:, cot\__nmc_deg:,
+ asin\__nmc_deg:, acos\__nmc_deg:, atan\__nmc_deg:,
+ acsc\__nmc_deg:, asec\__nmc_deg:, acot\__nmc_deg:,
+ sinh, cosh, tanh, csch, sech, coth,
+ asinh, acosh, atanh, acsch, asech, acoth,
+ exp, ln, (0.4342944819032518)ln, (1.442695040888963)ln, sign,
+ max, min, sqrt, sqrt,
+ abs, floor, ceil,
+ abs, floor, ceil,
+ *, *, /,
+ !, !, &&, &&, ||, ||,
+ <=, <=, <=, <=,
+ >=, >=, >=, >=,
+ !=, !=, !<, !>,
+ !<=, !<=, !<=, !>=, !>=, !>=
+ }
+\__nmc_subst:NNN \g__nmc_subst_fpfn_prop \l_tmpa_seq \l_tmpb_seq
+
+% property list for some miscellaneous substitutions
+\prop_new:N \g__nmc_subst_misc_prop
+\seq_set_from_clist:Nn \l_tmpa_seq
+ {
+ \cosh, \sinh, \tanh, \csch, \sech, \coth,
+ (, [, \{, \lparen, \lbrack, \lbrace,
+ \left, \bigl, \Bigl, \biggl, \Biggl,
+ \lvert, \lceil, \lfloor,
+ sum, prod, iter, solve
+ }
+\seq_set_from_clist:Nn \l_tmpb_seq
+ {
+ acosh, asinh, atanh, acsch, asech, acoth,
+ ), ], \}, \rparen, \rbrack, \rbrace,
+ \right, \bigr, \Bigr, \biggr, \Biggr,
+ \rvert, \rceil, \rfloor,
+ term, factor, iteration, step
+ }
+\__nmc_subst:NNN \g__nmc_subst_misc_prop \l_tmpa_seq \l_tmpb_seq
+
+% for variables & \reuse \cs
+\prop_new:N \g__nmc_subst_var_prop
+%---------------------------------------------------------------------------
+% base function variants
+\cs_generate_variant:Nn \tl_if_head_eq_meaning:nNTF { V }
+\cs_generate_variant:Nn \tl_if_head_eq_meaning:nNF { V }
+\cs_generate_variant:Nn \tl_if_head_eq_meaning_p:nN { V }
+\cs_generate_variant:Nn \tl_if_head_eq_charcode:nNTF { V }
+\cs_generate_variant:Nn \tl_if_head_eq_charcode:nNT { V }
+\cs_generate_variant:Nn \tl_if_head_eq_charcode:nNF { V }
+\cs_generate_variant:Nn \tl_if_head_is_group:nTF { V }
+\cs_generate_variant:Nn \tl_if_in:NnTF { NV }
+\cs_generate_variant:Nn \tl_if_in:NnT { NV }
+\cs_generate_variant:Nn \tl_if_in:nnTF { nV }
+\cs_generate_variant:Nn \tl_if_eq:nnTF { V }
+\cs_generate_variant:Nn \tl_if_eq:nnT { V }
+\cs_generate_variant:Nn \tl_if_eq:nnF { V }
+\cs_generate_variant:Nn \tl_replace_once:Nnn { Nnx }
+\cs_generate_variant:Nn \tl_count_tokens:n { V }
+\cs_generate_variant:Nn \int_compare:nNnTF { v }
+\cs_generate_variant:Nn \int_compare:nNnT { v }
+\cs_generate_variant:Nn \int_case:nnTF { v }
+\cs_generate_variant:Nn \clist_pop:NN { cc }
+\cs_generate_variant:Nn \prop_put_if_new:Nnn { No }
+\cs_generate_variant:Nn \iow_open:Nn { NV }
+\cs_generate_variant:Nn \iow_now:Nn { NV }
+\cs_generate_variant:Nn \keys_set_known:nn { x }
+\cs_generate_variant:Nn \file_get:nnN { V }
+\cs_generate_variant:Nn \file_get:nnNT { V }
+\cs_generate_variant:Nn \file_get:nnNTF { V }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% general purpose routines
+\tl_new:N \l__nmc_A_tl
+\tl_new:N \l__nmc_B_tl
+\tl_new:N \l__nmc_C_tl
+\tl_new:N \l__nmc_L_tl
+\tl_new:N \l__nmc_R_tl
+% prune head from A, place in B; record new head of A in C
+\cs_new_protected:Npn \__nmc_next:
+ { \exp_last_unbraced:NV\__nmc_next:w \l__nmc_A_tl\q_stop }
+\cs_new_protected:Npn \__nmc_next:w #1#2#3\q_stop
+ { \__nmc_assign_vals:nnnN {#1} {#2} {#3} \l__nmc_B_tl }
+% get |arg|
+\cs_new_protected:Npn \__nmc_absval_arg:N #1
+ { \exp_last_unbraced:NV \__nmc_absval_arg:wN \l__nmc_A_tl\q_stop #1 }
+\cs_new_protected:Npn \__nmc_absval_arg:wN #1|#2#3\q_stop #4
+ { \__nmc_assign_vals:nnnN {#1} {#2} {#3} #4 }
+\cs_new_protected:Npn \__nmc_assign_vals:nnnN #1#2#3#4
+ {
+ \tl_if_single:nTF { #2 }
+ { \tl_set:Nn \l__nmc_A_tl { #2#3 } }
+ { \tl_set:Nn \l__nmc_A_tl { {#2}#3 } }
+ \tl_set:Nn #4 { #1 }
+ \tl_set:Nn \l__nmc_C_tl { #2 }
+ }
+\cs_new_protected:Npn \__nmc_parenth:N #1
+ { \tl_set:Nx #1 { ( \exp_not:o #1 ) } }
+\cs_new_protected:Npn \__nmc_accum_fn_parenth:NNn #1#2#3
+ { \tl_put_right:Nx #1 { #3( \exp_not:o #2 ) } }
+
+\cs_new_protected:Npn \__nmc_accum_fpify_parenth:NN #1#2
+ { % fpify #2, wrap in ( ), append to #1
+ \group_begin:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN #2 \l_tmpa_tl
+ \exp_args:NNNV
+ \group_end:
+ \__nmc_accum_fn_parenth:NNn #1\l_tmpa_tl{}
+ }
+\cs_new_protected:Npn \__nmc_fpify_next_braced:N #1
+ { % fp-ify next braced argument, parenthesize, append to #1
+ \__nmc_next:
+ \__nmc_accum_fpify_parenth:NN #1 \l__nmc_B_tl
+ }
+% wrap #3 in #2 & #4, & append to #1
+\cs_new_protected:Npn \__nmc_put_right_wrap:NnNn #1#2#3#4
+ { \tl_put_right:Nx #1 { #2 \exp_not:o #3 #4 } }
+
+\cs_new_protected:Npn \__nmc_get_arg_L:nN #1#2
+ {
+ \tl_set:Nn \l__nmc_L_tl { #1 }
+ \prop_get:NnN \g__nmc_subst_misc_prop { #1 } \l__nmc_R_tl
+ \__nmc_get_arg_L_aux:NNV #1#2 \l__nmc_R_tl
+ }
+\cs_generate_variant:Nn \__nmc_get_arg_L:nN { V }
+
+\cs_new_protected:Npn \__nmc_get_arg_L_aux:NNn #1#2#3
+ { \__nmc_get_arg_LR:NNN #1#2#3 }
+\cs_generate_variant:Nn \__nmc_get_arg_L_aux:NNn { NNV }
+
+% #1 left delim; #2 <-- arg (delims *not* included)
+% #3 right delim (*must* differ from #1)
+\cs_new_protected:Npn \__nmc_get_arg_LR:NNN #1#2#3
+ {
+ \int_set:Nn \l_tmpa_int { 1 }
+ \tl_map_inline:Nn \l__nmc_A_tl
+ {
+ \str_case:nn { ##1 }
+ {
+ { #1 } { \int_incr:N \l_tmpa_int }
+ { #3 } { \__nmc_get_arg_LR_rdelim: }
+ { \q_nil } { \__nmc_get_arg_LR_qnil:N #1 }
+ }
+ \__nmc_next:
+ \tl_if_single:nTF { ##1 }
+ { \tl_put_right:Nn #2 { ##1 } }
+ { \tl_put_right:Nn #2 { {##1} } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_get_arg_LR_rdelim:
+ {
+ \int_decr:N \l_tmpa_int
+ \int_if_zero:nT { \l_tmpa_int }
+ { \__nmc_next: \tl_map_break: }
+ }
+\cs_new_protected:Npn \__nmc_get_arg_LR_qnil:N #1
+ {
+ \__nmc_error_what:n { Unmatched~\__nmc_verb:n { #1 }{}~in }
+ \tl_map_break:
+ }
+% #1 <== multi-char num poss. in sci notation
+% #2 delim only bool
+\cs_new_protected:Npn \__nmc_get_dec:NN #1#2
+ {
+ \tl_set:NV #1 \l__nmc_B_tl
+ \__nmc_get_dec_digits:NN #1 \c__nmc_decimals_tl
+ \bool_lazy_and:nnT { \l__nmc_sci_num_in_bool }
+ { \tl_if_eq_p:NN \l__nmc_C_tl \l__nmc_sci_num_in_tl }
+ { \__nmc_get_dec_aux:NN #1#2 }
+ }
+\cs_new_protected:Npn \__nmc_get_dec_aux:NN #1#2
+ { % l3fp uses "e"
+ \__nmc_next:
+ \tl_if_eq:VnTF \l__nmc_C_tl { - }
+ {
+ \__nmc_next:
+ \__nmc_get_dec_aux:NNn #1#2 { e- }
+ }
+ { \__nmc_get_dec_aux:NNn #1#2 { e } }
+ }
+ \cs_new_protected:Npn \__nmc_get_dec_aux:NNn #1#2#3
+ {
+ \tl_if_in:NVTF \c__nmc_digits_tl \l__nmc_C_tl
+ {
+ \bool_if:NTF #2
+ { \tl_put_right:NV #1 \l__nmc_sci_num_in_tl }
+ { \tl_put_right:Nn #1 { #3 } }
+ \__nmc_get_dec_digits:NN #1 \c__nmc_sdigits_tl
+ }
+ {
+ \tl_set_eq:NN \l__nmc_C_tl \l__nmc_sci_num_in_tl
+ \str_if_in:nnT { #3 } { - }
+ { \tl_put_left:Nn \l__nmc_A_tl { - } }
+ \tl_put_left:NV \l__nmc_A_tl \l__nmc_sci_num_in_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_get_dec_digits:NN #1#2
+ {
+ \tl_map_inline:Nn \l__nmc_A_tl
+ {
+ \tl_if_in:NnTF #2 { ##1 }
+ {
+ \tl_put_right:Nn #1 { ##1 }
+ \__nmc_next:
+ }
+ { \tl_map_break: }
+ }
+ }
+% for factorial, binom, n-th root, sum/prod
+\cs_new_protected:Npn \__nmc_intify:Nn #1#2
+ { % #1 value; #2 lower bound (e.g. 0,1,-1)
+ \int_set:Nn \l_tmpa_int { \l__nmc_intify_round_tl }
+ \fp_compare:nTF { { #2 } <= round(#1+0,\l_tmpa_int) = round(#1+0) }
+ { \tl_set:Nx #1 { \fp_eval:n { round(#1+0) } } }
+ {
+ \fp_compare:nNnTF { #2 } = { -inf }
+ { \__nmc_error_what:n { Integer~required~in } }
+ { \__nmc_error_what:n { Integer~$\ge#2$~required~in } }
+ }
+ }
+%------------------------------------------------
+% explicit/implicit calc. of fn vals (sum/prod, tables, etc.)
+% #1 var; #2 fn; #3 var val (fp); #4 <= fn val at #3
+\cs_new_protected:Npn \__nmc_calc_fn_val:nNnN #1#2#3#4
+ {
+ \prop_put:Nnx \g__nmc_subst_var_prop { #1 } { \fp_eval:n { #3 } }
+ \__nmc_fpify_set:NN #4 #2
+ }
+\cs_new_protected:Npn \__nmc_calc_mode:n #1
+ {
+ \int_set:Nn \l__nmc_mode_int { #1 }
+ \int_case:nn { #1 }
+ {
+ { 1 } { \cs_set_eq:NN \__nmc_calc_fn_val:nNnN
+ \__nmc_calc_fn_vali:nNnN }
+ { 2 } { \cs_set_eq:NN \__nmc_calc_fn_val:nNnN
+ \__nmc_calc_fn_valii:nNnN }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_calc_fn_val:nNnN { V }
+
+\cs_new_protected:Npn \__nmc_calc_fn_vali:nNnN #1#2#3#4
+ { % mode 1 (all vals in vv-list may change)
+ \prop_put:Nnx \l__nmc_vv_change_prop { #1 } { \fp_eval:n { #3 } }
+ \__nmc_vv_get_vars_vals:NN \l__nmc_calc_fn_seq \l__nmc_vv_change_prop
+ \__nmc_fpify_set:NN #4 #2
+ }
+\cs_new_protected:Npn \__nmc_calc_fn_valii:nNnN #1#2#3#4
+ { % mode 2 (some vals in vv-list held const.)
+ \prop_put:Nnx \l__nmc_vv_change_prop { #1 } { \fp_eval:n { #3 } }
+ \clist_map_inline:Nn \g__nmc_unchanged_clist
+ {
+ \prop_get:NnN \g__nmc_subst_var_prop { ##1 } \l__nmc_subst_tl
+ \prop_put:NnV \l__nmc_vv_change_prop { ##1 } \l__nmc_subst_tl
+ }
+ \__nmc_vv_get_vars_vals:NN \l__nmc_calc_fn_seq \l__nmc_vv_change_prop
+ \__nmc_fpify_set:NN #4 #2
+ }
+\cs_new_protected:Npn \__nmc_fpify_set:NN #1#2
+ { % fp-set #1 to fpified #2
+ \group_begin:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN #2 \l_tmpa_tl
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_fp_exprn_tl { \l_tmpa_tl }
+ \fp_set:Nn #1 { \l__nmc_fp_exprn_tl }
+ \__nmc_error_fpflag:
+ }
+\cs_generate_variant:Nn \__nmc_fpify_set:NN { cc }
+\cs_new_protected:Npn \__nmc_verb:n #1
+ { \texttt{ \tl_trim_spaces:n { \tl_to_str:n { #1 } } } }
+\cs_generate_variant:Nn \__nmc_verb:n { V }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% Error handling
+\seq_new:N \g__nmc_error_where_seq
+\bool_new:N \g__nmc_error_bool
+%--------------------------------------
+\cs_new_protected:Npn \__nmc_error_where:n #1
+ { \seq_gpush:Nn \g__nmc_error_where_seq { #1 } }
+
+\cs_new_protected:Npn \__nmc_error_what:n #1
+ {
+ \bool_gset_true:N \g__nmc_error_bool
+ \tl_gclear:N \g__nmc_reuse_tl
+ \__nmc_error_msg:n { #1 }
+ }
+\cs_new_protected:Npn \__nmc_error_msg:n #1
+ {
+ \seq_if_empty:NF \g__nmc_error_where_seq
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l_tmpa_tl
+ \hbox:n { !!!~ #1 :~\l_tmpa_tl.~!!! }
+ }
+ }
+% l3fp exceptions; invalid op triggers a LaTeX *error*, hence:
+\fp_trap:nn { invalid_operation } { flag }
+\cs_new_protected:Npn \__nmc_error_fpflag:
+ { % ln(1),cos(90),sin(360) => underflow. Ignored!
+ \flag_if_raised:nTF { fp_overflow } { \__nmc_flag:n { 1 } }
+ { \flag_if_raised:nTF { fp_division_by_zero } { \__nmc_flag:n { 2 } }
+ { \flag_if_raised:nT { fp_invalid_operation }
+ { \__nmc_flag:n { 3 } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_flag:n #1
+ {
+ \__nmc_error_what:n
+ {
+ \__nmc_verb:n{l3fp}~error~
+ \int_case:nnT { #1 }
+ {
+ { 1 } { `Overflow' }
+ { 2 } { `Division~by~zero' }
+ { 3 } { `Invalid~operation' }
+ }
+ { ~in }
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% multi-tok vars (primed, subscripted, ...) to single tok
+% assumes math delims absent
+\int_new:N \g__nmc_prep_multitok_int
+\tl_new:N \l_nmc_multitoka_tl
+\tl_new:N \l_nmc_multitokb_tl
+\cs_new_protected:Npn \__nmc_prep_multitok:NN #1#2
+ { % #1 vv-list (seq); #2 formula (tl)
+ \seq_clear:N \l_tmpa_seq
+ \seq_map_inline:Nn #1
+ {
+ \seq_set_split:Nnn \l_tmpb_seq { = } { ##1 }
+ \seq_pop:NN \l_tmpb_seq \l_tmpb_tl
+ \int_compare:nNnT { \tl_count_tokens:V \l_tmpb_tl } > { 1 }
+ { \seq_push:NV \l_tmpa_seq \l_tmpb_tl }
+ }
+ \seq_sort:Nn \l_tmpa_seq
+ { % big --> small
+ \int_compare:nNnTF { \tl_count:n {##2} } > { \tl_count:n { ##1 } }
+ { \sort_return_swapped: }
+ { \sort_return_same: }
+ }
+ \seq_map_inline:Nn \l_tmpa_seq
+ { % multitoks => \_nmca, \_nmcb, etc
+ \int_gincr:N \g__nmc_prep_multitok_int
+ \tl_set:Nn \l_nmc_multitoka_tl { ##1 }
+ \tl_set:Nx \l_nmc_multitokb_tl { \cs:w _nmc\int_to_alph:n
+ { \g__nmc_prep_multitok_int } \cs_end: }
+ \tl_set:cn { _nmc\int_to_alph:n { \g__nmc_prep_multitok_int } }
+ { ##1 }
+ \regex_replace_all:nnN
+ { \u{l_nmc_multitoka_tl} } { \u{l_nmc_multitokb_tl} } #1
+ \regex_replace_all:nnN
+ { \u{l_nmc_multitoka_tl} } { \u{l_nmc_multitokb_tl} } #2
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \nmcInfo (no. of terms/loops/steps in "infinite" processes)
+\tl_new:N \g__nmc_sum_info_tl
+\tl_new:N \g__nmc_prod_info_tl
+\NewDocumentCommand \nmcInfo { s m }
+ {
+ \prop_if_in:NnTF \g__nmc_subst_misc_prop { #2 }
+ {
+ \tl_if_empty:cT { g__nmc_#2_info_tl }
+ { \tl_set:cn { g__nmc_#2_info_tl } { 0 } }
+ \tl_use:c { g__nmc_#2_info_tl }
+ \IfBooleanF { #1 }
+ {
+ \prop_get:NnN \g__nmc_subst_misc_prop { #2 } \l_tmpb_tl
+ \ \l_tmpb_tl % pluralise?
+ \int_if_zero:nF { \tl_use:c { g__nmc_#2_info_tl } - 1 } { s }
+ }
+ }
+ {
+ \__nmc_error_where:n { info~command }
+ \__nmc_error_what:n { Unknown~argument~\__nmc_verb:n { #2 }~in }
+ }
+ }
+\ProvideDocumentCommand \info {} { \nmcInfo }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\int_new:N \l__nmc_mode_int
+\tl_new:N \l__nmc_formula_tl
+\tl_new:N \l__nmc_formula_dup_tl
+\tl_new:N \l__nmc_fp_expr_tl
+\tl_new:N \l__nmc_fp_exprn_tl
+\tl_new:N \l_nmc_result_tl
+\bool_new:N \l__nmc_vv_val_only_bool
+\clist_new:N \l__nmc_vv_clist
+\seq_new:N \l__nmc_vv_visible_seq
+\seq_new:N \l__nmc_vv_all_seq
+\seq_new:N \l__nmc_calc_fn_seq
+\prop_new:N \l__nmc_vv_change_prop
+\tl_new:N \l__nmc_eq_var_tl
+\tl_new:N \l__nmc_eq_val_tl
+\tl_new:N \l__nmc_vv_fp_expr_tl
+\int_new:N \l__nmc_round_int
+\int_new:N \l__nmc_sci_num_int
+\int_new:N \l__nmc_prime_int
+\tl_new:N \l__nmc_bool_F_tl
+\tl_new:N \l__nmc_bool_T_tl
+\bool_new:N \l__nmc_TF_out_bool
+\bool_new:N \l__nmc_allow_TF_out_bool
+\bool_new:N \l__nmc_sci_num_x_bool
+\bool_new:N \l__nmc_sci_num_table_bool
+\tl_new:N \l__nmc_punc_tl
+\tl_new:N \l__nmc_math_delimi_tl
+\tl_new:N \l__nmc_math_delimii_tl
+\bool_new:N \l__nmc_wraps_math_bool
+% ##1 = number-only switch, ##2 = keyval settings,
+% ##3 = formula, ##4 = vv-list, ##5 = number format,
+% #1 = \nmcCommand, #2 = id, #3 =short-name cmd
+\cs_new_protected:Npn \nmc_define:nnN #1#2#3
+ {
+ \NewDocumentCommand { #1 } { s O{} m O{} O{} }
+ {
+ \bool_gset_false:N \g__nmc_error_bool
+ \group_begin:
+ \bool_set_eq:NN \l__nmc_num_only_bool ##1
+ \__nmc_get_inputs:nnnnn { #2 } { ##2 } { ##3 } {##4 } {##5 }
+ \bool_if:NF \g__nmc_error_bool
+ { \use:c { __nmc_#2_process: } }
+ \int_if_zero:nTF { \l__nmc_debug_int }
+ {
+ \bool_if:NF \g__nmc_error_bool
+ { \use:c { __nmc_#2_display: } }
+ }
+ { \__nmc_debug_display:nn { \l__nmc_debug_int } { #2 } }
+ \group_end:
+ }
+ \ProvideDocumentCommand { #3 } { } { #1 }
+ }
+\nmc_define:nnN { \nmcEvaluate } { eval } \eval
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_get_inputs:nnnnn #1#2#3#4#5
+ {
+ \__nmc_settings_get:nn { #1 }{ #2 }
+ \use:c { __nmc_#1_settings_digest: }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \mode_if_math:TF
+ { \tl_set:Nn \l__nmc_formula_tl { #3 } }
+ { \__nmc_math_delims:n { #3 } }
+ \tl_set_eq:NN \l__nmc_formula_dup_tl \l__nmc_formula_tl
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \tl_if_in:nnTF { #4 } { = }
+ { % vv-list
+ \__nmc_vv_extract_visible:nN { #4 } \l__nmc_vv_clist
+ \use:c { __nmc_#1_vv_digest:N } \l__nmc_vv_clist
+ \bool_if:NF \g__nmc_error_bool
+ { % number format
+ \tl_if_empty:nF { #5 }
+ { \__nmc_get_num_format:nn { #5 } { #1 } }
+ }
+ }
+ { % if empty vv-list
+ \tl_clear:N \l__nmc_vv_inline_tl
+ \tl_clear:N \l__nmc_vv_display_tl
+ \__nmc_get_num_format:nn { #4#5 } { #1 }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_settings_get:nn #1#2
+ { % #1 = iter,solve,recur,table; #2 = settings
+ \__nmc_error_where:n { settings }
+ \tl_if_empty:nF { #2 }
+ {
+ \keys_set_known:nn { numerica/generic } { #2 }
+ \keys_set_known:xn { numerica/#1 }{ #2 }
+ }
+ }
+\cs_new_protected:Npn \__nmc_debug_display:nn #1#2
+ { \__nmc_debug_display:xnn { \mode_if_math:TF { ed }{ * } } { #1 } { #2 } }
+\cs_new_protected:Npn \__nmc_debug_display:nnn #1#2#3
+ { % #2 dbg int #3 fn id
+ \__nmc_if_mod_zero:nnT { #2 } { 5 }
+ { \__nmc_debug_stored:N \l_tmpb_tl }
+ \raggedright
+ \begin{ align#1 }
+ \__nmc_debug_display:xnnn { \seq_use:Nn \l__nmc_vv_all_seq { ,~ } }
+ { #2 } { 2 } { vv-list }
+ \__nmc_debug_display:Vnnn \l__nmc_formula_tl { #2 } { 3 } { formula }
+ \__nmc_debug_display:Vnnn \l_tmpb_tl { #2 } { 5 } { stored }
+ \__nmc_debug_display:Vnnn \l__nmc_fp_expr_tl { #2 } { 7 } { fp-form }
+ \bool_if:nT
+ {
+ !\g__nmc_error_bool &&
+ \__nmc_if_mod_zero_p:nn { #2 } { 7 } &&
+ !\str_if_eq_p:nn { #3 } { table }
+ }
+ { \text{result:} & \quad \l_nmc_result_tl }
+ \__nmc_debug_display:Vnnn \l__nmc_reuse_retrieved_tl
+ { #2 } { 99991 } { Saved }
+ \end{ align#1 }
+ }
+\cs_generate_variant:Nn \__nmc_debug_display:nnn { x }
+
+\cs_new_protected:Npn \__nmc_debug_stored:N #1
+ {
+ \tl_clear:N \l_tmpa_tl
+ \tl_if_empty:NF \l__nmc_reuse_retrieved_tl
+ {
+ \clist_map_inline:Nn \l__nmc_reuse_retrieved_tl
+ { \tl_put_left:Nx \l_tmpa_tl { \exp_not:o { \use_i:nn ##1 }; } }
+ \tl_replace_all:Nnn \l_tmpa_tl { ~; } { ; }
+ }
+ \tl_clear:N #1
+ \prop_map_inline:Nn \g__nmc_subst_var_prop
+ {
+ \str_if_in:NnF \l_tmpa_tl { ##1; }
+ { \tl_put_right:Nn #1 { ##1=##2,~ } }
+ }
+ \tl_set:Nx #1 { \tl_range:Nnn #1 { 1 } { -2 } }
+ }
+\cs_new:Npn \__nmc_debug_display:nnnn #1#2#3#4
+ {
+ \__nmc_if_mod_zero:nnT { #2 } { #3 }
+ { \hbox:n { #4:} & \quad
+ \int_compare:nNnTF { #2 } > { 0 }
+ { \vbox_top:n } { \hbox:n } { \tl_to_str:n { #1 } } \\ }
+ }
+\cs_generate_variant:Nn \__nmc_debug_display:nnnn { V, x }
+%%%%%%%%% formula, delimiters %%%%%%%%%%%
+% delimiters
+% #1(tl) = formula
+\cs_new_protected:Npn \__nmc_math_delims:n #1
+ {
+ \tl_set:Nn \l__nmc_formula_tl { #1 }
+ \tl_trim_spaces:N \l__nmc_formula_tl
+ \bool_set_true:N \l__nmc_wraps_math_bool
+ % $ ?
+ \tl_if_head_eq_meaning:VNTF \l__nmc_formula_tl $
+ { \__nmc_math_delims_aux:n { 1 } }
+ { % \[ ?
+ \tl_if_head_eq_meaning:VNTF \l__nmc_formula_tl \[
+ { \__nmc_math_delims_aux:n { 1 } }
+ { %\begin{environ} ?
+ \tl_if_head_eq_meaning:VNTF \l__nmc_formula_tl \begin
+ { \__nmc_math_delims_aux:n { 2 } }
+ {
+ \__nmc_math_delims_aux:nn { \[ } { \] }
+ \bool_set_false:N \l__nmc_wraps_math_bool
+ }
+ }
+ }
+ \tl_trim_spaces:N \l__nmc_formula_tl
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \__nmc_math_delims_aux:nn {}{} }
+ {
+ \bool_if:NT \l__nmc_vv_multline_bool
+ { \__nmc_math_delims_mliner:V \l__nmc_eq_num_tl }
+ }
+ }
+\cs_new:Npn \__nmc_math_delims_mliner:n #1
+ {
+ \bool_if:nT { \l__nmc_wraps_math_bool || !\mode_if_math_p: }
+ {
+ \tl_set:Nn \l__nmc_math_delimi_tl { \begin{multline#1} }
+ \tl_set:Nn \l__nmc_math_delimii_tl { \end{multline#1} }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_math_delims_mliner:n { V }
+\cs_new_protected:Npn \__nmc_math_delims_aux:n #1
+ {
+ \__nmc_math_delims_aux:xx
+ { \tl_range:Nnn \l__nmc_formula_tl { 1 } { #1 } }
+ { \tl_range:Nnn \l__nmc_formula_tl { -#1 } { -1 } }
+ \int_set:Nn \l_tmpa_int { #1 + 1 }
+ \tl_set:Nx \l__nmc_formula_tl
+ { \tl_range:Nnn \l__nmc_formula_tl { \l_tmpa_int }
+ { -\l_tmpa_int } }
+ }
+\cs_new_protected:Npn \__nmc_math_delims_aux:nn #1#2
+ {
+ \tl_set:Nn \l__nmc_math_delimi_tl { #1 }
+ \tl_set:Nn \l__nmc_math_delimii_tl { #2 }
+ }
+\cs_generate_variant:Nn \__nmc_math_delims_aux:nn { xx }
+%%%%%%%%%% vv-list routines %%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_vv_extract_visible:nN #1#2
+ { % #2 reverses #1
+ \clist_clear:N #2
+ \seq_clear:N \l__nmc_vv_visible_seq
+ \clist_set:Nn \l_tmpa_clist { #1 }
+ \mode_if_math:F
+ {
+ \tl_remove_all:Nn \l_tmpa_clist { $ }
+ \tl_if_head_eq_charcode:VNT \l_tmpa_clist \[
+ {
+ \tl_remove_once:Nn \l_tmpa_clist { \[ }
+ \tl_remove_once:Nn \l_tmpa_clist { \] }
+ }
+ \tl_trim_spaces:N \l_tmpa_clist
+ }
+ \clist_map_inline:Nn \l_tmpa_clist
+ {
+ \tl_if_head_is_group:nTF { ##1 }
+ { % to remove { }
+ \clist_put_left:Nx #2
+ { \tl_head:n { ##1 } \tl_tail:n { ##1 } }
+ }
+ {
+ \seq_put_right:Nn \l__nmc_vv_visible_seq { ##1 }
+ \tl_if_head_eq_meaning:nNF { ##1 } \\
+ { \clist_put_left:Nn #2 { ##1 } }
+ }
+ }
+ \tl_replace_once:Nnx \l__nmc_vv_inline_tl { vv }
+ { \seq_use:Nn \l__nmc_vv_visible_seq { , } }
+ \tl_replace_once:Nnx \l__nmc_vv_display_tl { vv }
+ { \seq_use:Nn \l__nmc_vv_visible_seq { , } }
+ }
+\cs_new_protected:Npn \__nmc_vv_digest:N #1
+ {
+ \seq_set_from_clist:NN \l__nmc_vv_all_seq #1
+ \__nmc_error_where:n { variable\,=\,value~list }
+ \bool_if:NT \l__nmc_multitok_bool
+ { \__nmc_prep_multitok:NN \l__nmc_vv_all_seq \l__nmc_formula_tl }
+ \__nmc_vv_get_vars_vals:NN \l__nmc_vv_all_seq \c_empty_prop
+ }
+\cs_new_protected:Npn \__nmc_vv_get_vars_vals:NN #1#2
+ { % #1(seq) vv-list;#2(prop) changed vars (implicit-mode)
+ \seq_set_eq:NN \l_tmpa_seq #1
+ \seq_clear:N \l__nmc_calc_fn_seq
+ \seq_map_inline:Nn \l_tmpa_seq
+ {
+ \__nmc_vv_split_item:n { ##1 }
+ \bool_if:NT \g__nmc_error_bool { \seq_map_break: }
+ \__nmc_vv_record:NVN \l__nmc_eq_var_tl \l__nmc_eq_val_tl #2
+ \bool_if:NTF \g__nmc_error_bool { \seq_map_break: }
+ { \seq_put_right:Nn \l__nmc_calc_fn_seq { ##1 } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_vv_get_vars_vals_lims:NN #1#2
+ { % for summation, integration limits
+ \__nmc_vv_split_item:V #1
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_vv_record:NVN \l__nmc_eq_var_tl \l__nmc_eq_val_tl #2 }
+ }
+\cs_new_protected:Npn \__nmc_vv_record:NnN #1#2#3
+ {
+ \bool_if:NF \l__nmc_vv_val_only_bool
+ { % record the var
+ \prop_put_if_new:Non \g__nmc_class_prop #1
+ { \__nmc_fpify_var: \__nmc_delim_var: }
+ }
+ \prop_get:NVNF #3 #1 \l__nmc_vv_fp_expr_tl
+ { % fp-ify & calc. (changed `var=val's in prop #3)
+ \tl_clear:N \l__nmc_vv_fp_expr_tl
+ \__nmc_fpify:nN { #2 } \l__nmc_vv_fp_expr_tl
+ }
+ \bool_if:nF
+ {
+ \l__nmc_vv_val_only_bool ||
+ \g__nmc_error_bool
+ }
+ { \__nmc_vv_write:VN #1 \l__nmc_vv_fp_expr_tl }
+ }
+\cs_generate_variant:Nn \__nmc_vv_record:NnN { NV }
+
+\cs_new_protected:Npn \__nmc_vv_write:nN #1#2
+ { % #1 = var; #2 = val as fp expr.
+ \tl_if_empty:NF #2
+ {
+ \tl_if_eq:VnTF #2 { inf }
+ { \prop_put:Nnn \g__nmc_subst_var_prop { #1 } { inf } }
+ {
+ \prop_put:Nnx \g__nmc_subst_var_prop
+ { #1 } { \fp_eval:n { #2 } }
+ \__nmc_error_fpflag:
+ }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_vv_write:nN { V }
+% split var=val
+\cs_new_protected:Npn \__nmc_vv_split_item:n #1
+ {
+ \bool_set_false:N \l__nmc_vv_val_only_bool
+ \tl_if_in:nnTF { #1 } { = }
+ {
+ \__nmc_split_eq:w #1\q_stop
+ \tl_if_empty:NT \l__nmc_eq_val_tl
+ { \__nmc_error_what:n { No~value~for~$\l__nmc_eq_var_tl$~in } }
+ }
+ {
+ \bool_set_true:N \l__nmc_vv_val_only_bool
+ \tl_set:Nn \l__nmc_eq_val_tl { #1 }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_vv_split_item:n { V }
+\cs_new_protected:Npn \__nmc_split_eq:w #1=#2\q_stop
+ {
+ \tl_set:Nn \l__nmc_eq_var_tl { #1 }
+ \tl_trim_spaces:N \l__nmc_eq_var_tl
+ \tl_set:Nn \l__nmc_eq_val_tl { #2 }
+ }
+%%%%%%%%%% number-formatting routines %%%%%%%%%%
+\cs_new_protected:Npn \__nmc_get_num_format:nn #1#2
+ {
+ \__nmc_error_where:n { result~format~spec. }
+ \tl_clear:N \l_tmpa_tl
+ \str_map_inline:nn { #1 }
+ {
+ \str_if_in:NnTF \c__nmc_sdigits_tl { ##1 }
+ { \tl_put_right:Nn \l_tmpa_tl { ##1 } }
+ {
+ \str_if_eq:nnTF { ##1 } { * }
+ { \bool_set_true:N \l__nmc_pad_zeros_bool }
+ {
+ \bool_set_true:N \l__nmc_sci_num_out_bool
+ \str_case:nnF { ##1 }
+ {
+ { x } { \bool_set_true:N \l__nmc_sci_num_x_bool
+ \tl_set:Nn \l__nmc_sci_num_out_tl { ##1 } }
+ { t } { \bool_set_true:N \l__nmc_sci_num_table_bool }
+ { ' } { \int_incr:N \l__nmc_prime_int }
+ }
+ { \tl_set:Nn \l__nmc_sci_num_out_tl { ##1 } }
+ }
+ }
+ }
+ \str_if_in:nnT { #1 } { ? }
+ {
+ \bool_set_eq:NN \l__nmc_TF_out_bool \l__nmc_allow_TF_out_bool
+ \bool_set_false:N \l__nmc_sci_num_out_bool
+ }
+ \tl_if_empty:NF \l_tmpa_tl
+ { \int_set:Nn \l__nmc_round_int \l_tmpa_tl }
+ \bool_if:NT \l__nmc_TF_out_bool
+ {
+ \bool_if:NTF \l__nmc_num_only_bool
+ { \__nmc_num_TF_format:n { 1 } }
+ { \__nmc_num_TF_format:n { 1 + \l__nmc_prime_int } }
+ }
+ \bool_if:NF \g__nmc_error_bool
+ { \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl }
+ }
+%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_num_TF_format:n #1
+ {
+ \tl_set:Nx \l__nmc_bool_F_tl { \seq_item:Nn \c__nmc_F_out_seq { #1 } }
+ \tl_set:Nx \l__nmc_bool_T_tl { \seq_item:Nn \c__nmc_T_out_seq { #1 } }
+ }
+\cs_new_protected:Npn \__nmc_num_format:nNnN #1#2#3#4
+ { % #1=fpf'ied in; #2=tl fmt'ed out; #3=int rnd; #4 sci bool
+ \bool_if:NTF \l__nmc_TF_out_bool
+ {
+ \tl_set:Nx #2 { \fp_compare:nNnTF { 0 } = { #1 }
+ { \exp_not:o \l__nmc_bool_F_tl }
+ { \exp_not:o \l__nmc_bool_T_tl } }
+ }
+ {
+ \bool_if:NTF #4
+ {
+ \exp_last_unbraced:Nx \__nmc_sci_output_aux:wNn
+ { \fp_to_scientific:n { #1 } } \q_stop #2 { #3 }
+ }
+ { \__nmc_num_format_decimal:nNn { #1 } #2 { #3 } }
+ }
+ }
+\cs_new_protected:Npn \__nmc_num_format_decimal:nNn #1#2#3
+ {
+ \tl_set:Nx #2 { \fp_eval:n { 0 + round( #1, #3 ) } }
+ \__nmc_error_fpflag:
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_set:Nn \l__nmc_num_sgn_int { \fp_sign:n { #2 } }
+ \bool_lazy_and:nnT { \l__nmc_pad_zeros_bool }
+ { \int_compare_p:nNn { #3 } > { 0 } }
+ {
+ \tl_if_in:NnTF #2 { . }
+ { \__nmc_num_pad:Nnn #2 {} { #3 } }
+ { \__nmc_num_pad:Nnn { #2 } {.} { #3 } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_num_pad:Nnn #1#2#3
+ { % pad with 0s: #1 number; #2 point #3 rounding
+ \tl_set:Nx #1 { \exp_last_unbraced:NV
+ \__nmc_num_pad:wnN #1#2\q_stop{ #3 } }
+ }
+\cs_new:Npn \__nmc_num_pad:wnN #1.#2\q_stop#3
+ {
+ \int_if_zero:nTF { #3 } { #1 }
+ { #1.#2\prg_replicate:nn { #3 - \tl_count:n { #2 } } { 0 } }
+ }
+\cs_new_protected:Npn \__nmc_sci_output_aux:wNn #1e#2\q_stop#3#4
+ { % #1e#2=fp input; #3=formatted tl out; #4 = rounding
+ \tl_set:Nx #3
+ { \fp_eval:n { 0 + round( #1, #4 ) } }
+ \__nmc_error_fpflag:
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_set:Nn \l__nmc_sci_num_int { #2 }
+ \tl_if_eq:VnT #3 { 10 }
+ {
+ \int_incr:N \l__nmc_sci_num_int
+ \tl_set:Nn #3 { 1 }
+ }
+ \bool_if:NT \l__nmc_pad_zeros_bool
+ {
+ \tl_if_in:NnTF #3 {.}
+ { \__nmc_num_pad:Nnn #3 {} { #4 } }
+ { \__nmc_num_pad:Nnn #3 {.} { #4 } }
+ }
+ \bool_lazy_or:nnT { !\int_if_zero_p:n { \l__nmc_sci_num_int } }
+ { \int_compare_p:nNn { \l__nmc_prime_int } > { 0 } }
+ { \__nmc_sci_write:Nx #3 { \int_use:N \l__nmc_sci_num_int } }
+ }
+ \int_set:Nn \l__nmc_num_sgn_int { \fp_sign:n { #1 } }
+ }
+\cs_new_protected:Npn \__nmc_sci_write:Nn #1#2
+ { % #1 significand #2 exponent
+ \bool_if:NTF \l__nmc_sci_num_x_bool
+ { \tl_put_right:Nn #1 { \times 10^{#2} } }
+ {
+ \bool_if:NTF \l__nmc_sci_num_table_bool
+ { \tl_set:Nx #1 { \__nmc_sci_num_table:nn { #1 } { #2 } } }
+ { \tl_put_right:Nx #1 { \hbox:n { \l__nmc_sci_num_out_tl {#2} } } }
+ }
+ }
+\cs_generate_variant:Nn \__nmc_sci_write:Nn { Nx }
+\cs_new:Npn \__nmc_sci_num_table:nn #1#2
+ { (#2)\exp_not:n {\,}{#1} }
+%%%%%%%%%%%%%% eval-specific routines %%%%%%%%%%%%%%%
+\cs_new:Npn \__nmc_eval_settings_digest:
+ { \bool_set_true:N \l__nmc_allow_TF_out_bool }
+\cs_set_eq:NN \__nmc_eval_vv_digest:N \__nmc_vv_digest:N
+\cs_new_protected:Npn \__nmc_eval_process:
+ {
+ \__nmc_error_where:n { formula }
+ \tl_clear:N \l__nmc_fp_expr_tl
+ \__nmc_fpify:VN \l__nmc_formula_tl \l__nmc_fp_expr_tl
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_num_format:nNnN { \l__nmc_fp_expr_tl } \l_nmc_result_tl
+ { \l__nmc_round_int } \l__nmc_sci_num_out_bool }
+ }
+\cs_new_protected:Npn \__nmc_eval_display:
+ {
+ \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl
+ \bool_if:NF \l__nmc_num_only_bool
+ {
+ \tl_set:Nx \l_nmc_result_tl
+ {
+ \exp_not:o \l__nmc_math_delimi_tl
+ \bool_if:NT \l__nmc_wraps_math_bool
+ {
+ \exp_not:o \l__nmc_formula_dup_tl
+ \bool_if:NTF \l__nmc_TF_out_bool
+ { \rightarrow } { = }
+ }
+ \exp_not:o \l_nmc_result_tl
+ \bool_if:NTF \l__nmc_vv_multline_bool
+ { \exp_not:o \l__nmc_vv_display_tl }
+ {
+ \mathchoice{ \exp_not:o \l__nmc_vv_display_tl }
+ { \exp_not:o \l__nmc_vv_inline_tl }{}{}
+ }
+ \l__nmc_punc_tl
+ \exp_not:o \l__nmc_math_delimii_tl
+ }
+ \int_if_zero:nT { \l__nmc_eval_reuse_int }
+ { \tl_gset_eq:NN \g__nmc_reuse_tl \l_nmc_result_tl }
+ }
+ \l_nmc_result_tl
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \__nmc_fpify:nN
+\bool_new:N \l__nmc_insert_aster_bool
+\tl_new:N \l__nmc_subst_tl
+\tl_new:N \l__nmc_accum_tl
+\tl_new:N \l__nmc_arg_tl
+% unary functions
+\tl_new:N \l__nmc_fn_tl
+\tl_new:N \l__nmc_fn_arg_tl
+\tl_new:N \l__nmc_power_arg_tl
+\bool_new:N \l__nmc_hyperbolic_bool
+% factorial
+\fp_new:N \l__nmc_fact_fp
+\tl_new:N \l__nmc_fact_arg_tl
+% n-ary
+\int_new:N \l_tmpc_int
+\int_new:N \l_tmpd_int
+\bool_new:N \l__nmc_comma_nary_bool
+% sqrt
+\tl_new:N \l__nmc_ubrace_n_tl
+% binom
+\tl_new:N \l__nmc_binom_top_tl
+\tl_new:N \l__nmc_binom_bot_tl
+% sum/prod
+\tl_new:N \l__nmc_sum_var_tl
+\tl_new:N \l__nmc_summand_tl
+\tl_new:N \l__nmc_sum_op_tl
+\tl_new:N \l__nmc_sum_type_tl
+\tl_new:N \g__nmc_sum_A_tl
+\bool_new:N \l__nmc_sum_bool
+\int_new:N \l__nmc_sum_index_int
+\int_new:N \l__nmc_sum_end_int
+\int_new:N \l__nmc_sum_round_int
+\int_new:N \l__nmc_suma_int
+\int_new:N \l__nmc_sumb_int
+\fp_new:N \l__nmc_sum_total_fp
+\fp_new:N \l__nmc_sum_prev_fp
+\fp_new:N \l__nmc_sum_rounded_fp
+% #1 = latex expr #2 = fp-ified result
+\cs_new_protected:Npn \__nmc_fpify:nN #1#2
+ {
+ \group_begin:
+ \bool_set_false:N \l__nmc_superscript_bool
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ \tl_clear:N \l__nmc_accum_tl
+ \tl_clear:N \l__nmc_C_tl
+ \tl_set:Nn \l__nmc_A_tl { #1\q_nil }
+ \tl_if_empty:nT { #1 }
+ { \__nmc_error_what:n { Empty~argument~to~fp-ify~in } }
+ \bool_until_do:nn
+ {
+ \quark_if_nil_p:N \l__nmc_C_tl
+ || \g__nmc_error_bool
+ }
+ {
+ \tl_clear:N \l__nmc_arg_tl
+ \__nmc_next:
+ \prop_get:NVNTF \g__nmc_class_prop \l__nmc_B_tl \l_tmpb_tl
+ { \exp_last_unbraced:NV \use_i:nn \l_tmpb_tl }
+ { \__nmc_fpify_unknown: }
+ }
+ \exp_args:NNNV
+ \group_end:
+ \tl_put_right:Nn #2 \l__nmc_accum_tl
+ }
+\cs_generate_variant:Nn \__nmc_fpify:nN { V }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\cs_new_protected:Npn \__nmc_fpify_unknown:
+ {
+ \tl_if_in:nVTF { )]\}\rbrace } \l__nmc_B_tl
+ { \__nmc_error_what:n { Unmatched~\__nmc_verb:V \l__nmc_B_tl{}~in } }
+ { \__nmc_error_what:n { Unknown~token~\__nmc_verb:V \l__nmc_B_tl{}~in } }
+ }
+\cs_new_protected:Npn \__nmc_insert_aster:
+ {
+ \bool_if:NT \l__nmc_insert_aster_bool
+ { \tl_put_right:Nn \l__nmc_accum_tl { * } }
+ }
+\cs_new_protected:Npn \__nmc_fpify_dec:
+ {
+ \__nmc_get_dec:NN \l__nmc_fact_arg_tl \c_false_bool
+ \tl_if_eq:VnF \l__nmc_C_tl { ! }
+ {
+ \__nmc_insert_aster:
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_fact_arg_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_var:
+ { % ( ) in case var < 0 raised to power
+ \prop_get:NVNTF \g__nmc_subst_var_prop \l__nmc_B_tl \l__nmc_subst_tl
+ {
+ \tl_set_eq:NN \l__nmc_fact_arg_tl \l__nmc_subst_tl
+ \tl_if_in:noTF { !' } \l__nmc_C_tl
+ {
+ % \tl_if_eq:VnT \l__nmc_C_tl { ' }
+ % { \__nmc_deriv_dot_from_prime: }
+ }
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l__nmc_subst_tl {}
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+ { \__nmc_fpify_unknown: }
+ }
+\cs_new_protected:Npn \__nmc_fpify_const:
+ { % if const used as var
+ \prop_get:NVNTF \g__nmc_subst_var_prop \l__nmc_B_tl \l__nmc_subst_tl
+ { \__nmc_fpify_var: }
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \tl_clear:N \l__nmc_fact_arg_tl }
+ { \tl_put_right:NV \l__nmc_accum_tl \l__nmc_subst_tl }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_arith:
+ {
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_B_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_slash:
+ {
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_B_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_uny_int }
+ \__nmc_accum_fpify_parenth:NN \l__nmc_accum_tl \l__nmc_arg_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_arith_alt:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_B_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_B_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_comparison:
+ {
+ \tl_if_in:NVTF \c__nmc_comparisons_tl \l__nmc_C_tl
+ { \__nmc_error_what:n { Multi-token~comparison~in } }
+ {
+ \prop_get:NVNF \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l_tmpa_tl
+ { \tl_set_eq:NN \l_tmpa_tl \l__nmc_B_tl } % < = >
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_cmp_int }
+ \tl_clear:N \l_tmpb_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpb_tl
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpb_tl { - }
+ \tl_set:Nx \l__nmc_accum_tl
+ { round( \l__nmc_accum_tl, \int_use:N \l__nmc_round_int ) }
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { 0 }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_andor:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+% \tl_show:N \l__nmc_A_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_and_int }
+% \tl_show:N \l__nmc_arg_tl
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+
+ }
+\cs_new_protected:Npn \__nmc_fpify_lparen:
+ {
+ \__nmc_get_arg_L:VN \l__nmc_B_tl \l__nmc_arg_tl
+ \tl_set_eq:NN \l__nmc_fact_arg_tl \l__nmc_arg_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \bool_set_false:N \l__nmc_insert_aster_bool }
+ {
+ \__nmc_accum_fpify_parenth:NN \l__nmc_accum_tl \l__nmc_arg_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_lvert:
+ {
+ \tl_if_eq:VnTF \l__nmc_B_tl { | }
+ {
+ \tl_set:Nn \l__nmc_L_tl { \lvert }
+ \__nmc_absval_arg:N \l__nmc_arg_tl
+ \tl_set:Nn \l__nmc_R_tl { \rvert }
+ }
+ {
+ \tl_set_eq:NN \l__nmc_L_tl \l__nmc_B_tl
+ \__nmc_get_arg_L:VN \l__nmc_B_tl \l__nmc_arg_tl
+ }
+ \__nmc_fpify_lvert_aux:
+ }
+\cs_new_protected:Npn \__nmc_fpify_lvert_aux:
+ {
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ {
+ \tl_set:NV \l__nmc_fact_arg_tl \l__nmc_L_tl
+ \tl_put_right:NV \l__nmc_fact_arg_tl \l__nmc_arg_tl
+ \tl_put_right:NV \l__nmc_fact_arg_tl \l__nmc_R_tl
+ }
+ {
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \str_case:onF \l__nmc_L_tl
+ {
+ { \lvert } { \tl_put_right:Nn \l__nmc_accum_tl { abs( } }
+ { \lceil } { \tl_put_right:Nn \l__nmc_accum_tl { ceil( } }
+ { \lfloor } { \tl_put_right:Nn \l__nmc_accum_tl { floor( } }
+ { | } { \tl_put_right:Nn \l__nmc_accum_tl { abs( } }
+ { / } { \tl_put_right:Nn \l__nmc_accum_tl { / } }
+ }
+ { \tl_put_right:Nn \l__nmc_accum_tl { ( } }
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \tl_if_eq:VnF \l__nmc_R_tl { . }
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_lmod:
+ {
+ \tl_set_eq:NN \l__nmc_L_tl \l__nmc_B_tl
+ \prop_get:NVN \g__nmc_subst_misc_prop \l__nmc_L_tl \l__nmc_R_tl
+ \str_case:on \l__nmc_C_tl
+ {
+ { | } { \__nmc_fpify_lmod_abs:V \l__nmc_L_tl }
+ { . } { \__nmc_fpify_lmod_aux:V \l__nmc_L_tl }
+ { / }
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { / }
+ \__nmc_fpify_lmod_aux:V \l__nmc_L_tl
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_lmod_abs:n #1
+ {
+ \__nmc_next:
+ \tl_put_right:Nn \l__nmc_accum_tl { abs( }
+ \exp_args:NNNV\__nmc_get_arg_LR:NNN { #1 } \l__nmc_arg_tl \l__nmc_R_tl
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_generate_variant:Nn \__nmc_fpify_lmod_abs:n { V }
+
+\cs_new_protected:Npn \__nmc_fpify_lmod_aux:n #1
+ {
+ \__nmc_next:
+ \exp_args:NNNV\__nmc_get_arg_LR:NNN #1 \l__nmc_arg_tl \l__nmc_R_tl
+ \tl_if_eq:VnT \l__nmc_C_tl { . }
+ { \__nmc_next: }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_generate_variant:Nn \__nmc_fpify_lmod_aux:n { V }
+
+\cs_new_protected:Npn \__nmc_fpify_rmod:
+ {
+ \tl_if_eq:VnT \l__nmc_C_tl { . }
+ { \__nmc_next: }
+ }
+\cs_new_protected:Npn \__nmc_fpify_unary:n #1
+ {
+ \tl_set:NV \l__nmc_fn_tl \l__nmc_B_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ^ }
+ { \__nmc_fpify_unary_superscript: }
+ { \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_fn_tl \l__nmc_fn_tl }
+ \str_if_in:NnTF \l__nmc_fn_tl { h }
+ { \bool_set_true:N \l__nmc_hyperbolic_bool }
+ { \bool_set_false:N \l__nmc_hyperbolic_bool }
+ % get fn arg
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { #1 }
+ \tl_clear:N \l__nmc_fn_arg_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l__nmc_fn_arg_tl
+ \bool_if:NT \l__nmc_hyperbolic_bool
+ {
+ \tl_set:Nx \l__nmc_fn_tl
+ { \__nmc_fpify_unary_hyperbolic:N \l__nmc_fn_arg_tl }
+ }
+ % append fn
+ \bool_if:NTF \l__nmc_hyperbolic_bool
+ { \tl_put_right:NV \l__nmc_accum_tl \l__nmc_fn_tl }
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl
+ \l__nmc_fn_arg_tl \l__nmc_fn_tl
+ }
+ \bool_if:NT \l__nmc_superscript_bool
+ { % e.g. \sin^{2}
+ \tl_put_right:Nn \l__nmc_accum_tl { ^ }
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_power_arg_tl
+ \bool_set_false:N \l__nmc_superscript_bool
+ }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_unary_superscript:
+ {
+ \__nmc_next:
+ \tl_clear:N \l__nmc_power_arg_tl
+ \__nmc_fpify_next_braced:N \l__nmc_power_arg_tl
+ \tl_if_eq:VnTF \l__nmc_power_arg_tl { (-1) }
+ { % inverse trig/hyp. function?
+ \prop_get:NVNF \g__nmc_subst_misc_prop \l__nmc_fn_tl \l__nmc_fn_tl
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop
+ \l__nmc_fn_tl \l__nmc_subst_tl
+ \tl_concat:NNN \l__nmc_fn_tl a \l__nmc_subst_tl
+ }
+ }
+ { % not inverse fn
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_fn_tl \l__nmc_fn_tl
+ \bool_set_true:N \l__nmc_superscript_bool
+ }
+ }
+\cs_new:Npn \__nmc_fpify_unary_hyperbolic:N #1
+ {
+ \str_case:on \l__nmc_fn_tl
+ {
+ { sinh } { ( 0.5( exp(#1)- exp(-(#1)) ) ) }
+ { cosh } { ( 0.5( exp(#1) + exp(-(#1)) ) ) }
+ { tanh } { ( (exp(2(#1))-1)/(exp(2(#1))+1) ) }
+ { asinh } { ln( #1 + sqrt((#1)^2+1) ) }
+ { acosh } { ln( #1 + sqrt((#1)^2-1) ) }
+ { atanh } { ( 0.5 ln( (1+(#1))/(1-(#1)) ) ) }
+ { csch } { 1/( 0.5( exp(#1)- exp(-(#1)) ) ) }
+ { sech } { 1/( 0.5( exp(#1) + exp(-(#1)) ) ) }
+ { coth } { ( (exp(2(#1))+1)/(exp(2(#1))-1) ) }
+ { acsch } { ( ln(1/(#1)+sqrt(1/(#1)^2+1)) ) }
+ { asech } { ( ln(1/(#1)+sqrt(1/(#1)^2-1)) ) }
+ { acoth } { ( 0.5 ln( ((#1)+1)/((#1)-1) ) ) }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_power:
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ^ }
+ \__nmc_fpify_next_braced:N \l__nmc_accum_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_frac:
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ( }
+ \__nmc_fpify_next_braced:N \l__nmc_accum_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { / }
+ \__nmc_fpify_next_braced:N \l__nmc_accum_tl
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_unarybrace:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \tl_if_eq:VnT \l__nmc_C_tl { * }
+ { \__nmc_next: }
+ \tl_if_eq:VnTF \l__nmc_C_tl { [ }
+ { % n-th root
+ \tl_if_eq:VnTF \l__nmc_subst_tl { sqrt }
+ {
+ \__nmc_next:
+ \__nmc_fpify_ubrace_n:
+ }
+ {
+ \__nmc_next: \__nmc_next: \__nmc_next:
+ \__nmc_fpify_ubrace_aux:
+ }
+ }
+ { \__nmc_fpify_ubrace_aux: }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_ubrace_aux:
+ {
+ \tl_set_eq:NN \l_tmpa_tl \l__nmc_subst_tl
+ \__nmc_fpify_next_braced:N \l_tmpa_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \__nmc_fpify_ubrace_fact: }
+ { \tl_put_right:NV \l__nmc_accum_tl \l_tmpa_tl }
+ }
+\cs_new_protected:Npn \__nmc_fpify_ubrace_n:
+ {
+ \__nmc_error_where:n { [~]~argument~of~\__nmc_verb:n { \sqrt } }
+ \tl_clear:N \l__nmc_arg_tl
+ \tl_clear:N \l__nmc_ubrace_n_tl
+ \__nmc_get_arg_LR:NNN [ \l__nmc_arg_tl ]
+ \__nmc_fpify:VN \l__nmc_arg_tl \l__nmc_ubrace_n_tl
+ \__nmc_intify:Nn \l__nmc_ubrace_n_tl { 1 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_B_tl \l_tmpa_tl
+ \bool_if:nTF
+ {
+ \fp_compare_p:nNn { \l_tmpa_tl } < { 0 }
+ && \int_if_even_p:n { \l__nmc_ubrace_n_tl }
+ }
+ { \flag_raise:n { fp_invalid_operation } \__nmc_error_fpflag: }
+ {
+ \tl_put_right:Nx \l__nmc_accum_tl
+ { \__nmc_fpify_ubrace_n_aux:VV {\l_tmpa_tl}{ \l__nmc_ubrace_n_tl }}
+ }
+ }
+ }
+\cs_new:Npn \__nmc_fpify_ubrace_n_aux:nn #1#2
+ { sign( #1 )( abs( #1 ) )^( 1/( #2 ) ) }
+\cs_generate_variant:Nn \__nmc_fpify_ubrace_n_aux:nn { VV }
+
+\cs_new_protected:Npn \__nmc_fpify_ubrace_fact:
+ {
+ \tl_if_eq:VnF \l__nmc_subst_tl { sqrt }
+ {
+ \str_case:onT \l__nmc_subst_tl
+ {
+ { abs } { \tl_set:Nn \l__nmc_fact_arg_tl { \abs } }
+ { ceil } { \tl_set:Nn \l__nmc_fact_arg_tl { \ceil } }
+ { floor} { \tl_set:Nn \l__nmc_fact_arg_tl { \floor } }
+ }
+ {
+ \tl_put_right:Nx \l__nmc_fact_arg_tl
+ { { \exp_not:o \l__nmc_B_tl } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_unarysub:
+ { % e.g. \log_{10}
+ \str_if_eq:VnTF \l__nmc_C_tl { _ }
+ {
+ \__nmc_next: \__nmc_next:
+ \__nmc_fpify_unarysub:N \l__nmc_B_tl
+ }
+ { \__nmc_fpify_unarysub:N \l__nmc_log_base_tl }
+ }
+\cs_new_protected:Npn \__nmc_fpify_unarysub:N #1
+ {
+ \__nmc_fpify:VN #1 \l__nmc_arg_tl
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \fp_compare:nF { 0 < \l__nmc_arg_tl != 1 }
+ {
+ \__nmc_error_what:n
+ { Valid~base~required~for~\__nmc_verb:n { \log } in }
+ }
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \tl_set:Nx \l_tmpa_tl
+ { \fp_eval:n { 1 / ( ln ( \l__nmc_arg_tl ) ) } }
+ \__nmc_error_fpflag:
+ }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpa_tl {}
+ \tl_put_left:Nn \l__nmc_A_tl { \ln }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_surd:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \tl_put_right:NV \l__nmc_accum_tl \l__nmc_subst_tl
+ \tl_clear:N \l__nmc_arg_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_srd_int }
+ \__nmc_parenth:N \l__nmc_arg_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l__nmc_accum_tl
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_fact:
+ {
+ \__nmc_error_where:n { factorial }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_fact_arg_tl \l_tmpa_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \__nmc_fpify_fact_repeated: }
+ {
+ \__nmc_intify:Nn \l_tmpa_tl { 0 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpa_tl { fact }
+ }
+ }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_fact_repeated:
+ {
+ \__nmc_intify:Nn \l_tmpa_tl { -1 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_fpify_double_fact:N \l_tmpa_tl
+ \tl_set:Nx \l_tmpa_tl { \fp_use:N \l__nmc_fact_fp }
+ \tl_if_eq:VnTF \l__nmc_C_tl { ! }
+ { \tl_set_eq:NN \l__nmc_fact_arg_tl \l_tmpa_tl }
+ { \__nmc_accum_fn_parenth:NNn \l__nmc_accum_tl \l_tmpa_tl {} }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_double_fact:N #1
+ {
+ \__nmc_next:
+ \int_compare:nNnTF { #1 } > { 0 }
+ {
+ \int_if_even:nTF { #1 }
+ { \__nmc_fpify_fact_do:nNn { 2 } #1 { 2 } }
+ { \__nmc_fpify_fact_do:nNn { 1 } #1 { 2 } }
+ }
+ { \fp_set:Nn \l__nmc_fact_fp { 1 } }
+ }
+% multiply nos from #1 to #2, put in \l__nmc_fact_fp
+% #3 = 1 (!), = 2 (!!)
+\cs_new_protected:Npn \__nmc_fpify_fact_do:nNn #1#2#3
+ {
+ \fp_set:Nn \l_tmpa_fp { #1 }
+ \fp_set:Nn \l_tmpb_fp { #2 }
+ \int_set:Nn \l_tmpb_int { #3 }
+ \fp_set_eq:NN \l__nmc_fact_fp \l_tmpa_fp
+ \fp_while_do:nNnn { \l_tmpa_fp } < { \l_tmpb_fp }
+ {
+ \fp_add:Nn \l_tmpa_fp { \l_tmpb_int }
+ \fp_set:Nn \l__nmc_fact_fp { \l__nmc_fact_fp * \l_tmpa_fp }
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_nary:
+ {
+ \bool_set_true:N \l__nmc_comma_nary_bool
+ \str_case:on \l__nmc_B_tl
+ {
+ { \gcd } { \__nmc_fpify_nary_gcd: }
+ { \max } { \__nmc_fpify_nary_maxmin: }
+ { \min } { \__nmc_fpify_nary_maxmin: }
+ }
+ \bool_set_false:N \l__nmc_comma_nary_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_nary_gcd:
+ {
+ \__nmc_insert_aster:
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c_max_int }
+ \tl_set:Nx \l__nmc_arg_tl { \tl_range:Nnn \l__nmc_arg_tl { 2 } { -2 } }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ \seq_set_split:NnV \l_tmpa_seq {,} \l_tmpa_tl
+ % eval. 1st expr.
+ \seq_pop:NN \l_tmpa_seq \l_tmpb_tl
+ \int_set:Nn \l_tmpb_int { \fp_to_int:n \l_tmpb_tl }
+ \seq_map_inline:Nn \l_tmpa_seq
+ {
+ \int_set:Nn \l_tmpd_int { \fp_to_int:n { ##1 } }
+ \int_set:Nn \l_tmpa_int { \int_max:nn { \l_tmpd_int } { \l_tmpb_int } }
+ \int_set:Nn \l_tmpb_int { \int_min:nn { \l_tmpd_int } { \l_tmpb_int } }
+ \int_set:Nn \l_tmpc_int { \int_mod:nn { \l_tmpa_int } { \l_tmpb_int } }
+
+ \int_until_do:nNnn { \l_tmpc_int } = { 0 }
+ {
+ \int_set_eq:NN \l_tmpa_int \l_tmpb_int
+ \int_set_eq:NN \l_tmpb_int \l_tmpc_int
+ \int_set:Nn \l_tmpc_int { \int_mod:nn
+ { \l_tmpa_int } { \l_tmpb_int } }
+ }
+ }
+ \tl_put_right:Nx \l__nmc_accum_tl { \int_use:N \l_tmpb_int }
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_nary_maxmin:
+ {
+ \prop_get:NVN \g__nmc_subst_fpfn_prop \l__nmc_B_tl \l__nmc_subst_tl
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c_max_int }
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_fpify:VN \l__nmc_arg_tl \l_tmpa_tl
+ % em-brace in case nested
+ \tl_put_right:Nx \l__nmc_accum_tl
+ { { \l__nmc_subst_tl \exp_not:o \l_tmpa_tl } }
+ \bool_set_true:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_comma:
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { , }
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+\cs_new_protected:Npn \__nmc_fpify_binom:
+ {
+ \__nmc_fpify_next_braced:N \l__nmc_binom_top_tl
+ \__nmc_fpify_next_braced:N \l__nmc_binom_bot_tl
+ \__nmc_error_where:n { 2nd~argument~of~\__nmc_verb:n { \binom } }
+ \__nmc_intify:Nn \l__nmc_binom_bot_tl { 0 }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \seq_gpop:NN \g__nmc_error_where_seq \l__nmc_toss_tl
+ \__nmc_fpify_binom_calc:NN
+ \l__nmc_binom_top_tl \l__nmc_binom_bot_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_binom_calc:NN #1 #2
+ {
+ \tl_put_right:Nn \l__nmc_accum_tl { ( }
+ \int_if_zero:nTF { #2 }
+ { \tl_put_right:Nn \l__nmc_accum_tl { 1 } }
+ { % calc. top
+ \__nmc_fpify_binom_calc_do:nN { #1 - #2 + 1 } #1
+ \tl_put_right:Nn \l__nmc_accum_tl { / }
+ \__nmc_fpify_binom_calc_do:nN { 1 } #2 % calc. bot.
+ }
+ \tl_put_right:Nn \l__nmc_accum_tl { ) }
+ }
+\cs_new_protected:Npn \__nmc_fpify_binom_calc_do:nN #1 #2
+ {
+ \__nmc_fpify_fact_do:nNn { #1 } #2 { 1 }
+ \tl_put_right:Nx \l__nmc_accum_tl { \fp_use:N \l__nmc_fact_fp }
+ }
+\cs_new_protected:Npn \__nmc_fpify_BE:
+ {
+ \__nmc_next:
+ \tl_if_in:VnT \l__nmc_B_tl { alignat }
+ { \__nmc_next: }
+ }
+% \cs { m }
+\cs_new_protected:Npn \__nmc_fpify_absorbm:
+ { \__nmc_next: }
+% \cs { o m }
+\cs_new_protected:Npn \__nmc_fpify_absorbom:
+ {
+ \str_if_eq:VnT \l__nmc_C_tl { [ }
+ {
+ \__nmc_next:
+ \__nmc_get_arg_LR:NNN [ \l__nmc_toss_tl ]
+ }
+ \__nmc_next:
+ }
+\cs_new_protected:Npn \__nmc_fpify_font:
+ { \__nmc_fpify_meta: }
+
+\cs_new_protected:Npn \__nmc_fpify_meta:
+ {
+ \__nmc_next:
+ \__nmc_fpify:VN \l__nmc_B_tl \l__nmc_accum_tl
+ }
+\cs_new_protected:Npn \__nmc_fpify_metamm:
+ { \prg_replicate:nn { 2 } { \__nmc_fpify_meta: } }
+%%%%%%%%%%%%%%%%%%%
+% sum/prod
+\cs_new_protected:Npn \__nmc_fpify_sum:
+ {
+ \__nmc_insert_aster:
+ \group_begin:
+ \tl_if_eq:VnTF \l__nmc_B_tl { \sum }
+ { % sum
+ \tl_set:Nn \l__nmc_sum_op_tl { + }
+ \fp_zero:N \l__nmc_sum_total_fp
+ \fp_set:Nn \l__nmc_sum_prev_fp { 1 }
+ \tl_set:Nn \l__nmc_sum_type_tl { sum }
+ \bool_set_true:N \l__nmc_sum_bool
+ }
+ { % product
+ \tl_set:Nn \l__nmc_sum_op_tl { * }
+ \fp_set:Nn \l__nmc_sum_total_fp { 1 }
+ \fp_set:Nn \l__nmc_sum_prev_fp { 2 }
+ \tl_set:Nn \l__nmc_sum_type_tl { product }
+ \bool_set_false:N \l__nmc_sum_bool
+ }
+ \__nmc_error_where:n { \l__nmc_sum_type_tl }
+
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \__nmc_error_where:n { \l__nmc_sum_type_tl~\ limits }
+ \__nmc_sum_get_limits:NNN \l__nmc_sum_var_tl
+ \l__nmc_sum_index_int \l__nmc_sum_end_int
+ }
+ \bool_if:NTF \g__nmc_error_bool
+ { \group_end: }
+ { % get the summand/multiplicand
+ \__nmc_delim_arg:Nnn \l__nmc_arg_tl { 1 } { \c__nmc_sum_int }
+ \tl_if_eq:VnF \l__nmc_C_tl { ( }
+ { \__nmc_parenth:N \l__nmc_arg_tl }
+ \tl_set_eq:NN \l__nmc_summand_tl \l__nmc_arg_tl
+ % do the sum/prod; result in \l__nmc_sum_total_fp
+ \__nmc_error_where:n { \l__nmc_sum_type_tl }
+ \__nmc_sum_do:nnN { \l__nmc_sum_index_int }
+ { \l__nmc_sum_end_int } \l__nmc_sum_total_fp
+ \fp_gset_eq:NN \g_tmpa_fp \l__nmc_sum_total_fp
+ \tl_gset_eq:NN \g_tmpa_tl \l__nmc_A_tl
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_C_tl \l__nmc_C_tl
+ \tl_put_right:Nx \l__nmc_accum_tl { \fp_use:N \g_tmpa_fp }
+ \tl_set_eq:NN \l__nmc_A_tl \g_tmpa_tl
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_get_limits:NNN #1#2#3
+ { % #1 (tl) sum var #2, #3 (ints) lower, upper limits
+ \str_if_eq:VnTF \l__nmc_C_tl { _ }
+ {
+ \__nmc_sum_limit_lower:NN #1#2
+ \bool_if:NF \g__nmc_error_bool
+ { \__nmc_sum_limit_upper:N #3 }
+ }
+ {
+ \__nmc_sum_limit_upper:N #3
+ \__nmc_sum_limit_lower:NN #1#2
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_limit_lower:NN #1#2
+ {
+ \__nmc_next: \__nmc_next:
+ \bool_if:NT \l__nmc_multitok_bool
+ { % may not be in vv-list
+ \seq_clear:N \l_tmpc_seq
+ \seq_push:NV \l_tmpc_seq \l__nmc_B_tl
+ \__nmc_prep_multitok:NN \l_tmpc_seq \l__nmc_A_tl
+ \seq_pop:NN \l_tmpc_seq \l__nmc_B_tl
+ }
+ \str_if_in:NnTF \l__nmc_B_tl { = }
+ {
+ \__nmc_vv_get_vars_vals_lims:NN \l__nmc_B_tl \c_empty_prop
+ \tl_set_eq:NN #1 \l__nmc_eq_var_tl
+ \__nmc_sum_limit_assign:NN #2 \l__nmc_vv_fp_expr_tl
+ \seq_if_in:NVF \l__nmc_calc_fn_seq \l__nmc_B_tl
+ { \seq_put_left:NV \l__nmc_calc_fn_seq \l__nmc_B_tl }
+ }
+ {
+ \__nmc_error_what:n
+ { No~\__nmc_verb:V \l__nmc_sum_type_tl\ variable~in }
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_limit_upper:N #1
+ {
+ \__nmc_next: \__nmc_next:
+ \__nmc_vv_get_vars_vals_lims:NN \l__nmc_B_tl \c_empty_prop
+ \__nmc_sum_limit_assign:NN #1 \l__nmc_vv_fp_expr_tl
+ }
+\cs_new_protected:Npn \__nmc_sum_limit_assign:NN #1#2
+ {
+ \tl_if_in:VnTF #2 { inf }
+ { \int_set:Nn #1 { \fp_sign:n { #2 } * \c_max_int } }
+ {
+ \fp_compare:nNnTF { abs(#2) } > { \c_max_int }
+ { \int_set:Nn #1 { \fp_sign:n { #2 } * \c_max_int } }
+ {
+ \__nmc_intify:Nn #2 { -inf }
+ \bool_if:NF \g__nmc_error_bool
+ { \int_set:Nn #1 { #2 } }
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_sum_do:nnN #1#2#3
+ { % #1,#2 lower, upper limits (int), #3 <-- total (fp)
+ \int_compare:nNnTF { \c_max_int } =
+ { \int_max:nn { \int_abs:n { #1 } } { \int_abs:n { #2 } } }
+ {
+ \bool_if:NTF \l__nmc_sum_bool
+ { \__nmc_sum_do_infinite:Nn #1 { sum } }
+ { \__nmc_sum_do_infinite:Nn #1 { prod } }
+ }
+ { \int_step_function:nnnN { #1 } { 1 } { #2 } \__nmc_sum_eval:n }
+ \bool_if:NF \g__nmc_error_bool
+ {
+ \int_compare:nTF { #1 <= #2 }
+ { \fp_set:Nn #3 { \l__nmc_sum_total_fp } }
+ {
+ \tl_if_eq:VnTF \l__nmc_sum_op_tl { * }
+ { \fp_set:Nn #3 { 1 } }
+ { \fp_set:Nn #3 { 0 } }
+ }
+ }
+ }
+\cs_set:Npn \__nmc_sum_do_infinite:Nn #1#2
+ {
+ \int_set:Nn \l__nmc_sum_round_int
+ { \l__nmc_round_int + \use:c { l__nmc_#2_extra_int } }
+ \int_set:Nn \l__nmc_suma_int { #1 - 1 }
+ % check loop
+ \bool_do_while:nn { \l__nmc_sum_cont_bool }
+ { % main loop
+ \bool_set_false:N \l__nmc_sum_cont_bool
+ \fp_until_do:nNnn { \l__nmc_sum_prev_fp }
+ = { \l__nmc_sum_rounded_fp }
+ {
+ \int_incr:N \l__nmc_suma_int
+ \fp_set_eq:NN \l__nmc_sum_prev_fp \l__nmc_sum_rounded_fp
+ \__nmc_sum_eval:n { \l__nmc_suma_int }
+ \fp_set:Nn \l__nmc_sum_rounded_fp
+ { round( \l__nmc_sum_total_fp, \l__nmc_sum_round_int ) }
+ }
+ % query terms
+ \int_set:Nn \l__nmc_sumb_int
+ { \l__nmc_suma_int + \use:c { l__nmc_#2_query_int } }
+ \int_while_do:nNnn { \l__nmc_suma_int } < { \l__nmc_sumb_int }
+ {
+ \int_incr:N \l__nmc_suma_int
+ \__nmc_sum_query:n { \l__nmc_suma_int }
+ }
+ }
+ \tl_gset:cx { g__nmc_ \tl_range:Nnn \l__nmc_sum_type_tl{1}{4} _info_tl }
+ { \int_eval:n { \l__nmc_suma_int - #1
+ - \use:c { l__nmc_#2_query_int } } }
+ }
+\cs_new_protected:Npn \__nmc_sum_eval:n #1
+ {
+ \__nmc_calc_fn_val:VNnN \l__nmc_sum_var_tl \l__nmc_summand_tl
+ { #1 } \l_tmpb_fp
+ \fp_set:Nn \l__nmc_sum_total_fp
+ { \l__nmc_sum_total_fp \l__nmc_sum_op_tl \l_tmpb_fp }
+ }
+\cs_new_protected:Npn \__nmc_sum_query:n #1
+ {
+ \__nmc_sum_eval:n { #1 }
+ \fp_set:Nn \l__nmc_sum_rounded_fp
+ { round( \l__nmc_sum_total_fp, \l__nmc_sum_round_int ) }
+ \fp_compare:nNnF { \l__nmc_sum_prev_fp } = { \l__nmc_sum_rounded_fp }
+ {
+ \bool_set_true:N \l__nmc_sum_cont_bool
+ \int_set_eq:NN \l__nmc_sumb_int \l__nmc_suma_int
+ }
+ }
+\cs_new_protected:Npn \__nmc_fpify_cmd:
+ {
+ \__nmc_insert_aster:
+ \str_case:onT \l__nmc_B_tl
+ {
+ { \eval } { \__nmc_cmd:n { eval } }
+ { \iter } { \__nmc_cmd:n { iter } }
+ { \solve } { \__nmc_cmd:n { solve } }
+ { \recur } { \__nmc_cmd:n { recur } }
+ { \tabulate } { \__nmc_cmd:n { table } }
+ { \nmcInfo } { \nmcInfo }
+ { \nmcEvaluate } { \__nmc_cmd:n { eval } }
+ { \nmcIterate } { \__nmc_cmd:n { iter } }
+ { \nmcSolve } { \__nmc_cmd:n { solve } }
+ { \nmcRecur } { \__nmc_cmd:n { recur } }
+ { \nmcTabulate } { \__nmc_cmd:n { table } }
+ }
+ {
+ \bool_set_false:N \l__nmc_insert_aster_bool
+ \bool_set_false:N \l__nmc_wraps_math_bool
+ }
+ }
+\cs_new_protected:Npn \__nmc_cmd:n #1
+ { % \cm { s O m O O }
+ \tl_set:Nn \l__nmc_delim_arg_tl { { #1 } } % cmd
+ \tl_if_eq:VnT \l__nmc_C_tl { * }
+ { \__nmc_next: }
+ \__nmc_append_sqbrarg: % settings
+ \__nmc_next:
+ \tl_put_right:Nx \l__nmc_delim_arg_tl { { \exp_not:o \l__nmc_B_tl } }
+ \__nmc_append_sqbrarg: % vv-list
+ \__nmc_append_sqbrarg: % num format
+ \tl_put_right:Nn \l__nmc_accum_tl { ( }
+ \group_begin:
+ \bool_set_true:N \l__nmc_num_only_bool
+ \exp_last_unbraced:No \__nmc_get_inputs:nnnnn \l__nmc_delim_arg_tl
+ \bool_set_false:N \l__nmc_sci_num_out_bool
+ \bool_if:NF \g__nmc_error_bool
+ { \use:c { __nmc_#1_process: } }
+ \exp_args:NNNV
+ \group_end:
+ \tl_put_right:Nn \l__nmc_accum_tl { \l_nmc_result_tl ) }
+ }
+\cs_new_protected:Npn \__nmc_append_sqbrarg:
+ {
+ \tl_if_eq:VnTF \l__nmc_C_tl { [ }
+ { \__nmc_append_sqbrarg_aux: }
+ { \tl_put_right:Nx \l__nmc_delim_arg_tl { {} } }
+ }
+\cs_new_protected:Npn \__nmc_append_sqbrarg_aux:
+ {
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_get_arg_LR:NNN [ \l_tmpa_tl ]
+ \tl_put_right:Nx \l__nmc_delim_arg_tl { { \exp_not:o \l_tmpa_tl } }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% \nmc_delim_arg:Nnn
+\tl_new:N \l__nmc_delim_arg_tl
+\tl_new:N \l__nmc_delim_argi_tl
+\int_new:N \l__nmc_delim_pas_int
+\int_new:N \l__nmc_delim_pos_int
+\bool_new:N \l__nmc_delim_stop_bool
+\bool_new:N \l__nmc_delim_lmod_bool
+
+\prg_new_conditional:Npnn \__nmc_if_pospas_gtr:nn #1#2 { p, TF }
+ {
+ \bool_if:nTF
+ {
+ \int_compare_p:nNn { \l__nmc_delim_pos_int } > { #1 }
+ &&
+ \int_compare_p:nNn { \l__nmc_delim_pas_int } > { #2 }
+ }
+ { \prg_return_true: }
+ { \prg_return_false: }
+ }
+%--------------------------------------------------------------------
+% #1(tlvar) receives arg (\l__nmc_delim_arg_tl)
+% #2 (int) initial pos
+% #3 (int) parsing state
+\cs_new_protected:Npn \__nmc_delim_arg:Nnn #1#2#3
+ {
+ \group_begin:
+ \bool_set_false:N \__nmc_delim_stop_bool
+ \bool_set_false:N \l__nmc_delim_lmod_bool
+ \int_set:Nn \l__nmc_delim_pos_int { #2 }
+ \int_set:Nn \l__nmc_delim_pas_int { #3 }
+ \tl_clear:N \l__nmc_delim_arg_tl
+ \bool_until_do:nn
+ {
+ \l__nmc_delim_stop_bool ||
+ \quark_if_nil_p:N \l__nmc_C_tl
+ }
+ {
+ \prop_get:NVNTF \g__nmc_class_prop \l__nmc_C_tl \l__nmc_class_tl
+ { \exp_last_unbraced:NV \use_ii:nn \l__nmc_class_tl }
+ { \__nmc_delim_var: } % unknown; treat as var
+ \int_incr:N \l__nmc_delim_pos_int
+ }
+ \tl_gset_eq:NN \g_tmpa_tl \l__nmc_A_tl
+ \tl_gset_eq:NN \g_tmpb_tl \l__nmc_delim_arg_tl
+ \exp_args:NNNV
+ \group_end:
+ \tl_set:Nn \l__nmc_C_tl \l__nmc_C_tl
+ \tl_set_eq:NN \l__nmc_A_tl \g_tmpa_tl
+ \tl_set_eq:NN #1 \g_tmpb_tl
+ }
+%%%%%%%%% delimit arg routines %%%%%%%%
+\cs_new_protected:Npn \__nmc_delim_stop:
+ {
+ \tl_if_eq:VnTF \l__nmc_C_tl { \q }
+ { \bool_set_false:N \l__nmc_delim_stop_bool }
+ { \bool_set_true:N \l__nmc_delim_stop_bool }
+ }
+\cs_new_protected:Npn \__nmc_delim_stop_if_gtr:n #1
+ {
+ \int_compare:nNnT { \l__nmc_delim_pas_int } > { #1 }
+ { \__nmc_delim_stop: }
+ }
+\cs_new_protected:Npn \__nmc_delim_append:
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ }
+\cs_new_protected:Npn \__nmc_delim_append_braced:
+ {
+ \__nmc_next:
+ \tl_put_right:Nx \l__nmc_delim_arg_tl { { \exp_not:o\l__nmc_B_tl } }
+ }
+\cs_new_protected:Npn \__nmc_delim_append_twobraced:
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_append_braced:
+ }
+\cs_new_protected:Npn \__nmc_delim_append_sqbra:
+ {
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_get_arg_LR:NNN [ \l_tmpa_tl ]
+ \__nmc_put_right_wrap:NnNn \l__nmc_delim_arg_tl
+ { [ } \l_tmpa_tl { ] }
+ }
+\cs_new_protected:Npn \__nmc_delim_dec:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \__nmc_get_dec:NN \l_tmpa_tl \c_true_bool
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l_tmpa_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_var:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_const:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_arith:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_pm:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_comparison:
+ {
+ \int_compare:nNnT { \l__nmc_delim_pas_int } = { \c__nmc_cmp_int }
+ {
+ \tl_put_left:NV \l__nmc_A_tl \l__nmc_delim_arg_tl
+ \tl_put_left:Nn \l__nmc_A_tl { \land }
+ \tl_put_left:Nn \l__nmc_C_tl { \land }
+ }
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_and_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_cmp_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_andor:
+ { \__nmc_delim_stop: }
+
+\cs_new_protected:Npn \__nmc_delim_lparen:
+ {
+ \bool_lazy_or:nnTF { \__nmc_if_pospas_gtr_p:nn { 1 } { \c__nmc_trg_int } }
+ { \__nmc_if_pospas_gtr_p:nn { \l__nmc_posi_int } { \c__nmc_sum_int } }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \str_case:on \l__nmc_B_tl
+ {
+ { ( } { \__nmc_delim_lparen:nn { ( } { ) } }
+ { [ } { \__nmc_delim_lparen:nn { [ } { ] } }
+ { \{ } { \__nmc_delim_lparen:nn {\{ } {\} } }
+ { \lparen } { \__nmc_delim_lparen:nn { \lparen } { \rparen } }
+ { \lbrack } { \__nmc_delim_lparen:nn { \lbrack } { \rbrack } }
+ { \lbrace } { \__nmc_delim_lparen:nn { \lbrace } { \rbrace } }
+ }
+ \__nmc_delim_lparen_auxi:n { \l__nmc_posi_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_lparen:nn #1#2
+ {
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { ( }
+ \__nmc_get_arg_LR:NNN #1 \l__nmc_delim_arg_tl #2
+ \bool_if:NT \l__nmc_delim_lmod_bool
+ {
+ \tl_set:Nx \l__nmc_delim_arg_tl
+ { \tl_range:Nnn \l__nmc_delim_arg_tl { 1 } { -2 } }
+ }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { ) }
+ \int_incr:N \l__nmc_delim_pos_int
+ }
+\cs_generate_variant:Nn \__nmc_delim_lparen:nn { VV }
+
+\cs_new_protected:Npn \__nmc_delim_lparen_auxi:n #1
+ { % #1 is first pos.
+ \int_compare:nNnTF { \l__nmc_delim_pas_int } = { \c__nmc_trg_int }
+ { % +1 to prevent ( )( )
+ \__nmc_delim_arg:Nnn
+ \l__nmc_delim_argi_tl { #1 + 1 } { \c__nmc_trg_int }
+ }
+ {
+ \int_compare:nNnT { \l__nmc_delim_pas_int } = { \c__nmc_uny_int }
+ {
+ \__nmc_delim_arg:Nnn
+ \l__nmc_delim_argi_tl { 2 } { \c__nmc_prn_int }
+ }
+ }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+\cs_new_protected:Npn \__nmc_delim_rparen:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_lvert:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ \str_case:on \l__nmc_B_tl
+ {
+ { \lvert } { \__nmc_delim_lvert:nn { \lvert } { \rvert } }
+ { \lceil } { \__nmc_delim_lvert:nn { \lceil } { \rceil } }
+ { \lfloor } { \__nmc_delim_lvert:nn { \lfloor } { \rfloor } }
+ { | }
+ {
+ \__nmc_absval_arg:N \l__nmc_delim_arg_tl
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { | }
+ }
+ }
+ \__nmc_delim_lparen_auxi:n { 0 }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_lvert:nn #1#2
+ {
+ \__nmc_delim_lparen:nn { #1 } { #2 }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { #2 }
+ }
+\cs_new_protected:Npn \__nmc_delim_lmod:
+ {
+ \bool_lazy_and:nnT { \int_compare_p:nNn { \l__nmc_posi_int } < { 3 } }
+ { \int_compare_p:nNn { \l__nmc_delim_pas_int } = { \l__nmc_trg_int } }
+ { \int_incr:N \l__nmc_delim_pas_int }
+ \__nmc_if_pospas_gtr:nnTF { \l__nmc_posi_int } { \l__nmc_trg_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next:
+ \tl_if_in:NVTF \c__nmc_modds_tl \l__nmc_C_tl
+ {
+ \prop_get:NVN \g__nmc_subst_misc_prop \l__nmc_B_tl \l_tmpb_tl
+ \__nmc_delim_lparen:VV \l__nmc_B_tl \l_tmpb_tl
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ \tl_if_eq:VnTF \l__nmc_C_tl { | }
+ { \__nmc_delim_lparen_auxi:n { 0 } }
+ {
+ \__nmc_next:
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_B_tl
+ \__nmc_delim_stop:
+ }
+ }
+ {
+ \int_decr:N \l__nmc_delim_pos_int
+ \bool_set_true:N \l__nmc_delim_lmod_bool
+ }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_unary:n #1
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \tl_if_eq:VnT \l__nmc_C_tl { ^ }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ }
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { #1 }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_power:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_prn_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_tfrac:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append_twobraced: }
+ }
+\cs_new_protected:Npn \__nmc_delim_frac: % also \(d)binom
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append_twobraced:
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_unarybrace:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \str_case:on \l__nmc_C_tl
+ {
+ { * } { \__nmc_delim_append: }
+ { [ } { \__nmc_delim_append_sqbra: }
+ }
+ \__nmc_delim_append_braced:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 2 }
+ { \c__nmc_prn_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_log:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_uny_int }
+ \__nmc_accum_fn_parenth:NNn \l__nmc_delim_arg_tl
+ \l__nmc_delim_argi_tl {}
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_surd:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_uny_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \tl_clear:N \l__nmc_delim_argi_tl
+ \tl_if_eq:VnT \l__nmc_B_tl { \prod }
+ {
+ \prg_replicate:nn { 2 }
+ { \__nmc_delim_append: \__nmc_delim_append_braced: }
+ }
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_srd_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_fact:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_prn_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \tl_if_eq:VnT \l__nmc_C_tl { ! }
+ { \__nmc_delim_append: }
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_qleave:
+ {
+ \__nmc_next:
+ \tl_if_eq:VnTF \l__nmc_B_tl { \q }
+ { \int_zero:N \l__nmc_delim_pos_int } % adhere
+ { \bool_set_true:N \l__nmc_delim_stop_bool } % sever
+ \tl_if_eq:VnT \l__nmc_C_tl { ' }
+ { \__nmc_next: }
+ }
+\cs_new_protected:Npn \__nmc_delim_nary:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_get_arg_LR:NNN ( \l__nmc_B_tl )
+ \__nmc_put_right_wrap:NnNn \l__nmc_delim_arg_tl
+ { ( } \l__nmc_B_tl { ) }
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_sum_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_comma:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ { \__nmc_delim_append: }
+ }
+\cs_new_protected:Npn \__nmc_delim_sum:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_delim_append:
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_append:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_arg:Nnn \l__nmc_delim_argi_tl { 1 } { \c__nmc_sum_int }
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_delim_argi_tl
+ \__nmc_delim_stop_if_gtr:n { \c__nmc_cmp_int }
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_absorb:
+ { % \cs
+ \__nmc_next:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_absorbm:
+ { % \cs { m }
+ \__nmc_next: \__nmc_next:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_absorbom:
+ { % \cs { o m }
+ \__nmc_next:
+ \str_if_eq:VnT \l__nmc_C_tl { [ }
+ {
+ \__nmc_next:
+ \__nmc_get_arg_LR:NNN [ \l__nmc_toss_tl ]
+ }
+ \__nmc_next:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_BE:
+ {
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_cmp_int }
+ { \__nmc_delim_stop: }
+ {
+ \__nmc_next: \__nmc_next:
+ \str_if_in:VnTF \l__nmc_B_tl { alignat }
+ { \__nmc_next: }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { \c_group_begin_token }
+ \__nmc_get_arg_LR:NNN \begin \l__nmc_delim_arg_tl \end
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { \c_group_end_token }
+ \__nmc_next:
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_font:
+ {
+ \__nmc_next:
+ \tl_put_left:NV \l__nmc_A_tl \l__nmc_B_tl
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_metamm:
+ {
+ \__nmc_next:
+ \__nmc_delim_append_braced:
+ \__nmc_delim_append_braced:
+ \int_decr:N \l__nmc_delim_pos_int
+ }
+\cs_new_protected:Npn \__nmc_delim_cmd:
+ { % \cmd { s O m O O }
+ \__nmc_if_pospas_gtr:nnTF { 1 } { \c__nmc_sum_int }
+ { \__nmc_delim_stop: }
+ {
+ \tl_put_right:NV \l__nmc_delim_arg_tl \l__nmc_C_tl
+ \__nmc_next:
+ \tl_if_eq:VnT \l__nmc_C_tl { * }
+ { \__nmc_next: }
+ \tl_put_right:Nn \l__nmc_delim_arg_tl { * }
+ \__nmc_delim_nmcfn_aux:NN [ ]
+ \__nmc_next:
+ \tl_put_right:Nx
+ \l__nmc_delim_arg_tl { { \exp_not:o \l__nmc_B_tl } }
+ \__nmc_delim_nmcfn_aux:NN [ ]
+ \__nmc_delim_nmcfn_aux:NN [ ]
+ }
+ }
+\cs_new_protected:Npn \__nmc_delim_nmcfn_aux:NN #1#2
+ {
+ \tl_if_eq:VnT \l__nmc_C_tl { [ }
+ {
+ \__nmc_next:
+ \tl_clear:N \l_tmpa_tl
+ \__nmc_get_arg_LR:NNN #1 \l_tmpa_tl ]
+ \__nmc_put_right_wrap:NnNn \l__nmc_delim_arg_tl
+ { #1 } \l_tmpa_tl { #2 }
+ }
+ }
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% save & retrieve
+\tl_new:N \g__nmc_reuse_tl
+\tl_new:N \g__nmc_reuse_filename_tl
+\tl_new:N \l__nmc_reuse_retrieved_tl
+\bool_new:N \l__nmc_reuse_retrieved_bool
+\bool_new:N \g__nmc_lyx_bool
+\ior_new:N \g__nmc_ior
+\iow_new:N \g__nmc_iow
+\tl_gset:NV \g__nmc_reuse_filename_tl \jobname
+\tl_gput_right:Nn \g__nmc_reuse_filename_tl { .nmc }
+%
+\NewDocumentCommand \nmcReuse { s >{ \TrimSpaces } O{} }
+ { % save \eval result to csname #2
+ \tl_if_empty:nTF { #2 }
+ { \__nmc_reuse_retrieve: }
+ {
+ \tl_if_empty:NTF \g__nmc_reuse_tl
+ {
+ \__nmc_error_where:n { \__nmc_verb:n { \nmcReuse }~command }
+ \__nmc_error_what:n { No~value~to~save~with }
+ }
+ { \__nmc_reuse_result:c { #2 } }
+ }
+ \IfBooleanT #1 { \__nmc_debug_display:nn { 99991 } { reuse } }
+ }
+\ProvideDocumentCommand\reuse {} { \nmcReuse }
+\cs_new_protected:Npn \__nmc_reuse_retrieve:
+ {
+ \file_get:VnNT \g__nmc_reuse_filename_tl {} \l__nmc_reuse_retrieved_tl
+ {
+ \bool_lazy_or:nnF
+ { \tl_if_blank_p:V \l__nmc_reuse_retrieved_tl }
+ { \tl_if_head_eq_meaning_p:VN \l__nmc_reuse_retrieved_tl \par }
+ {
+ \seq_set_split:NnV \l_tmpa_seq {,} \l__nmc_reuse_retrieved_tl
+ \seq_map_inline:Nn \l_tmpa_seq
+ { \__nmc_reuse_defcmds:Nnn ##1 { _props } }
+ }
+ }
+ \bool_set_true:N \l__nmc_reuse_retrieved_bool
+ }
+\cs_new_protected:Npn \__nmc_reuse_result:N #1
+ {
+ \tl_set:Nn \l_tmpb_tl { #1 }
+ \tl_put_right:Nx \l_tmpb_tl { { \tl_to_str:N \g__nmc_reuse_tl } }
+ \bool_if:NF \l__nmc_reuse_retrieved_bool
+ { \__nmc_reuse_retrieve: }
+ \clist_put_right:NV \l__nmc_reuse_retrieved_tl \l_tmpb_tl
+ \exp_last_unbraced:NV\__nmc_reuse_defcmds:Nnn \l_tmpb_tl {}
+ \tl_gclear:N \g__nmc_reuse_tl
+ }
+\cs_generate_variant:Nn \__nmc_reuse_result:N { c }
+
+\cs_new_protected:Npn \__nmc_reuse_defcmds:Nnn #1#2#3
+ {
+ \prop_get:NnNTF \g__nmc_subst_var_prop { #1 } \l__nmc_toss_tl
+ { \RenewDocumentCommand #1{}{#2} }
+ { \NewDocumentCommand #1{}{#2} }
+ \tl_set_rescan:Nnn #1 { \ExplSyntaxOn } { #2 }
+ \use:c { __nmc_reuse_write#3:Nn } #1{#2}
+ }
+\cs_new_protected:Npn \__nmc_reuse_write:Nn #1#2
+ {
+ \__nmc_reuse_write_props:Nn #1{#2}
+ \file_get:VnNTF \g__nmc_reuse_filename_tl {} \l_tmpa_tl
+ {
+ \tl_if_head_eq_meaning:VNTF \l_tmpa_tl \par
+ { \tl_clear:N \l_tmpa_tl }
+ {
+ \__nmc_reuse_delete:NN #1 \l_tmpa_tl
+ \tl_if_blank:VF \l_tmpa_tl
+ {
+ \tl_trim_spaces:N \l_tmpa_tl
+ \tl_put_right:Nn \l_tmpa_tl {,}
+ }
+ }
+ }
+ { \tl_clear:N \l_tmpa_tl }
+ \tl_put_right:Nn \l_tmpa_tl { #1 {#2} }
+ \iow_open:NV \g__nmc_iow \g__nmc_reuse_filename_tl
+ \iow_now:NV \g__nmc_iow \l_tmpa_tl
+ \iow_close:N \g__nmc_iow
+ }
+\cs_new_protected:Npn \__nmc_reuse_delete:NN #1#2
+ {
+ \seq_clear:N \l_tmpb_seq
+ \seq_set_split:NnV \l_tmpa_seq {,} #2
+ \seq_map_inline:Nn \l_tmpa_seq
+ {
+ \tl_if_head_eq_meaning:nNF { ##1 } #1
+ { \seq_put_right:Nn \l_tmpb_seq { ##1 } }
+ }
+ \tl_set:Nx #2 { \seq_use:Nn \l_tmpb_seq {,} }
+ }
+\cs_new_protected:Npn \__nmc_reuse_write_props:Nn #1#2
+ {
+ \prop_put_if_new:Nnn \g__nmc_class_prop { #1 }
+ { \__nmc_fpify_var: \__nmc_delim_var: }
+ \prop_put:Nnn \g__nmc_subst_var_prop { #1 } { #2 }
+ }
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% settings
+\bool_new:N \l__nmc_multitok_bool
+\bool_new:N \l__nmc_vv_multline_bool
+\bool_new:N \l__nmc_deg_bool
+\tl_new:N \l__nmc_eq_num_tl
+\int_new:N \l__nmc_posi_int
+\int_new:N \l__nmc_eval_reuse_int
+\int_new:N \l__nmc_debug_int
+\tl_new:N \l__nmc_reuse_sep_tl % numerica-plus
+\int_set_eq:NN \l__nmc_posi_int \c_one_int
+
+\cs_new_protected:Npn \__nmc_set_trig_parsing:n #1
+ { % #1=1,2,3
+ \int_set:Nn\l__nmc_posi_int { #1 }
+ \int_compare:nNnTF { #1 } > { 1 }
+ { % #1=2,3
+ \int_set_eq:NN \l__nmc_trg_int \c__nmc_trg_int
+ \int_compare:nNnT { #1 } > { 2 }
+ { % #1=3
+ \__nmc_fill_class:nn { \frac, \dfrac }
+ { \__nmc_fpify_frac: \__nmc_delim_tfrac: }
+ }
+ }
+ { \int_set_eq:NN \l__nmc_trg_int \c__nmc_uny_int } % #1=1
+ }
+\cs_new:Npn \__nmc_deg:
+ { \bool_if:NT { \l__nmc_deg_bool } { d } }
+\cs_new:Npn \__nmc_int_to_bool:Nn #1#2
+ { \bool_set:Nn #1 { !\int_if_zero_p:n { #2 } } }
+\keys_define:nn { numerica }
+ {
+ rounding .int_set:N = \l__nmc_round_int,
+ pad .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_pad_zeros_bool { #1 },
+ output-sci-notation .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_sci_num_out_bool { #1 },
+ output-exponent-char .tl_set:N = \l__nmc_sci_num_out_tl,
+ input-sci-notation .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_sci_num_in_bool { #1 },
+ input-exponent-char .tl_set:N = \l__nmc_sci_num_in_tl,
+ multitoken-variables .code:n = \__nmc_int_to_bool:Nn
+ \l__nmc_multitok_bool { #1 },
+ logarithm-base .tl_set:N = \l__nmc_log_base_tl,
+ intify-rounding .tl_set:N = \l__nmc_intify_round_tl,
+ vv-display .tl_set:N = \l__nmc_vv_display_tl,
+ vv-inline .tl_set:N = \l__nmc_vv_inline_tl,
+ sum-extra-rounding .int_set:N = \l__nmc_sum_extra_int,
+ sum-query-terms .int_set:N = \l__nmc_sum_query_int,
+ prod-extra-rounding .int_set:N = \l__nmc_prod_extra_int,
+ prod-query-terms .int_set:N = \l__nmc_prod_query_int,
+ eval-reuse .int_set:N = \l__nmc_eval_reuse_int,
+%
+ rounding .default:n = 6,
+ pad .default:n = 0,
+ output-sci-notation .default:n = 0,
+ output-exponent-char .default:n = e,
+ input-sci-notation .default:n = 0,
+ input-exponent-char .default:n = e,
+ multitoken-variables .default:n = 1,
+ logarithm-base .default:n = 10,
+ intify-rounding .default:n = 14,
+ vv-display .default:n = {,}\mskip 36mu minus 24mu(vv),
+ vv-inline .default:n = {,}\mskip 12mu plus 6mu minus 9mu(vv),
+ sum-extra-rounding .default:n = 2,
+ sum-query-terms .default:n = 0,
+ prod-extra-rounding .default:n = 2,
+ prod-query-terms .default:n = 0,
+ eval-reuse .default:n = 0,
+ }
+\keys_set_known:nn { numerica }
+ {
+ rounding,pad,output-sci-notation,output-exponent-char,
+ input-sci-notation,input-exponent-char,
+ multitoken-variables,logarithm-base,intify-rounding,
+ vv-display,vv-inline,sum-extra-rounding,sum-query-terms,
+ prod-extra-rounding,prod-query-terms,eval-reuse
+ }
+\keys_define:nn { numerica/generic }
+ {
+ dbg .code:n = \int_set:Nn \l__nmc_debug_int
+ { \int_compare:nNnTF { \int_abs:n { #1 } } = { 1 }
+ { \int_sign:n { #1 }*2*3*5*7 } { #1 } },
+ ^ .code:n = \bool_set_true:N \l__nmc_sci_num_in_bool
+ \tl_set:Nn \l__nmc_sci_num_in_tl { #1 },
+ xx .code:n = \__nmc_int_to_bool:Nn \l__nmc_multitok_bool { #1 },
+ () .code:n = \__nmc_set_trig_parsing:n { 2+\int_sign:n { #1-1 } },
+ () .initial:n = 0,
+ o .code:n = \bool_set_true:N \l__nmc_deg_bool,
+ log .tl_set:N = \l__nmc_log_base_tl,
+ vvmode .code:n = \__nmc_calc_mode:n { \int_sign:n { #1*#1 } },
+ vvd .code:n = \tl_set:Nn \l__nmc_vv_display_tl { #1 }
+ \tl_if_in:NnT \l__nmc_vv_display_tl { \\ }
+ { \bool_set_true:N \l__nmc_vv_multline_bool },
+ vvi .tl_set:N = \l__nmc_vv_inline_tl,
+ * .code:n = \tl_set:Nn \l__nmc_eq_num_tl { * },
+ p .tl_set:N = \l__nmc_punc_tl,
+ p .default:n = {,},
+ S+ .int_set:N = \l__nmc_sum_extra_int,
+ S? .int_set:N = \l__nmc_sum_query_int,
+ P+ .int_set:N = \l__nmc_prod_extra_int,
+ P? .int_set:N = \l__nmc_prod_query_int,
+ }
+\keys_define:nn { numerica/eval }
+ { reuse .int_set:N = \l__nmc_eval_reuse_int }
+\keys_define:nn { numerica/package }
+ {
+ plus .code:n = \file_input:n { numerica-plus.def },
+ tables .code:n = \file_input:n { numerica-tables.def },
+ % calculus .code:n = \file_input:n { numerica-calculus.def },
+ lyx .code:n = \file_input:n { numerica-lyx.def }
+ }
+\ProcessKeysPackageOptions { numerica/package }
+
+\file_if_exist:nT { numerica.cfg }
+ {
+ \file_get:nnN { numerica.cfg } {} \l_tmpa_tl
+ \exp_args:NnV \keys_set_known:nn { numerica } \l_tmpa_tl
+ }
+% end of `numerica.sty' \ No newline at end of file
diff --git a/macros/latex/contrib/verifiche/Esempio.pdf b/macros/latex/contrib/verifiche/Esempio.pdf
deleted file mode 100644
index 509d6e67be..0000000000
--- a/macros/latex/contrib/verifiche/Esempio.pdf
+++ /dev/null
Binary files differ
diff --git a/macros/latex/contrib/verifiche/README.md b/macros/latex/contrib/verifiche/README.md
index 348e4c9b58..75f142ca02 100644
--- a/macros/latex/contrib/verifiche/README.md
+++ b/macros/latex/contrib/verifiche/README.md
@@ -1,7 +1,7 @@
# Package Verifiche
## Abstract
-The purpose of the package is to mange the exercises for a test, their points, the difficulty level and their solutions.
+The purpose of the package is to manage the exercises for a test, their points, the difficulty level and their solutions.
Some typical format of exercises are already implemented:
1. Plain exercise
2. Complete the Text - exercise
@@ -10,6 +10,11 @@ Some typical format of exercises are already implemented:
5. Open questtions
6. Find the Error - exercise
+## Informations
+Author: Francesco Raccanello
+Mail: raccanello.francesco@gmail.com
+License: LPPL 1.3
+
## Installation
To install the package run:
- `latex verifiche.ins`
@@ -21,7 +26,7 @@ To install the package run:
tree (e.g. \tex\latex\verifiche).
## Example
-Run `pdflatex Esempio.tex` to see some examples.
+Run `pdflatex verifiche-example.tex` to see some examples.
### To do
diff --git a/macros/latex/contrib/verifiche/Esempio.tex b/macros/latex/contrib/verifiche/verifiche-example.tex
index b84e48eef6..361c7cd52a 100644
--- a/macros/latex/contrib/verifiche/Esempio.tex
+++ b/macros/latex/contrib/verifiche/verifiche-example.tex
@@ -1,6 +1,6 @@
% compile 2 times
\documentclass[a4paper]{article}
-\usepackage[solutions
+\usepackage[solutions%show/hide the solutions
]{verifiche}
\usepackage[margin=3cm]{geometry}
\usepackage{lipsum}
@@ -16,6 +16,8 @@
\instruction{Risolvere il seguente esercizio nel più breve tempo possibile.\\}
\printheading
+\pagestyle{verifiche}
+
\begin{esercizio}
Risolvere il seguente problema.
@@ -69,6 +71,34 @@ Calcolare
\]
\end{esercizio*}
+\ptprefix{.../}
+\ptdelimiters{[]}
+\ptlabel{punti}
+\begin{esercizio}[pt=2]
+Calcolare la seguente derivata:
+\[
+\frac{d e^x}{dx} =
+\]
+\end{esercizio}
+
+\partialptprefix{.../}
+\partialptdelimiters{{}{}}
+\partialptlabel[punto]{punti}
+\begin{esercizio}
+Risolvere i seguenti quesiti:
+\begin{enumerate}[(i)]
+\item\partialpt{2}\noindent Calcolare il seguente integrale indefinito:
+\[
+\int e^x dx =
+\]
+\item\partialpt{1}\noindent Calcolare il seguente integrale definito:
+\[
+\int_0^1 e^x dx =
+\]
+\end{enumerate}
+\end{esercizio}
+
+
\begin{soluzione}
Questa è la soluzione dell'esercizio \ref{es:ferropiuma} a pagina \pageref{es:ferropiuma}.
\end{soluzione}
diff --git a/macros/latex/contrib/verifiche/verifiche.dtx b/macros/latex/contrib/verifiche/verifiche.dtx
index 8bad537777..405d7ec732 100644
--- a/macros/latex/contrib/verifiche/verifiche.dtx
+++ b/macros/latex/contrib/verifiche/verifiche.dtx
@@ -16,7 +16,7 @@
%
% \iffalse
%<package>\NeedsTeXFormat{LaTeX2e}
-%<package>\ProvidesPackage{verifiche} [2020/07/01 v2.2 .dtx verifiche file]
+%<package>\ProvidesPackage{verifiche} [2021/02/14 v4.0 .dtx verifiche file]
%<*driver>
\documentclass{ltxdoc}
\usepackage[utf8]{inputenc}
@@ -25,10 +25,10 @@
\usepackage{listings}
\lstset{
language=[LaTeX]TeX,
- morekeywords={\institute, \asyear, \testtype, \instruction, \duration, \printheading,
- \institutefont, \asyearfont, \testtypefont, \instructiondelimiter, \instrunctionfont,
+ morekeywords={\setlength, \institute, \asyear, \testtype, \instruction, \duration, \printheading,
+ \institutefont, \asyearfont, \testtypefont, \instructiondelimiter, \headerfont, \instrunctionfont,
\durationfont, \eserciziolabel, \solutionfont, \diffsymb, \closedquestionitem,
- \inlinesol, \completetext, \truefalse, \openquestion, \linedopenquestion, \squaredopenquestion, \checked, \finderror, \textandimage, \partialpt},
+ \inlinesol, \completetext, \truefalse, \openquestion, \linedopenquestion, \squaredopenquestion, \headercandidatenamelabel, \checked, \finderror, \textandimage, \partialpt, \ptprefix, \ptlabel, \partialptlabel, \partialptdelimiters, \ptdelimiters},
keywordstyle=\color{blue},
escapeinside={*@}{@*},
inputencoding=utf8,
@@ -66,11 +66,13 @@
%</driver>
% \fi
%
-% \CheckSum{537}
+% \CheckSum{736}
% \changes{v1.0}{2020/06/14}{Initial version.}
% \changes{v2.0}{2020/06/26}{Better performance in exercise environment.}
% \changes{v2.1}{2020/06/30}{Introduced closedquestionitem to control |item| symbol.}
% \changes{v2.2}{2020/07/01}{Modify reference to points of an exercise.}
+% \changes{v3.0}{2021/02/14}{Remove fancyhdr and introduce a new page style. Improve the printheading command.}
+% \changes{v4.0}{2021/02/15}{Introduce comando to change the aspect of the exercise score.}
% \GetFileInfo{verifiche.sty}
% \DoNotIndex{\newcommand,\newenvironment,\def,\begin,\end,\par}
% \DoNotIndex{\DeclareOption,\ExecuteOptions,\ProcessOptions,\RequirePackage}
@@ -115,10 +117,11 @@
% \maketitle
%\begin{abstract}
%Il presente pacchetto si propone di gestire gli elementi che generalmente compongono
-%una prova scritta per le scuole secondarie di primo e secondo grado.
-%I comandi e gli ambienti implementati permettono di gestire l'intestazione della pagina
-% e alcuni strumenti utili alla stesura di test scolastici.
-% Il pacchetto, inoltre, permette di riportare soluzioni degli esercizi riportandoli, o meno, nello stesso documento in base ad un'apposita opzione.
+%una prova scritta per le scuole secondarie di primo e secondo grado:
+%l'intestazione della pagina, gli esercizi della verifica e le loro soluzioni.
+%Gli esercizi possono presentare una numerazione, un punteggio e un grado di difficoltà.
+%Alcuni comandi permettono di generare alcune tipologie di test: esercizi semplici, esercizi di completamento, vero o falso, domande aperte e chiuse e trova l'errore.
+%Ciascun esercizio può presentare la soluzione corretta che viene riportata nel documento in base ad un'apposita opzione del pacchetto.
%\end{abstract}
%
%\tableofcontents
@@ -141,7 +144,6 @@
%\item |amsmath|, |amssymb| per la gestione della matematica;
%\item |siunitx| per la gestione delle unità di misura;
%\item |graphicx|, |xcolor| e |tikz| per la gestione grafica;
-%\item |famcyhdr| per la gestione delle testatine.
%\end{itemize}
%
%\section{Come usare il pacchetto \textsf{verifiche}}
@@ -275,6 +277,17 @@
%\end{tabular}
%\end{center}
%
+% Il pacchetto definisce due stili di pagina \emph{ad hoc}: |mainverifiche| e |verifiche|. Il primo è evocato quando si definisce il comando |\institute| e stampa sulla prima pagina l'argomento di questo comando. \DescribeMacro{verifiche}Il secondo, invece, può essere utilizzato dall'utente tramite il comando:
+%\iffalse
+%<*example>
+%\fi
+\begin{lstlisting}
+\pagestyle{verifiche}
+\end{lstlisting}
+%\iffalse
+%</example>
+%\fi
+%e riproduce, nelle pagine dispari diversa dalla prima, l'intestazione per l'identificazione del candidato: il nome, la classe e la data.
%\section{Comandi ed ambienti per le \textsf{verifiche}}
%Nel seguito verranno presentati i comandi che permettono di strutturare un test.
%Il pacchetto fornisce un ambiente di esercizio in cui racchiudere le domande.
@@ -441,6 +454,8 @@ che voglio somministrare
%</example>
% \fi
%
+%Quando usato negli ambienti |enumerate| o |itemize|, il comando introduce uno spazio spurio che può essere rimosso posponendo il comando |\noindent|.
+%
%Quando viene esplicitata l'\emph{opzione} |partialpt| nella definizione dell'ambiente esercizio, il pacchetto si occupa della somma complessiva dei singoli punteggi parziali riportandola come punteggio totale del quesito. In questo caso il documento dev'essere compilato \emph{due volte} per ottenere correttamente il risultato finale.\\
%
% \iffalse
@@ -741,13 +756,24 @@ dell'esercizio somministrato
% \fi
%
%\section{Personalizzazione}
-%Il pacchetto prevede un certo numero di comandi definiti al fine di gestire l'aspetto del documento finale.\par
+%Il pacchetto prevede un certo numero di comandi definiti al fine di gestire l'aspetto del documento finale.
+%\subsection*{Font}
%Per l'intestazione tutti i font sono definiti secondo lo schema:
%\begin{center}
%\meta{comando intestazione}|font|
%\end{center}
%ed oltre a gestire l'aspetto del carattere, alcuni di questi, specificano la posizione occupata del testo ed eventuali prefissi.
-
+%
+%\DescribeMacro{\headerfont} Il font delle intestazioni quando viene usato lo stile di pagina |verifiche|.
+%\iffalse
+%<*example>
+%\fi
+\begin{lstlisting}
+\renewcommand{\headerfont}[1]*@\marg{...}@*
+\end{lstlisting}
+%\iffalse
+%</example>
+%\fi
%\DescribeMacro{\institutefont}Il font dell'istituto, generato dal comando |\institute|, è gestito dal comando |\institutefont|. Per variare le caratteristiche del font è necessario ridefinire il comando:
%\iffalse
%<*example>
@@ -792,6 +818,18 @@ dell'esercizio somministrato
%</example>
%\fi
%
+%\DescribeMacro{\solutionfont}Le soluzioni degli esercizi vengono stampate, quando viene esplicitata l'opzione |solutions| quando viene caricato il pacchetto, con un font più piccolo e in corsivo. Per modificare il font delle soluzioni è sufficiente ridefinire il comando |\solutionfont|:
+%\iffalse
+%<*example>
+%\fi
+\begin{lstlisting}
+\renewcommand{\solutionfont}*@\marg{...}@*
+\end{lstlisting}
+%\iffalse
+%</example>
+%\fi
+%
+%\subsection*{Intestazione}
%\DescribeMacro{\instructiondelimiter}Le istruzioni e la durata della prova sono racchiuse tra due delimitatori che normalmente sono due righe orizzontali. Per cambiare, o rimuovere, questi delimitatori è sufficiente ridefinire il seguente comando:
%\iffalse
%<*example>
@@ -803,6 +841,30 @@ dell'esercizio somministrato
%</example>
%\fi
%
+%\DescribeMacro{\headercandidatenamelabel}Quando viene composta l'intestazione principale nella prima pagina, compare una riga in cui il candidato può scrivere il proprio nome e cognome. L'etichetta di questa parte controllata dal comando |\headercandidatenamelabel|, che può essere opportunamente cambiata tramite il comando:
+%\iffalse
+%<*example>
+%\fi
+\begin{lstlisting}
+\renewcommand{\headercandidatenamelabel}*@\marg{...}@*
+\end{lstlisting}
+%\iffalse
+%</example>
+%\fi
+%Allo stesso modo è possibile modificare la descrizione della classe \DescribeMacro{\headerclasslabel} e della data\DescribeMacro{\headerdatelabel}.
+% Tutte queste tre voci prevedono, per ciascuna, una riga vuota ove scrivere l'informazione richiesta.\DescribeMacro{\candidatenamerulerlength}\DescribeMacro{\classrulerlength}\DescribeMacro{\daterulerlength} Le lunghezze delle tre righe possono essere modificate impostando le lunghezze desiderate
+%\iffalse
+%<*example>
+%\fi
+\begin{lstlisting}
+\setlength{\...rulerlength}*@\marg{dim}@*
+\end{lstlisting}
+%\iffalse
+%</example>
+%\fi
+%Attraverso questi comandi è possibile anche rimuovere una (o più) delle tre voci. Se ad esempio non si vuole visualizzare la classe d'appartenenza del candidato, sarà sufficiente ridefinire gli opportuni comandi nel seguente modo: |\renewcommand{\headerclasslabel}{}|, così da non visualizzare alcuna scritta, e |\setlength{\classrulerlength}{0pt}| per rimuovere la riga associata.
+%
+%\subsection*{Ambiente \texttt{esercizio}}
%\DescribeMacro{\eserciziolabel}Gli esercizi vengono inseriti anteponendo al loro testo la dicitura ‘‘Quesito’’ seguito (eventualmente) da un numero. Per cambiare questa dicitura, a livello globale, ovvero per ogni esercizio, è sufficiente ridefinire il comando |\eserciziolabel|.
%\iffalse
%<*example>
@@ -814,6 +876,34 @@ dell'esercizio somministrato
%</example>
%\fi
%
+%L'ambiente |esercizio| permette di riportare anche il punteggio complessivo e parziale per le sotto-domande. Di default il punteggio totale del quesito viene riportato anteponendo un tratto orizzontale seguito da uno slash, il valore del punteggio ed un'etichetta indicante la sigla \emph{pt}.
+% Per quanto riguarda il punteggio parziale, questo viene riportato racchiuso tra parentesi tonde.
+% È possibile modificare l'aspetto di entrambi i punteggio che sono strutturati nel seguente modo: \texttt{prefisso delimitatore punteggio label delimitatore}.
+%
+%\DescribeMacro{\ptprefix} Il comando |\ptprefix| permette di modificare il \texttt{prefisso} del punteggio. Di default traccia una linea di 1 cm ed uno slash. Se si vuole modificarne l'aspetto è sufficiente evocare il comando.
+%\iffalse
+%<*example>
+%\fi
+\begin{lstlisting}
+\ptprefix*@\marg{...}@*
+\end{lstlisting}
+%\iffalse
+%</example>
+%\fi
+%
+%\DescribeMacro{\partialptprefix} Allo stesso modo il comando |\partialptprefix| definisce un \texttt{prefisso} per il punteggio parziale.
+%
+%\DescribeMacro{\ptlabel}\DescribeMacro{\partialptlabel} Il punteggio viene riportato indicando anche una \texttt{label}. Questa di default è "pt" sia per il punteggio parziale che per quello totale. È possibile definire un nome alternativo usando il comando |\ptlabel| per il punteggio totale o |\partialptlabel| per quello parziale. I comandi prevedono un argomento obbligatorio che è valido per tutti i punteggi, ma anche uno facoltativo che viene usato quanto il valore del punteggio è pari a 1 e che necessita quindi di un sostantivo singolare al posto di quello plurale (i.e. "punto" al posto di "punti").
+%\iffalse
+%<*example>
+%\fi
+\begin{lstlisting}
+\ptlabel*@\oarg{sing.}\marg{plur.}@*
+\partialptlabel*@\oarg{sing.}\marg{plur.}@*
+\end{lstlisting}
+%\iffalse
+%</example>
+%\fi
%\DescribeMacro{\diffsymb}L'ambiente esercizi prevede di poter descrivere la difficoltà tramite un simbolo grafico. Di default questo è una stella ($\bigstar$), tuttavia è possibile cambiare il tipo di simbolo ridefinendo il comando |\diffsymb|:
%\iffalse
%<*example>
@@ -825,24 +915,24 @@ dell'esercizio somministrato
%</example>
%\fi
%
-%\DescribeMacro{\closedquestionitem}Il simbolo delle domande chiuse è controllato dal comando |\closedquestionitem| che di default è un quadrato ($\square$). Per modificare questo simbolo è sufficiente ridefinire il comando:
+%\DescribeMacro{\ptdelimiters}\DescribeMacro{\partialptdelimiters} Infine, i punteggi sono racchiusi tra due delimitatori. Il punteggio totale non presenta alcun simbolo, mentre quello parziale è racchiuso tra parentesi tonde. È possibile definire dei nuovi delimitatori utilizzando i comandi |\partialptdelimiters| e |\ptdelimiters| i quali argomenti contengono due caratteri che fungono da delimitatore dentro e sinistro.
%\iffalse
%<*example>
%\fi
\begin{lstlisting}
-\renewcommand{\closedquestionitem}*@\marg{codice del simbolo}@*
+\ptdelimiters*@\marg{coppia di caratteri}@*
+\partialptdelimiters*@\marg{coppia di caratteri}@*
\end{lstlisting}
%\iffalse
%</example>
%\fi
-
%
-%\DescribeMacro{\solutionfont}Le soluzioni degli esercizi vengono stampate, quando viene esplicitata l'opzione |solutions| quando viene caricato il pacchetto, con un font più piccolo e in corsivo. Per modificare il font delle soluzioni è sufficiente ridefinire il comando |\solutionfont|:
+%\DescribeMacro{\closedquestionitem}Il simbolo delle domande chiuse è controllato dal comando |\closedquestionitem| che di default è un quadrato ($\square$). Per modificare questo simbolo è sufficiente ridefinire il comando:
%\iffalse
%<*example>
%\fi
\begin{lstlisting}
-\renewcommand{\solutionfont}*@\marg{...}@*
+\renewcommand{\closedquestionitem}*@\marg{codice del simbolo}@*
\end{lstlisting}
%\iffalse
%</example>
@@ -865,7 +955,6 @@ dell'esercizio somministrato
\RequirePackage{xcolor}
\RequirePackage{tikz}
\RequirePackage{booktabs}
-\RequirePackage{fancyhdr}
\RequirePackage[normalem]{ulem}
\RequirePackage{xstring}
% \end{macrocode}
@@ -941,6 +1030,20 @@ dell'esercizio somministrato
\newcounter{exercisenumber}
\newcounter{partialpoints}[exercisenumber]
% \end{macrocode}
+%\subsection{Length}
+%\iffalse
+%%newlenght
+%\fi
+%In order to fine-tuning the dimension of the rulers in the headers and to leave the possibility to the user to redefine their dimensions, some length are introduced.
+% \begin{macrocode}
+\newlength{\candidatenamerulerlength}
+\newlength{\classrulerlength}
+\newlength{\daterulerlength}
+\setlength{\candidatenamerulerlength}{.35\textwidth}
+\setlength{\classrulerlength}{.1\textwidth}
+\setlength{\daterulerlength}{.15\textwidth}
+% \end{macrocode}
+%
%\subsection{Macro and environment}
%\iffalse
%%macro and environment
@@ -968,6 +1071,10 @@ dell'esercizio somministrato
%%fonts and aspects
%\fi
% \begin{macrocode}
+\newcommand{\headerfont}[1]{%
+ \sffamily\color{darkgray}#1}
+% \end{macrocode}
+% \begin{macrocode}
\newcommand{\institutefont}[1]{%
\large \sffamily\color{darkgray}#1}
% \end{macrocode}
@@ -992,6 +1099,15 @@ dell'esercizio somministrato
\hspace*{\fill}\@durationpreamble\space\bfseries #1\hspace*{\fill}}
% \end{macrocode}
% \begin{macrocode}
+\newcommand{\headercandidatenamelabel}{Nome e Cognome}
+\newcommand{\headerclasslabel}{Classe}
+\newcommand{\headerdatelabel}{Data}
+\newcommand{\@header@candidatename}{\headercandidatenamelabel\space\underline{\hspace{\candidatenamerulerlength}}}
+\newcommand{\@header@class}{\headerclasslabel\space\underline{\hspace{\classrulerlength}}}
+\newcommand{\@header@date}{\headerdatelabel\space\underline{\hspace{\daterulerlength}}}
+\newcommand{\@header}{\@header@candidatename\hfill\@header@class\hfill\@header@date}
+% \end{macrocode}
+% \begin{macrocode}
\newcommand{\eserciziolabel}{%
Quesito}
% \end{macrocode}
@@ -1007,6 +1123,106 @@ dell'esercizio somministrato
\newcommand{\closedquestionitem}{%
$\square$}
% \end{macrocode}
+%\iffalse
+%%pt prefix
+%\fi
+% \begin{macrocode}
+\newcommand{\@ptprefix}{\underline{\hspace{1cm}}/}
+\newcommand{\ptprefix}[1]{%
+ \renewcommand{\@ptprefix}{#1}}
+% \end{macrocode}
+%\iffalse
+%%partialpt prefix
+%\fi
+% \begin{macrocode}
+\newcommand{\@partialptprefix}{}
+\newcommand{\partialptprefix}[1]{%
+ \renewcommand{\@partialptprefix}{#1}}
+% \end{macrocode}
+%\iffalse
+%%pt label
+%\fi
+% \begin{macrocode}
+\newcommand{\@ptsinglabel}{pt}
+\newcommand{\@ptplurlabel}{pt}
+\NewDocumentCommand{\ptlabel}{o m}{%
+ \renewcommand{\@ptplurlabel}{#2}%
+ \renewcommand{\@ptsinglabel}{\IfNoValueTF{#1}{#2}{#1}}}%
+\newcommand{\@useptlabel}[1]{%
+ \ifnum#1=1 \@ptsinglabel \else \@ptplurlabel\fi}
+% \end{macrocode}
+%\iffalse
+%%partialpt label
+%\fi
+% \begin{macrocode}
+\newcommand{\@partialptsinglabel}{pt}
+\newcommand{\@partialptplurlabel}{pt}
+\NewDocumentCommand{\partialptlabel}{o m}{%
+ \renewcommand{\@partialptplurlabel}{#2}%
+ \renewcommand{\@partialptsinglabel}{\IfNoValueTF{#1}{#2}{#1}}}%
+\newcommand{\@usepartialptlabel}[1]{%
+ \ifnum#1=1 \@partialptsinglabel \else \@partialptplurlabel\fi}
+% \end{macrocode}
+%\iffalse
+%%partialpt delimiter
+%\fi
+% \begin{macrocode}
+\newcommand{\@lpartialpt@delimiter}{(}
+\newcommand{\@rpartialpt@delimiter}{)}
+\newcommand{\@definepartialptdelimiter}[2]{
+ \renewcommand{\@lpartialpt@delimiter}{#1}
+ \renewcommand{\@rpartialpt@delimiter}{#2}}
+\newcommand{\partialptdelimiters}[1]{%
+ \@definepartialptdelimiter#1}
+% \end{macrocode}
+%\iffalse
+%%pt delimiter
+%\fi
+% \begin{macrocode}
+\newcommand{\@lpt@delimiter}{}
+\newcommand{\@rpt@delimiter}{}
+\newcommand{\@defineptdelimiter}[2]{%
+ \renewcommand{\@lpt@delimiter}{#1}%
+ \renewcommand{\@rpt@delimiter}{#2}}
+\newcommand{\ptdelimiters}[1]{%
+ \@defineptdelimiter#1}
+% \end{macrocode}
+%\iffalse
+%% convert ref to num
+%\fi
+% \begin{macrocode}
+\newcommand*{\convertreftonum}[1]{%
+ \romannumeral
+ \@ifundefined{r@#1}{%
+ \expandafter\ltx@zero
+ \rc@default}{%
+ \expandafter\expandafter\expandafter\rc@extract@
+ \expandafter\expandafter\expandafter!%
+ \csname r@#1\expandafter\endcsname
+ \expandafter{\rc@default}\@nil}%
+}
+\def\rc@default{0}%
+\long\def\rc@extract@#1#2#3\@nil{%
+ \ltx@zero
+ #2}
+\chardef\ltx@zero=0%
+% \end{macrocode}
+
+%\paragraph{Page styles}
+%The package define two new pagestyle: \texttt{mainverifiche} and \texttt{verifiche}. The former is used if \verb|\@institute| is defined (or if \verb|noinstitute| is not declared) as \verb|\thispagestyle| mode. The latter can be used by the user and print, on the odd pages, the name, date and class of the candidate.
+%\iffalse
+%%pagestyles
+%\fi
+% \begin{macrocode}
+\newcommand{\ps@mainverifiche}{%
+ \renewcommand\@oddhead{\parbox{\textwidth}{\centering \institutefont{\@institute}}}%
+ \renewcommand\@evenhead{}%
+}
+\newcommand{\ps@verifiche}{%
+ \renewcommand\@oddhead{
+ \ifodd\thepage\parbox{\textwidth}{\headerfont\@header}\fi}%
+}
+% \end{macrocode}
%
%\paragraph{Print the heading}The purpose of |\printheading| macro is to typeset the headings of the page, printing the name of the institute (if included), the type of the test (if included) and so on.
%\iffalse
@@ -1014,14 +1230,12 @@ dell'esercizio somministrato
%\fi
% \begin{macrocode}
\newcommand{\printheading}{%
- \ifinstitute\pagestyle{fancy}\fi%
- {\ifinstitute\chead{\institutefont{\@institute}}\fi}%
- {\ifcandidatename\vspace*{.25em}Nome e Cognome \underline{\hspace{6cm}}%
- \hfill Classe \underline{\hspace{1.5cm}} \hfill Data \underline{\hspace{2.5cm}}\\[1em]\fi}%
- {\ifasyear \asyearfont{\@asyearpreamble\space\@asyear}\fi}%
- {\iftesttype\testtypefont{\@testtype}\fi}\\%
- {\ifinstruction\instructiondelimiter\\{\instrunctionfont{\@instruction}}\\%
- \ifduration{\durationfont\@duration\\}\fi\instructiondelimiter\fi}%
+ {\ifdefined\@institute\ifinstitute\thispagestyle{mainverifiche}\fi\else\thispagestyle{plain}\fi}%
+ {\ifcandidatename\@header\\[1em]\fi}%
+ {\ifdefined\@asyear\ifasyear \asyearfont{\@asyearpreamble\space\@asyear}\fi\fi}%
+ {\ifdefined\@testtype\iftesttype\testtypefont{\@testtype}\fi\fi}\\%
+ {\ifdefined\@instruction\ifinstruction\instructiondelimiter\\{\instrunctionfont{\@instruction}}\\%
+ \ifdefined\@duration\ifduration{\durationfont\@duration\\}\fi\fi\instructiondelimiter\fi\fi}%
}
% \end{macrocode}
@@ -1038,10 +1252,11 @@ dell'esercizio somministrato
\showmarginexercisetrue}%
\define@key{esercizio}{pt}[none]{
\def\pointes{#1}%
- \def\espoint{\hfill\underline{\hspace{1cm}}/\pointes pt}%
+ \def\espoint{\hfill\@ptprefix\@lpt@delimiter\pointes\space\@useptlabel{\pointes}\@rpt@delimiter}
\showmarginexercisetrue}%
\define@key{esercizio}{partialpt}[none]{%
- \def\sumpartialpt{\underline{\hspace{1cm}}/\ref{ptes@\theexercisenumber} pt}%
+ \def\@totalpartialpt{\convertreftonum{ptes@\theexercisenumber}}%
+ \def\sumpartialpt{\@ptprefix\@lpt@delimiter\@totalpartialpt\space\@useptlabel{\@totalpartialpt}\@rpt@delimiter}%
\showmarginexercisetrue}%
% \end{macrocode}
%
@@ -1125,8 +1340,9 @@ dell'esercizio somministrato
%\begin{macro}{\partialpt}The |partialpt| macro define the content of the margin note where it's used.
% \begin{macrocode}
\NewDocumentCommand{\partialpt}{m}{%
- \leavevmode\reversemarginpar\marginpar{\hfill(#1 pt)\addtocounter{partialpoints}{#1}}%
-}
+ \leavevmode\reversemarginpar\marginpar{%
+ \hfill\@partialptprefix\@lpartialpt@delimiter#1\space\@usepartialptlabel{#1}\@rpartialpt@delimiter%
+ \addtocounter{partialpoints}{#1}}}%
% \end{macrocode}
%\end{macro}
%\paragraph{Soluzione}
diff --git a/macros/latex/contrib/verifiche/verifiche.ins b/macros/latex/contrib/verifiche/verifiche.ins
index 8063f393f6..de63841f73 100644
--- a/macros/latex/contrib/verifiche/verifiche.ins
+++ b/macros/latex/contrib/verifiche/verifiche.ins
@@ -16,7 +16,7 @@
\usedir{tex/latex/verifiche}
\preamble
This is a generated file.
-Copyright (C) 2020 by Francesco Raccanello
+Copyright (C) 2021 by Francesco Raccanello
This file may be distributed and/or modified under the
conditions of the LaTeX Project Public License, either
version 1.3 of this license or (at your option) any later
diff --git a/macros/latex/contrib/verifiche/verifiche.pdf b/macros/latex/contrib/verifiche/verifiche.pdf
deleted file mode 100644
index 25838a476d..0000000000
--- a/macros/latex/contrib/verifiche/verifiche.pdf
+++ /dev/null
Binary files differ
diff --git a/macros/luatex/generic/chinese-jfm/README.md b/macros/luatex/generic/chinese-jfm/README.md
index 9ed15b670a..c3d52d5d31 100644
--- a/macros/luatex/generic/chinese-jfm/README.md
+++ b/macros/luatex/generic/chinese-jfm/README.md
@@ -10,11 +10,11 @@ For detailed useages, please read the [documentation](chinese-jfm.pdf) (in Chine
* v1.0.0
-First published to CTAN.
+ First published to CTAN.
* v1.0.1
-Update documentation.
+ Update documentation.
* v1.1.0
@@ -22,5 +22,9 @@ Update documentation.
- Add `fzpr` feature for zh_CN.
+* v1.1.1
+
+ Fix bugs of `fzpr` under `vert` .
+
---
Copyright (c) 2020 -- 2021 ListLee.
diff --git a/macros/luatex/generic/chinese-jfm/chinese-jfm.pdf b/macros/luatex/generic/chinese-jfm/chinese-jfm.pdf
index f3436b5fd1..3b90865938 100644
--- a/macros/luatex/generic/chinese-jfm/chinese-jfm.pdf
+++ b/macros/luatex/generic/chinese-jfm/chinese-jfm.pdf
Binary files differ
diff --git a/macros/luatex/generic/chinese-jfm/chinese-jfm.tex b/macros/luatex/generic/chinese-jfm/chinese-jfm.tex
index c2ebd81c81..37b66c7034 100644
--- a/macros/luatex/generic/chinese-jfm/chinese-jfm.tex
+++ b/macros/luatex/generic/chinese-jfm/chinese-jfm.tex
@@ -14,7 +14,7 @@
\title{\textbf{\textsf{Chinese JFM}}\thanks{\url{https://github.com/tanukihee/ChineseJFM}}}
\author{ListLee}
-\date{\zhtoday \qquad v1.1.0}
+\date{\zhtoday \qquad v1.1.1}
\begin{document}
\maketitle
@@ -83,13 +83,13 @@
\nocite{*}
\begin{thebibliography}{9}
-\bibitem{LuaTeX-ja} Lua\TeX-jaプロジェクトãƒãƒ¼ãƒ ï¼Ž\textit{Lua\TeX-ja パッケージ} ï¼»EB/OL].version 20210103.0,(2021-01-03)\\\mbox{}
+\bibitem{LuaTeX-ja} Lua\TeX-jaプロジェクトãƒãƒ¼ãƒ ï¼Ž\textit{Lua\TeX-jaパッケージ}ï¼»EB/OL].version 20210214.0,(2021-02-14)\\\mbox{}
\hfill\url{https://ctan.org/pkg/luatexja}
-\bibitem{CTeX} CTEX.ORG.\textit{\CTeX{} å®é›†æ‰‹å†Œ} ï¼»EB/OL].version 2.5.5,(2020-10-19)\\\mbox{}
+\bibitem{CTeX} CTEX.ORG.\textit{\CTeX{}å®é›†æ‰‹å†Œ}ï¼»EB/OL].version 2.5.5,(2020-10-19)\\\mbox{}
\hfill\url{https://ctan.org/pkg/ctex}
-\bibitem{孔雀计划} 刘庆(Eric Q L).\textit{孔雀计划:中文字体排å°çš„æ€è·¯} ï¼»EB/OL].\\\mbox{}
+\bibitem{孔雀计划} 刘庆(Eric Q L).\textit{孔雀计划:中文字体排å°çš„æ€è·¯}ï¼»EB/OL].\\\mbox{}
\hfill\url{https://thetype.com/kongque/}
\end{thebibliography}
diff --git a/macros/luatex/generic/chinese-jfm/jfm-ja_JP.lua b/macros/luatex/generic/chinese-jfm/jfm-ja_JP.lua
index e1e7527140..10518d1641 100644
--- a/macros/luatex/generic/chinese-jfm/jfm-ja_JP.lua
+++ b/macros/luatex/generic/chinese-jfm/jfm-ja_JP.lua
@@ -1,5 +1,5 @@
-- Chinese JFM for Japanese fonts (ja_JP)
--- v1.1.0
+-- v1.1.1, 2021/02/15
-- Copyright (c) 2020 -- 2021 ListLee.
local is_vt, is_qj, is_bj, is_km, is_pr
diff --git a/macros/luatex/generic/chinese-jfm/jfm-zh_CN.lua b/macros/luatex/generic/chinese-jfm/jfm-zh_CN.lua
index d7213c227a..a1affbb65b 100644
--- a/macros/luatex/generic/chinese-jfm/jfm-zh_CN.lua
+++ b/macros/luatex/generic/chinese-jfm/jfm-zh_CN.lua
@@ -1,5 +1,5 @@
-- Chinese JFM for Simplified Chinese fonts (zh_CN)
--- v1.1.0
+-- v1.1.1, 2021/02/15
-- Copyright (c) 2020 -- 2021 ListLee.
local is_vt, is_qj, is_bj, is_km, is_hc, is_fz
@@ -94,7 +94,7 @@ luatexja.jfont.define_jfm {
'‘',
'“'
},
- align = 'right',
+ align = is_vt and (is_fz and 'middle' or 'right') or 'right',
left = 0,
down = 0,
width = 0.5,
@@ -145,7 +145,7 @@ luatexja.jfont.define_jfm {
'’',
'â€'
},
- align = 'left',
+ align = is_vt and (is_fz and 'middle' or 'left') or 'left',
left = 0,
down = 0,
width = 0.5,
diff --git a/macros/luatex/generic/chinese-jfm/jfm-zh_TW.lua b/macros/luatex/generic/chinese-jfm/jfm-zh_TW.lua
index 5a90144b41..16a9a711e6 100644
--- a/macros/luatex/generic/chinese-jfm/jfm-zh_TW.lua
+++ b/macros/luatex/generic/chinese-jfm/jfm-zh_TW.lua
@@ -1,5 +1,5 @@
-- Chinese JFM for Traditional Chinese fonts (zh_TW)
--- v1.1.0
+-- v1.1.1, 2021/02/15
-- Copyright (c) 2020 -- 2021 ListLee.
local is_vt, is_qj, is_bj, is_km
diff --git a/systems/texlive/tlnet/tlpkg/texlive.tlpdb b/systems/texlive/tlnet/tlpkg/texlive.tlpdb
index ac1709acc9..14390d104b 100644
--- a/systems/texlive/tlnet/tlpkg/texlive.tlpdb
+++ b/systems/texlive/tlnet/tlpkg/texlive.tlpdb
@@ -26,11 +26,11 @@ depend container_split_src_files/1
depend frozen/0
depend minrelease/2016
depend release/2020
-depend revision/57736
+depend revision/57747
name 00texlive.image
category TLCore
-revision 57716
+revision 57744
shortdesc TeX Live files only in the source repository
longdesc The files here are not copied by the installer and containers
longdesc are not built for them; they exist only in the source
@@ -183346,7 +183346,7 @@ catalogue-topics maths luatex
name luatex
category TLCore
-revision 56291
+revision 57746
shortdesc The LuaTeX engine
longdesc LuaTeX is an extended version of pdfTeX using Lua as an
longdesc embedded scripting language. The LuaTeX project's main
@@ -183370,10 +183370,10 @@ depend unicode-data
execute AddFormat name=dviluatex engine=luatex options="dviluatex.ini" patterns=language.def,language.dat.lua fmttriggers=cm,etex,hyphen-base,knuth-lib,plain,tex-ini-files,unicode-data,hyph-utf8
execute AddFormat name=luatex engine=luatex options="luatex.ini" patterns=language.def,language.dat.lua fmttriggers=cm,etex,hyphen-base,knuth-lib,plain,tex-ini-files,unicode-data,hyph-utf8
containersize 12912
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-doccontainersize 1898588
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-docfiles size=695
+containerchecksum b82ddb2176b627f0e62034b8a8fc7b47e0fd721463530262cd4a7afd7809d2e3906efecf761ea37bd1b0f2b25998d93b86a9586fb0dc407aa94194141253a9dd
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+docfiles size=660
texmf-dist/doc/luatex/base/graphics/luaharfbuzz.pdf
texmf-dist/doc/luatex/base/luatex-backend.tex
texmf-dist/doc/luatex/base/luatex-callbacks.tex
@@ -183706,18 +183706,18 @@ catalogue-version 1.3
name luatexja
category Package
-revision 57316
+revision 57741
shortdesc Typeset Japanese with Lua(La)TeX
relocated 1
longdesc The package offers support for typesetting Japanese documents
longdesc with LuaTeX. Either of the Plain and LaTeX2e formats may be
longdesc used with the package.
depend luatexbase
-containersize 189336
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-docfiles size=965
+containersize 190120
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+docfiles size=966
RELOC/doc/luatex/luatexja/COPYING
RELOC/doc/luatex/luatexja/README details="Readme"
RELOC/doc/luatex/luatexja/jfm-test.lua
@@ -183734,8 +183734,8 @@ docfiles size=965
RELOC/doc/luatex/luatexja/luatexja-ruby.tex
RELOC/doc/luatex/luatexja/luatexja.dtx
RELOC/doc/luatex/luatexja/luatexja.ins
-srccontainersize 83292
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+srccontainersize 83288
+srccontainerchecksum 71f17ca90f14d923c56364ca31ff924c34c255e11c31254bf1d0586b50c57450ec6bbbc09eca287a8f2033481cc2037d1c70090402eb7600b22900c3797360dd
srcfiles size=118
RELOC/source/luatex/luatexja/ltjclasses.dtx
RELOC/source/luatex/luatexja/ltjclasses.ins
@@ -183759,7 +183759,7 @@ srcfiles size=118
RELOC/source/luatex/luatexja/tool/table_ivd_aj1.lua
RELOC/source/luatex/luatexja/tool/table_jisx0208.tex
RELOC/source/luatex/luatexja/tool/unicodeBlocks.tex
-runfiles size=405
+runfiles size=406
RELOC/tex/luatex/luatexja/addons/luatexja-adjust.sty
RELOC/tex/luatex/luatexja/addons/luatexja-ajmacros.sty
RELOC/tex/luatex/luatexja/addons/luatexja-fontspec-27c.sty
@@ -183859,7 +183859,7 @@ catalogue-contact-repository https://osdn.net/projects/luatex-ja/scm/git/luatexj
catalogue-ctan /macros/luatex/generic/luatexja
catalogue-license bsd
catalogue-topics japanese luatex class
-catalogue-version 20210103.0
+catalogue-version 20210214.0
name luatexko
category Package
@@ -259807,7 +259807,7 @@ catalogue-version 1.0h
name reledmac
category Package
-revision 57487
+revision 57739
shortdesc Typeset scholarly editions
relocated 1
longdesc A package for typesetting scholarly critical editions,
@@ -259816,10 +259816,10 @@ longdesc itself was a LaTeX port of the plain TeX EDMAC macros. The
longdesc package supports indexing by page and by line numbers, and
longdesc simple tabular- and array-style environments. The package is
longdesc distributed with the related reledpar package.
-containersize 54768
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+containersize 54756
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docfiles size=3324
RELOC/doc/latex/reledmac/README.md details="Readme"
RELOC/doc/latex/reledmac/doc-include/migrate-mac.dtx
@@ -259926,8 +259926,8 @@ docfiles size=3324
RELOC/doc/latex/reledmac/migration.pdf
RELOC/doc/latex/reledmac/reledmac.pdf details="Package documentation"
RELOC/doc/latex/reledmac/reledpar.pdf
-srccontainersize 207660
-srccontainerchecksum 47994cf180cf1a2a26f84d8475d75fb8c2b91e78c40a8e20a666e6336e76833b74609931c695280a3b7c55f3710830554beace23b57442fabe666821653e0854
+srccontainersize 207768
+srccontainerchecksum f6db52e365852a24eb7e474a8a52a62845406c92385d6c48b287efb24607fa21c9b4680c0707b023f95a69bbf884e849009c54702993f67692bdbfffe529023c
srcfiles size=296
RELOC/source/latex/reledmac/migration.dtx
RELOC/source/latex/reledmac/reledmac.dtx
@@ -259944,7 +259944,7 @@ catalogue-contact-support http://geekographie.maieul.net/146
catalogue-ctan /macros/latex/contrib/reledmac
catalogue-license lppl1.3
catalogue-topics crit-ed
-catalogue-version 2.36.0
+catalogue-version 2.36.1
name relenc
category Package
@@ -270557,7 +270557,7 @@ catalogue-version 1.05
name shtthesis
category Package
-revision 55712
+revision 57740
shortdesc An unofficial LaTeX thesis template for ShanghaiTech University
relocated 1
longdesc This package, forked from ucasthesis, is an unofficial LaTeX
@@ -270566,11 +270566,11 @@ longdesc format requirements of the school. The user just needs to set
longdesc \documentclass{shtthesis} and to set up mandatory information
longdesc via \shtsetup, then his or her thesis document will be typeset
longdesc properly.
-containersize 11480
-containerchecksum 0c29bfcb175a6525eee4407834f2284f82eb9ccaa0619b7118d426f1129048aba212da63ea1b5e32aa3b436d10c47d01deb9856d421345c36cba9fb9dbc8c549
-doccontainersize 2168668
-doccontainerchecksum bb71b50bc56eba75a1d5755179c72dc7c8ea78f1463c42482a04007b581c6fbd6ba83c82d442b129dc46c8442dd4938e599e9259621072f2c68df63c2a99ebf3
-docfiles size=557
+containersize 11540
+containerchecksum 50a81503c33b2d38f04315335c4184f8949b23e0a535297b611c1c28248a87871ca288c1953db0edefa50cc4de238b270400a47e530339601207bb9cc2560429
+doccontainersize 2179340
+doccontainerchecksum e4817b159e7b449f440750bb9bf2fc479f61e0532bfb832b34af989bddaada5d45be5a1be7dc24baca11c0338c4fb91219454b93f9024eef5dc861f2caf844f4
+docfiles size=558
RELOC/doc/latex/shtthesis/CHANGELOG.md
RELOC/doc/latex/shtthesis/LICENSE
RELOC/doc/latex/shtthesis/README.md details="Readme"
@@ -270584,7 +270584,7 @@ catalogue-contact-repository https://github.com/lirundong/shtthesis
catalogue-ctan /macros/latex/contrib/shtthesis
catalogue-license gpl3
catalogue-topics dissertation class chinese
-catalogue-version 0.3.1
+catalogue-version 0.3.2
name shuffle
category Package
@@ -287534,7 +287534,7 @@ binfiles arch=x86_64-solaris size=1
name tex4ht
category Package
-revision 57704
+revision 57742
shortdesc Convert (La)TeX to HTML/XML
longdesc A converter from TeX and LaTeX to SGML-based formats such as
longdesc (X)HTML, MathML, OpenDocument, and DocBook, providing a
@@ -287549,10 +287549,10 @@ longdesc restricted-syntax systems such as hyperlatex and gellmu. Note
longdesc that CTAN no longer holds the definitive sources of the
longdesc package: see the 'Readme' file.
depend tex4ht.ARCH
-containersize 904160
-containerchecksum 8f6dc6e4b46f1daf0221c1ba41809cf13bdc3f02a878eb0a06a2ef431ce562538759efae6ab17ec3cb41472395af21b430ec8c7196cdd9a4c448583da19a0c75
+containersize 904228
+containerchecksum 23bb542c842ce4d5dd0bdb64aac3d48c0a17b952bbf37afce41bc01837840143a9c39155daa092febd23580e8f34e2997b27a1a431025c7b208cc1ed38811eab
doccontainersize 136156
-doccontainerchecksum 3bd7ab485d8c9cbc0cb57a8e3c90f0a60248feefed3975f796c18ed5da5b9e5752130d991d811a80db21383eb79b00c32623b29601db5dbfa8416abbc2872637
+doccontainerchecksum e730f3de1604b157e8498b6cf3050b2423c37f8daf8927ef2b67dff27269fe6be4e9897017a5f46ee5dccd56f219d019fde7cc3d06c10021b6cd169dcad314b9
docfiles size=259
texmf-dist/doc/generic/tex4ht/Makefile
texmf-dist/doc/generic/tex4ht/TeX4ht-doc.tex
@@ -287623,7 +287623,7 @@ docfiles size=259
texmf-dist/doc/generic/tex4ht/mn56.html
texmf-dist/doc/generic/tex4ht/mn5x.png
texmf-dist/doc/generic/tex4ht/mn6x.png
-runfiles size=9790
+runfiles size=9791
texmf-dist/scripts/tex4ht/ht.sh
texmf-dist/scripts/tex4ht/htcontext.sh
texmf-dist/scripts/tex4ht/htcopy.pl
@@ -288000,6 +288000,7 @@ runfiles size=9790
texmf-dist/tex/generic/tex4ht/spanish.4ht
texmf-dist/tex/generic/tex4ht/splitidx.4ht
texmf-dist/tex/generic/tex4ht/srcltx.4ht
+ texmf-dist/tex/generic/tex4ht/subcaption.4ht
texmf-dist/tex/generic/tex4ht/subeqnarray.4ht
texmf-dist/tex/generic/tex4ht/subfigure.4ht
texmf-dist/tex/generic/tex4ht/subscript.4ht
diff --git a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.md5 b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.md5
index 74dd280512..45def453df 100644
--- a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.md5
+++ b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.md5
@@ -1 +1 @@
-7ba6a01d1f87ee2e4f46919e182011cf texlive.tlpdb
+b7dc2726110bd35c833e755ad6e54404 texlive.tlpdb
diff --git a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512 b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512
index b104c3d715..f41f079a20 100644
--- a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512
+++ b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512
@@ -1 +1 @@
-22ce86b34306ba90e1ec1596afc0b66a7537865c38cd0c8eb0ba46a9e247546ef42a3aee7e3f49e10d8018bbff33909a628c1ef07115668e3160cb1eb43c1c16 texlive.tlpdb
+6f5cf50faf702239ea8e266c5bcbf624f10cf371c3ef2efdc066c9a13d71ed0ff14c82f26f86b13febc726256575ddcf7d5f3a5232ad9f58e0a91a73287b97e5 texlive.tlpdb
diff --git a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512.asc b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512.asc
index 4efe7c3a5d..fa12bef5ad 100644
--- a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512.asc
+++ b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.sha512.asc
@@ -1,10 +1,10 @@
-----BEGIN PGP SIGNATURE-----
-iQEcBAEBCgAGBQJgKHW/AAoJEEzhh34ZQ4xwdtIIAKwS3s0d2/aRIrgKQCICS77x
-4263In2s9IXjkbZFQA24kGVmtH3ZMl3LIW/Barmc5KN0sWF1FXndl396xKO2rYM5
-hsxsqrcRWTmMoUPLnVvhWHrChUnzhs9+7ymb5rFLMvu59ZYPT6HFtdVJPbVCaPCq
-M1p+4wzdYwIbGrtNwNt6Yqcb8tCnyX+pOID5GW4zgPITMVapqWHCuYmzwLRt+Xlq
-gA4elVcWcNLX8rjOE25jDZXt+lRYC9Cew7FLruKW0htq6fw9XVMX0rp9PTMB6ULr
-dY98b9YKfN+q+05gs8StBY4Mwyq/bTejSWYquUW9JUrczozZEcnO0fsGHNUY71o=
-=QCQs
+iQEcBAEBCgAGBQJgKceCAAoJEEzhh34ZQ4xwLwgIAKtZuLU2YN21NDqY1MZ87+EX
+u+mzG0FliQlp3BwMyjqn4+8jLsz3YbV+EKDSr4CnfMAYd705e12uLdfptgIMMrV9
+Kr+dgA+HMgZTanX4BfjuwHZs13LSZoM9FqxbQN80yqv/d7Fu9q8Aut1CtXHtXJiq
+M+uTb6Xpr1BFY3fY7dg+uuVMS62WHdfIeyPyk9v9Q2OA67ZeJk5z6vRtuo+rxFqo
+mhduvdcvJljsCGX2RcrkURJb5yb01ReSHj6Uo5D7tsIjkr2xCYsFdx2joCrt9nfi
+amB2YCCzDlMU1n5+5IKAJMMB5LUYPvPZyrb0J63rjLdKmqTr6k9bQR2Njrcit+Q=
+=fu4M
-----END PGP SIGNATURE-----
diff --git a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.xz b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.xz
index 6d27ba0ea9..7d877c463d 100644
--- a/systems/texlive/tlnet/tlpkg/texlive.tlpdb.xz
+++ b/systems/texlive/tlnet/tlpkg/texlive.tlpdb.xz
Binary files differ
diff --git a/systems/unix/tex-fpc/MFT/logmacfonts b/systems/unix/tex-fpc/MFT/logmacfonts
index 24c12db460..3c6326f544 100755
--- a/systems/unix/tex-fpc/MFT/logmacfonts
+++ b/systems/unix/tex-fpc/MFT/logmacfonts
@@ -1,3 +1,3 @@
-# fonts for logmac
+# create .tfm files for logmac
mkfont cmsltt9
mkfont logosl9
diff --git a/systems/unix/tex-fpc/MFT/manfonts b/systems/unix/tex-fpc/MFT/manfonts
index ba37849e82..cd84126c14 100755
--- a/systems/unix/tex-fpc/MFT/manfonts
+++ b/systems/unix/tex-fpc/MFT/manfonts
@@ -1,9 +1,9 @@
+# This script creates and installs .tfm files for the TeX/MF books.
mkfont cmtex10
mkfont cmssi9
mkfont cminch
mkfont cmssdc10 4
-mkpkfont cmr5 2
mkfont cmman # font used for miscellaneous Computer Modern variations
- # from ctan:/systems/knuth/local/mf/cmman.mf
+ # from ctan:/systems/knuth/local/mf/cmman.mf
mkfont random # from ctan:/systems/knuth/local/lib/random.mf
mkfont cmss9
diff --git a/systems/unix/tex-fpc/MFT/mfwebfonts b/systems/unix/tex-fpc/MFT/mfwebfonts
index a66d9a6be8..c9dfdf854a 100755
--- a/systems/unix/tex-fpc/MFT/mfwebfonts
+++ b/systems/unix/tex-fpc/MFT/mfwebfonts
@@ -1,4 +1,4 @@
-# fonts used by mf.web
+# create and install .tfm fonts used by mf.web
mkfont logo10 # font used for the METAFONT logo
mkfont logosl10
mkfont logo8
diff --git a/systems/unix/tex-fpc/MFT/plainfonts b/systems/unix/tex-fpc/MFT/plainfonts
index d2d18bd490..b7f25e8f55 100755
--- a/systems/unix/tex-fpc/MFT/plainfonts
+++ b/systems/unix/tex-fpc/MFT/plainfonts
@@ -1,4 +1,4 @@
-# This script builds and installs .tfm and .pk fonts for plain.tex.
+# create and install .tfm files for plain.tex.
rm -f trouble # clean list of trouble makeing fonts
mkfont cmr10 # roman text
mkfont cmr9
@@ -46,7 +46,4 @@ mkfont cmbsy10 # bold math symbols
mkfont cmcsc10 # caps and small caps
mkfont cmssbx10 # sans serif bold extended
mkfont cmdunh10 # Dunhill style
-mkpkfont cmr7 magstep4 # for titles
-mkpkfont cmtt10 magstep2
-mkpkfont cmssbx10 magstep2
mkfont manfnt # METAFONT logo and dragon curve and special symbols
diff --git a/systems/unix/tex-fpc/MFT/tripmanfonts b/systems/unix/tex-fpc/MFT/tripmanfonts
deleted file mode 100755
index 1d4e4356c4..0000000000
--- a/systems/unix/tex-fpc/MFT/tripmanfonts
+++ /dev/null
@@ -1,2 +0,0 @@
-# font for tripman
-mkpkfont cmr7 magstep5 # for titles
diff --git a/systems/unix/tex-fpc/MFT/webfonts b/systems/unix/tex-fpc/MFT/webfonts
index 86676e2c70..a089db2772 100755
--- a/systems/unix/tex-fpc/MFT/webfonts
+++ b/systems/unix/tex-fpc/MFT/webfonts
@@ -1,2 +1,2 @@
-# fonts for the web macros
+# create and install font for the web macros
mkfont cmtex10
diff --git a/systems/unix/tex-fpc/README b/systems/unix/tex-fpc/README
index b227918699..1d9cd8dcfa 100644
--- a/systems/unix/tex-fpc/README
+++ b/systems/unix/tex-fpc/README
@@ -1,9 +1,9 @@
-TeX-FPC, second edition, 2020-11-24
+TeX-FPC 3rd ed. 2021-02-14
This directory contains change files for Donald E. Knuth's suite
of TeX related programs. The change files (.ch) and Don Knuth's WEB
-files (.web) from January 2014 are the key components of a construction
-kit for "TeX-FPC"---a complete TeX system based on the Free Pascal
+files (.web) from January 2021 are the key components of a construction
+kit for "TeX-FPC"---a complete TeX system based purely on the Free Pascal
Compiler (FPC), and running in any Unix environment.
These building bricks are glued together by shell scripts in
@@ -36,22 +36,19 @@ In fact, I only used stuff from:
Kernighan, Pike: The Unix programming environment (1984)
I have to admit, now and then I used vi instead of ed, the Unix Standard
Editor.
-If it doesn't work on your system, it is not Unix, not even remotely so.
+If TeX-FPC doesn't work on your system, it is not Unix, not even remotely so.
-Any version of (FPC) from 3.0.0 on will suffice. To compile TeX and
-METAFONT, three additional units are needed:
-SysUtils, BaseUnix, Unix
-They have to be specified on the command line:
- $ fpc -Fasysutils,baseunix,unix
+Ane version of FPC from 3.0.0 (Nov. 2015) to 3.2.0 (June 2020)
+works fine.
To build the DVI-driver xdvi and dvips a C-compiler is necessary.
To run xdvi, you need X-Window and to print your books, you either
need a postscript printer or you use ghostscript to translate
-postscript to your printers language.
+postscript to your printers language. The DVI-drivers are written in C.
Everything you need existed in the mid eighties of the last century.
All the great books produced with TeX demonstrates how much can be
-done with how little.
+done without the latest "innovations".
Overview:
=========
@@ -83,7 +80,7 @@ TeX files:
.dvi device independent file, "TeX: The Program" (tex.web), part 31
.log a log file
-Dependencies of files and programs:
+Dependencies of files:
File depends on
---- ----------
xxx xxx.p
@@ -98,51 +95,45 @@ plain.base plain.mf
local.mf
mf.pool
+Dependencies of programs
Program depends on
------- ----------
initex TeXformats/tex.pool
-tex TeXformats/tex.fmt
+tex TeXformats/plain.fmt
inimf MFbases/mf.pool
mf MFbases/plain.base
Files and their processors:
--------------------------
WEB:
- tangle
+ tangle
x.web, x.ch --------> x.p, x.pool
- fpc
- x.p --------> x
-
+ fpc
+ x.p --------> x (to compile tex and mf you need some units:
+ fpc -FaUnix,sysutils,baseunix)
tgl
- x.web, x.ch --------> x (combines the last two steps, no pool)
+ x.web, x.ch --------> x (combines the last two steps, no pool)
itgl
x.web, x.ch --------> inix.p, inix, x.pool (like tgl, but with pool)
weave
- .web, .ch --------> .tex (description of the program)
+ .web, .ch --------> .tex (description of the program)
- wve
- .web, .ch --------> .dvi (linking the above step and tex)
+ wve
+ .web, .ch --------> .dvi (linking the above step and tex)
-METAFONT:
+METAFONT and TeX metric fonts:
inimf
mf.pool, local.mf plain.mf --------> plain.base
mf
- plain.base, .mf --------> .nnngf, .tfm
-
- gftopk
- .nnngf --------> .nnnpk
+ plain.base, .mf --------> .tfm
mkfont
- .mf --------> TeXfonts/.tfm, PKfonts/.nnnpk
-
- mkpkfont
- .mf --------> PKfonts/.nnnpk
-
+ .mf --------> TeXfonts/.tfm
TeX:
initex
@@ -151,18 +142,29 @@ TeX:
tex
plain.fmt, .tfm, .tex --------> .dvi
-DVI drivers:
+DVI drivers and bitmap fonts:
+ mf
+ plain.base, .mf --------> .nnngf
+ gftopk
+ .nnngf --------> .nnnpk
+
+ MKTeXPK
+ .mf --------> PKfonts/.nnnpk (combining the above steps)
+
+
dvips
.tfm, .nnnpk, .dvi --------> .ps, PostScript printer
xdvi
.tfm, .nnnpk, .dvi --------> X-Window
+The dvi-drivers dvips and xdvi will call MKTeXPK to install .nnnpk
+on demand.
+
File names:
-----------
-TeX-FPC searches files in directories which are local to the working
-directory:
+TeX-FPC searches files in subdirectories of the working directory:
TeXinputs .tex
TeXfonts .tfm
TeXformats .pool, .fmt
@@ -171,23 +173,29 @@ directory:
PKfonts .nnnpk
DVIPSconf .pro, .ps, .map (Header and configuration files for dvips)
-Create one set of "master directories" and used symbolic links
-to your working directory like:
+To run tex or initex the working directory must contain the first
+three directories from the list above. Likewise for mf or inimf and
+the last four directories from the list. The dvi driver xdvi and
+dvips will invoke mf to create PK fonts whenever needed. So you
+might need to install the mf-directories when printing.
+
+To avoid copying the directories to each directory where you want
+to run TeX, create one set of "master directories" and use symbolic
+links to your working directory like:
ln -s ~/TeXformats .
-The scripts mk_TeX_dir and mk_MF_dir help to prepare a directory
-for running TeX resp. METAFONT. Adapt these scripts to the locations
-of your master directories.
+The scripts mk_TeX_dir and mk_MF_dir help do just this automatically.
+Adapt these scripts to the locations of your master directories.
TeX-FPC's path searching applies to file names without directory
components, i.e. without a /:
- .tex local directory and then TeXinputs
- .mf local directory and then MFinputs
- .tfm local directory and the TeXfonts
- .fmt local directory and then TeXformats
+ .tex working directory and then TeXinputs
+ .mf working directory and then MFinputs
+ .tfm working directory and the TeXfonts
+ .fmt working directory and then TeXformats
plain.fmt hard coded (!) TeXformats/plain.fmt
- .base local directory and MFbases
+ .base working directory and then MFbases
plain.base hard coded (!) MFbases/plain.base
.pool hard coded (!) in TeXformats, MFbases
@@ -224,8 +232,9 @@ creates a list of the font names used by plain.tex. I edited
plainfonts to convert it to a shell script that creates all fonts
of the plain format. In the MFT directory you'll find other examples.
If the generation of a font failed, the failing command is written
-to the file trouble in the working directory. You might want to fix the
-error and rerun the file trouble, (sh trouble).
+to the file trouble in the working directory. You might want to fix
+the error and rerun the file trouble, (sh trouble). If the error
+persists though, you might end up in a loop.
Installing new fonts from CTAN
------------------------------
@@ -251,36 +260,35 @@ Both tests are passed by TeX-FPC. See the trip and the trap directories..
Hints to build and install TeX-FPC from Knuth's sources.
========================================================
+Create the master directories if you haven't done so allready.
+Create links to the master directories in tex-fpc:
+ mk_TeX_dir
+ mk_MF_dir
Get Knuth's source files:
------------------------
-Get Knuth's distribution from
- ftp://ftp.dante.de/pub/tex/systems/knuth/dist.zip
-At the DANTE server, passive ftp mode does not work. Try
- tug.org/tex-archive/systems/knuth
-or ftp ftp.cs.stanford.edu/tex/dist
+Get some directories Knuth's distribution from
+ tug.org/tex-archive/systems/knuth/dist.zip
+Unpack dist.zip in the parent directory of tex-fpc and move the files from
+Knuth's distribution to tex-fpc.
+ $ mv dist/* tex-fpc
-This gives you the WEB files, manuals, macros and some .mf files.
-At least you need the subdirectories lib, tex, mfware, mf and web.
+This directory, i.e. tex-fpc will be our stage directory, that is
+all compiling etc. will be done in tex-fpc and it's subdirectories.
-You need another set of source files, namely the METAFONT source
-files (.mf) for Knuth's font Computer Modern.
-
-Copy the .mf files from
- ftp://ftp.dante.de/pub/tex/fonts/cm/mf/
-or ftp.cs.stanford.edu/tex/fonts/
-to MFinputs.
+Install the files from cm to MFinputs:
+ mv cm/* MFinputs
And copy the .mf files from the lib directory to MFinputs.
cp -p lib/*.mf MFinputs
-All path names are given relative to directories of Knuth's distribution.
+All path names are given relative to tex-fpc.
Building from WEB sources, (xxx.web and xxx.ch --> xxx, xxx.pool)
----------------------------------------------------------------
The general procedure to build a binary xxx from its WEB files is:
Move xxx.ch to the directory that contains xxx.web, apply tangle
-to xxx.web and xxx.ch to generate xxx.p and xxx.pool, and compile the
+to xxx.web and xxx.ch to generate xxx.p, and compile the
Pascal source with fpc. The command line
tgl xxx.web xxx.ch
will do all that. It produces xxx that you need to move to your binary
@@ -295,17 +303,17 @@ activate the change blocks marked by "ini" and use tgl.
A change block is active, if each of the web commands @x, @y and
@z are at the start of their lines, inactive if none of the web
-commands are at the start of their lines and defect otherwise.
-See webman.tex, section~11. for details on change files.
+commands are at the start of their lines.
+See webman.tex, for a short description of change files (look for @x).
Step by step procedure to bootstrap TeX-FPC
-------------------------------------------
-Install the Free Pascal Compiler
+Install the Free Pascal Compiler (3.0.0+)
https://freepascal.org/download.html
Create the master directories (see above).
-Copy the shell scripts from the tex-fpc/shell to your binary directory.
+Copy the shell scripts from tex-fpc/shell to your binary directory.
0. Compile tangle tangle.p --> tangle
@@ -321,48 +329,48 @@ binary directory.
tex.web, tex.ch --> initex,initex.p tex.pool
--------------------------------------------
-Make sure, the ini-change blocks in mf.ch and tex.ch are deactivated.
-Build the ini-programs and save them as initex, resp. inimf.
-Save tex.pool in TeXformats and mf.pool in MFbases, use itgl instead
-of tgl to keep inimf.p and initex.p and the .pool files.
+Build the ini-programs itgl and save them as initex, resp. inimf in your
+bin directory. Save tex.pool in TeXformats and mf.pool in MFbases.
2. Make plain.base lib/plain.mf --> MFbases/plain.base
------------------------------------------------------------------------
-Go to the directory tex-fpc/mf and use inimf to create plain.base.
+Go to tex-fpc/mf and use inimf to create plain.base.
The file local.mf contains mode definitions for my HP LaserJet 1320
-printer. Adapt local.mf to your printer and copy it to the mf directory.
- inimf ../lib/plain input local dump
+printer. Adapt local.mf to your printer.
+ inimf ../lib/plain input ../MFT/local dump
Move plain.base to MFbases.
3. Make mf mf.web, mf.ch --> mf
------------------------------------------------------------------------
-Activate the the inimf-change block in mf.ch and build it with tgl.
+goto mf and use ../ch.ch/mkprod mf to activate the inimf-change
+block in mf.ch, build mf with tgl, save mf in your bin directory.
-4. Make gftopk gftopk.web, gftopk.ch --> gftopk
-------------------------------------------------------------------------
-Build gftopk in the mfware directory. You need gftopk to install
-packed font files (.nnnpk) in the next step.
-5. Install fonts: MFinputs/*.mf --> TeXfonts/*.tfm, PKfonts/*.pk
+4. Install TFM-fonts: MFinputs/*.mf --> TeXfonts/*.tfm
------------------------------------------------------------------------
-Prepare the stage directory to run both METAFONT (mk_MF_dir) and
-TeX (mk_TeX_dir) and run the shell script MFT/plainfonts from there.
+Run the shell script MFT/plainfonts.
Check the file trouble that might contain commands that failed.
-6. Make plain.fmt lib/plain.tex --> TeXformats/plain.fmt
+5. Make plain.fmt lib/plain.tex --> TeXformats/plain.fmt
------------------------------------------------------------------------
-Copy lib/hyphen.tex to TeXinputs and go to the directory tex
-to build plain.fmt.
+Copy lib/hyphen.tex to TeXinputs and go to tex to build plain.fmt.
This command line will do
initex ../lib/plain \\dump # one backslash is consumend by sh!
Move plain.fmt to TeXformats.
-7. Finally build tex tex.web, tex.ch --> tex
+6. build tex tex.web, tex.ch --> tex
------------------------------------------------------------------------
-Activate the initex-change block in tex.ch and build it with tgl.
+Use ../ch.ch/mkprod to activate the initex-change block in tex.ch
+and build tex with tgl.
+
+Step 6 completes the minimal installation of TeX-FPC, which enables
+you to study, change, build, and test the six programs tangle,
+weave, inimf, mf, initex, and tex. Run them to typeset your book,
+create your font, and last but not least to try literate programming
+and write your own web programs.
-Happy weaving and happy type setting with your system!
+Happy weaving and typesetting,
Wolfgang Helbig,
Comments welcome: helbig@mailbox.org
diff --git a/systems/unix/tex-fpc/dvips/Makefile b/systems/unix/tex-fpc/dvips/Makefile
index cbdf300d17..05c6431438 100644
--- a/systems/unix/tex-fpc/dvips/Makefile
+++ b/systems/unix/tex-fpc/dvips/Makefile
@@ -1,6 +1,6 @@
#
# Makefile for dvips. Edit this first part of the file.
-# Adapted to TeX-GPC in Aug. 2008.
+# Adapted to TeX-FPC in Jan. 2019.
#
# First, the things that absolutely must be edited for your system.
# Please, don't forget to edit MakeTeXPK in the same way!
diff --git a/systems/unix/tex-fpc/dvips/README b/systems/unix/tex-fpc/dvips/README
index 06119ad5e1..6425406d8d 100644
--- a/systems/unix/tex-fpc/dvips/README
+++ b/systems/unix/tex-fpc/dvips/README
@@ -16,6 +16,9 @@ See xdvi/README for hints on the man directory.
Create a master directory DVIPSconf if you haven't done so already, goto
dvips576/dvips, run mk_TeX_dir, mk_MF_dir, and make install.
+Build and install the program mfware/gftopk. This program is called
+by MakeTeXPK to create bitmap fonts .nnnpk on demand.
+
Edit DVIPSconf/config.ps to reflect paper size and resolution of
your printer. To make dvips generate missing .pk fonts automatically,
the resolutions in config.ps and in the mf mode "localfont" must
diff --git a/systems/unix/tex-fpc/dvips/dvipsmanfonts b/systems/unix/tex-fpc/dvips/dvipsmanfonts
index aa55b0d360..44cd73544c 100644
--- a/systems/unix/tex-fpc/dvips/dvipsmanfonts
+++ b/systems/unix/tex-fpc/dvips/dvipsmanfonts
@@ -1,4 +1,6 @@
# This script builds and installs .pk fonts for dvips.tex
+# you do not need to run it, since dvips will call MakeTeXPK to generate
+# and install .pk fonts on demand.
rm -f trouble # clean list of trouble makeing fonts
mkpkfont logo10 magstep0.5
mkpkfont cminch magstep0.5
diff --git a/systems/unix/tex-fpc/dvitype.ch b/systems/unix/tex-fpc/dvitype.ch
index 7216ae63e0..bcd7d95a81 100644
--- a/systems/unix/tex-fpc/dvitype.ch
+++ b/systems/unix/tex-fpc/dvitype.ch
@@ -1,4 +1,4 @@
-This is a change file of DVItype for FPC, Wolfgang Helbig, Oct. 2020
+This is a change file of DVItype for FPC, Wolfgang Helbig, Oct. 2020 - Jan 2021
[0] About DVItype-FPC
@x
@@ -31,7 +31,7 @@ Comments and questions are welcome!
@x
@d banner=='This is DVItype, Version 3.6' {printed when the program starts}
@y
-@d banner=='This is DVItype-FPC, 2nd ed.' {printed when the program starts}
+@d banner=='This is DVItype-FPC, 3rd ed.' {printed when the program starts}
@z
[2] othercases
diff --git a/systems/unix/tex-fpc/gftopk.ch b/systems/unix/tex-fpc/gftopk.ch
index 2732b87db7..38a66fd4a0 100644
--- a/systems/unix/tex-fpc/gftopk.ch
+++ b/systems/unix/tex-fpc/gftopk.ch
@@ -1,4 +1,4 @@
-This is a change file of GFtoPK for FPC, Wolfgang Helbig, Oct. 2020
+This is a change file of GFtoPK for FPC, Wolfgang Helbig, Oct. 2020 - Feb. 2021
[0] About GFtoPK-FPC
@x
@@ -32,7 +32,7 @@ Comments and questions are welcome!
@x
@d banner=='This is GFtoPK, Version 2.4' {printed when the program starts}
@y
-@d banner=='This is GFtoPK-FPC, 2nd ed.'
+@d banner=='This is GFtoPK-FPC, 3rd ed.'
{printed when the program starts}
@z
diff --git a/systems/unix/tex-fpc/gftype.ch b/systems/unix/tex-fpc/gftype.ch
index 216f9b653e..d3e01980c8 100644
--- a/systems/unix/tex-fpc/gftype.ch
+++ b/systems/unix/tex-fpc/gftype.ch
@@ -32,7 +32,7 @@ Comments and questions are welcome!
@x
@d banner=='This is GFtype, Version 3.1' {printed when the program starts}
@y
-@d banner=='This is GFtype-FPC, 2nd ed.' {printed when the program starts}
+@d banner=='This is GFtype-FPC, 3rd ed.' {printed when the program starts}
@z
[2] othercases
diff --git a/systems/unix/tex-fpc/history b/systems/unix/tex-fpc/history
index 45b9840773..29b8229088 100644
--- a/systems/unix/tex-fpc/history
+++ b/systems/unix/tex-fpc/history
@@ -63,14 +63,14 @@
2015-09-30: Update to macOS 10.11 broke GPC and therefore TeX-GPC
as well. Helpless, sad, hoping for a revival of GPC.
-2019-02-22: Emergency edition of TeX-FPC. Date fixed at 1776-07-14.
+2019-02-22: 1st edition of TeX-FPC. Date fixed at 1776-07-14.
-2019-03-04: same as above, fixing three bugs, all reported by Joachim.
+2019-03-04: fixing three bugs of 1st. edition, all reported by Joachim.
The shell script tgl tried to launch good old GPC. An
extra space in tex.ch corrupted the change file. In
- tex.ch a '\marke G' still waits to be replaced by '\marke X'.
+ tex.ch a '\marke G' waited to be replaced by '\marke X'.
-now: 2nd. edition. All features recreated! Finally arrived the state
+2020-11-25: 2nd edition. All features recreated! Finally arrived the state
of 2008-09-07. Exhausted but happy!
Igor Liferenko reported another bug in TeX: An extra space
@@ -81,13 +81,13 @@ now: 2nd. edition. All features recreated! Finally arrived the state
Realizing that TeX-FPC is the only pure Pascal distribution,
I decided that my personal preferences are not worthwhile
to clutter up TeX-FPC, since I can reinstall them
- whenever I want them. My wishes are: Don't ignore tailing
+ whenever I want them. My wishes are: Don't ignore trailing
spaces of input lines, don't ignore leading spaces of
the first input line. Don't print extra empty lines on
the terminal, when entering a file interactively. These
regressions leads us back to IBM's fixed length records
and punching cards, when the user could not control the
- length of a line or a punching card.
+ length of a line.
future: Bugs that remain to be fixed:
This one-liner loops:
@@ -96,4 +96,10 @@ future: Bugs that remain to be fixed:
Whatever you enter during error recovery, you cannot convince
TeX to forgive your faux pas and continue processing the rest
of the file.
+2021-02: 3rd edition. Will Senn and Dave Jarvis pointed out some ambiguities
+ in the README file. I'll took their comments and questions to
+ improve the wordings in the README file and some of TeX's error
+ messages.
+ The minimal installation does not care for the .pk fonts. They will
+ be created on demand by the dvi drivers. (dvips and xdvi)
diff --git a/systems/unix/tex-fpc/license b/systems/unix/tex-fpc/license
index 1783216cf9..ab23d3f7db 100644
--- a/systems/unix/tex-fpc/license
+++ b/systems/unix/tex-fpc/license
@@ -1,11 +1,16 @@
-Copyright (c) Wolfgang Helbig, All rights reserved.
+Copyright (c) 2021, Wolfgang Helbig
+All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:
-Redistributions of source code must retain the above copyright
-notice, this list of conditions and the following disclaimer.
+1. Redistributions of source code must retain the above copyright
+ notice, this list of conditions and the following disclaimer.
+
+2. Redistributions in binary form must reproduce the above copyright
+ notice, this list of conditions and the following disclaimer in the
+ documentation and/or other materials provided with the distribution
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
diff --git a/systems/unix/tex-fpc/local.mf b/systems/unix/tex-fpc/local.mf
index 491430d781..90ee274d78 100644
--- a/systems/unix/tex-fpc/local.mf
+++ b/systems/unix/tex-fpc/local.mf
@@ -14,7 +14,7 @@
mode_def lj = % HP LaserJet 1320 ProRes (1200 dpi)
proofing :=0;
fontmaking := 1;
- tracingtitels := 1;
+ tracingtitles := 1;
pixels_per_inch := 1200;
blacker := .7;
fillin := .15;
@@ -24,7 +24,7 @@ enddef;
mode_def ljfr = % HP LaserJet 1320 FastRes (600dpi)
proofing :=0;
fontmaking := 1;
- tracingtitels := 1;
+ tracingtitles := 1;
pixels_per_inch := 600;
blacker := .25;
fillin := .3;
diff --git a/systems/unix/tex-fpc/mf.ch b/systems/unix/tex-fpc/mf.ch
index 0b697ec3da..1a5a2eab18 100644
--- a/systems/unix/tex-fpc/mf.ch
+++ b/systems/unix/tex-fpc/mf.ch
@@ -13,8 +13,8 @@
\N0\*. \[0] About \namefpc.
-This is an adaption of Donald~E. Knuth's \MF, version 2.7182815
-from January 2014, to Free~Pascal (FPC) and Unix.
+This is an adaption of Donald~E. Knuth's \MF, version 2.71828182
+from January 2021, to Free~Pascal (FPC) and Unix.
The features added include treating the command line as the first
input line and invoking a system editor, in this case \.{vi}, to
@@ -36,9 +36,9 @@ Comments and questions are welcome!
[2] Change the banner line
@x
-@d banner=='This is METAFONT, Version 2.7182818' {printed when \MF\ starts}
+@d banner=='This is METAFONT, Version 2.71828182' {printed when \MF\ starts}
@y
-@d banner=='This is METAFONT-FPC, 2nd ed.' {printed when \MF\ starts}
+@d banner=='This is METAFONT-FPC, 3rd ed.' {printed when \MF\ starts}
@z
[4] terminal output and input
@@ -85,15 +85,15 @@ cursor to the @x line and enter >} or <} to (un)indent a paragraph.
@z
[7] shift left to turn on statistics, shift rigth to turn off statistics
- @x stat
- @d stat==@{ {change this to `$\\{stat}\equiv\null$' when gathering
- usage statistics}
- @d tats==@t@>@} {change this to `$\\{tats}\equiv\null$' when gathering
- usage statistics}
- @y
- @d stat== {change this to `$\\{stat}\equiv\.{@@\{}$' to turn off statistics}
- @d tats== {change this to `$\\{tats}\equiv\.{@@\}}$' to turn off statistics}
- @z
+@x stat
+@d stat==@{ {change this to `$\\{stat}\equiv\null$' when gathering
+ usage statistics}
+@d tats==@t@>@} {change this to `$\\{tats}\equiv\null$' when gathering
+ usage statistics}
+@y
+@d stat== {change this to `$\\{stat}\equiv\.{@@\{}$' to turn off statistics}
+@d tats== {change this to `$\\{tats}\equiv\.{@@\}}$' to turn off statistics}
+@z
[8] shift left to build MF and right it to build INIMF.
@x inimf
@@ -417,9 +417,17 @@ loop@+begin wake_up_terminal; write(term_out,'**'); update_terminal;
@x
else bad_pool('! I can''t read MF.POOL.')
@y
-else bad_pool('! I can''t read mf.pool.')
+else bad_pool('! I can''t read MFbases/mf.pool.')
+@z
+
+[53] tangle would'nt help, installation would
+@x
+done: if a<>@$ then bad_pool('! MF.POOL doesn''t match; TANGLE me again.');
+@y
+done: if a<>@$ then bad_pool('! MFbases/mf.pool doesn''t match. Not installed?');
@z
+
[74] start editor
@x
@!use_err_help:boolean; {should the |err_help| string be shown?}
@@ -438,14 +446,9 @@ start_edit := false; { don't start \.{vi}}
[79] set edit_cmd
@x
- begin print_nl("You want to edit file ");
-@.You want to edit file x@>
- slow_print(input_stack[file_ptr].name_field);
- print(" at line "); print_int(line);@/
interaction:=scroll_mode; jump_out;
end;
@y
- begin {save values to be passed to the system editor}
start_edit := true;
interaction:=scroll_mode; jump_out;
end;
@@ -471,10 +474,12 @@ if c="Q" then decr(selector); return;
[194] date and time
@x
@p procedure fix_date_and_time;
-begin internal[time]:=12*60*unity; {minutes since midnight}
-internal[day]:=4*unity; {fourth day of the month}
-internal[month]:=7*unity; {seventh month of the year}
-internal[year]:=1776*unity; {Anno Domini}
+begin sys_time:=12*60;
+sys_day:=4; sys_month:=7; sys_year:=1776; {self-evident truths}
+internal[time]:=sys_time*unity; {minutes since midnight}
+internal[day]:=sys_day*unity; {day of the month}
+internal[month]:=sys_month*unity; {month of the year}
+internal[year]:=sys_year*unity; {Anno Domini}
end;
@y
@p procedure fix_date_and_time;
@@ -482,11 +487,15 @@ var yy,mm,dd: word;
hh,ss,ms: word;
begin
decodedate(now,yy,mm,dd); {current date}
-internal[day] := dd*unity;
-internal[month] := mm*unity;
-internal[year] := yy*unity;@/
+sys_day := dd;
+sys_month := mm;
+sys_year := yy;
+internal[day] := sys_day*unity;
+internal[month] := sys_month*unity;
+internal[year] := sys_year*unity;@/
decodetime(now,hh,mm,ss,ms); {current time}
-internal[time] := (hh*60+mm)*unity; {minutes since midnight}
+sys_time := hh*60+mm; {minutes since midnight}
+internal[time] := sys_time*unity; {minutes since midnight}
end;
@z
@@ -535,15 +544,15 @@ int(4096))|, and |abs(int(a+b-4096))| will throw an overflow exception.
@z
[679] spurious empty line on terminal
- @x chatty
- print_ln; first:=start;
- prompt_input("*"); {input on-line into |buffer|}
- @y
- @.Please type...@>
- print_nl("");
- first:=start; {avoid empty lines on terminal and log file}
- prompt_input("*"); {input on-line into |buffer|}
- @z
+@x chatty
+ print_ln; first:=start;
+ prompt_input("*"); {input on-line into |buffer|}
+@y
+@.Please type...@>
+ print_nl("");
+ first:=start; {avoid empty lines on terminal and log file}
+ prompt_input("*"); {input on-line into |buffer|}
+@z
[769] Path separator in Unix
@x
@@ -561,27 +570,9 @@ else begin if c="/" then
[774]
@x
-@d append_to_name(#)==begin c:=#; incr(k);
-@y
-@d fpc_set_length==s@&e@&t@&l@&e@&n@&g@&t@&h
-@d append_to_name(#)==begin c:=#; incr(k);
-@z
-
-[774]
-@x
-@!j:pool_pointer; {index into |str_pool|}
-@y
-@!j:pool_pointer; {index into |str_pool|}
-rbs: fpc_string;
-@z
-
-[774]
-@x
for k:=name_length+1 to file_name_size do name_of_file[k]:=' ';
@y
-rbs := name_of_file;
-fpc_set_length(rbs, name_length);
-name_of_file := rbs;
+for k:=name_length+1 to file_name_size do name_of_file[k]:=chr(0);
@z
[776]
@@ -593,19 +584,16 @@ MF_base_default:='MFbases/plain.base';
[778]
@x
-@!j:integer; {index into |buffer| or |MF_base_default|}
+for k:=name_length+1 to file_name_size do name_of_file[k]:=' ';
@y
-@!j:integer; {index into |buffer| or |MF_base_default|}
-rbs: fpc_string;
+for k:=name_length+1 to file_name_size do name_of_file[k]:=chr(0);
@z
-[778]
+[779] report the file you can't read
@x
-for k:=name_length+1 to file_name_size do name_of_file[k]:=' ';
+ wterm_ln('I can''t find the PLAIN base file!');
@y
-rbs := name_of_file;
-fpc_set_length(rbs, name_length);
-name_of_file := rbs;
+ wterm_ln('I can''t find MFbases/plain.base!');
@z
[786]
@@ -617,7 +605,7 @@ print_nl("Please type another "); print(s); print(" or Control-D");
[793] name might be needed to be passed to vi
@x
-if name=str_ptr-1 then {we can conserve string pool space now}
+if name=str_ptr-1 then {conserve string pool space (but see note above)}
begin flush_string(name); name:=cur_name;
end;
@y
@@ -766,11 +754,6 @@ begin
line_arg := chr(0);
iv
@#
- ed
- selector := term_only;
- write('You want to edit at line '); print(s); print_ln;
- de
- @#
selector := sel;
j := str_start[edit_file];
l := length(edit_file);
diff --git a/systems/unix/tex-fpc/shell/mkfont b/systems/unix/tex-fpc/shell/mkfont
index 34b2eb7849..735ca9ad6f 100755
--- a/systems/unix/tex-fpc/shell/mkfont
+++ b/systems/unix/tex-fpc/shell/mkfont
@@ -1,4 +1,4 @@
-# Install .tfm and . pk files from .mf files for TeX-FPC
+# Install .tfm files from .mf files for TeX-FPC
f=`basename $1 .mf`
case $# in
1) mag=1;;
@@ -6,9 +6,8 @@ case $# in
*) echo Usage: $0 mf-file [ magnification ]; exit 1
esac
rm -f $f.*gf
-mf "\\mode=localfont; mag=$mag; batchmode; input $1"\
-&& gftopk $f.*gf `echo $f.*gf | sed 's/gf$/pk/'` \
-&& mv $f.*pk PKfonts/ \
+mf "\\mode=localfont; batchmode; input $1"\
&& mv $f.tfm TeXfonts/ \
&& rm $f.log && rm $f.*gf \
-|| echo $0 $1 $mag >>trouble
+&& echo $1.tfm installed \
+|| (echo $0 $1 $mag >>trouble; echo $f failed, see file trouble)
diff --git a/systems/unix/tex-fpc/tangle.ch b/systems/unix/tex-fpc/tangle.ch
index ee73c2a3c8..661dcd49bb 100644
--- a/systems/unix/tex-fpc/tangle.ch
+++ b/systems/unix/tex-fpc/tangle.ch
@@ -1,4 +1,4 @@
-% This is a change file for TANGLE-FPC, Wolfgang Helbig, Nov. 2007
+% This is a change file for TANGLE-FPC, Wolfgang Helbig, Nov. 2007 - Jan 2021
[0] About TANGLE-FPC
@x
@@ -30,9 +30,9 @@ Comments and questions are welcome!
[1] Change the banner line
@x
-@d banner=='This is TANGLE, Version 4.5'
+@d banner=='This is TANGLE, Version 4.6'
@y
-@d banner=='This is TANGLE-FPC, 2nd ed.'
+@d banner=='This is TANGLE-FPC, 3rd ed.'
@z
[3] turn debugging on
diff --git a/systems/unix/tex-fpc/tangle.p b/systems/unix/tex-fpc/tangle.p
index a34f81d7dd..cc0d9b7ad3 100644
--- a/systems/unix/tex-fpc/tangle.p
+++ b/systems/unix/tex-fpc/tangle.p
@@ -1,5 +1,5 @@
{2:}{4:}{$MODE ISO}{$Q+}{$R+}{:4}PROGRAM TANGLE(OUTPUT);LABEL 9999;
-CONST{8:}BUFSIZE=100;MAXBYTES=45000;MAXTOKS=50000;MAXNAMES=4000;
+CONST{8:}BUFSIZE=100;MAXBYTES=45000;MAXTOKS=65000;MAXNAMES=4000;
MAXTEXTS=2000;HASHSIZE=353;LONGESTNAME=400;LINELENGTH=72;OUTBUFSIZE=144;
STACKSIZE=50;MAXIDLENGTH=12;UNAMBIGLENGT=7;
{:8}TYPE{11:}ASCIICODE=0..255;{:11}{12:}TEXTFILE=TEXT;
@@ -502,9 +502,9 @@ END ELSE IF BUFFER[1]=122 THEN BEGIN PRIMETHECHAN;
CHANGING:=NOT CHANGING;TEMPLINE:=OTHERLINE;OTHERLINE:=LINE;
LINE:=TEMPLINE;END;END;END{:137};
IF NOT CHANGING THEN BEGIN{136:}BEGIN LINE:=LINE+1;
-IF NOT INPUTLN(WEBFILE)THEN INPUTHASENDE:=TRUE ELSE IF LIMIT=CHANGELIMIT
-THEN IF BUFFER[0]=CHANGEBUFFER[0]THEN IF CHANGELIMIT>0 THEN CHECKCHANGE;
-END{:136};IF CHANGING THEN GOTO 20;END;LOC:=0;BUFFER[LIMIT]:=32;END;
+IF NOT INPUTLN(WEBFILE)THEN INPUTHASENDE:=TRUE ELSE IF CHANGELIMIT>0
+THEN CHECKCHANGE;END{:136};IF CHANGING THEN GOTO 20;END;LOC:=0;
+BUFFER[LIMIT]:=32;END;
{:135}{139:}FUNCTION CONTROLCODE(C:ASCIICODE):EIGHTBITS;
BEGIN CASE C OF 64:CONTROLCODE:=64;39:CONTROLCODE:=12;
34:CONTROLCODE:=13;36:CONTROLCODE:=125;32,9:CONTROLCODE:=136;
@@ -525,9 +525,9 @@ WHILE TRUE DO BEGIN IF LOC>LIMIT THEN BEGIN GETLINE;
IF INPUTHASENDE THEN BEGIN BEGIN WRITELN(OUTPUT);
WRITE('! Input ended in mid-comment');ERROR;END;GOTO 10;END;END;
C:=BUFFER[LOC];LOC:=LOC+1;{142:}IF C=64 THEN BEGIN C:=BUFFER[LOC];
-IF(C<>32)AND(C<>9)AND(C<>42)AND(C<>122)AND(C<>90)THEN LOC:=LOC+1 ELSE
-BEGIN BEGIN WRITELN(OUTPUT);WRITE('! Section ended in mid-comment');
-ERROR;END;LOC:=LOC-1;GOTO 10;
+IF(C<>32)AND(C<>9)AND(C<>42)THEN LOC:=LOC+1 ELSE BEGIN BEGIN WRITELN(
+OUTPUT);WRITE('! Section ended in mid-comment');ERROR;END;LOC:=LOC-1;
+GOTO 10;
END END ELSE IF(C=92)AND(BUFFER[LOC]<>64)THEN LOC:=LOC+1 ELSE IF C=123
THEN BAL:=BAL+1 ELSE IF C=125 THEN BEGIN IF BAL=0 THEN GOTO 10;
BAL:=BAL-1;END{:142};END;10:END;{:141}{145:}FUNCTION GETNEXT:EIGHTBITS;
@@ -743,11 +743,11 @@ END;10:END;}{:181}{182:}BEGIN INITIALIZE;{134:}OPENINPUT;LINE:=0;
OTHERLINE:=0;CHANGING:=TRUE;PRIMETHECHAN;CHANGING:=NOT CHANGING;
TEMPLINE:=OTHERLINE;OTHERLINE:=LINE;LINE:=TEMPLINE;LIMIT:=0;LOC:=1;
BUFFER[0]:=32;INPUTHASENDE:=FALSE;{:134};
-WRITELN(OUTPUT,'This is TANGLE-FPC, 2nd ed.');{183:}PHASEONE:=TRUE;
+WRITELN(OUTPUT,'This is TANGLE-FPC, 3rd ed.');{183:}PHASEONE:=TRUE;
MODULECOUNT:=0;REPEAT NEXTCONTROL:=SKIPAHEAD;UNTIL NEXTCONTROL=136;
WHILE NOT INPUTHASENDE DO SCANMODULE;
-{138:}IF CHANGELIMIT<>0 THEN BEGIN FOR II:=0 TO CHANGELIMIT DO BUFFER[II
-]:=CHANGEBUFFER[II];LIMIT:=CHANGELIMIT;CHANGING:=TRUE;LINE:=OTHERLINE;
+{138:}IF CHANGELIMIT<>0 THEN BEGIN FOR II:=0 TO CHANGELIMIT-1 DO BUFFER[
+II]:=CHANGEBUFFER[II];LIMIT:=CHANGELIMIT;CHANGING:=TRUE;LINE:=OTHERLINE;
LOC:=CHANGELIMIT;BEGIN WRITELN(OUTPUT);
WRITE('! Change file entry did not match');ERROR;END;END{:138};
PHASEONE:=FALSE;{:183};{FOR II:=0 TO 3 DO MAXTOKPTR[II]:=TOKPTR[II];}
diff --git a/systems/unix/tex-fpc/tex-fpc.pdf b/systems/unix/tex-fpc/tex-fpc.pdf
index d57450f062..fab992d408 100644
--- a/systems/unix/tex-fpc/tex-fpc.pdf
+++ b/systems/unix/tex-fpc/tex-fpc.pdf
Binary files differ
diff --git a/systems/unix/tex-fpc/tex.ch b/systems/unix/tex-fpc/tex.ch
index 491307d948..c6f0087062 100644
--- a/systems/unix/tex-fpc/tex.ch
+++ b/systems/unix/tex-fpc/tex.ch
@@ -1,4 +1,4 @@
-% This is a change file of TeX, Wolfgang Helbig, Nov. 2007-Nov. 2020
+% This is a change file of TeX, Wolfgang Helbig, Nov. 2007-Feb. 2021
[0] About TeX-FPC
Strangely, the first page is page 3
@@ -19,15 +19,15 @@ Strangely, the first page is page 3
\let\maybe=\iffalse % maybe -> print all, notmaybe -> print changed modules only
% put a mark in the left margin. (see Exercise 14.28 in The TeXbook)
-\def\marke#1{\strut\vadjust{\kern-\dp\strutbox\vbox to 0pt{\vss
- \llap{\bf #1\/\quad\strut}}}\ignorespaces }
+\def\marke#1{\strut\vadjust{\vbox to 0pt
+ {\vskip-\baselineskip\moveleft 2em\hbox{\bf #1\strut}\vss}}\ignorespaces}
\def\name{\TeX}
\N0\*. \[0] About \namefpc.
-\namefpc\ is an adaption of Donald~E. Knuth's \TeX82 in
-the version 3.14159265 from January 2014 to Free~Pascal (FPC) and Unix.
-To help you identify the differences of \TeX82 and
+\namefpc\ is a port of Donald~E. Knuth's typesetting program \TeX,
+version 3.141592653 from February 2021 to Free~Pascal (FPC) and Unix.
+To help you identify the differences of \TeX\ and
\namefpc, the numbers of modified modules carry an asterisk. Letters
in the left margin indicate the reason for a change. They mean:
\medskip
@@ -47,22 +47,21 @@ recovery.
new line until it reaches a nondiscardable node. This might leave
you with an empty box resulting in an \.{Underfull box} warning.
Btw, I discovered this bug while trying to prove the line breaking
-algorithm, not through by testing. In this regard, I go with Dijkstra, not
-with Knuth.
+algorithm, not while plain testing it. If you have time, prove, if
+not test.
\item{(3)} \TeX\ fails to respect end of file (Control-D) from
-terminal oinput during debug dialog.
+terminal input during debug dialog.
-\item{(3)}
-Igor Liferenko reported an extra space in the transcript file and
-a missing last line terminator on terminal output after user switched
-to \.{/batchmode} during error recovery.
+\item{(4)}
+Igor Liferenko reported an extra space in the transcript file
+after the user switched to \.{/batchmode} during error recovery.
\medskip
\item{(1)}\marke F\namefpc\ treats the command line as the first input line;
-\item{(2)}\namefpc\ starts \.{vi} if the user types `\.{E}' during
-error recovery.
+\item{(2)}\namefpc\ starts \.{ed}, the unix system editor, if the
+user types `\.{E}' during error recovery.
\item{(3)}You can interrupt \namefpc\ by typing `\.{Control-C}'.
@@ -129,7 +128,7 @@ I'd rather maintain invariants---not software.
known as `\TeX' [cf.~Stanford Computer Science report CS1027,
November 1984].
-@d banner=='This is TeX, Version 3.14159265' {printed when \TeX\ starts}
+@d banner=='This is TeX, Version 3.141592653' {printed when \TeX\ starts}
@y
known as `\TeX' [cf.~Stanford Computer Science report CS1027,
November 1984].
@@ -137,7 +136,7 @@ November 1984].
Even though \namefpc\ does not differ from \TeX\ I proudly change
the banner! And take responsibility for any error.
-@d banner=='This is TeX-FPC, 2nd ed.'
+@d banner=='This is TeX-FPC, 3rd ed.'
@z
[4] program header
@@ -150,7 +149,7 @@ the banner! And take responsibility for any error.
@d term_in==i @& n @& p @& u @& t
@d term_out==o @& u @& t @& p @& u @& t
-@d mtype==t @& y @& p @& e
+@d mtype==t@&y@&p@&e
@z
[4] program header
@@ -183,15 +182,15 @@ to the @x line and type >} or <} to (un)indent a paragraph.
@z
[7] shift left to turn stats on, right to turn them off.
- @x stats
- @d stat==@{ {change this to `$\\{stat}\equiv\null$' when gathering
- usage statistics}
- @d tats==@t@>@} {change this to `$\\{tats}\equiv\null$' when gathering
- usage statistics}
- @y
- @d stat== {change this to `$\\{stat}\equiv\.{@@\{}$' to turn off statistics}
- @d tats== {change this to `$\\{tats}\equiv\.{@@\}}$' to turn off statistics}
- @z
+@x stats
+@d stat==@{ {change this to `$\\{stat}\equiv\null$' when gathering
+ usage statistics}
+@d tats==@t@>@} {change this to `$\\{tats}\equiv\null$' when gathering
+ usage statistics}
+@y
+@d stat== {change this to `$\\{stat}\equiv\.{@@\{}$' to turn off statistics}
+@d tats== {change this to `$\\{tats}\equiv\.{@@\}}$' to turn off statistics}
+@z
[8] shift left to build TEX and right to build INITEX.
@@ -256,9 +255,9 @@ to Pascal.
@z
The changes headed by trip need to be made for the trip test suite.
- For trip activate all changeblocks with the first line @x trip.
+For trip activate all changeblocks with the first line @x trip.
- [11]
+[11]
@x trip
@!mem_max=30000; {greatest index in \TeX's internal |mem| array;
@y
@@ -638,7 +637,9 @@ while argc <= fpc_param_count do
begin
if last+1>=buf_size then
@<Report overflow of the input buffer, and abort@>;
- buffer[last] := xord[arg[cc]]; incr(last); incr(cc)
+ if xord[arg[cc]] <> invalid_code then
+ buffer[last] := xord[arg[cc]];
+ incr(last); incr(cc)
end;
if (argc <= fpc_param_count) then begin
buffer[last] := " "; incr(last) {insert a space between arguments}
@@ -695,11 +696,19 @@ loop@+begin write(term_out,'**');
get_strings_started:=false; return;
@z
-[51] TANGLE would'nt help, but update
+[51] print more sensible what you can't read
+@x
+else bad_pool('! I can''t read TEX.POOL.')
+@y
+else bad_pool('! I can''t read TeXformats/tex.pool.'){\marke u Unix file name}
+@z
+
+[53] TANGLE would'nt help. Don't guess but report what does not match.
@x
done: if a<>@$ then bad_pool('! TEX.POOL doesn''t match; TANGLE me again.');
@y
-done: if a<>@$ then bad_pool('! TeXformats/tex.pool doesn''t match.');
+done: if a<>@$ then bad_pool
+('! TeXformats/tex.pool doesn''t match. Not installed?'); {\marke u Unix file name}
@z
[79] start editor
@@ -722,22 +731,9 @@ want_edit := false; {\marke F don't start \.{ed}}
[84] set edit_cmd
@x
- begin print_nl("You want to edit file ");
-@.You want to edit file x@>
- slow_print(input_stack[base_ptr].name_field);
- print(" at line "); print_int(line);
interaction:=scroll_mode; jump_out;
- end;
@y
-@^Editor@>
- begin print_nl("You want to edit file ");
-@.You want to edit file x@>
- slow_print(input_stack[base_ptr].name_field);
- print(" at line "); print_int(line);
- interaction:=scroll_mode;
- want_edit := true;
- jump_out;
- end;
+ interaction:=scroll_mode; want_edit := true; jump_out;
@z
[86] extra space in log and missing last line terminator on terminal
@@ -745,8 +741,7 @@ want_edit := false; {\marke F don't start \.{ed}}
"Q":begin print_esc("batchmode"); decr(selector);
end;
@y
-"Q":print_esc("batchmode");
-{ \marke E don't turn off terminal now }
+"Q":print_esc("batchmode"); {\marke E don't turn off terminal now}
@z
[86] continued
@@ -806,10 +801,12 @@ macros are simplified in the obvious way when |min_quarterword=0|.
[241]
@x
@p procedure fix_date_and_time;
-begin time:=12*60; {minutes since midnight}
-day:=4; {fourth day of the month}
-month:=7; {seventh month of the year}
-year:=1776; {Anno Domini}
+begin sys_time:=12*60;
+sys_day:=4; sys_month:=7; sys_year:=1776; {self-evident truths}
+time:=sys_time; {minutes since midnight}
+day:=sys_day; {day of the month}
+month:=sys_month; {month of the year}
+year:=sys_year; {Anno Domini}
end;
@y
\marke X The functions |now|, |decodedate|, and |decodetime| are provided
@@ -828,11 +825,11 @@ var yy,mm,dd: word;
hh,ss,ms: word;
begin
fpc_decode_date(fpc_now,yy,mm,dd); {current date}
-day := dd;
-month := mm;
-year := yy;@/
+sys_day := dd; day:=sys_day;
+sys_month := mm; month:=sys_month;
+sys_year := yy; year := sys_year;@/
fpc_decode_time(fpc_now,hh,mm,ss,ms); {current time}
-time := hh*60+mm; {minutes since midnight}
+sys_time := hh*60+mm; time := sys_time; {minutes since midnight}
end;
@z
@@ -897,25 +894,11 @@ to place.
@d append_to_name(#)==begin c:=#; incr(k);
@y
-\marke X With Free~Pascal you cannot set the length of a |packed
-array|. And with Unix, it does not help to blank out the rest of
-the |array|. But assigning the |name_of_file| to an |fpc_string|,
-setting the length of the |fpc_string| and then assigning it back
-to the |name_of_file| seems to work. @^FPC Pascal@>
-
-@d fpc_set_length==s@&e@&t@&l@&e@&n@&g@&t@&h
+\marke U In Unix strings are terminated by chr(0).
@d append_to_name(#)==begin c:=#; incr(k);
@z
-[519]
-@x
-@!j:pool_pointer; {index into |str_pool|}
-@y
-@!j:pool_pointer; {index into |str_pool|}
-@!s: fpc_string;
-@z
-
[519] pack file name
@x
if k<=file_name_size then name_length:=k@+else name_length:=file_name_size;
@@ -923,9 +906,7 @@ for k:=name_length+1 to file_name_size do name_of_file[k]:=' ';
end;
@y
if k<=file_name_size then name_length:=k@+else name_length:=file_name_size;
-s := name_of_file;
-fpc_set_length(s, name_length);
-name_of_file := s;
+for k:=name_length+1 to file_name_size do name_of_file[k]:=chr(0);
end;
@z
@@ -939,18 +920,17 @@ TEX_format_default:='TeXformats/plain.fmt';
[523] packed buffered name
@x
-@!j:integer; {index into |buffer| or |TEX_format_default|}
+for k:=name_length+1 to file_name_size do name_of_file[k]:=' ';
@y
@^FPC Pascal@>
-@!j:integer; {index into |buffer| or |TEX_format_default|}
-s: fpc_string;
+for k:=name_length+1 to file_name_size do name_of_file[k]:=chr(0);
@z
-[523] packed buffered name
+[524 specify better, what you cannot read:
@x
-for k:=name_length+1 to file_name_size do name_of_file[k]:=' ';
+ wterm_ln('I can''t find the PLAIN format file!');
@y
-s := name_of_file; fpc_set_length(s, name_length); name_of_file := s;
+ wterm_ln('I can''t find TeXformats/plain.fmt!'); {\marke u Unix file name}
@z
[530] Control D will help
@@ -964,26 +944,38 @@ print_nl("Please type another "); print(s); print(" or Control-D");
@x
@ Let's turn now to the procedure that is used to initiate file reading
when an `\.{\\input}' command is being processed.
-
+Beware: For historic reasons, this code foolishly conserves a tiny bit
+of string pool space; but that can confuse the interactive `\.E' option.
+@^system dependencies@>
@y
@ Let's turn now to the procedure that is used to initiate file reading
when an `\.{\\input}' command is being processed.
+Beware: For historic reasons, this code foolishly conserves a tiny bit
+of string pool space; but that can confuse the interactive `\.E' option.
+
+\marke E In fact, it breaks the `\.E' option whenever the file to be
+edited was opened after the log file.
+In that case, the last string constructed is the name of the log file,
+otherwise, the last string constructed is the name of the input file.
+If the name of the input file is the last string constructed,
+\TeX\ strips off area and extension to conserve string
+pool space. The user is shown the base name of the file he wants
+to edit to fix a bug.
+
+Sadly, Knuth doesn't dare to fix this bug, which is known for at
+least twelve years. Are we approaching the limits of ``Literate Programming''.
+It looks beautiful but does it really help to cope with complexity?
+
@^Editor@>
@^Fix@>
-
-\marke E Keep the complete file name since it might be needed to
-be passed to the system editor.
-(\TeX82 strips off area and extension to conserve string
-pool space.)
@z
[537] never flush file name
@x
-if name=str_ptr-1 then {we can conserve string pool space now}
+if name=str_ptr-1 then {conserve string pool space (but see note above)}
begin flush_string; name:=cur_name;
end;
@y
-@^Editor@>
@z
[575] Don't check eof (fpc bug: eof off by one)
@@ -1177,9 +1169,11 @@ fpc_halt(history);
[1333] print last end-of-line marker if needed
@x
-This program doesn't bother to close the input files that may still be open.
+If |final_cleanup| is bypassed, this program doesn't bother to close
+the input files that may still be open.
@y
-This program doesn't bother to close the input files that may still be open.
+If |final_cleanup| is bypassed, this program doesn't bother to close
+the input files that may still be open.
\marke u
Terminate the last line on the terminal.
@z
@@ -1235,7 +1229,7 @@ itself will get a new section number.
@y
itself will get a new section number.
-@ \marke F If the user typed |'E'| to edit a file after confronted
+@ \marke F If the user typed \.{E} to edit a file after confronted
with an error message, \TeX\ will clean up and then call |edit|
as its last feat.
@^Editor@>
@@ -1302,15 +1296,15 @@ unit. @^unix@>
@d fpc_pchar == pchar
@d edit_file == input_stack[base_ptr].name_field
@#
-@d vi==@{ {change this to `$\\{vi}\equiv\null$' when you think \.{vi}
+@d vi== {change this to `$\\{vi}\equiv\null$' when you think \.{vi}
is the system editor}
-@d iv==@t@>@} {change this to `$\\{iv}\equiv\null$' when you think \.{vi}
+@d iv== {change this to `$\\{iv}\equiv\null$' when you think \.{vi}
is the sytem editor}
@f vi == begin
@f iv == end
-@d ed== {change this to `$\\{ed}\equiv\.{@@\{}$' when you think \.{vi}
+@d ed==@{ {change this to `$\\{ed}\equiv\.{@@\{}$' when you think \.{vi}
is the system editor}
-@d de== {change this to `$\\{de}\equiv\.{@@\}}$' when you think \.{vi}
+@d de==@} {change this to `$\\{de}\equiv\.{@@\}}$' when you think \.{vi}
is the system editor}
@f ed == begin
@f de == end
diff --git a/systems/unix/tex-fpc/trap/report b/systems/unix/tex-fpc/trap/report
index 639eccab79..47bc7b12c1 100644
--- a/systems/unix/tex-fpc/trap/report
+++ b/systems/unix/tex-fpc/trap/report
@@ -1,4 +1,5 @@
How to perform the trap test according to the trapman file.
+report of the trap test (cf. trapman)
Step 0.
Move the trap directory to the mf directory, and cd to the trap directory.
@@ -18,38 +19,29 @@ $ cd -
Step 2. run inimf
$ inimf
-This is METAFONT-FPC, 2nd ed. (INIMF)
+This is METAFONT-FPC, 3rd ed. (INIMF)
**
Please type the name of your input file or Control-D.
**\input trap
...
+mv trap.log trapin.log
+See diff_trapin.log for the differences of ../trapin.log and trapin.log
Step 3. run inimf again
$ inimf
-This is METAFONT-FPC, 2nd ed. (INIMF)
+This is METAFONT-FPC, 3rd ed. (INIMF)
** &trap trap
...
-See diff_trapin.log for the differences of ../trapin.log and trapin.log
-
-Step 4. trap.log differs from the master ../trap.log more than
-expected. Reason: Don Knuth changed the log files from the 2008
-distribution manually. He didn't want to wait for a reference installation
-of TeX and friends. He could have taken a slightly modified TeX-GPC!
-But from 2008 to 2014 the one and only change in mf resulted in
-changed addresses of the capsules. Which are not harmful. I undid
-the change of the 2014 distribution and voila, the difference of
-the addresses was gone.
+Copy the screen output to the file ../trap.fot
See diff_trap.log for the differences of ../trap.log and trap.log
-
See diff_trap.fot for the differences of ../trap.fot and trap.fot
Step 5. Check the generic font file:
$ gftype trap.72270gf trap.typ
-This is GFtype-FPC, 2nd ed.
+This is GFtype-FPC, 3rd ed.
Mnemonic output? (default=no, ? for help): y
Pixel output? (default=yes, ? for help): y
-
See diff_trap.typ for the differences of ../trap.typ and trap.typ.
Step 6.
diff --git a/systems/unix/tex-fpc/trip/diff_trip.fot b/systems/unix/tex-fpc/trip/diff_trip.fot
index 0d141edb62..e87a1d87a4 100644
--- a/systems/unix/tex-fpc/trip/diff_trip.fot
+++ b/systems/unix/tex-fpc/trip/diff_trip.fot
@@ -1,4 +1,4 @@
1c1
-< This is TeX-FPC, 2nd ed. (INITEX)
+< This is TeX, Version 3.141592653 (INITEX)
---
-> This is TeX, Version 3.14159265 (INITEX)
+> This is TeX-FPC, 3rd ed. (INITEX)
diff --git a/systems/unix/tex-fpc/trip/diff_trip.log b/systems/unix/tex-fpc/trip/diff_trip.log
index 0a1a5f42ca..1091ee177a 100644
--- a/systems/unix/tex-fpc/trip/diff_trip.log
+++ b/systems/unix/tex-fpc/trip/diff_trip.log
@@ -1,10 +1,12 @@
1c1
-< This is TeX-FPC, 2nd ed. (preloaded format=trip 2020.11.14) 14 NOV 2020 20:58
+< This is TeX, Version 3.141592653 (preloaded format=trip 1776.7.4) 4 JUL 1776 12:00
---
-> This is TeX, Version 3.14159265 (preloaded format=trip 2014.1.7) 7 JAN 2014 09:59
-7296,7297c7296,7297
-< 49 strings out of 1676
-< 273 string characters out of 8422
+> This is TeX-FPC, 3rd ed. (preloaded format=trip 2021.2.6) 6 FEB 2021 15:02
+1838a1839
+>
+7297,7298c7298,7299
+< 47 strings out of 1674
+< 253 string characters out of 8367
---
-> 47 strings out of 1674
-> 253 string characters out of 8354
+> 49 strings out of 1673
+> 273 string characters out of 8354
diff --git a/systems/unix/tex-fpc/trip/diff_trip.typ b/systems/unix/tex-fpc/trip/diff_trip.typ
index 00ee02fdb5..39ed777eb3 100644
--- a/systems/unix/tex-fpc/trip/diff_trip.typ
+++ b/systems/unix/tex-fpc/trip/diff_trip.typ
@@ -1,8 +1,8 @@
1c1
-< This is DVItype-FPC, 2nd ed.
+< This is DVItype, Version 3.6
---
-> This is DVItype, Version 3.6
+> This is DVItype-FPC, 3rd ed.
9c9
-< ' TeX output 2020.11.14:2058'
+< ' TeX output 1776.07.04:1200'
---
-> ' TeX output 2014.01.07:0959'
+> ' TeX output 2021.02.06:1502'
diff --git a/systems/unix/tex-fpc/trip/diff_tripin.log b/systems/unix/tex-fpc/trip/diff_tripin.log
index ed7e3fec2f..63567c0aa8 100644
--- a/systems/unix/tex-fpc/trip/diff_tripin.log
+++ b/systems/unix/tex-fpc/trip/diff_tripin.log
@@ -1,14 +1,10 @@
-1,3c1,3
-< This is TeX-FPC, 2nd ed. (INITEX) 14 NOV 2020 20:09
-< **\input ../trip
-< (../trip.tex
+1c1
+< This is TeX, Version 3.141592653 (INITEX) 4 JUL 1776 12:00
---
-> This is TeX, Version 3.14159265 (INITEX) 7 JAN 2014 09:09
-> **\input trip
-> (trip.tex
+> This is TeX-FPC, 3rd ed. (INITEX) 6 FEB 2021 15:01
450,451c450,451
-< (preloaded format=trip 2020.11.14)
-< 1325 strings of total length 23584
+< (preloaded format=trip 1776.7.4)
+< 1326 strings of total length 23633
---
-> (preloaded format=trip 2014.1.7)
-> 1326 strings of total length 23646
+> (preloaded format=trip 2021.2.6)
+> 1327 strings of total length 23646
diff --git a/systems/unix/tex-fpc/trip/report b/systems/unix/tex-fpc/trip/report
index dfe6472d82..b58628b260 100644
--- a/systems/unix/tex-fpc/trip/report
+++ b/systems/unix/tex-fpc/trip/report
@@ -3,7 +3,8 @@ tripman file.
Step 0.
mv the trip directory to the tex directory, and cd to the trip directory.
-This protects the original files like trip.log to be overwritten.
+Copy the file trip.tex from the tex directory to the trip directory:
+cp ../trip.tex .
Step 1. make and test trip.tfm
$ pltotf ../trip.pl trip.tfm
@@ -25,45 +26,46 @@ $ mktrip
37493
...
$ itgl tex.web tex.ch
-This is TANGLE-FPC, 2nd ed.
+This is TANGLE-FPC, 3rd ed.
...
-(No errors were found.)
-$ mv initex ~/bin/
+
+Save initex to you bin directory:
+mv initex ~/bin/
+
$ cd -
Step 3
$ initex
-This is TeX-FPC, 2nd ed. (INITEX)
+This is TeX-FPC, 3rd ed. (INITEX)
**
Please type the name of your input file or Control-D.
-**\input ../trip
+**\input trip
...
+Save trip.log:
+$ mv trip.log tripin.log
+
+Report the differences:
+$ diff ../tripin.log tripin.log >diff_tripin.log
Step 4
-This is TeX-FPC, 2nd ed. (INITEX)
-** &trip ../trip
-(../trip.tex ##
+$ initex
+This is TeX-FPC, 3rd ed. (INITEX)
+** &trip trip
! Bad number (-7).
...
-Step 5; compare with original files:
-$ diff tripin.log ../tripin.log >mydiff_tripin.log
-... etc. for trip.log, trip.fot,
+Copy the screen output to trip.fot.
+Report the differences:
+$ diff ../trip.log trip.log >diff_trip.log
+$ diff ../trip.fot trip.fot >diff_trip.fot
Note: The files diff_.. are distributed in this directory.
-Note: I performed the trip test with two versions of initex: One compiled
-by FPC 3.0.0 and one compiled by FPC 3.2.0. They seemed to use different
-floating point implementations as the glue set differences demonstrate. See
-the file diff_trip.log300 (fpc 3.0.0) and diff_trip.log (fpc 3.2.0).
-fpc 3.2.0 seems to have the same floating point implementation as Knuth's
-implementation.
Note: I copied the terminal window to create the trip.fot. It contains
-both what I typed in and what tex printed.
+both what I typed and what tex printed.
Step 6
The file options relieves you from typing the options manually.
$ dvitype trip.dvi trip.typ <options
-This is DVItype-FPC, 2nd ed.
+This is DVItype-FPC, 3rd ed.
Output level (default=4, ? for help): Starting page (default=*): Maximum number of pages (default=1000000): Assumed device resolution in pixels per inch (default=300/1): New magnification (default=0 to keep the old one): $
-
The file diff_trip.typ contains the differences of my run to Knuth's.
-
+$ diff ../trip.typ trip.typ >diff_trip.typ
diff --git a/systems/unix/tex-fpc/weave.ch b/systems/unix/tex-fpc/weave.ch
index e476736c28..a59132fbed 100644
--- a/systems/unix/tex-fpc/weave.ch
+++ b/systems/unix/tex-fpc/weave.ch
@@ -1,7 +1,4 @@
-% This is a change file for WEAVE-FPC, Wolfgang Helbig, Nov. 2007
-% Apr. 2008 get file names from command line
-% Jul. 2008 pass the exit code
-% Jul. 2016 change from gpc to fpc
+% This is a change file for WEAVE-FPC, Wolfgang Helbig, Nov. 2007 - Feb. 2021
[0] About WEAVE-FPC
@x
\pageno=\contentspagenumber \advance\pageno by 1
@@ -11,12 +8,12 @@
\def\name{\tt WEAVE}
\tracingstats=1
-\let\maybe=\iftrue % print changed modules only.
+\let\maybe=\iffalse % print changed modules only.
\N0\*. About \namefpc.\fi
-This is an adaption of Donald~E. Knuth's \.{WEAVE}, version 4.4
-from December 2002, to Unix. \namefpc\ is based on FPC~Pascal,
-version 3.2.0.
+This is an adaption of Donald~E. Knuth's \.{WEAVE}, version 4.5
+from January 2021, to Unix. \namefpc\ is based on FPC~Pascal,
+version 3.0.0.
This program expects three file names on the command line: A
web~file (\.{.web}), a change~file (\.{.ch}), and a \TeX~file
@@ -48,9 +45,9 @@ Comments and questions are welcome!
[1] Change the banner line
@x
-@d banner=='This is WEAVE, Version 4.4'
+@d banner=='This is WEAVE, Version 4.5'
@y
-@d banner=='This is WEAVE-FPC, 2nd. ed'
+@d banner=='This is WEAVE-FPC, 3rd ed.'
@z
[2] program header
diff --git a/systems/unix/tex-fpc/webmac-fpc.tex b/systems/unix/tex-fpc/webmac-fpc.tex
index 5d1e3af2ef..b33454ed14 100644
--- a/systems/unix/tex-fpc/webmac-fpc.tex
+++ b/systems/unix/tex-fpc/webmac-fpc.tex
@@ -3,7 +3,7 @@
\def\namefpc{$\hbox{\name}{}_{\fam0 FPC}$}
\def\TeXfpc{\TeX$_{\fam0 FPC}$}
\def\address{{\parindent0pt\sl
- November 2020
+ February 2021
Wolfgang Helbig, Programmierer\hfill
diff --git a/systems/unix/tex-fpc/xdvi/README b/systems/unix/tex-fpc/xdvi/README
index 5e1c79750d..779142c2a5 100644
--- a/systems/unix/tex-fpc/xdvi/README
+++ b/systems/unix/tex-fpc/xdvi/README
@@ -35,6 +35,11 @@ Ignore the install-data error from make install.
The file $HOME/Xdvi will be read by xdvi.
An example configuration file is Xdvi.
+Install the directory mfware from Knuth's distribution in tex-fpc.
+Build and install mfware/gftopk and cp dvips/MakeTeXPK to your bin
+directory. xdvi uses these programs to create bitmap pixel fonts~.nnnpk
+on demand.
+
You might want to change /etc/man.conf to include your private man pages.
I boldly replaced the line
_default /usr/{share,X11R6,pkg,local}/man/
diff --git a/systems/win32/miktex/tm/packages/files.csv.lzma b/systems/win32/miktex/tm/packages/files.csv.lzma
index 5328697c62..6d884042e3 100644
--- a/systems/win32/miktex/tm/packages/files.csv.lzma
+++ b/systems/win32/miktex/tm/packages/files.csv.lzma
Binary files differ
diff --git a/systems/win32/miktex/tm/packages/next/files.csv.lzma b/systems/win32/miktex/tm/packages/next/files.csv.lzma
index f328ed2bc6..87d0c56440 100644
--- a/systems/win32/miktex/tm/packages/next/files.csv.lzma
+++ b/systems/win32/miktex/tm/packages/next/files.csv.lzma
Binary files differ
diff --git a/systems/win32/miktex/tm/packages/next/pr.ini b/systems/win32/miktex/tm/packages/next/pr.ini
index 500318cfb1..fa057fde54 100644
--- a/systems/win32/miktex/tm/packages/next/pr.ini
+++ b/systems/win32/miktex/tm/packages/next/pr.ini
@@ -1,13 +1,13 @@
[repository]
-date=1613337255
+date=1613423781
lastupd=miktex-arctrl-bin-2.9 miktex-arctrl-bin-x64-2.9 miktex-asymptote-bin-2.9 miktex-asymptote-bin-x64-2.9 miktex-autosp-bin-2.9 miktex-autosp-bin-x64-2.9 miktex-axohelp-bin-2.9 miktex-axohelp-bin-x64-2.9 miktex-bibarts-bin-2.9 miktex-bibarts-bin-x64-2.9 miktex-bibtex-bin-2.9 miktex-bibtex-bin-x64-2.9 miktex-bibtex8bit-bin-2.9 miktex-bibtex8bit-bin-x64-2.9 miktex-bzip2-bin-2.9 miktex-bzip2-bin-x64-2.9 miktex-cairo-bin-2.9 miktex-cairo-bin-x64-2.9 miktex-chktex-bin-2.9 miktex-chktex-bin-x64-2.9
-lstdigest=896e56a3cce56d84fb73979c0c21207b
+lstdigest=353a24ce183080f4365e81c28e0fdb49
numpkg=4049
relstate=next
-version=7715
+version=7716
;;;;This configuration file is signed by a MiKTeX maintainer. The signature follows.
;;;;-----BEGIN MIKTEX SIGNATURE-----
-;;;; signature/miktex: NdB1z4Dtkk4ycwubytVVpFQ1s8Zs9DyBh4zugyfrtyJhCRE9+m1I34F4iBsOa30XgVdJBCec6E67GvnK0urF4UYpnVlf406j3q9S+2ElQm5iVEjvF5VIg1jiVy/q/rwdrPa3eLoIJSHhnZy9lQWlhBb7pThzNrxhSquThvyLvM6h6ak90KX40ZyseO8gq8GlGAhFTKuNxgGSWLTA6ekDIOR0Jt1iD7BhvbCjvxmW3Sg4gXvW03a68WLXcHim9tPE+kYm91fhwqtI/fZzjQ2Iy78zB3QL0QK1AFkVPpQdeOlnlyaNbiRSNG1xeoDqv+0uhfazsSy6AD9CGSieXwWthw==
+;;;; signature/miktex: WZUN6B/H0rkdmZuwNpGLScSmoF1jnUXj5uS+GEr7TA9AYe2PXt10PDs4rottrlkyEaqrcHQO6hEMosHAP3cKDEdi54ksgC8sRjxFnJxsfNX3WqTELWpBOUphp7LBx2bP+uxlHelhfI3gd5aWgdx6O4/1SPKXtGtY0/vt2L9OOb2Y0m/0VCVTSJhf1PekR1qzvzmEK2DbRs+emtJOPSJLXRcOHcN0R4tuYObqkZflX24NRbgYtp8t2xehZk5R/Hf9Y91wDkpVFDFoyuXFZ3y6Mj6Q0NFzvN/R0kWVzGKHZcKZGPAO7aZjgEX1A2ccHzIvycDyRR54VhwewBeuMm6pQQ==
;;;;-----END MIKTEX SIGNATURE-----
diff --git a/systems/win32/miktex/tm/packages/pr.ini b/systems/win32/miktex/tm/packages/pr.ini
index 02b606ea10..467b2546c9 100644
--- a/systems/win32/miktex/tm/packages/pr.ini
+++ b/systems/win32/miktex/tm/packages/pr.ini
@@ -1,13 +1,13 @@
[repository]
-date=1613337068
-lastupd=beamer easybook eczar fontsize hyperref mcf2graph acrotex aeb-pro asmejour atkinson babel-danish babel-finnish babel-portuges babel-slovenian babel-sorbian babel-swedish bxjscls cellprops cmap europecv
-lstdigest=a1380bd269f451777aa2399823ae537a
+date=1613423520
+lastupd=aalok babel-japanese easybook komacv luatexja pdfpc platex pxjahyper reledmac shtthesis tex4ht twemojis beamer eczar fontsize hyperref mcf2graph acrotex aeb-pro asmejour
+lstdigest=5f745d8c5765f4f00c46de03dea2d379
numpkg=4049
relstate=stable
-version=7715
+version=7716
;;;;This configuration file is signed by a MiKTeX maintainer. The signature follows.
;;;;-----BEGIN MIKTEX SIGNATURE-----
-;;;; signature/miktex: BAYdKm2SD7Z7CowS05HrHgjx3pnPH71zYdL7VcRLISIR3t+ROG+HNI+/vKd26QFP/xUApi3aEnikI+pw8zTLv5P5hOU/q3HUb8ik6mCj+XzoRg8utCgyEq5nKHOIjXGWtRFfg4zcgJFV3wQHt5hY/AqOwsMcIp66o+JF2UnFouG/m2meR9935C67pQOVu8/iYEtHYsW9SOjyNeT9URt/akBjROEFK0tATlBRz/+OQCD/bidGj6vHhaZE6MUfoqdn6aL/OpO7+ciye9QQYtIdZKhO6WPe6LDzPzykKTdOyB3jmB/0pX/rb8nWPuAyb1GMSB4LbYeuRJUmk4OlfKrs3A==
+;;;; signature/miktex: TDSlgLRg7BB5vS+PZh737QkjxQuKXarQlwYLmpOdYa0IBo+N7Fv+sbzljWhBWHy72jqo7FGAJTuvW8I7slqzPyWsVI5KvJmnbGwRbhrOo9oxt9lh1igMd3f33ag3oc0W+7Bj+1J7G+ZXZuvjn1723JUtRVneSwQ3UQtD3OdUCsSQbSYqGvWTqYxTZgf5TBnXDdJoImwP86zynpQ2xJnRuLs4zQt+RZrsJhoE5pOdaRbV4qMYaacf9Fl/7m1YieUnKWw0hhD3woHbMh4lFX9fobc1acU5MG4OQh1BEuqnEWwdoeSf05GbHfv9aG8oiP04RDKkZdPUYzlZjIw49npfFA==
;;;;-----END MIKTEX SIGNATURE-----
diff --git a/systems/win32/w32tex/ChangeLog b/systems/win32/w32tex/ChangeLog
index 9dd35aefbb..f1333218dc 100644
--- a/systems/win32/w32tex/ChangeLog
+++ b/systems/win32/w32tex/ChangeLog
@@ -1,5 +1,41 @@
W32TeX (x86 & x64) ChangeLog
+[2021/02/14]
+(01) cjkzr.tar.xz
+ Update pxjahyper.
+(02) context.tar.xz
+ Update format files.
+(03) latex-dev.tar.xz
+ Update format files.
+(04) pdftex-w32.tar.xz
+ Rebuild pdftex.dll.
+(05) platex.tar.xz
+ Update platex-tools, plautopatch and platex.
+(06) ptex-ng-w32.tar.xz
+ Update ptex-ng.exe (a new primitive \tracingstacklevels.)
+(07) ptex-w32.tar.xz
+ Update ptex.dll and eptex.dll.
+ (a new primitive \tracingstacklevels.)
+(08) uptex-w32.tar.xz
+ Update uptex.dll and euptex.dll.
+ (a new primitive \tracingstacklevels.)
+(09) xetex-w32.tar.xz
+ Rebuild xetex.dll.
+(10) web2c-lib.tar.xz
+ babel/japanese.
+(11) win64/pdftex-w64.tar.xz
+ Rebuild pdftex.dll.
+(12) win64/ptex-ng-w64.tar.xz
+ Update ptex-ng.exe (a new primitive \tracingstacklevels.)
+(13) win64/ptex-w64.tar.xz
+ Update ptex.dll and eptex.dll.
+ (a new primitive \tracingstacklevels.)
+(14) win64/uptex-w64.tar.xz
+ Update uptex.dll and euptex.dll.
+ (a new primitive \tracingstacklevels.)
+(15) win64/xetex-w64.tar.xz
+ Rebuild xetex.dll.
+
[2021/02/13]
(01) context.tar.xz
ConTeXt version: 2021.02.12 18:45.
diff --git a/timestamp b/timestamp
index 3bdc5d5e08..fb8a7548da 100644
--- a/timestamp
+++ b/timestamp
@@ -2,4 +2,4 @@
# The source CTAN of this site's material:
dante.ctan.org
# The year-month-day-hour-minute of this site's material:
-2021-02-15-02-02
+2021-02-16-02-02