diff options
Diffstat (limited to 'Master/texmf-dist/tex/plain/siam')
-rw-r--r-- | Master/texmf-dist/tex/plain/siam/docuptex.tex | 456 | ||||
-rw-r--r-- | Master/texmf-dist/tex/plain/siam/pexample.tex | 523 | ||||
-rw-r--r-- | Master/texmf-dist/tex/plain/siam/siamptex.sty | 373 |
3 files changed, 1352 insertions, 0 deletions
diff --git a/Master/texmf-dist/tex/plain/siam/docuptex.tex b/Master/texmf-dist/tex/plain/siam/docuptex.tex new file mode 100644 index 00000000000..2f673f9b7bf --- /dev/null +++ b/Master/texmf-dist/tex/plain/siam/docuptex.tex @@ -0,0 +1,456 @@ +%% +%% SIAM Plain TeX macro documentation. +%% Paul Duggan +%% 9-15-94 + +\input siamptex.sty + +% The definitions are to provide a verbatim text environment + +\def\uncatcodespecials{\def\do##1{\catcode`##1=12 }\dospecials} +\def\setupverbatim{\tt% + \def\par{\leavevmode\endgraf}% + \obeylines\uncatcodespecials\obeyspaces} +{\obeyspaces\global\let =\ } +\def\doverbatim#1{\def\next##1#1{##1\endgroup}\next} +\def\verbatim{\begingroup\setupverbatim\doverbatim} + + +\overfullrule=0pt + +\topmatter +\vol{1} +\no{1, pp.~000--000} +\SIMAC +\date{October 1994} +\copyyear{1994} +\code{000} + +\title Using SIAM's \TeX\ Macros\endtitle + +\shorttitle{USING SIAM'S \TeX\ MACROS} + +\recdate{*}{August 9, 1994. +This work was supported by the Society for Industrial +and Applied Mathematics} + +\author Paul Duggan\fnmark{$^{\dag}$}\endauthor + +\address{$^{\dag}$}{Society for Industrial and Applied Mathematics, +Philadelphia, Pennsylvania ({\tt duggan@siam.org}). Questions, comments, +or corrections to this document may be directed to that e-mail address} + +\abstract{Documentation is given for use of the SIAM \TeX\ macros. +Instructions and suggestions for compliance with SIAM style +standards are also included.} +\subjclass\endsubjclass +\keywords\endkeywords + + +\endtopmatter + + +\heading{1}{Introduction} +This file is documentation for the SIAM \TeX\ macros and provides +instruction for submission of files formatted in \TeX. + +To accommodate authors who electronically typeset their manuscripts, +SIAM supports the use of \TeX. To ensure quality typesetting according +to SIAM style standards, SIAM provides a \TeX\ macro style file. +Using \TeX\ to format a manuscript should simplify the editorial process +and lessen the author's proofreading burden. However, +it is still necessary to proofread the galley proofs with care. + +Electronic files should not be submitted until the paper has been +accepted, and then not until requested to do so by someone in the SIAM +office. Once an article is slated for an issue, +someone from the SIAM office will contact the author about any or all +of the following: editorial and stylistic queries, +supplying the source files (and any supplementary macros) +for the properly formatted article, and handling figures. + +When submitting electronic files (electronic submissions) +(to {\tt tex@siam.org}) write the journal, issue, and author's +name in the subject line of the message. +Authors are responsible for ensuring that the paper generated +from the source files exactly matches the paper that +was accepted for publication by the review editor. If it does not, +information on how it differs should be indicated in the transmission +of the file. When submitting a file, please be sure to include any +additional macros (other than those provided by SIAM) that will be +needed to run the paper. + +SIAM uses MS-DOS-based computers for \TeX\ processing. Therefore +all filenames should be restricted to eight characters or less, +plus a three character extension. + +Once the files are corrected here at SIAM, we will mail the revised +proofs to be read against the original edited hardcopy +manuscript. We are not +set up to shuttle back and forth varying electronic versions of each +paper, so we must rely on hard copy of the galleys. The author's proofreading +is an important but easily overlooked step. Even if SIAM were not +to introduce a single editorial change into your manuscript, there +would still be a need to check, because electronic transmission +can introduce errors. + + +This distribution contains the following items: {\tt siamptex.sty}, the +main macro package; this documentation file; and a sample file {\tt +pexample.tex}. The sample file is representative of the standard way +to apply the macros. The rest of this paper emphasizes some aspects +of applying the macros, points out options and special cases, and +describes the SIAM style standards. +The SIAM macros make use of the small caps font, {\tt cmcsc}, which +is not installed in some systems. This font along with other +AMS-\TeX\ fonts can be retrieved from the American Mathematical +Society via anonymous FTP to {\tt e-math.ams.com}. + +\heading{2}{Headings} +The top matter of a journal is in a standard format. The macro +and initial definitions should be included as follows. + +\verbatim: +\input siamptex.sty +\overfullrule=0pt + +: + +The \verbatim:\overfullrule: should be set wider than zero points +while still being edited by the author to assist in locating +lines that extend beyond the margins. + + +The ``slugline,'' which contains the journal and copyright information, is +created by special commands following the \verbatim:\topmatter:. + +\verbatim: +\SIMAX +\vol{0} +\no{0, pp.~000--000} +\date{April 1995} +\copyyear{1995} +\code{000} + +: +Because authors will probably not know exact +volume, number, date, or code, insert zeros in their place as +above. SIAM will place the appropriate information in +the commands in production; however, the commands must be used. +\verbatim:\SIMAX:, \verbatim:\SIAP:, \verbatim:\SICOMP:, +\verbatim:\SICON:, \verbatim:\SIDMA:, \verbatim:\SIMA:, +\verbatim:\SIMAX:, \verbatim:\SINUM:, \verbatim:\SIOPT:, +\verbatim:\SISC:, and \verbatim:\SIREV: are the commands provided +to include the journal title in the heading. + +The title and author(s) of the paper are indicated by the +\verbatim:\title\endtitle: and \verbatim:\author\endauthor: commands. +Author support and address information is indicated as follows. + +\verbatim: +\author A.~U. Thorone\fnmark{$^{\dag}$} + \and A.~U. Thortwo\fnmark{$^{\ddag}$}\endauthor + +\address{$^{\dag}$}{Address of the first author and support} +\address{$^{\ddag}$}{Address of the second author and support} + +: + +If more than two authors are included, each should be separated +by a comma after the \verbatim:\fnmark{}: command. If more than one +author shares common footnote information, then a common footnote +and footnote mark should be used. + +The commands \verbatim:\shorttitle{}: and \verbatim:\shortauthor{}: +are used to designate the running heads for the paper. + +The \verbatim:\abstract{}:, \verbatim:\keywords\endkeywords:, and +\verbatim:\subjclass\endsubjclass:\break commands are used to include the +abstract, key words, and AMS subject classification numbers, respectively. +If there is to be only one subject classification number, the +\verbatim:\subjclass: command should be preceded by +\verbatim:\oneclass:. (If the AMS numbers are unknown, leave that +information blank.) Authors are responsible for providing AMS numbers. +They can be found in the Annual Index of Math Reviews or +through {\tt e-Math} ({\tt telnet e-math.ams.com}; login +and password are both {\tt e-math}). +Complete the topmatter section by including \verbatim:\endtopmatter:. + + + +\heading{3}{Equations and mathematics} +Equations and mathematics +are handled by standard \TeX\ commands. SIAM style is +for numbered equations to appear flush with the left margin. The +\verbatim:\leqno: and \verbatim:\leqalignno{}: commands are used for +this purpose. If any letters indicating subequations are to be used +with the numbers, they should be set in roman type. + +Clear equation formatting using \TeX\ can be challenging. Aside from +the regular \TeX\ documentation, authors will find Nicholas +J. Higham's book {\it Handbook of Writing for the Mathematical +Sciences\/} [1] useful for guidelines and tips on formatting with +\TeX. The book covers many other topics related to article +writing as well. + +Authors commonly make mistakes by using + \verbatim:<:, \verbatim:>:, \verbatim:\mid:, and +\verbatim:\parallel: as delimiters, instead of +\verbatim:\langle:, \verbatim:\rangle:, \verbatim:|:, +and \verbatim:\|:. The incorrect symbols have particular +meanings distinct from the correct ones and should not be confused. + +\bigskip +\eightpoint +\halign{#\hfil\quad&&#\hfil\quad\cr +&{\bf Wrong} &&&& {\bf Right}&\cr +&\verbatim:<x, y>: & $<x, y>$ &&& + \verbatim:\langle x, y\rangle: & $\langle x, y\rangle$ \cr +&\verbatim:5 < \mid A \mid: & $5 < \mid A \mid$ &&& + \verbatim:5 < |A|: & $5 < |A|$ \cr +& \verbatim:6x = \parallel x: \cr +&\verbatim: - 1\parallel_{i}: & $6x = \parallel x - 1\parallel_{i}$ &&& + \verbatim:6x = \|x - 1\|_{i}: & $6x = \| x - 1\|_{i}$\cr} + \tenpoint\rm +\bigskip + +Another common author error is to put large (and even medium sized) +matrices in-line with the text, rather than displaying them. This +creates unattractive line spacing problems, and should be assiduously +avoided. Text-sized matrices (like $({a \atop b} {b \atop c})$) might +be used but anything much more complex than the example cited will +not be easy to read and should be displayed. + +More information on the formatting of equations and aligned +equations is found in Knuth [2]. Authors bear primary responsibility +for formatting their equations within margins and in an aesthetically +pleasing and informative manner. + +The SIAM macros include additional roman math words, or ``log-like" +functions, to those provided in standard \TeX. The following +commands are added: \verbatim:\const:, \verbatim:\diag:, \verbatim:\grad:, +\verbatim:\Range:, \verbatim:\rank:, and \verbatim:\supp:. +These commands produce the same word as the command name +in math mode, in roman type. + + +Groups of equations that are not directly related to each other +should normally be centered independently. This may be done through +the \TeX\ math command \verbatim:\displaylines{}:. Numbering +independently centered equations can be difficult, so Seroul and +Levy's [3] macro \verbatim:\ldisplaylinesno{}: has been included +in {\tt siamptex.sty}. \verbatim:\ldisplaylinesno{}: works just +like \verbatim:\leqalignno:, except no ampersand is used to align +the equations, since they are to be centered. + +\heading{4}{Text formatting} +Section and subsection headings are both included using the +\verbatim:\heading{}{}: command, which requires two arguments. The +first argument is for the number, and the second +is the title of the section or +subsection. + +No extra spacing should be placed between paragraphs. The +\verbatim:\heading: command inserts the required spacing between +sections. + +SIAM style does not normally make use of plain \TeX's +\verbatim:\item: command. The \verbatim:\meti: command is preferred for +lists of items beginning with, for instance, bullets ($\bullet$) or +roman numerals (iv). The \verbatim:\meti: command retains normal +paragraph shape but places all labels aligned flush right. For +example: + +\verbatim: +\meti{(i)} This is the first item. +\meti{(ii)} This is the second item of the series. + +: + +produces + +\meti{(i)} This is the first item. +\meti{(ii)} This is the second item of the series. +\medskip + +The \verbatim:\meti: macro was adapted from Seroul and Levy [3]. + +\heading{4.1}{Punctuation} +All standard punctuation and all numerals should be set in roman type +(upright) even within italic text. +The only exceptions are periods and commas. They may +be set to match the surrounding text. + +References to sections should use the symbol \S, generated by +\verbatim:\S:. (If the reference begins a sentence, the term ``Section'' +should be spelled out in full.) Authors should not redefine +\verbatim:\S:, say, to be a calligraphic S, because \verbatim:\S: +must be reserved for use as the section symbol. + +Authors sometimes confuse the use of various types of dashes. +Hyphens (\verbatim:-:, -) are used for some compound words (many +such words should have no hyphen but must be run together, +like ``nonzero,'' or split apart, like ``well defined.'' +Minus signs (\verbatim:$-$:, $-$) +should be used in math to represent subtraction or negative numbers. +En dashes (\verbatim:--:, --) are used for ranges (like 3--5, +June--August), or for joined names (like Runge--Kutta). Em dashes +(\verbatim:---:, ---) are used to set off a clause---such as this +one---from the rest of the sentence. + +\heading{4.2}{Theorems, lemmas, and proofs} +Theorems, lemmas, propositions, and so forth, have macros included +for correct formatting. Below is an example. + +\verbatim: +\thm{Theorem 4.1} +Sample theorem included for illustration. +Numbers and parentheses, like equation $(3.2)$, should be set +in roman type. Note that words in displayed equations, such as +$$ x^2 = Y^2 \sin z^2 \hbox{ for all } x $$ +will appear in italic type in a theorem, though normally +they should appear in roman.\endthm + +: + +This sample produces Theorem 4.1 below. + +\thm{Theorem 4.1} +Sample theorem included for illustration. +Numbers and parentheses, like equation $(3.2)$, should be set +in roman type. Note that words in displayed equations, such as +$$ x^2 = Y^2 \sin z^2 \hbox{ for all } x $$ +will appear in italic type in a theorem, though normally +they should appear in roman.\endthm + + +The \verbatim:\cor:, \verbatim:\dfn:, \verbatim:\lem:, and +\verbatim:\prop: commands +all work similarly. Named theorems should be designated with the +title in roman type, enclosed in parentheses. + +\verbatim: +\thm{Theorem 3.2 {\rm (sample theorem with title)}} + +: + +Proofs are illustrated in the following example: + +\verbatim: +\prf{Proof} +The body of the proof. +\qquad\endproof + +: + +If the proof ends with a displayed equation, the \verbatim:\endproof: +box \endproof\ should appear two ems (\verbatim:\qquad:) +from the end of the equation on line with it horizontally. + +\heading{5}{Figures and tables} +Figures and tables are best handled in \TeX\ by putting them within a +\verbatim:\topinsert \endinsert: or \verbatim:\midinsert \endinsert: +environment. The appropriate amount +of space should be left for the figure, and the caption should be +formatted to be centered or as a paragraph if more than one line. Text +should be italic, eight-point type, with the words ``Fig.~\#'' in small +caps. A sample follows. + +\verbatim: +\midinsert +\vskip 22pc +\centerline{\eightpoint{\smc Fig.~5.1}. \it Italic caption text.} +\endinsert + +: + +SIAM tables should be formatted in eight-point type, with enough +space left between entries and surrounding lines so that they do not +touch. Take particular care with super- and subscript characters. +See Knuth [2] or Seroul and Levy [3] for more information on +the formatting of tables in \TeX. Table captions are similar to +figure captions, but the word ``Table'' and the number appear on +a separate line from the caption text. + +SIAM supports the use of {\tt psfig} for including {\smc PostScript} +figures. All {\smc Post\-Script} figures should be sent in separate +files. See the {\tt psfig} documentation (from wherever you acquired +{\tt psfig}) for more details on the use +of this style option. It is a good idea to submit high-quality +hardcopy of all {\smc Post\-Script} figures just in case there +is difficulty in the reproduction of the figure. Figures produced +by other non-\TeX\ methods should be included as high-quality +hardcopy when the manuscript is submitted. + +\heading{6}{Bibliographies} +References are handled using the \verbatim:\Refs: command. +All names are to be keyed initial upper case cap and small +caps. Only the first and middle initials, followed by the +last name, are to be used. Last names should never be listed first. +Some representative sample entries are illustrated below: + +\verbatim| +\Refs + +\ref 1\\ {\smc A.~U Thorone}, {\it Title of paper with lower case +letters}, SIAM J. Abbrev. Correctly, 2 (1992), pp.~000--000.\endref + +\ref 2\\ \sameauthor, % generates a 3-em rule +{\it Title of paper appearing in book}, in Book Title: +With All Initial Caps, Publisher, Location, 1992.\endref + +\ref 3\\ {\smc W. Riter}, +{\it Title of another paper appearing in a book}, in The Book +Title, E.~D. One, E.~D. Two, and A.~N. Othereditor, eds., +Publisher, Location, 1992, pp.~000--000.\endref + +\ref 4\\ {\smc A.~U. Thorone, A.~U. Thortwo, and A.~U. Thorthree}, +{\it Title of Book{\rm III:} Note Initial Caps}, +Publisher, Location, pp.~000--000, 1994.\endref + +\ref 5\\ {\smc A. Notherauth}, {\it Title of paper that's not yet +published}, SIAM J. Abbrev. Correctly, to appear.\endref +| + +Other types of references fall into the same general pattern. See the +sample file or any SIAM journal for other examples. Authors must +correctly format their bibliography to be considered as having used +the macros correctly. An incorrectly formatted bibliography is not +only time-consuming for SIAM to process +but it is possible that errors may be introduced by +keyboarders/copy editors. + +As an alternative to the above style of reference, an alphanumeric +code may be used in place of the number (e.g., [AUTh90]). The same +\verbatim:\Refs: and \verbatim:\ref: commands are used, but the +command \verbatim:\resetrefindent{}: must be used before the +\verbatim:\Refs: command, with the widest expected alphanumeric code +as an argument. + +Another alternative is no number, simply the authors' names and +the year of publication following in parentheses. The rest of the +format is identical. To get an entry with no number in brackets +before it, use the \verbatim:\xref: command. This method is acceptable +but not encouraged. + +\heading{7}{Conclusion} Many other style suggestions and tips +could be given to help authors but are beyond the scope of this +document. Simple mistakes can be avoided by increasing your familiarity +with how \TeX\ functions. The books referred to throughout this document +are also useful to the author who wants clear, beautiful typography +with minimal mistakes. + +\Refs + +\ref 1\\ {\smc N.~J. Higham}, {\it Handbook of Writing for +the Mathematical Sciences}, Society for Industrial and Applied +Mathematics, Philadelphia, PA, 1993.\endref + +\ref 2\\ {\smc D.~E. Knuth}, {\it The \TeX book}, Addison Wesley, +Reading, MA, 1986.\endref + +\ref 3\\ {\smc R. Seroul and S. Levy}, {\it A Beginner's Book of +{\rm \TeX}}, Springer-Verlag, Berlin, New York, 1991.\endref + + +\bye diff --git a/Master/texmf-dist/tex/plain/siam/pexample.tex b/Master/texmf-dist/tex/plain/siam/pexample.tex new file mode 100644 index 00000000000..eb2c2f50101 --- /dev/null +++ b/Master/texmf-dist/tex/plain/siam/pexample.tex @@ -0,0 +1,523 @@ +% Sample file for SIAM's plain TeX macro package. +% 9-14-94 Paul Duggan +\input siamptex.sty + +% author defined macros included for illustrative purposes only. +% symbols for real numbers, complex, ... (\Bbb font from AMS-TeX +% fonts v2.x also usable) + +\def\fR{{\bf R}} +\def\fC{{\bf C}} +\def\fK{{\bf K}} + +% misc. operators +\def\Span {\mathop{\hbox{\rm span}}\nolimits} +\def\Range{\mathop{\hbox{\rm Range}}\nolimits} +\def\Det {\mathop{\hbox{\rm det}}} +\def\Re {\mathop{\hbox{\rm Re}}} +\def\Im {\mathop{\hbox{\rm Im}}} +\def\Deg {\mathop{\hbox{\rm deg}}} + +% misc. + +\def\Kr{\hbox{\bf K}} +\def\K { { K}} +\def\sT{\hbox{$\cal T$}} +\def\sB{\hbox{$\cal B$}} + +\def\bmatrix#1{\left[ \matrix{#1} \right]} + +% Each of the following commands have to be filled in with +% something. If the data is unknown, the arguments can be +% left blank. + +\topmatter +\journal{SIAM J. E{\smc XAMPLE} F{\smc ILES}} +\vol{1} +\no{1, pp.~000--000} +\date{October 1994} +\copyyear{1994} +\code{000} + + +\title SAMPLE FILE FOR SIAM PLAIN \TeX\ MACRO +PACKAGE\endtitle + +\shorttitle{SIAM MACRO EXAMPLE} + +\recdate{*}{October 1, 1994; accepted by the editors Month, x, +xxxx. This work was supported by the Society for Industrial +and Applied Mathematics, Philadelphia, Pennsylvania} + +\author Paul Duggan\fnmark{$^{\dag}$} \and Various A.~U. +Thors\fnmark{$^{\ddag}$}\endauthor + +\address{$^{\dag}$}{Composition Department, Society for +Industrial and Applied Mathematics, 3600 University City +Science Center, Philadelphia, Pennsylvania, 19104-2688 +({\tt duggan@siam.org})} + +\address{$^{\ddag}$}{Various affiliations, supported by +various foundation grants} + +\abstract{An example of SIAM \TeX\ macros is presented. +Various aspects of composing manuscripts for SIAM's journals +are illustrated with actual examples from accepted +manuscripts. SIAM's stylistic standards are adhered to +throughout, and illustrated.} + +\keywords polynomials, SI model\endkeywords + +\subjclass 33H40, 35C01\endsubjclass + +% if there is only one AMS subject number, the +% command \oneclass should precede the \subjclass command. + +\endtopmatter + +\heading{1}{Introduction and examples} +This paper presents a sample file for the use of SIAM's +\TeX\ macro package. It illustrates the features of the +macro package, using actual examples culled from various +papers published in SIAM's journals. This sample will provide +examples of how to use the +macros to generate standard elements of journal papers, +e.g., equations, theorems, or figures. This paper also +serves as an exmple of SIAM's stylistic preferences for the +formatting of such elements as bibliographic references, +displayed equations, and aligned equations, among others. +Some special circumstances are not dealt with this the +sample file; for that information, please see the +associated documentation file. + +{\it Note}. This paper is not to be read in any form for +content. The conglomeration of equations, lemmas, and other +text elements were put together solely for typographic +illustrative purposes. + +For theoretical reasons, it is desirable to find characterizations of the +conditions of breakdown of the algorithms that are based on the key {\it +spaces} $\Kr_n(r^{(0)},A)$ and $\Kr_n(\tilde r^{(0)},A^*)$ rather than +the {\it formulas} for the algorithms. In particular, we will +characterize breakdown of the three Lanczos algorithms in terms of the +{\it moment matrices} $\K_n(\tilde r^{(0)},A^*)^*\K_n(r^{(0)},A)$ and +$\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$. Here we define the matrix +$\K_n(v,A)=\bmatrix{v&Av&\cdots&A^{n-1}v\cr}$, a matrix whose columns span +the Krylov space $\Kr_n(v,A)$. + +The following three theorems give exact conditions for breakdown of the +above algorithms. Detailed proofs may be found in [3]. A +result similar to Theorem 2 is found in [1]; see also [5]. + + +\thm{Theorem 1 {\rm (Lanczos--Orthodir breakdown)}} +Suppose Lanczos/Orthodir has successfully generated +$u^{(n-1)}\not=u$. Then the following are equivalent: + +\meti{$\bullet$} The algorithm does not break down at step $n$. + +\meti{$\bullet$} The matrix $\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$ +is nonsingular. + +\meti{$\bullet$} There exists a unique iterate $u^{(n)}$ satisfying $(2)$. +\endthm + + +\thm{Theorem 2 {\rm (Lanczos--Orthomin breakdown)}} +Suppose Lanczos/Orthomin has successfully generated $u^{(n-1)}\not=u$. +Then the following are equivalent: + +\meti{$\bullet$} The algorithm breaks down at step $n$. + +\meti{$\bullet$} Either +$\K_{n-1}(\tilde r^{(0)},A^*)^*\K_{n-1}(r^{(0)},A)$ or +$\K_n(\tilde r^{(0)},A^*)^*A\K_n(r^{(0)},A)$ is singular. +\endthm + + +\prop{Proposition 3 {\rm (zero sets of polynomials)}} +Let $\fK=\fR$ or $\fC$. If $P$ is a complex nonzero polynomial in the +variables $x_1,x_2,\ldots ,x_N\in\fK$, then $P(x)\not=0$ for almost every +$x=(x_1,x_2,\ldots,x_N)\in \fK^N$. +\endprop + +\prf{Proof} +If $\fK=\fR$ and $P$ is nonzero, then either $\Re P(z)$ or $\Im P(z)$ +is a nonzero (real) polynomial; if $\fK=\fC$, we may decompose each $x_i$ +into real and imaginary parts, giving $2N$ variables, and consider the +real polynomial $P(x)^*P(x)$. In any case, we may assume without loss of +generality that $P$ is a nonzero real polynomial of real variables. + +We know that for any point $x$, the polynomial $P$ is the zero polynomial +if and only if the polynomial and all its derivatives are zero at $x$. +Let $V_0$ denote the set of zeros of $P$ in $\fR^N$. Suppose the set +$V_0$ has nonzero measure. We know from integration theory (see, for +example, [6, pp.\ 128f]) that almost every point of $V_0$ is +a point of density in each of the $N$ coordinate directions. We recall +that $x\in\fR$ is a point of density of a measurable subset +$S\subseteq\fR$ if for any sequence of intervals $I_n$ such that +$x\in I_n$ with measure $m(I_n)\rightarrow 0$ we have +$m(S\cap I_n)/m(I_n)\rightarrow 1$. + +It is easily seen that at such points in $V_0$, the first +partial derivatives of $P$ must necessarily be zero. Let $V_1$ be the +points of $V_0$ where all first derivatives are also zero. We have just +shown that $V_0$ and $V_1$ both have the same nonzero measure. The +argument +may be repeated for $V_1$ to show all second partial derivatives of $f$ +are zero at almost every point of $V_0$, and so forth, resulting in the +fact that $P$ and all its derivatives are zero on a set which has nonzero +measure. The proof is completed by selecting any one of these points. +\qquad\endproof + +\thm{Theorem 4 {\rm (Lanczos breakdown, iterate $n$)}} +Let $\fK=\fR$ or $\fC$, $A, \tilde Z\in\fK^{N\times N}$, and $n\leq d(A)$. +Then exactly one of the following three conditions holds for the Lanczos +method with $\tilde r^{(0)}=\tilde Z^* r^{(0)}$. + +\meti{\rm (i)} Hard breakdown at step $n$ occurs for every vector +$r^{(0)}\in\sT_n(A)\cap\fK^N$ $($and thus at least for almost every +$r^{(0)}\in\fK^N)$. + +\meti{\rm (ii)} Hard breakdown at step $n$ occurs for a nonempty measure-zero +set of vectors $r^{(0)}\in\sT_n(A)\cap\fK^N$ +$($and thus a nonempty measure-zero set of vectors in $\fK^N)$. + +\meti{\rm (iii)} Hard breakdown at step $n$ occurs for no vectors +$r^{(0)}\in\sT_n(A)\cap\fK^N$ $($and thus for at most a measure-zero set of +vectors in $\fK^N)$. + +Furthermore, the same result holds if ``hard breakdown'' is replaced by +``soft breakdown'' in the statement of this theorem. +\endthm + + +\prf{Proof} +For vectors $r^{(0)}\in\sT_n(A)\cap\fK^N$, breakdown is equivalent to +singularity of an appropriate moment matrix. The set $\sT_n(A)\cap\fK^N$ +amounts to almost +every vector in $\fK^N$. Now, by Corollary 5, the set $S_n$ of vectors in +$\fK^N$ for which the moment matrix of dimension $n$ is singular is either +the set of all vectors or a subset of measure zero. If the moment matrix +is singular for every vector (i.e., $S_n=\fK^N$), then it is singular for +every vector in $\sT_n(A)\cap\fK^N$, giving case (i) above. Otherwise the +set $S_n$ is measure zero in $\fK^N$. Thus +$\sB_n\equiv S_n\cap(\sT_n(A)\cap\fK^N)$ is of measure zero +and is either empty or nonempty. +\qquad\endproof + +\heading{2}{Tables and figures} +In Tables 1 and 2 we consider the unpreconditioned problem and also the (left) +ILU- and MILU-preconditioned problem (see [2] and [4]). Runs for which +convergence was not possible in ITMAX iterations are labeled by (--). + + +\topinsert +\hbox{\vbox{ \eightpoint +{\parindent 0pt +\centerline{\smc Table 1} +\centerline{\it Model problem, $h^{-1}=128$, {\rm ITMAX=3000}. + Number of iterations.}\vskip 6pt +\hfil\vbox{\offinterlineskip +\hrule +\halign{&\vrule#&\strut\ \hfil#\ \cr +height2pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +&{\rm method $\backslash$ Dh: } & + &0&&2${}^{-3}$&&2${}^{-2}$&&2${}^{-1}$&&2${}^{0}$& + &2${}^{1}$&&2${}^{ 2}$&&2${}^{ 3}$&&2${}^{ 4}$&&2${}^{5}$&\cr +height2pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +\noalign{\hrule} +height2pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +&{GMRES}($\infty$) \hfill & +& 290&& 269&& 245&& 220&& 200&& 189&& 186&& 189&& 207&& 249&\cr +&{BCG} \hfill & +& 308&& 341&& 299&&1518&& -- && -- && -- && -- && 533&& -- &\cr +&{BCG}{\rm, random $u^{(0)}$} \hfill & +& 309&& 354&& 300&& 310&& 313&& 301&& 299&& 302&& 290&& 293&\cr +&{BCGNB} \hfill & +& 308&& 353&& 284&& 338&& 253&& 240&& 243&& 240&& 302&& 962&\cr +&{CGS} \hfill & +& 272&& 254&& 222&& -- && -- && -- && -- && -- && -- && -- &\cr +&{CGS}{\rm, random $u^{(0)}$} \hfill & +& 193&& 189&& 200&& 192&& 193&& 175&& 225&& 212&& 216&& 197&\cr +&{CGSNB} \hfill & +& 272&& 284&& 212&& 196&& 151&& 162&& 158&& 173&& 156&& 256&\cr +height1pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +} \hrule}\hfil}}} +\endinsert + + +\topinsert + +\hbox{\vbox{ \eightpoint +{\parindent 0pt +\centerline{\smc Table 2} + +\centerline{\it Model Problem, $h^{-1}=128$,} +\centerline{\it {\rm MILU}-preconditioning, {\rm ITMAX=500.} +Number of iterations.} + +\medskip + +\hfil\vbox{\offinterlineskip +\hrule +\halign{&\vrule#&\strut\ \hfil#\ \cr +height2pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +&{\rm Method $\backslash$ Dh: } & + &0&&2${}^{-3}$&&2${}^{-2}$&&2${}^{-1}$&&2${}^{0}$& + &2${}^{1}$&&2${}^{ 2}$&&2${}^{ 3}$&&2${}^{ 4}$&&2${}^{5}$&\cr +height2pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +\noalign{\hrule} +height2pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +&{\rm {GMRES}($\infty$)} \hfill & +& 27&& 25&& 24&& 26&& 28&& 28&& 25&& 19&& 14&& 10&\cr +&{\rm {GMRES}($\infty$), random $u^{(0)}$} \hfill & +& 33&& 29&& 28&& 29&& 31&& 31&& 29&& 24&& 19&& 14&\cr +&{\rm {BCG}} \hfill & +& 31&& 27&& 29&& 33&& 30&& 37&& 30&& 23&& 15&& 10&\cr +% &{BCG}, random $u^{(0)}$, $\gamma=.1$ \hfill & +% & 35&& 30&& 31&& 35&& 40&& 37&& 34&& 27&& 20&& 15&\cr +&{\rm {BCG}, random $u^{(0)}$} \hfill & +& 38&& 34&& 33&& 37&& 44&& 40&& 38&& 29&& 23&& 18&\cr +&{\rm {BCGNB}} \hfill & +& 28&& 27&& 29&& 30&& 34&& 35&& 30&& 23&& 15&& 10&\cr +&{\rm {CGS}} \hfill & +& 21&& 18&& 17&& 20&& 22&& 22&& 19&& 15&& 9&& 6&\cr +&{\rm {CGS}, random $u^{(0)}$} \hfill & +& 24&& 18&& 20&& 22&& 22&& 23&& 21&& 16&& 12&& 9&\cr +&{\rm {CGSNB}} \hfill & +& 21&& 18&& 17&& 20&& 22&& 27&& 20&& 15&& 9&& 6&\cr +height1pt&\omit&&\omit&&\omit&&\omit&&\omit + &&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr +} \hrule}\hfil}}} +\endinsert + +We make the following observations about these runs. + +\meti{$\bullet$} For the unpreconditioned problem, the standard +{BCG} and {CGS} algorithms break down in a number of cases, but the use +of random $u^{(0)}$ or the use of {BCGNB} or {CGSNB} +resulted in convergence. Furthermore, the iteration counts for the +algorithms {BCG} and {BCGNB} are in +general comparatively close to those of the ``best'' method, +{GMRES}($\infty$), while these algorithms have short economical +recurrences, unlike {GMRES}($\infty$). This underscores the +importance of the Lanczos algorithms as economical solution techniques. + +\meti{$\bullet$} For the ILU-preconditioned problems, in most cases +all methods worked well. For the case of $Dh=1$, {BCG} gave +an excessive number of iterations, but this was remedied significantly +by {BCGNB} and much more so +by the use of random $u^{(0)}$. Similarly, {CGS} could +not converge, but {CGSNB} and {CGS} with random +$u^{(0)}$ both converged. + +\meti{$\bullet$} For all of the MILU-preconditioned problems, all of +the Lanczos-type algorithms performed quite well. In particular, the +{BCG} algorithm gave approximately the same number of +iterations as {GMRES}($\infty$). + +Figures 1 and 2 give representative plots of the convergence behavior of the +algorithms for the case of $h^{-1}=128$, $Dh=4$, and no preconditioning. +These results show that the new algorithms keep the residual size +better behaved than the standard {BCG} and {CGS} +algorithms over the course of the run. + +\topinsert +\vskip 3.2in +\centerline{\eightpoint\smc Fig.~1. \it Residual +behavior: $h^{-1}=128$, $Dh=4$.} +\endinsert + + +\topinsert + \vskip 3.2in +\centerline{\eightpoint\smc Fig.~2. \it Residual +behavior: $h^{-1}=128$, $Dh=4$.} +\endinsert + +We now consider a more difficult class of finite difference problems, +namely, central finite differencing applied to the Dirichlet problem +$$ -u_{xx}(x,y) - u_{yy}(x,y) + + D[(y-\textstyle{1\over 2}\displaystyle) u_x(x,y) + + (x-\textstyle{1\over 3}\displaystyle) + (x-\textstyle{2\over 3}\displaystyle) u_y(x,y)], $$ +$$ - 43\pi^2u(x,y) = G(x,y) \quad {\rm on}\ \Omega=[0,1]^2,$$ +$$u(x,y) = 1 + xy \quad \hbox{\rm on}\ \partial\Omega,$$ +with $G(x,y)$ chosen as before so that the true solution is $u(x,y)=1+xy$. +Again, we let $h$ denote the mesh size in each direction. For $D=0$ +and $h$ small, the matrix generated by this problem is a symmetric +indefinite matrix with 16 distinct negative eigenvalues and the rest +of the spectrum positive. + +The standard conjugate residual algorithm applied to this problem with +$h^{-1}=128$ and $D=0$ requires 766 iterations to converge to +$||r^{(n)}||/||b||<\zeta=10^{-6}$. In any case, this is a difficult +problem to solve. + + \def\qed{\vrule height8pt width4pt depth0pt\par\medskip} + \def\Zero{{\bf 0}} + \def\dis{\displaystyle} + \def\b{\beta} + \def\r{\rho} + \def\X{{\bf X}} + \def\Y{{\bf Y}} + \def\bb{{\bar \beta}} + \def\tbcr{\theta\bb c_h \rho_h} + \def\ep{\varepsilon} + + + +Figures 3(a) and 3(b) show the compartmental diagrams for SI models without +and with deaths due to the disease, for the situation in which the infectious +period has only one stage. Figures 4(a) and 4(b) give the corresponding models +with $m$ stages of infection. Venereal +warts, caused by the human papilloma virus, and ordinary herpes are examples +of sexually transmitted diseases without deaths +due to the disease, although both are not quite SI diseases because of +partial immunity. AIDS is the example of an SI disease with death due to +the disease. Although our main focus is on the latter, we present results +on SI models without deaths due to the +disease because the simplification in the dynamics of such models + throws light on the case with disease-related deaths. + +\topinsert +\vskip 2in +\centerline{\eightpoint {\smc Fig.} 3(a). SI {\it model for subgroup $i$, without death +due to the disease.}} +\vskip 2in +\centerline{\eightpoint {\smc Fig.} 3(b). SI {\it model with death due to the disease.}} +\endinsert + +\topinsert +\vskip 2in +\centerline{\eightpoint {\smc Fig.} 4(a). SI {\it model without deaths due to the +disease with $m$ stages of infection.}} +\vskip 2in +\centerline{\eightpoint {\smc Fig.} 4(b). SI {\it model with deaths due to the disease, +with $m$ stages of infection.}} +\endinsert + +\heading{3}{Equations and alignments} +The equations for the system follow directly from the definitions and the +compartmental diagrams. For one infected stage with no disease-related +deaths, the equations are +$$ \dot X_i=-X_ig_i-\mu X_i+U_i, \leqno(1)$$ +$$ \dot Y_i=X_ig_i-\mu Y_i. \leqno(2)$$ +If there are multiple stages to the infection, (2) is replaced by +(3)--(5) as follows: +$$\leqalignno{\dot Y_{i1}&=X_ig_i-(k+\mu)Y_{i1}, &(3)\cr +\dot Y_{ir}&=kY_{i,r-1}-(k+\mu)Y_{ir},\qquad r=2,\ldots,m-1 &(4)\cr +\dot Y_{im}&=kY_{i,m-1}-\mu Y_{im}. &(5)\cr }$$ + + + +\heading{3.1}{The SI model with structured mixing} +In this subsection we write the equations for the SI model with +structured mixing, with one infected stage and with deaths due to the +disease. The equations for multiple infected stages follow easily, as do +those for SI models without death due to the disease. Recall that $f_{is}$ +gives the fraction of population subgroup $i$'s +contacts that are made in activity group $s$. The total contact rate of +susceptibles from population subgroup $i$ in activity group $s$ must be +$c_iX_if_{is}$. Let $\rho_{ij}(s)$ be the fraction of the contacts of group +$i$ that are with members of group $j$, within activity group $s$. +Assuming random allocation of the susceptibles and infecteds from each +population subgroup to the activity groups, the fraction infected in group +$j$ in activity group $s$ must be $Y_j/N_j$, giving +$$ c_iX_if_{is}\rho_{ij}(s)\beta_j{Y_j \over N_j}\leqno(6) $$ +for the rate at which susceptibles in $i$ are infected by contacts +with infecteds from $j$ in activity group $s$. Thus, in this case, $g_i$ is +given by +$$ + g_i=c_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over N_j}, +\leqno(7) + $$ +and (1a) and (1b) become +$$ \dot X_i=-c_iX_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over +N_j}-\mu X_i+U_i, \leqno(8) $$ +$$ \dot Y_i=c_iX_i\sum_sf_{is}\sum_j\rho_{ij}(s)\beta_j{Y_j \over +N_j}-(\mu+k)Y_i. \leqno(9) $$ + +\heading{3.2}{Structured mixing within activity groups} +If the mixing within activity groups is proportional mixing, then +$\rho_{ij}(s)$ is given by (10): +$$\rho_{ij}(s)={f_{js}c_jN_j\over \sum_pf_{ps}c_pN_p}, \leqno(10)$$ +and (8) and (9) become (11) and (12): +$$\dot X_i=-c_iX_i\sum_sf_{is}{\sum_jf_{js}c_j\beta_jY_j \over +\sum_jf_{js}c_jN_j}-\mu X_i+U_i \leqno(11)$$ +$$\dot Y_i=c_iX_i\sum_sf_{is}{\sum_jf_{js}c_j\beta_jY_j \over +\sum_jf_{js}c_jN_j}-(k+\mu)Y_i. \leqno(12)$$ + +Expressions (11) and (12) show an important consequence of death due +to the disease. If there are no deaths due to the disease, $N_j$ is +constant on the asymptotically stable invariant subspace $U_j=\mu +N_j$ for all $j$, and the first term, the nonlinear term, in +(11) and (12) is a sum of {\it quadratic} terms. If there are deaths +due to the disease, $N_j$ is no longer constant and the first term is +a sum of rational expressions, each homogeneous of degree one. This +observation extends to SIS, SIR, and SIRS models. + + + +\Refs + + +\ref 1\\ +{\smc R. Fletcher}, {\it Conjugate gradient methods for indefinite +systems}, in Numerical Analysis Dundee 1975, G.~A. Watson, ed., +Springer-Verlag, New York, Lecture Notes in Math. 506, +1976, pp. 73--89. +\endref + + +\ref 2\\ +{\smc I. Gustafsson}, {\it Stability and rate of convergence of +modified incomplete Cholesky factorization methods}, Ph.D. thesis, +Chalmers University of Technology and the University of Goteborg, +Goteborg, Sweden, 1979. +\endref + + +\ref 3\\ +{\smc W.~D. Joubert}, {\it Generalized conjugate gradient and +Lanczos methods for the solution of nonsymmetric systems of linear +equations}, Ph.D. thesis and Report +CNA-238, Center for Numerical Analysis, University of Texas, +Austin, TX, January 1990. +\endref + + +\ref 4\\ +{\smc J.~A. Meijerink and H.~A. van der Vorst}, {\it An iterative +solution method for linear systems of which the coefficient matrix is +a symmetric $M$-matrix}, Math. Comp., 31 (1977), pp.~148--162. +\endref + + + +\ref 5\\ +{\smc Y.~Saad}, {\it The Lanczos biorthogonalization algorithm and +other oblique projection methods for solving large unsymmetric systems}, +SIAM J. Numer. Anal., 19 (1982), pp. 485--506. +\endref + + +\ref 6\\ +{\smc S. Saks}, {\it The Theory of the Integral}, G.~E. Stechert, +New York, 1937. +\endref + +\ref 7\\ +{\smc M. Tinkham}, {\it Introduction to +Superconductivity}, McGraw-Hill, New York, 1975. +\endref + +\bye diff --git a/Master/texmf-dist/tex/plain/siam/siamptex.sty b/Master/texmf-dist/tex/plain/siam/siamptex.sty new file mode 100644 index 00000000000..3ffce26af64 --- /dev/null +++ b/Master/texmf-dist/tex/plain/siam/siamptex.sty @@ -0,0 +1,373 @@ +% SIAMPTEX.STY; 12-11-92; Paul Duggan, Society for Industrial +% and Applied Mathematics. From +% Ralph Youngen's siamtex.sty +% 9-14-94 ; Paul Duggan Revised. +% This file may be freely distributed but may not be altered in +% any way. +% +% This is a file of macros and definitions for creating an article +% for publication in a SIAM journal. +% +% Make the @ sign a letter for internal control sequences. +\catcode`\@=11 +% +% Report the version number of this package. +\message{ *** SIAM Plain TeX macro package, version + 2.1.1, SEPTEMBER-1994 ***} +% +%%% +%%% FONTS +%%% +\font\tenrm=cmr10 +\font\tenbf=cmbx10 +\font\tenit=cmti10 +\font\tentt=cmtt10 +\font\tensmc=cmcsc10 +\def\tenpoint{% + \def\rm{\fam0\tenrm}\def\bf{\fam\bffam\tenbf}% + \def\it{\fam\itfam\tenit}\def\smc{\tensmc}\def\tt{\tentt} + \textfont0=\tenrm \scriptfont0=\sevenrm + \textfont1=\teni \scriptfont1=\seveni + \textfont2=\tensy \scriptfont2=\sevensy + \textfont3=\tenex \scriptfont3=\tenex +\baselineskip=12pt\rm}% + +\font\ninerm=cmr9 +\font\ninebf=cmbx9 +\font\nineit=cmti9 +\def\ninepoint{% + \def\rm{\ninerm}\def\bf{\ninebf}% + \def\it{\nineit}\baselineskip=11pt\rm}% + +\font\eightrm=cmr8 +\font\eightbf=cmbx8 +\font\eightit=cmti8 +\font\eighti=cmmi8 +\font\eightsy=cmsy8 +\font\eightsmc=cmcsc10 at 8pt +\font\eighttt=cmtt8 + \def\eightpoint{% + \def\rm{\fam0\eightrm}\def\bf{\fam\bffam\eightbf}% + \def\it{\fam\itfam\eightit}\def\smc{\eightsmc}\def\tt{\eighttt} + \textfont0=\eightrm \scriptfont0=\sixrm + \textfont1=\eighti \scriptfont1=\sixi + \textfont2=\eightsy \scriptfont2=\sixsy +\textfont3=\tenex \scriptfont3=\tenex +\baselineskip=10pt\rm}% + +\font\sixrm=cmr6 +\font\sixbf=cmbx6 +\font\sixi=cmmi6 +\font\sixsmc=cmr5 +\font\sixsy=cmsy6 +\def\sixpoint{% + \def\rm{\sixrm}\def\bf{\sixbf}% + \def\smc{\sixsmc}\baselineskip=8pt\rm}% + +\fontdimen13\tensy=2.6pt +\fontdimen14\tensy=2.6pt +\fontdimen15\tensy=2.6pt +\fontdimen16\tensy=1.2pt +\fontdimen17\tensy=1.2pt +\fontdimen18\tensy=1.2pt + +\def\rheadfont{\eightpoint\rm} +\def\titlefont{\tenpoint\bf} +\def\authorfont{\eightpoint\smc} +\def\dedicatefont{\ninepoint\it} + + +%%% +%%% ROMAN PUNCTUATION +%%% (WARNING: may cause incompatibility with psfig) +%%% +\def\othersmcolon{;} +\catcode`\;= \active +\def;{\relax\ifmmode\othersmcolon + \else\ifdim\fontdimen1\the\font>0pt\/{\rm \othersmcolon}% + \else\othersmcolon\fi\fi} + + +%%% DIMENSIONS +%%% +\hsize=31pc +\vsize=50pc +\parindent=2em + + +%%% TOPMATTER MACROS +%%% +\newif\ifkeywords@ +\newif\ifsubjclass@ +\newif\ifdedicate@ +\newif\ifrecdate@ +\newif\ifoneclass@ + +\keywords@false +\subjclass@false +\dedicate@false +\recdate@false +\oneclass@false + +\def\title#1\endtitle{\def\thetitle{\uppercase{#1}}% + \def\\{\relax\ }\xdef\rightrh{\uppercase{#1}}} + +\def\shorttitle#1{\xdef\rightrh{\uppercase{#1}}} + +\let\protect\relax + +\def\author#1\endauthor{% + \def\and{\hbox{\sixrm AND }}\def\\{\break} + % prevent \uppercase when typesetting + \def\theauthor{\uppercase{#1}}% + {% + \def\\{\relax\ }% change \\ to be a space in the running head + \def\fnmark##1{}% turn address off in the running head + \def\and{and }% "and" is same size as names in running head + \def\protect{\noexpand\protect\noexpand} + \xdef\leftrh{\expandafter{\uppercase{#1}}}% + }} + +\def\shortauthor#1{\def\and{and }\xdef\leftrh{\uppercase{#1}}} + +\gdef\fnmark#1{#1} +\gdef\address#1#2{\makefootnote@{\textfont2=\eightsy #1}{#2\unskip.}} +\def\journal#1{\def\thejournal{#1}} +\def\SIMAC{\journal{SIAM J{\smc OURNAL} M{\smc ACRO}}} +\def\SIAP{\journal{SIAM J. A{\smc PPL.} M{\smc ATH.}}} +\def\SICOMP{\journal{SIAM J. C{\smc OMPUT.}}} +\def\SICON{\journal{SIAM J. C{\smc ONTROL} {\smc AND} O{\smc PTIMIZATION}}} +\def\SIDMA{\journal{SIAM J. D{\smc ISC.} M{\smc ATH.}}} +\def\SIMA{\journal{SIAM J. M{\smc ATH.} A{\smc NAL.}}} +\def\SIMAX{\journal{SIAM J. M{\smc ATRIX} A{\smc NAL.} A{\smc PPL.}}} +\def\SINUM{\journal{SIAM J. N{\smc UMER.} A{\smc NAL.}}} +\def\SIOPT{\journal{SIAM J. O{\smc PTIMIZATION}}} +\def\SISC{\journal{SIAM J. S{\smc CI.} C{\smc OMPUT.}}} +\def\SIREV{\journal{SIAM R{\smc EVIEW}}} + + +\def\copyyear#1{\def\thecopyyear{#1}} +\copyyear{1989} +\def\vol#1{\def\thevol{#1}} +\def\no#1{\def\theno{#1}} +\def\date#1{\def\thedate{#1}} +\def\code#1{\def\thecode{#1}} +\def\dedicate#1{\dedicate@true\def\thededicate{#1}} +\def\keywords#1\endkeywords{\keywords@true\def\thekeywords{#1}} +\def\subjclass#1\endsubjclass{\subjclass@true\def\thesubjclass{#1}} +\def\oneclass{\oneclass@true} +\long\def\abstract#1{\def\theabstract{#1}} + +\def\raggedcenter{\leftskip=0pt plus 1fill \rightskip=0pt plus 1fill} + +\def\copyright#1{{\ooalign{\hfil\raise.07ex\hbox{c}\hfil\crcr#1\char"0D}}} + +\def\recdate#1#2{\global\recdate@true + \gdef\recdate@{#1Received by the editors \ignorespaces#2\unskip.}% + \makefootnote@{}{\recdate@}} + +\let\topmatter=\relax +\def\endtopmatter{% + \global\headline={\hss\vtop to \z@ {% + \sixpoint\rm\noindent\thejournal \hfill% + \rm\copyright{\sevensy}\rm% + \ \thecopyyear\ Society for Industrial and Applied Mathematics\par + \noindent Vol. \thevol , No. \theno , \thedate\hfill\thecode\par + \vss}\hss}% + \leavevmode\vskip8pt + \vbox{\noindent\raggedcenter\let\\=\break\titlefont\thetitle + \ifrecdate@*\fi} + \vskip6pt + \vbox{\noindent\raggedcenter\let\\=\break\authorfont\theauthor} + \ifdedicate@ + \vskip6pt + \vbox{\noindent\raggedcenter\let\\=\break\dedicatefont + \thededicate} + \fi + \vskip14pt + {\eightpoint{\bf Abstract.\ }\theabstract\par + \ifkeywords@ + \vskip10pt + {\bf Key words.} \thekeywords\par + \fi + \ifsubjclass@ + \vskip10pt + {\bf AMS subject classification\ifoneclass@\else s\fi. }% + \thesubjclass\par + \fi} + \vskip8pt + \tenpoint} + + +%%% THEOREM AND OTHER MACROS +%%% +\def\heading#1#2{% + \vskip6pt{\bf #1.\enspace\ignorespaces#2.\enspace}\ignorespaces} + +\def\thm#1{{\smc #1.}\begingroup\it\enspace\ignorespaces} +\let\lem=\thm +\let\cor=\thm +\let\prop=\thm +\let\dfn=\thm + +\def\endthm{\endgroup} +\let\endlem=\endthm +\let\endcor=\endthm +\let\endprop=\endthm +\let\enddfn=\endthm + +\def\prf#1{{\it #1}.\rm\enspace\ignorespaces} +\let\rem=\prf +\let\case=\prf + +\def\endproof{\vbox{\hrule height0.8pt\hbox{% + \vrule height1.3ex width0.8pt\hskip0.8ex + \vrule width0.8pt}\hrule height0.8pt + }} + + +%%% FORMATTING AND MATH +%%% +%%% NEW ROMAN MATH OPERATORS +%%% +\def\const{\mathop{\rm const}\nolimits} +\def\diag{\mathop{\rm diag}\nolimits} +\def\grad{\mathop{\rm grad}\nolimits} +\def\Range{\mathop{\rm Range}\nolimits} +\def\rank{\mathop{\rm rank}\nolimits} +\def\supp{\mathop{\rm supp}\nolimits} + +%%% Centered displays, with numbering +%%% + +\def\ldisplaylinesno#1{\displ@y\halign{ + \hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}& + \kern-\displaywidth\rlap{$##$}\kern\displaywidth\crcr + #1\crcr}} + +%%% \meti to be used instead of \item +%%% +\def\meti#1{\parindent=2\parindent +\par\indent\llap{#1\enspace}\ignorespaces\parindent=.5\parindent} +\def\metimeti#1{\parindent=2\parindent +\par\indent\llap{#1\enspace}\ignorespaces\parindent=.5\parindent} + + +%%% Miscellaneous + +%%% REFERENCES + +\newdimen\refindent@ +\newdimen\refhangindent@ +\newbox\refbox@ +\setbox\refbox@=\hbox{\eightpoint\rm [00]}% Default 2 digits +\refindent@=\wd\refbox@ + +\def\resetrefindent#1{% + \setbox\refbox@=\hbox{\eightpoint\rm [#1]}% + \refindent@=\wd\refbox@} + +\def\Refs{% + \unskip\vskip2pc + \centerline{\eightpoint\rm REFERENCES}% + \penalty10000 + \vskip4pt + \penalty10000 + \refhangindent@=\refindent@ + \global\advance\refhangindent@ by 2em + \global\everypar{\hangindent\refhangindent@}% + \parindent=0pt\eightpoint\rm} + +\def\XRefs{% + \unskip\vskip2pc + \centerline{\eightpoint\rm REFERENCES}% + \penalty10000 + \vskip4pt + \penalty10000 + \refhangindent@=\refindent@ + \global\everypar{\hangindent\refhangindent@}% + \parindent=0pt\eightpoint\rm} + +\def\sameauthor{\leavevmode\vbox to 1ex{\vskip 0pt plus 100pt + \hbox to 3em{\leaders\hrule\hfil}\vskip 0pt plus 300pt}} + +\def\ref#1\\#2\endref{\leavevmode\hbox to \refindent@{\hfil[#1]}\enspace #2\par} + +\def\xref\\#1\endref{\leavevmode #1\par} + +%%% OUTPUT + +\global\footline={\hss\eightpoint\rm\folio\hss}% first page +\output{\plainoutput} +\def\plainoutput{\shipout\vbox{\makeheadline\pagebody\makefootline}% + \advancepageno + \ifnum\pageno>1 + \global\footline={\hfill}% + \fi + \ifodd\pageno + \global\headline={\hss\rightrh\hss{\tenpoint\rm\folio}}% + \else + \global\headline={\hskip-8pt{\tenpoint\rm\folio}\hss\leftrh\hss}% + \fi + \ifnum\outputpenalty>-\@MM \else\dosupereject\fi} +\def\pagebody{\vbox to\vsize{\boxmaxdepth\maxdepth \pagecontents}} +\def\makeheadline{\vbox to\z@{\vskip-22.5\p@ + \line{\vbox to8.5\p@{}\rheadfont\the\headline}\vss}% + \nointerlineskip} +\def\makefootline{\baselineskip24\p@\vskip-8\p@\line{\the\footline}} +\def\dosupereject{\ifnum\insertpenalties>\z@ % something is being held over + \line{}\kern-\topskip\nobreak\vfill\supereject\fi} + +\def\relaxnext@{\let\next\relax} +\def\footmarkform@#1{\ifmmode {}^{#1}\else$^{#1}$\fi } +\let\thefootnotemark\footmarkform@ +\def\makefootnote@#1#2{\insert\footins + {\interlinepenalty\interfootnotelinepenalty + \eightpoint \splittopskip=\ht\strutbox + \splitmaxdepth=\dp\strutbox + \floatingpenalty=\@MM + \leftskip=\z@ \rightskip=\z@ + \spaceskip=\z@ \xspaceskip=\z@ + \leavevmode{#1}\footstrut\ignorespaces#2\unskip + \lower\dp\strutbox\vbox to\dp\strutbox{}}} +\newcount\footmarkcount@ + \footmarkcount@=\z@ % Initialization +\def\footnotemark{\let\@sf=\empty \relaxnext@ + \ifhmode \edef\@sf{\spacefactor=\the\spacefactor}\/\fi + \def\next@{\ifx[\next \let\next=\nextii@ \else + \ifx"\next \let\next=\nextiii@ \else + \let\next=\nextiv@ \fi\fi\next}% + \def\nextii@[##1]{\footmarkform@{##1}\@sf}% + \def\nextiii@"##1"{{##1}\@sf}% + \def\nextiv@{\global\advance\footmarkcount@\@ne + \footmarkform@{\number\footmarkcount@}\@sf}% + \futurelet\next\next@} +\def\footnotetext{\relaxnext@ + \def\next@{\ifx[\next \let\next=\nextii@ \else + \ifx"\next \let\next=\nextiii@ \else + \let\next=\nextiv@ \fi\fi\next}% + \def\nextii@[##1]##2{\makefootnote@{\footmarkform@{##1}}{##2}}% + \def\nextiii@"##1"##2{\makefootnote@{##1}{##2}}% + \def\nextiv@##1{\makefootnote@{\footmarkform@{% + \number\footmarkcount@}}{##1}}% + \futurelet\next\next@} +\def\footnote{\let\@sf=\empty \relaxnext@ + \ifhmode \edef\@sf{\spacefactor\the\spacefactor}\/\fi + \def\next@{\ifx[\next \let\next=\nextii@ \else + \ifx"\next \let\next=\nextiii@ \else + \let\next=\nextiv@ \fi\fi\next}% + \def\nextii@[##1]##2{\footnotemark[##1]\footnotetext[##1]{##2}}% + \def\nextiii@"##1"##2{\footnotemark"##1"\footnotetext"##1"{##2}}% + \def\nextiv@##1{\footnotemark\footnotetext{##1}}% + \futurelet\next\next@} +\def\adjustfootnotemark#1{\advance\footmarkcount@#1\relax} + +\skip\footins=18\p@ plus6\p@ minus6\p@ + +\def\footnoterule{\kern -4\p@\hrule width 3pc \kern 3.6\p@ } % rule = .4 pt high + + + +% Turn off @ as being a letter. +% +\catcode`\@=12 |