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authorKarl Berry <karl@freefriends.org>2021-12-12 22:29:10 +0000
committerKarl Berry <karl@freefriends.org>2021-12-12 22:29:10 +0000
commit693851fcb2698450b53dce0962991f803c52bc61 (patch)
tree95408be113c10a15c76f48a320f5e1f941c02d5a /Master/texmf-dist/doc/latex/numerica
parent25454e4e2e9e6412c42f9774d70f545efa5382e0 (diff)
numerica (12dec21)
git-svn-id: svn://tug.org/texlive/trunk@61283 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex/numerica')
-rw-r--r--Master/texmf-dist/doc/latex/numerica/README.txt30
-rw-r--r--Master/texmf-dist/doc/latex/numerica/numerica-basics.pdf13442
-rw-r--r--Master/texmf-dist/doc/latex/numerica/numerica-plus.pdf6722
-rw-r--r--Master/texmf-dist/doc/latex/numerica/numerica-plus.tex1973
-rw-r--r--Master/texmf-dist/doc/latex/numerica/numerica-tables.pdf6031
-rw-r--r--Master/texmf-dist/doc/latex/numerica/numerica-tables.tex1801
-rw-r--r--Master/texmf-dist/doc/latex/numerica/numerica.pdf18226
-rw-r--r--Master/texmf-dist/doc/latex/numerica/numerica.tex (renamed from Master/texmf-dist/doc/latex/numerica/numerica-basics.tex)4085
8 files changed, 21000 insertions, 31310 deletions
diff --git a/Master/texmf-dist/doc/latex/numerica/README.txt b/Master/texmf-dist/doc/latex/numerica/README.txt
index e31e824f7c9..844894f6095 100644
--- a/Master/texmf-dist/doc/latex/numerica/README.txt
+++ b/Master/texmf-dist/doc/latex/numerica/README.txt
@@ -1,7 +1,7 @@
numerica: a package to numerically evaluate mathematical
-expressions in the LaTeX form in which they are typeset.
+ expressions in their LaTeX form.
-Andrew Parsloe (ajparsloe@gmail.com) 05 February 2021
+Andrew Parsloe (ajparsloe@gmail.com)
This work may be distributed and/or modified under the conditions
of the LaTeX Project Public License, either version 1.3c of this
@@ -10,26 +10,18 @@ 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.
+This is version 2.0.0 of the numerica package. The packages
+l3kernel, l3packages, amsmath and mathtools are required. The
+commands \nmcIterate, \nmcSolve, \nmcRecur and \nmcTabulate
+available in version 1 of numerica by means of package options
+are now available as separate packages. See numerica.pdf for
+details on how to use the package.
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
+numerica.pdf documentation for use of numerica
+numerica.tex documentation source file
+
diff --git a/Master/texmf-dist/doc/latex/numerica/numerica-basics.pdf b/Master/texmf-dist/doc/latex/numerica/numerica-basics.pdf
deleted file mode 100644
index 347c5a5097d..00000000000
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-%%EOF
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-%% 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}
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-\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}
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diff --git a/Master/texmf-dist/doc/latex/numerica/numerica-tables.tex b/Master/texmf-dist/doc/latex/numerica/numerica-tables.tex
deleted file mode 100644
index b16f90e3b7c..00000000000
--- a/Master/texmf-dist/doc/latex/numerica/numerica-tables.tex
+++ /dev/null
@@ -1,1801 +0,0 @@
-%% 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}
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+trailer
+<< /Size 1906
+/Root 1904 0 R
+/Info 1905 0 R
+/ID [<CE1BE0628603F7565A1DFCEB767C24A3> <CE1BE0628603F7565A1DFCEB767C24A3>] >>
+startxref
+1124413
+%%EOF
diff --git a/Master/texmf-dist/doc/latex/numerica/numerica-basics.tex b/Master/texmf-dist/doc/latex/numerica/numerica.tex
index 6e58e348220..543ea46d492 100644
--- a/Master/texmf-dist/doc/latex/numerica/numerica-basics.tex
+++ b/Master/texmf-dist/doc/latex/numerica/numerica.tex
@@ -1,4 +1,4 @@
-%% LyX 2.4.0-alpha1 created this file. For more info, see https://www.lyx.org/.
+%% LyX 2.4.0-alpha3 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}
@@ -8,6 +8,7 @@
\usepackage{textcomp}
\usepackage[utf8]{inputenc}
\setcounter{secnumdepth}{3}
+\setcounter{tocdepth}{3}
\usepackage{color}
\definecolor{shadecolor}{rgb}{0.667969, 1, 1}
\usepackage{babel}
@@ -53,45 +54,40 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\usepackage{numerica}
+\usepackage{upquote}
\newcommand\rel{\,\varrho\;}
\DeclareMathOperator{\erf}{erf}
\DeclareMathOperator{\gd}{gd}
-\reuse
-
-\usepackage{upquote}
+\reuse{}
\makeatother
\begin{document}
-\title{\texttt{numerica-basics}}
+\title{\texttt{numerica}}
\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).\\
+The \verb`numerica` package defines a command to numerically evaluate
+mathematical expressions in the LaTeX form in which they are typeset.
+For programs like LyX with a preview facility, 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. \\
\\
\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 This document applies to version 2.0.0 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}
+and }{\normalsize\texttt{l3packages}}{\normalsize{} are required (although
+much of }{\normalsize\verb`l3kernel`}{\normalsize{} has
+been incorporated into \LaTeXe{} since February 2020).}{\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})
@@ -99,6 +95,29 @@ 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}
+\item {\normalsize Version 2.0.0 of }{\normalsize\texttt{numerica}}{\small\par}
+\begin{itemize}
+\item {\normalsize splits into distinct packages the additional functionality
+previously available with the }{\normalsize\texttt{plus}}{\normalsize{}
+and }{\normalsize\texttt{tables}}{\normalsize{} package options of version
+1;}{\small\par}
+\item {\normalsize allows for user-defined macros and constants (with the
+}{\normalsize\texttt{\textbackslash nmcMacros }}{\normalsize and }{\normalsize\texttt{\textbackslash nmcConstants}}{\normalsize{}
+commands) to be used in expressions to be evaluated;}{\small\par}
+\item {\normalsize rewrites the code and changes the behaviour of }{\normalsize\texttt{\textbackslash nmcReuse}}{\normalsize{}
+to maintain uniformity across all commands (}{\normalsize\texttt{\textbackslash nmcEvaluate}}{\normalsize ,
+}{\normalsize\texttt{\textbackslash nmcInfo}}{\normalsize , }{\normalsize\texttt{\textbackslash nmcMacros}}{\normalsize ,
+}{\normalsize\texttt{\textbackslash nmcConstants}}{\normalsize , }{\normalsize\texttt{\textbackslash nmcReuse}}{\normalsize );
+this command is no longer compatible with its use in v.1;}{\small\par}
+\item {\normalsize changes the behaviour of }{\normalsize\texttt{\textbackslash text}}{\normalsize{}
+and }{\normalsize\texttt{\textbackslash mbox}}{\normalsize{} in the
+}{\normalsize\texttt{\textbackslash eval}}{\normalsize{} command; adds
+}{\normalsize\texttt{\textbackslash textrm}}{\normalsize , }{\normalsize\texttt{\textbackslash textsf}}{\normalsize ,
+and }{\normalsize\texttt{\textbackslash texttt}}{\normalsize{} to compensate;}{\small\par}
+\item {\normalsize includes many adjustments to the code, including around
+nesting of commands;}{\small\par}
+\item {\normalsize adds to and amends documentation.}{\small\par}
+\end{itemize}
\end{itemize}
\end{shaded}%
\end{minipage}
@@ -112,8 +131,8 @@ by a number like 1.2.3 to locate the actual expression referenced.}{\small\par}
\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). }
+that was designed for use with the document processor \LyX{} (and available
+for download from a link on the \LyX{} wiki website but not 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),
@@ -135,11 +154,12 @@ For instance, given a formula
(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}):
+need to transform the formula, just wrap it in an \verb`\eval` command:
\begin{centred}
\verb`\eval{ \frac{\sin (3.5)}{2} + 2\cdot 10^{-3} }`.
\end{centred}
+(for the acutal calculation see §\ref{subsec:introSimple-examples}).\texttt{ }
+
\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`,
@@ -150,7 +170,7 @@ 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
+often however it \emph{is} possible. We shall see shortly that even
complicated formulas like
\[
@@ -179,83 +199,83 @@ 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,
+are built into \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).
+Associated packages provide for additional operations: iteration of
+functions, finding zeros of functions, recurrence relations, mathematical
+table building; others 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
+The package is invoked in the usual way: put
\begin{lyxcode}
-\textbackslash usepackage{[}<options>{]}\{numerica\}
+\textbackslash usepackage\{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}
+from the \texttt{amssymb} package provided that package is loaded
+by the user; see §\ref{subsec:evalBoolean-output}.
+
+\subsection{Package options}
+
+\label{subsec:introPackagesOptions}Currently there are none. With
+version 2.0.0 of \verb`numerica` a change has been made to how additional
+functionality for the package is invoked; see §\ref{subsec:Related-packages}
+below. This means that the options available in version 1 have been
+discontinued.
+
+\subsection{Associated packages}
+
+\label{subsec:Related-packages}In version 1 of \verb`numerica` some
+additional functionality for the package could be gained by specifying
+package options – for instance the ability to create tables of function
+values or to iterate or find fixed points of functions. However this
+manner of invoking the addtional functionality makes the maintaining
+of semantic version numbering across the whole \verb`numerica` package
+difficult. With version 2.0.0 of the package, the addtional functionality
+has been separated into separate \LaTeX{} packages. Currently there
+are two of these, \verb`numerica-plus` and \verb`numerica-tables`.
+They are loaded with the familiar \verb`\usepackage` command in the
+document preamble and require the availability of the \verb`numerica`
+package in your \TeX{} distribution. Neither package requires a \verb`\usepackage{numerica}`
+statement; they take care of that themselves. So, if you enter
+\begin{lyxcode}
+\textbackslash usepackage\{numerica-plus\}
+\end{lyxcode}
+in the preamble of your document you gain access not only to the commands
+in the \verb`numerica` package but also to the commands \verb`\nmcIterate`,
+\verb`\nmcSolve`, and \verb`\nmcRecur`. \verb`\nmcIterate` enables
+the iteration of functions of a single variable, including finding
+fixed points. \verb`\nmcSolve` enables the solving of equations of
+the form $f(x)=0$ (i.e. finding the zeros of $f$), or the finding
+of local maxima or minima of a function of one variable. \verb`\nmcRecur`
+enables the calculation of terms in recurrence relations, like the
+terms of the Fibonacci series, or othogonal polynomials defined recurrently.
+See the associated document \verb`numerica-plus.pdf` for details.
+
+If you enter
+\begin{lyxcode}
+\textbackslash usepackage\{numerica-tables\}
+\end{lyxcode}
+in the preamble of your document you gain access not only to the commands
+in the \verb`numerica` package but also to the command \verb`\nmcTabulate`
+which enables the creation of (possibly multi-column) tables of function
+values and makes available most of the table formats evident in \emph{HMF}.
+See the associated document \verb`numerica-tables.pdf` for details.
-\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\}}.
+A package \verb`numerica-calculus` is currently being developed.
\subsection{Simple examples of use}
\label{subsec:introSimple-examples}A simple example of use is provided
by the document
\begin{verbatim}
- \documentclass{minimal}
+ \documentclass{article}
\usepackage{numerica}
\begin{document}
@@ -284,7 +304,7 @@ male. Only a minute fraction is ever available.)
A second example is provided by the formula in earlier remarks:
\begin{verbatim}
- \documentclass{minimal}
+ \documentclass{article}
\usepackage{numerica}
\begin{document}
@@ -310,10 +330,14 @@ like (but generally more complicated than)
\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):
+($2$ in this instance). The reader will note that I have used dollar
+signs to delimit the math environment. I could (and perhaps should)
+have used the more \LaTeX -pure \verb`\( \)`, which will do equally
+well, but habit has won out. In the example 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 $}.
\end{centred}
@@ -324,9 +348,9 @@ only the \emph{result,} followed possibly by the \emph{variable~=~value
list} (see §\ref{subsec:evalVv-list}) is displayed.
\end{itemize}
Environments may include the various AMS environments as well as the
-standard \LaTeX{} inline ( \verb`$ $` ), \texttt{equation} ( \verb`\[ \]`
-) and \texttt{eqnarray} environments. For an example of \verb`\eval`
-within an \verb`align*` environment see §\ref{subsec:introExploring}
+standard \LaTeX{} inline ( \verb`$ $` or \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,
@@ -371,6 +395,42 @@ that case \texttt{numerica} acts as if \verb`\eval` had been used
within \verb`\[` and \verb`\]` and displays the result accordingly.
\end{itemize}
+\subsection{Checking}
+
+A question I found on the internet that caught my attention was to
+simplify $\sqrt{220-30\sqrt{35}}$. I found myself intrigued. After
+some bumbling and fumbling, I let
+\[
+x=\sqrt{220-30\sqrt{35}},\qquad y=\sqrt{220+30\sqrt{35}},
+\]
+(which seems an obvious thiing to do). Then
+\[
+xy=10\sqrt{484-315}=10\sqrt{13^{2}}=130.
+\]
+Since $x^{2}+y^{2}=440$ it was then easy to form both $(x+y)^{2}$
+and $(x-y)^{2}$ and by separating the resulting numbers into their
+prime factors, to work out that $x=5\sqrt{7}-3\sqrt{5}$. Was I right?
+\begin{centred}
+\verb`\eval{$ \sqrt{220-30\sqrt{35}} $}` $\Longrightarrow$ \eval{$ \sqrt{220-30\sqrt{35}} $},
+
+\verb`\eval{$ 5\sqrt{7}-3\sqrt{5} $}` $\Longrightarrow$ \eval{$ 5\sqrt{7}-3\sqrt{5} $}.
+\end{centred}
+Yes, the simplification was correct. And indeed $y=5\sqrt{7}+3\sqrt{5}$:
+\begin{centred}
+\verb`\eval{$ \sqrt{220+30\sqrt{35}} $}` $\Longrightarrow$ \eval{$ \sqrt{220+30\sqrt{35}} $},
+
+\verb`\eval{$ 5\sqrt{7}+3\sqrt{5} $}` $\Longrightarrow$ \eval{$ 5\sqrt{7}+3\sqrt{5} $}.
+\end{centred}
+As a final flourish,
+\begin{verbatim}
+ \eval{$ xy $}
+ [ x=5\sqrt{7}-3\sqrt{5},
+ y=5\sqrt{7}+3\sqrt{5} ]
+\end{verbatim}
+$\Longrightarrow$ \eval{$ xy $}
+ [x=5\sqrt{7}-3\sqrt{5},
+ y=5\sqrt{7}+3\sqrt{5}].
+
\subsection{Exploring}
\label{subsec:introExploring}When working on \texttt{numerica}'s
@@ -400,7 +460,6 @@ results:
\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],\\
@@ -430,7 +489,7 @@ 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}
+\subsection{Reassuring}
\label{subsec:introReassurance}In the course of some hobbyist investigations
in plane hyperbolic geometry I derived the formula
@@ -451,8 +510,8 @@ 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
+same numerical 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}
@@ -541,7 +600,7 @@ 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]`
+\noindent \verb`\nmcEvaluate*[settings]{expr.}[vv-list][num. format]`
\end{centred}
I discuss the various arguments in the referenced sections.
\begin{enumerate}
@@ -565,15 +624,17 @@ 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`%`).
+arguments. They do not need to be hard against their preceding arguments;
+intervening spaces are allowed. This means there is a possibility
+that should the \verb`\eval` command be followed by a square-bracketed
+mathematical expression \texttt{numerica} might confuse it 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
+with new lines and white space to aid clarity – 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
@@ -581,8 +642,9 @@ used in \LaTeX{} commands by consolidating such arguments into a single
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}.
+output) and placing them in a trailing argument is both convenient
+and intuitive for the kind of back-of-envelope calculations envisaged
+for \texttt{numerica}.
\section{The variable=value list}
@@ -593,23 +655,6 @@ or \emph{vv-list} for short. This is the fourth argument of the \texttt{\textbac
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
@@ -619,12 +664,12 @@ 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.
+(This criterion for what makes a variable name means a 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}
@@ -641,16 +686,20 @@ 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.
+\verb`numerica` works with single tokens. Variable names can be so
+different in structure, one from another, that to ease the parsing
+of formulas, all \emph{internal} variable names are assumed to be
+single tokens. Hence a necessary initial step for the package is to
+map 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`\nmc_a`,
+\verb`\nmc_b`, \verb`\nmc_c`, 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. (Doing the replacing
+in this order prevents \emph{parts} of longer names possibly being
+mistaken for shorter variable names.)
The conversion process uses computer resources. Even if there are
no multi-token variables present, \verb`numerica` still needs to
@@ -704,9 +753,9 @@ 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.
+on 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}
@@ -755,10 +804,13 @@ $\Longrightarrow$ \eval{$ \sqrt{s(s-a)(s-b)(s-c)} $}
\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 $\tfrac{1}{2}(1+\surd5)$,
+\label{subsec:Built-in-Constants}\verb`numerica` has five built-in
+constants and can also accept user-defined constants. For the latter,
+see §\ref{sec:supplConstants}. The five built-in constants known
+to \texttt{numerica} are \verb`\pi`, the ratio of circumference to
+diameter of a circle; \verb`e`, the base of natural logarithms; Euler's
+constant \verb`\gamma`, the limit of $\left(\sum_{1}^{N}1/n\right)-\ln N$
+as $N\to\infty$; the golden ratio \verb`\phi`, equal to $\tfrac{1}{2}(1+\surd5)$;
and the utilitarian constant \verb`\deg`, the number of radians in
a degree.
\begin{centred}
@@ -782,16 +834,15 @@ Let's combine two of these in a formula:
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}
-
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
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{}
+values to them in the vv-list. \cprotect\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]
@@ -801,17 +852,15 @@ $\Longrightarrow$
\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.
+\noindent Since this is the correct area we see that \cprotect\texttt{e}
+has been treated as a variable with the assigned value $5$, not as
+the constant. But if \cprotect\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 \cprotect\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`. \end{shaded}%
+\end{minipage}
\subsection{Display of the vv-list}
@@ -834,12 +883,14 @@ or simply the naked result:
\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:
+math delimiters around the \verb`\eval*` command as a whole. Wrapping
+them around the formula has no effect:
\begin{centred}
-\verb`\eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]` $\Longrightarrow$ \eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]
+\verb`\eval*{$ y $}[y=ax+b,x=2,a=-2,b=2]` $\Longrightarrow$ \eval*{$ y $}[y=ax+b,x=2,a=-2,b=2],
+
+\verb`$ \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.
+The star option delivers a number as result, pure and simple.
\subsubsection{Suppressing display of items}
@@ -859,6 +910,18 @@ 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{Empty vv-list suppressed}
+
+Should the vv-list be empty, or display of \emph{all} variables is
+suppressed by wrapping each in braces, then \emph{nothing} is displayed
+where the vv-list would normally be, not even any punctuation:
+\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}
+If you want a full stop after the result then you will need to add
+it by hand or use the \verb`p=.` setting of §\ref{subsec:settingsPunctuation}.
+
\subsubsection{Changing the display format}
\label{subsec:evalChanging-display-format}In two examples above,
@@ -896,17 +959,62 @@ $\Longrightarrow$ \eval{
}[s=\tfrac12(a+b+c+d),
a=\surd2,b=\surd2,c=\surd3,d=1]
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection{Abusing multi-token variable names}
+
+\label{subsec:evalDon't-do-this!}A variable name is what lies to
+the left of the equals sign of an item in the vv-list. Since multi-token
+variables are converted to single tokens (like \verb`\nmc_a`) before
+any calculating is done, it is possible to sin. Thus :
+\begin{centred}
+\verb`\eval{$ \sin\pi $}[{\sin\pi}=1]` $\Longrightarrow$ \eval{$ \sin\pi $}[{\sin\pi}=1];
+\end{centred}
+and (more?) egregiously,
+\begin{centred}
+\verb`\eval{$ 10 $}[{10}=20]` $\Longrightarrow$ \eval{$ 10 $}[{10}=20].
+\end{centred}
+What is happening here is that the multi-token `variables' \verb`\sin\pi`
+and \verb`10` are being converted, right at the start of proceedings,
+to single tokens like \verb`\nmc_a`, which in \TeX -speak are macros
+containing their respective multiple tokens. For display purposes
+they expand to those multiple tokens, but for calculating within \verb`numerica`
+the single token is used. By this means one can easily create further
+grotesqueries:
+\begin{centred}
+\verb`\eval{$ ++ + ++ $}[{++}=1]` $\Longrightarrow$ \eval{$ ++ + ++ $}[{++}=1],
+
+\verb`\eval{$ 2(1+1) $}[{2(1}=3,{+1)}=5]` $\Longrightarrow$ \eval{$ 2(1+1) $}[{2(1}=3,{+1)}=5],
+
+\verb`\eval{$ 1!! $}[{!!}=42]` $\Longrightarrow$ \eval{$ 1!! $}[{!!}=42].
+\end{centred}
+Should \verb`numerica` try to check variable names to avoid consequences
+like this? I don't see any reasonable way of doing that. Symbols like
+\verb`(` and \verb`+` can easily be part of valid variable names
+– $k^{+},\,k^{-}$, $C_{n}^{(0)}$ and so on. It is left to the user,
+in any \emph{public} document, to avoid such sins. (And they could
+easily construct the displayed expressions in \LaTeX{} if they so wished
+without recourse to \verb`\eval` at all.) See also §\ref{subsec:supplMacrosDisplay}
+where a similar issue arises with user-defined macros.\end{shaded}%
+\end{minipage}
+
\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
+\label{subsec:evalRoundingEtc}Internally, values are stored to $16$
+significant figures (if available), calculations are carried out to
+$16$ significant figures, but only rarely do we want to view the
+result to $16$ figures. Generally, we round to some smaller number
+of figures. The default rounding value is $6$, meaning by default
+at most $6$ decimal places are shown. So far, all results have been
+rounded to this figure, although not all digits are always 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:
+six possibilities:
\begin{itemize}
\item a question mark ?, which gives boolean output, or
\item an integer, the \emph{rounding value}, positive, negative or zero,
@@ -914,54 +1022,60 @@ 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
+`proper' scientific notation (a form like $1.2345\times10^{5}$
+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
+a bastardized scientific notation useful in tables (a form like $(5)1.2345$
+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.
+or an integer, usually one of the letters\texttt{ e d} \texttt{E
+D}, which presents the result in scientific notation with that character
+as the exponent mark (a form like $1.2345\text{e}5$ for $123450$).
\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.
+If you use \texttt{?} in the same specification as some other character,
+the \texttt{?} prevails; if you use \texttt{x} in the same specification
+as some other character except for \texttt{?}, the \texttt{x} prevails;
+if you use \texttt{t} in the same specification as some other character
+except for \texttt{?} or \texttt{x}, the \texttt{t} prevails.
+
+If you repeat the character serving as the exponent mark in scientific
+notation – say \verb`xx` or \verb`dd` – then scientific notation
+extends to numbers in the interval \verb`[1,10)`.
+
+If you repeat a question mark specifying boolean output, then the
+formatting of that output is changed from \verb`1/0` to $T/F$ or
+\verb`T/F` depending as there are two or three question marks used.
\subsection{Rounding value }
-\label{subsec:evalRounding-value}The rounding value specifies the
-number of decimal places displayed:
+\label{subsec:evalRounding-value}If the number is displayed as a
+decimal, the rounding value specifies the number of decimal places
+displayed. If a number is displayed in scientific notation (see below
+§\ref{subsec:evalScientificNotation}) that is still true, but it
+can mean differences in the overall number of digits displayed. For
+the moment, I show the effect of rounding in a purely decimal display:
\begin{centred}
\verb`$ \eval{ 1/3 }[4] $` $\Longrightarrow$ $ \eval{ 1/3 }[4] $
\end{centred}
-The default rounding value is $6$:
+In this case \verb`4` was entered in the number-format option and
+the result is displayed to four decimal places. 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 $6$:
+Following the default behaviour in \verb`l3fp`, the calculational
+engine which \verb`numerica` uses, `ties' are rounded to the nearest
+\emph{even} digit. Thus a number ending $55$ with a `choice' of
+rounding to $5$ or $6$ 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).
+\verb`l3fp` works to 16 significant figures and never displays more
+than that number (and often fewer).
\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
@@ -1022,22 +1136,40 @@ inserting an asterisk, {*}, into the final optional argument of the
\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:
+\label{subsec:evalScientificNotation} \verb`l3fp` 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}
+($3$). In scientific notation, the significand always has one \emph{non-zero}
+digit before the decimal point.\footnote{Except for $0$ itself.}
+
+For scientific notation rounding still means the number of decimal
+places displayed, but it can result in very different numbers of digits
+being shown from the number shown in decimal form. To switch on output
+in scientific notation in \verb`numerica` enter \verb`e` in the
+trailing optional argument:
+\begin{centred}
+\verb`$ \eval{ 123.456789 }[e] $` $\Longrightarrow$ $ \eval{ 123.456789 }[e] $.
+\end{centred}
+The default rounding value $6$ is in play here, with seven digits
+of the significand displayed overall, one preceding the decimal point,
+six following it. Compare this with the same number rounded in decimal
+form:
+\begin{centred}
+\verb`$ \eval{ 123.456789012345 } $` $\Longrightarrow$ $ \eval{ 123.456789012345 } $.
+\end{centred}
+In this instance, nine digits are displayed, three before the decimal
+point and six after. Similarly compare
+\begin{centred}
+\verb`$ \eval{ 0.0123456789 }[e] $` $\Longrightarrow$ $ \eval{ 0.0123456789 }[e] $
+\end{centred}
+with
\begin{centred}
-\verb`$ \eval{ 123456789 }[e] $` $\Longrightarrow$ $ \eval{ 123456789 }[e] $.
+\verb`$ \eval{ 0.0123456789 } $` $\Longrightarrow$ $ \eval{ 0.0123456789 } $.
\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.
+This time scientific notation has gained two extra decimal digits
+to display.
Negative rounding values are pointless for scientific notation. A
zero might on occasion be relevant:
@@ -1062,7 +1194,8 @@ form of notation:
\begin{centred}
\verb`\eval{$ 1/23456789 $}[4t]`\texttt{ $\Longrightarrow$ }\eval{$ 1/23456789 $}[4t].
\end{centred}
-This is discussed more fully in the associated document \texttt{numerica-tables.pdf}.
+This is discussed more fully in the documentation for the \texttt{numerica-tables}
+package.
In the next example three options are used in the trailing argument.
The order in which the items are entered does not matter:
@@ -1074,10 +1207,10 @@ into thinking the number-format option is the vv-list, and will certainly
confuse the reader.} save for \texttt{x} or \texttt{t} can be used to distinguish the
exponent, I use an @ character:
\begin{centred}
-\verb`\eval{$ 1/123 $}[@4]` $\Longrightarrow$ \eval{$ 1/123 $}[@4].
+\verb`\eval{$ 1/125 $}[@4]` $\Longrightarrow$ \eval{$ 1/125 $}[@4].
\end{centred}
-\subsubsection{Numbers in \texttt{{[}1,10)}}
+\subsubsection{Numbers in the interval \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,
@@ -1085,10 +1218,10 @@ $3.14159$ 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.
+to numbers in this range by repeating the letter chosen as the exponent
+mark in the trailing optional argument.
\begin{centred}
-\verb`\eval{$ \pi $}[4t']` $\Longrightarrow$ \eval{$ \pi $}[4t']
+\verb`\eval{$ \pi $}[4tt]` $\Longrightarrow$ \eval{$ \pi $}[4tt]
\end{centred}
\subsubsection{\textbackslash eval{*} and scientific notation}
@@ -1096,7 +1229,7 @@ the exponent mark in the trailing optional argument.
Scientific notation can be used for the numerical result output by
\verb`\eval*`:
\begin{centred}
-\verb`\eval*{$ \pi $}[e']` $\Longrightarrow$ \eval*{$ \pi $}[e']
+\verb`\eval*{ \pi }[ee]` $\Longrightarrow$ \eval*{ \pi }[ee]
\end{centred}
There is one catch: if you substitute \texttt{x} for \texttt{e} here,
\LaTeX{} will complain about a missing \verb`$`. An \texttt{x} in
@@ -1105,17 +1238,12 @@ requires a math environment. It is up to you, as the user, to provide
the necessary delimiters outside the \verb`\eval*` command. (This
applies even when \verb`\eval*` wraps around math delimiters.)
-(Because of the way \texttt{numerica} parses the number-format option,
-entering a prime with neither exponent character nor question mark
-specified will result in scientific output using \texttt{e} as the
-exponent mark. The last example could have been written \verb`\eval*{$ \pi $}[']`.)
-
\subsection{Boolean output}
-\label{subsec:evalBoolean-output}\texttt{l3fp} can evaluate comparisons,
+\label{subsec:evalBoolean-output}\verb`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
+you can force \verb`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.
@@ -1130,7 +1258,7 @@ exhibits a numerological curiosity:\footnote{The \texttt{{[}p=.{]}} of this and
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?]
+\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[5?]\smallskip{}
\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
@@ -1150,14 +1278,14 @@ places, say to $9$, we can see what is going on:
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.
+by adding one or two more question marks respectively in the number-format
+option.
\begin{centred}
-\verb`\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']` $\Longrightarrow$
-\eval[p=.]{\[ \frac1{0.0123456789}=81 \]}[6?'']
+\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
+\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.
@@ -1181,7 +1309,7 @@ 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;
+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
@@ -1195,8 +1323,8 @@ 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}
-
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
Mathematically, the claim that $X=Y$ at a rounding value $n$ is
the claim that
\[
@@ -1209,7 +1337,8 @@ 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.
+\emph{before }doing the rounding.\end{shaded}%
+\end{minipage}
\subsubsection{And, Or, Not}
@@ -1220,11 +1349,11 @@ both displaying as $\land$, but \texttt{numerica} adds thin spaces
\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 \wedge 2<3 $}[??]` $\Longrightarrow$ \eval{$ 1<2 \wedge 2<3 $}[??],
-\verb`\eval{$ 1<2 \land 2<3 $}[?'']` $\Longrightarrow$ \eval{$ 1<2 \land 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
+To my eye the second of these with its increased 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
@@ -1236,8 +1365,10 @@ both displaying as $\lnot$ . Not binds tightly to its argument:
\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 $1$.
-
+as a whole the result would have been different:
+\begin{centred}
+\verb`\eval{$ \lnot(A \land B) $}[A=0,B=0]` $\Longrightarrow$ \eval{$ \lnot(A \land B) $}[A=0,B=0].
+\end{centred}
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
@@ -1250,10 +1381,10 @@ 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][?'']
+ (\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][?'']
+ (\lnot A\lor\lnot B)$}[A=1,B=0,C=1][???].
\subsubsection{Chains of comparisons}
@@ -1286,7 +1417,7 @@ are also negations: \verb`\nless` ( $\nless$ ), \verb`\nleq` (~$\nleq$~),
\subsection{Arithmetic}
\label{subsec:evalArithmetic}Addition, subtraction, multiplication,
-division, square roots, \emph{$n$}-th roots, and exponentiating (raising
+division, square roots, \emph{$n$}th roots, and exponentiating (raising
to a power) are all available.
Multiplication can be rendered explicitly with an asterisk,
@@ -1322,7 +1453,12 @@ 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}
+Powers are indicated with the superscript symbol \verb`^`:
+\begin{centred}
+\verb` \eval{$ 3^{2^2} $}` $\Longrightarrow$ \eval{$ 3^{2^2} $} .
+\end{centred}
+
+\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
@@ -1333,7 +1469,7 @@ first, \verb`\surd` in the second):
\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
+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}:
@@ -1355,11 +1491,9 @@ 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]
+ \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=.]
{\[
@@ -1367,9 +1501,9 @@ $\Longrightarrow$\eval[p=.]
{\!\biggl(\!\left.\frac AD\right/\!\frac BC\biggr)}
\]}[A=729,B=81,C=9,D=3]
-As implemented in \texttt{numerica}, $n$-th roots found using \verb`\sqrt[n]`
+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,
+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
@@ -1383,7 +1517,7 @@ 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}
+\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`,
@@ -1395,14 +1529,15 @@ numbers are available with \verb`\sqrt`,
\subsubsection{Inverse integer powers }
-Of course to find an $n$-th root we can also raise to the inverse
+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}.
+an error even when, mathematically, it should not. This matter, which
+is a product of floating point representation of numbers, is discussed
+below in §\ref{subsec:errorsInverse-powers}.
\subsection{Precedence, parentheses}
@@ -1463,56 +1598,6 @@ $\Longrightarrow$ \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`,
@@ -1612,7 +1697,7 @@ but pedantic form
\verb`\eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234]`
$\Longrightarrow$ \eval{$ (\sin(\theta))^2+(\cos(\theta))^2 $}[\theta=1.234] .
\end{centred}
-A hyperbolic identity is confirmed in this example:
+A hyperbolic identity is corroborated in this example:
\begin{centred}
\verb`\eval{$ \sinh 3x $}[x=1]` $\Longrightarrow$ \eval{$ \sinh 3x $}[x=1],\medskip{}
@@ -1631,10 +1716,10 @@ the `sine of the square of pi', $\sin(\pi^{2})$, and \emph{not
\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 $\ln x^{n}=n\ln x=\ln(x^{n})$
+the logarithm in a natural way. Surely $\ln x^{n}=n\ln x$, i.e. $\ln x^{n}=\ln(x^{n})$
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 $\ln(x)^{n}=n\ln(x)=\ln(x^{n})$.
+avoid $\ln(x)^{n}=n\ln(x)=\ln(x^{n})$ too.
\subsection{\emph{n}-ary functions}
@@ -1642,8 +1727,8 @@ 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
+themselves can be expressions or numbers. For \verb`\gcd`, \emph{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}
@@ -1759,12 +1844,13 @@ $\Longrightarrow$ \eval{$ (\alpha+\beta)!-\alpha!-\beta! $}[\alpha=2,\beta=3].
\end{centred}
The examples illustrate how \texttt{numerica} interprets the argument
of the factorial symbol:\texttt{ }it `digests'
-\begin{enumerate}
+\begin{itemize}
\item a preceding (possibly multi-digit) integer, or
\item a preceding variable token, or
\item a bracketed expression, or
-\item a bracket-like expression – an absolute value, floor or ceiling function,
-\end{enumerate}
+\item a bracket-like expression.
+\end{itemize}
+A bracket-like expression is an absolute value, floor or ceiling function,
since they delimit arguments in a bracket-like way:
\begin{centred}
\verb`\eval{$ \abs{-4}!+\floor{\pi}!+\ceil{e}! $}` $\Longrightarrow$
@@ -1780,12 +1866,12 @@ Nesting of brackets for factorials is accepted:
\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.
+in \texttt{numerica}, and 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.
@@ -1812,7 +1898,7 @@ are defined for integers $\ge-1$.
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},
+and a display-style version \verb`\dbinom`. As implemented in {\ttfamily\verb`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}),
@@ -1824,12 +1910,12 @@ 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}.
+a non-negative integer, {\ttfamily\verb`numerica`} displays
+a message; see §\ref{subsec:errorsIntegerArgs}.
\subsection{Sums and products}
-\texttt{numerica} recognizes sums (\verb`\sum` displaying as $\sum$)
+\verb`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
@@ -1893,8 +1979,8 @@ 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:
+How many more? Let's `go the whole hog' and put $\infty$ in the
+upper limit of this product:
\begin{verbatim}
\eval[p=.]{\[
\prod_{k=1}^{\infty}
@@ -1918,10 +2004,10 @@ For this particular product the problem is that it converges slowly.
Any criterion for when to stop multiplying terms or, for an infinite
sum adding terms, seems bound to fail for some product or series.
Presumably any stopping criterion must measure smallness in some way.
-But terms of the divergent harmonic series, $\sum(1/n)$ can always
-be found smaller than any value we care to specify. It is not surprising
-that a sufficiently slowly converging product or series falls foul
-of a given criterion.
+But terms of, for example, the divergent harmonic series $\sum(1/n)$
+can always be found smaller than any value we care to specify. It
+is not surprising that a sufficiently slowly converging product or
+series falls foul of a given criterion.
The default criterion however can be changed. Because this involves
values assigned in the settings option of the \verb`\eval` command,
@@ -1942,6 +2028,189 @@ $\Longrightarrow$ \eval[p=.]
\sum_{n=0}^{\infty}\binom{\alpha}{n}x^{n}
\]}[\alpha=4.321,x=0.1234]
+\subsection{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 calculational content. \verb`numerica` copes with a great many
+of these formatting commands, although there will surely be some that
+it has overlooked and which will trigger an `Unknown token' message;
+see §\ref{sec:evalErrors}. \footnote{Please contact the author in that case: ajparsloe@gmail.com}
+
+\subsubsection{Spaces, phantoms, struts}
+
+These include cryptic forms like \verb`\,` \verb`\:` and \verb`\>`,
+\verb`\;` and the corresponding `verbose' forms, \verb`\thinspace`,
+\verb`\medspace` and \verb`\thickspace` and their negative equivalents
+\verb`\!` or \verb`\negthinspace`, \verb`\negmedspace` and \verb`\negthickspace`:
+\begin{centred}
+\verb`\eval{$ 1\negthickspace+\negthickspace 1 $}` $\Longrightarrow$
+\eval{$ 1\negthickspace+\negthickspace 1 $}
+\end{centred}
+which gives the text spacing of 1+1 as against the usual math spacing
+$1+1$ but doesn't affect the result of the calculation.
+
+Other spacing commands are \verb`\quad` and \verb`\qquad`, and \verb`\hspace{arg}`
+and \verb`\mspace{arg}`. For \verb`\hspace` there is also a starred
+form, \verb`\hspace*{arg}`. Phantoms similarly take an argument:
+\verb`\phantom{arg}`, \verb`\hphantom{arg}` and \verb`\vphantom{arg}`.
+\begin{centred}
+\verb`\eval{$ 1\hphantom{mmm}+\hphantom{mmm}1 $}` $\Longrightarrow$
+\eval{$ 1\hphantom{mmm}+ \hphantom{mmm}1 $}.
+\end{centred}
+Like \verb`\vphantom`, struts allow vertical spacing adjustments.
+\verb`numerica` should digest both \verb`\xmathstrut[optarg]{arg}`
+from \verb`mathtools` and its `baby cousin' \verb`\mathstrut`
+from \TeX . An example from \emph{The \TeX{} book} demonstrating the
+use of \verb`\mathstrut` is
+\begin{verbatim}
+ \eval{$\sqrt{\mathstrut a}+\sqrt{\mathstrut d}+
+ \sqrt{\mathstrut y}$}[a=4,d=9,y=16]
+\end{verbatim}
+$\Longrightarrow$ \eval{$\sqrt{\mathstrut a}+\sqrt{\mathstrut d}+\sqrt{\mathstrut y}$}[a=4,d=9,y=16],
+
+And here is an evaluation of an expression from the \verb`mathtools`
+documentation using \verb`\xmathstrut`:
+\begin{verbatim}
+ \eval{\[ \frac{ \frac{ \xmathstrut{0.1} x-1 }
+ { \xmathstrut{0.25} x-\sin{ x} } }
+ {\xmathstrut{0.4} \sqrt{ 10-x } } \]}
+ [x=\pi/6]
+\end{verbatim}
+$\Longrightarrow$ \eval{\[ \frac{ \frac{ \xmathstrut{0.1} x-1 }
+ { \xmathstrut{0.25} x-\sin{ x} } }
+ {\xmathstrut{0.4} \sqrt{ 1-x } } \]}
+ [x=\pi/6]
+
+\subsubsection{\texttt{\textbackslash splitfrac}}
+
+The \verb`mathtools` package provides \verb`\splitfrac` and \verb`\splitdfrac`
+to aid handling of clumsy fractions. The documentation gives an (artificial)
+example of use. I've mangled it to produce an even more ridiculous
+illustration, adding to the mess an enormous square root, the modifiers
+\verb`\left` and \verb`\right`, and the command-form alternatives
+to parentheses, \verb`\lparen` and \verb`\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]
+
+\subsubsection{Colour}
+
+\label{subsec:Colour}(Anglicised spelling at least for the heading!)
+If you add to the preamble of your document the line
+\begin{lyxcode}
+\textbackslash usepackage\{color\}
+\end{lyxcode}
+two commands become available, \verb`\textcolor[optarg]{arg1}{arg2}`
+and the declaration form of command, \verb`\color[optarg]{arg}`.
+\verb`numerica` readily accepts the former in a formula to be evaluated:
+\begin{centred}
+\verb`\eval{$ \sin \tfrac\pi6n\textcolor{red}{T}+1 $}[T=9,n=3]` $\Longrightarrow$
+\eval{$ \sin \tfrac\pi6n\textcolor{red}{T}+1 $}[T=9,n=3]
+\end{centred}
+(assuming you had some wish to highlight the time $T$).
+
+However there are restrictions on the use of \verb`\color` in \verb`\eval`
+commands. \verb`\color` is a \emph{declaration} form of command.
+It has effect until the end of the current group or environment. If
+you want to restrict it to only part of that group you need to em-brace
+the command and what it is to apply to,
+\begin{lyxcode}
+<pre-stuff>\{\textbackslash color\{red\}<red-stuff>\}<post-stuff\}
+\end{lyxcode}
+but that is where the problem arises. \verb`numerica` does not check
+for `unannounced' brace groups. It expects a brace group to be introduced
+by a preceding instruction like \verb`\sqrt` or \verb`\frac` or
+\verb`^`. When announced in this way, \verb`numerica` can handle
+the brace group appropriately. But the brace group \verb`{\color{red}<red-stuff>}`
+is not so announced. \verb`numerica`'s parsing routine will not recognize
+what it has just swallowed and a \LaTeX{} error will result. So, \verb`\color`
+cannot be used in a formula in a `naked' or unannounced brace group.
+Writing \verb`\eval{$ \color{red} \sin \tfrac\pi6nT+1 $}[T=9,n=3]`
+is fine, as is
+\begin{centred}
+\verb`\eval{$ \sin \tfrac\pi6nT+1 \color{red} $}[T=9,n=3]` $\Longrightarrow$
+\eval{$ \sin \tfrac\pi6nT+1 \color{red} $}[T=9,n=3].
+\end{centred}
+So too, because the \verb`\frac` introduces the confining brace group,
+is
+\begin{centred}
+\verb`\eval{$ \frac{\color[gray]{0.5}A}B $}[A=12,b=4]` $\Longrightarrow$\eval{$ \frac{\color[gray]{0.5}A}B $}[A=12,B=4],
+\end{centred}
+where both arguments of the \verb`\color` command are used for grayscale
+output.
+
+But trying something like \verb`\eval{$ 3{\color[gray]{0.5}x}+1 $}[x=2]`
+will cause a \LaTeX{} error and halt compilation since there is no
+command announcing the brace group confining the \verb`\color` command.
+
+\subsubsection{\texttt{\textbackslash text}, \texttt{\textbackslash mbox}, font
+commands}
+
+\label{subsec:Text-mbox-fonts}Following a rethink of the behaviour
+of a number of font and formatting commands, in version 2 of \verb`numerica`
+the content of a \verb`\text` or \verb`\mbox` command is \emph{invisible}
+to the \verb`\eval` command. \emph{This behaviour is different from
+that of version 1.} Now the content is ignored in a calculation,
+\begin{centred}
+\verb`\eval*{ 1/0.0123456789 \mbox{approx.} }[5]` $\Longrightarrow$
+\eval*{ 1/0.0123456789 \mbox{approx.}}[5],
+\end{centred}
+even when the \verb`\text` or \verb`\mbox` contains mathematical
+content.
+
+Conversely, the content of font commands (like \verb`\mathbf` or
+\verb`\mathcal`) is \emph{visible} to \verb`\eval`. This becomes
+useful should numbers be input in scientific notation (see §\ref{subsec:settingsInputtingSciNotation}).
+As well as the math font commands, \verb`\eval` also accepts \verb`\textrm`,
+\verb`\textsf` and \verb`\texttt`. Thus a number in scientific notation
+like \verb`2e-1` appearing in the formula or the vv-list can display
+correctly by wrapping it in a \verb`\textrm` or \verb`\texttt` command,
+rather than displaying inappropriately as the algebraic expression
+$2e-1$.
+
+\subsubsection{\texttt{\textbackslash ensuremath},\texttt{ \$},\texttt{ \textbackslash (},\texttt{
+\textbackslash )},\texttt{ \textbackslash{[}},\texttt{ \textbackslash{]}}}
+
+Should \verb`\ensuremath` be included in a formula for evaluation
+(but why?) it is digested without demur, irrespective of whether explicit
+math delimiters are present or not. More generally, should math delimiters
+(through some momentary oversight) be used both within and outside
+an \verb`\eval` command, the command is processed as if only the
+outside environment is involved; the inner delimiters are ignored:
+\begin{centred}
+\verb`$ \eval{\[ -4^2 \]} $` $\Longrightarrow$ $ \eval{\[ -4^2 \]} $.
+\end{centred}
+
\section{Error messages }
\label{sec:evalErrors}There are two kinds of error in \texttt{numerica}:
@@ -1952,7 +2221,7 @@ produce messages displayed in the pdf where one would expect the result
of the calculation to be. The original reason for doing things this
way was to enable \texttt{numerica} to be used effectively with the
instant preview facility of the document processor \LyX . More philosophically,
-one might view such errors as similar to grammatical errors or spelling
+one might view such errors as similar to errors of grammar or spelling
mistakes in text. It is not clear that they should halt compilation.
Hence strictly \texttt{numerica}-related errors leave brief messages
in the pdf at the offending places.
@@ -2021,51 +2290,34 @@ token message:
\begin{centred}
\verb`$\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{$y$}.
+Because of the missing comma, \verb`numerica` assumes \verb`a` has
+the `value' \verb`3y=2`, an expression which it then tries to evaluate,
+but the variable \verb`y` in this expression has not been assigned
+a value, which generates the message.
\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}
+The presence of multi-token variables can also cause an unknown token
+message if the check for such variables is turned off; see §\ref{subsec:settingsMultitokSwitch}.
-\subsection{Inverse powers of negative numbers}
+\subsection{Overlooked value assignments}
-\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:
+Perhaps if one is evaluating a formula with a number of variables
+and assigning different experimental values to them to see the effect,
+a variable might be overlooked:
\begin{centred}
-\verb`\eval{$ (-125)^{1/3} $}` $\Longrightarrow$ \eval{$ (-125)^{1/3} $}
+\verb`$\eval{axy}[a=3,y=,x=1]$` $\Longrightarrow$ \eval{axy}[a=3,y=,x=1]
\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.
+In the example the variable \verb`y` has been overlooked. The remedy
+is obvious – assign a value to it.
\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};
+\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.
@@ -2073,7 +2325,7 @@ 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
+is considered an integer (obviously, the value 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
@@ -2092,9 +2344,11 @@ can be changed: see §\ref{subsec:defaultsIntifyingRounding}.
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`$\eval{ \pi^e >= e^\pi }[?]$` $\Longrightarrow$ $\eval{ \pi^e >= e^\pi }[?]$
+\verb`$\eval{ e^\pi >= \pi^e }[?]$` $\Longrightarrow$ $\eval{ e^\pi >= \pi^e }[?]$
\end{centred}
-(The relation is false by the way.)
+(The relation is true by the way.) The same error is generated by
+other multi-token comparisons. They are used in programming languages,
+yes, but \emph{not} in mathematics.
\subsection{Invalid base for \texttt{\textbackslash log}}
@@ -2104,7 +2358,7 @@ 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`$\eval{ \log_12 1728 }$` $\Longrightarrow$ $\eval{ \log_12 1728 }$
+\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.
@@ -2113,30 +2367,49 @@ as the argument) of the logarithm and responds accordingly.
Some errors arising at the \texttt{l3fp} level are trapped and a message
displayed.
-\begin{itemize}
-\item Dividing by zero
-\end{itemize}
+
+\subsubsection{Dividing by zero}
\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}
+
+\subsubsection{Invalid operation}
+
+\label{subsec:errorsInverse-powers}Finding inverse integer powers
+of \emph{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.
+
+Trying to evaluate a function like a factorial or square root or inverse
+trig. function outside its domain of definition also produces this
+error:
\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:
+In this case the inverse cosine, which is defined only on the interval
+$[-1,1]$, has been fed the value $2$.
+
+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]
+\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}
+
+\subsubsection{Overflow/underflow}
+
The factorial (discussed in §\ref{subsec:evalFactorialsBinom}) provides
an example of overflow:
\begin{centred}
@@ -2150,9 +2423,11 @@ 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 $10^{-10001}$,
+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.
+accepts the zero value and the error is ignored.
\chapter{Settings}
@@ -2171,17 +2446,18 @@ 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
+nature 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}.
+if the packages \texttt{numerica-plus} or \texttt{numerica-tables}
+are loaded; see §\ref{subsec:Related-packages}.
\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
+result, \texttt{numerica} offers a means of examining various quantities
+at some intermediate stages on the way to the final result. To use
+the facility, enter
\begin{lyxcode}
dbg~=~<integer>
\end{lyxcode}
@@ -2191,8 +2467,6 @@ into the settings option. (White space around the equals sign is optional.)
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}
@@ -2202,132 +2476,118 @@ The `magic' integers are the following primes and their products:
{\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{view}} & & {\small equivalent to }{\small\texttt{dbg=1}} & \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{o}} & {\small int (0/1)} & {\small degree switch for trig. functions} & {\small\texttt{1}}\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{vv@}} & {\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 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
+{\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}
\end{table}
+The `magic' integers are the following primes and their products
\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.;
+been converted to their single token form, \texttt{\textbackslash nmc\_a},
+\texttt{\textbackslash nmc\_b}, 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);
+\item \texttt{dbg=5} displays the stored variables and their \emph{evaluated}
+values (\texttt{dbg=2} lists the values as expressions);
+\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.
+is also displayed (but without any formatting).
\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}).
+To display two or more of the debug 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 \verb`aligned`\texttt{ }or \verb`align*`
+environment to display its wares, depending on the presence or absence
+of math delimiters around the \verb`\eval` command. The following
+uses \verb`align*` and shows how multi-token variables are handled,
+how a chain of comparisons is evaluated (§\ref{subsec:evalBoolean-output})
+and how formatting instructions in the number-format option are ignored
+in the debug display:
\begin{verbatim}
- \eval[dbg=1]{ a_1<2a_2<3a_3<\pi+e }
- [a_1=\pi,a_2=\phi,a_3=e\gamma][6?'']
+ \eval[dbg=1]{ a < 2a' < 3a'' }
+ [a=\pi,a'=\phi,a''=e\gamma][4???]
\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:
+$\Longrightarrow$ \eval[dbg=1]{ a < 2a' < 3a'' }
+ [a=\pi,a'=\phi,a''=e\gamma][4???]
+
+\noindent The various items are displayed in chronological order.
+First comes the vv-list after conversion of multi-token to single-token
+variables, then the formula in those single-token variables; these
+are created essentially at the same time. 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. Often this can be a thicket of parentheses, especially
+if unary functions or fractions are involved. The final element of
+both the display and chronologically is the result from evaluating
+the formula. This is shown only if $7$ is a factor of the \texttt{dbg}
+integer, and there is no error. Despite the appearance of \verb`???`
+in the number-format option, the result displays as 1. Results are
+never rounded or formatted in the debug display, although as is apparent
+here, the rounding number \verb`4` is used in the comparisons.
+
+When interpreting the fp-form, differences in the ways \texttt{numerica}
+and \texttt{l3fp} read formulas can lead to more or less parentheses
+than seem strictly necessary. 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 $(\sin x)^{2}$. \verb`numerica` takes care of the former
+by inserting extra parentheses and exploits the latter by not inserting
+parentheses:
+\begin{verbatim}
+ \eval[dbg=1]{ \sin 2x \cos^2 y }
+ [x=\pi/12,y=\pi/4]
+\end{verbatim}
+$\Longrightarrow$ \eval[dbg=1]{ \sin 2x \cos^2 y }[x=\pi/12,y=\pi/4]Finally,
+note that those mathematical operations that have no direct representation
+in \verb`l3fp` contribute only their value to the fp-form. This applies
+to sums and products, double factorials, partly to binomial coefficients,
+and also to \verb`\eval` and other commands when nested one within
+another (see Chapter~\ref{chap:Nesting}). The following (ridiculous)
+example illustrates the matter:
\begin{verbatim}
\eval[dbg=1]{\[
- \sum_{n=1}^5 n + \binom{10}{m}
- + \prod_{n=2}^5 (1-1/n) + m!! \][m=6]
+ \sum_{n=1}^k n + \binom{2k}{m} - \frac1{4k} +
+ \prod_{n=2}^k (1-1/n) + m!! \]}[m=6,k=5]
\end{verbatim}
$\Longrightarrow$ \eval[dbg=1]{\[
- \sum_{n=1}^5 n+\binom{10}{m}
- +\prod_{n=2}^5(1-1/n) +m!! \]}[m=6]
+ \sum_{n=1}^k n + \binom{2k}{m} - \frac1{4k} +
+ \prod_{n=2}^k (1-1/n) + m!! \]}[m=6,k=5]
-\noindent The various contributions to the overall result are displayed
-simply as numbers because \texttt{l3fp} does not (at least as yet)
+\noindent ($0$\textdegree{}~C in kelvin!) In the \verb`fp-form`
+line, the various contributions to the overall result are displayed
+simply as numbers because \verb`l3fp` does not (at least as yet)
handle these elements natively.
-\subsubsection{Negative \texttt{dbg} values}
+\subsection{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
@@ -2341,18 +2601,19 @@ and the display occupies the text width. An example is presented in
\verb`\eval` is significantly improved with a negative \texttt{dbg}
value.
-\subsection{Reuse setting}
+\subsection{\texttt{view} 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}.
+Putting \texttt{dbg=1} may seem a little obscure in order to view
+internal values of \verb`numerica`. In that case, simply writing
+\verb`view` in the settings option will produce the same effect as
+entering \verb`dbg=1`.
\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}
+of the \texttt{\textbackslash eval} command. Such output 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}
@@ -2375,19 +2636,32 @@ 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).
+start with a digit: \verb`e-1` for instance does not, and will be
+treated as an algebraic expression involving the exponential constant:
\begin{centred}
-\verb`$ \eval[^=e]{ x+e-1 }[x=2e-1] $` $\Longrightarrow$ $ \eval[^=e]{ x+e-1 }[x=2e-1] $.
+\verb`$ \eval[^=e]{ x+e-1 }[x=1] $` $\Longrightarrow$ $ \eval[^=e]{ x+e-1 }[x=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:
+but
\begin{centred}
-\verb`$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $` $\Longrightarrow$
-$ \eval[^=e]{ x+e-1 }[x=\text{2e-1}] $.
+\verb`$ \eval[^=e]{ x+1e-1 }[x=1] $` $\Longrightarrow$ $ \eval[^=e]{ x+1e-1 }[x=1] $.
+\end{centred}
+A problem of appearance arises if scientific numbers appear in the
+vv-list or formula and either is displayed in the result. A number
+like \verb`2e-1` will display as $2e-1$, as if it were an algebraic
+expression. In version 1 of \verb`numerica` the cure was to wrap
+\verb`2e-1` in a \verb`\text` or \verb`\mbox` command. In version
+2 of \verb`numerica` the behaviour of \verb`\text` and \verb`\mbox`
+has been re-thought; see §\ref{subsec:Text-mbox-fonts}. Their contents
+are now invisible to the \verb`\eval` command. The solution is to
+wrap \verb`2e-1` in a \verb`\textrm` or \verb`\textsf` or \verb`\texttt`
+command. These commands were not recognized by \verb`\eval` in version
+1 but \emph{are} in version 2:
+\begin{centred}
+\verb`\eval[^=e]{$ 5x $ }[x=\texttt{2e-1}]` $\Longrightarrow$ \eval[^=e]{$ 5x $ }[x=\texttt{2e-1}]
+,
+
+\verb`\eval[^=e]{$ 5\texttt{2e-1} $ }` $\Longrightarrow$ \eval[^=e]{$ 5(\texttt{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
@@ -2430,22 +2704,33 @@ an error.
\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
+to be parsed (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
+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
+settings option. (The author hopes the charitable eye sees 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 $}.
\end{centred}
+This is a \verb`0/1` switch, \verb`0` signifying \verb`off` or
+`don't use degrees', \verb`1` signifying \verb`on` or `do use
+degrees'. Although the \verb`o` default is \verb`1`, out-of-the-box
+\verb`numerica` assumes radians are being used. Thus if \verb`o`
+is absent from the settings option of an \verb`\eval` command, the
+out-of-the-box setting prevails and radians are used, but if \verb`o`
+is present, it is equivalent to \verb`o=1`. To explicitly turn off
+the use of degrees requires the full setting, \verb`o=0`.
+
+If you want to change the out-of-the-box setting you need to put the
+line \verb`use-degrees = 1` into a configuration file; see §\ref{sec:settingsDefaults}.
\subsection{Specifying a logarithm base}
@@ -2453,14 +2738,15 @@ might see a degree symbol in the \texttt{o}.) Thus
a subscripted base in a particular calculation, then add an entry
like
\begin{lyxcode}
-log~=~<positive~number${}\,\mathtt{\ne1}$>
+log~=~<positive~number>
\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}.
+where \verb`<positive number>`~$\ne1$ 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}
@@ -2468,50 +2754,59 @@ of base to a configuration file; see §\ref{sec:settingsDefaults}.
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}
+a calculation. By entering either
+\begin{lyxcode}
+vv@~=~<integer>
+\end{lyxcode}
+or (as in version 1 of \verb`numerica`),
\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}
+possible values, \texttt{0} or \texttt{1}. \texttt{vv@=0} (or \texttt{vvmode=0})
+means the vv-list is evaluated once at the start of the calculation;
+\texttt{vv@=1} (or\texttt{ vvmode=1}) means the vv-list is recalculated
+every time the relevant variable changes.\footnote{In version 1 of \texttt{numerica} only the \texttt{vvmode} name for
+this setting was available. To the author's eye, the \texttt{@} sign
+seems sufficiently close to a symbol like $\circlearrowleft$, suggesting\texttt{
+}redo or recalculate, that \texttt{vv@} is now preferred. The \texttt{@}
+symbol is – universally? – available on keyboards and \texttt{vv@}
+is only half as many keypresses as \texttt{vvmode}.}
+
+For example, in a sum 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{vv@=1}
in the settings option. Repeating an earlier sum (the seting \verb`p=.`
-is discussed in §\ref{subsec:settingsPunctuation}) \medskip{}
+is discussed in §\ref{subsec:settingsPunctuation}), \medskip{}
\begin{verbatim}
- \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
+ \eval[p=.,vv@=1]{\[ \sum_{k=1}^N f(k) \]}
[N=100,f(k)=1/k^3,{k}=1][4]
\end{verbatim}
-$\Longrightarrow$ \eval[p=.,vvmode=1]{\[ \sum_{k=1}^N f(k) \]}
+$\Longrightarrow$ \eval[p=.,vv@=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.
+vv-list to avoid an unknown token message, 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{vv@=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{ }
+form of the function in the main argument, it seems natural to call
+\texttt{vv@=1} \emph{implicit }mode and \texttt{vv@=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}
@@ -2571,20 +2866,16 @@ specification, then the vv-list will not appear at all:
\verb`\eval[vvi=?!]{$ \pi $}[\pi=3]` $\Longrightarrow$ \eval[vvi=?!]{$ \pi $}[\pi=3]
\end{centred}
More relevantly, it may well be the case that all variables in the
-vv-list are suppressed (wrapped in braces). In that case the display
-would look something like \verb`, ()`. To prevent this enter \verb`vvi=`
-in the vv-list, in the inline case, or \verb`vvd=` in the display-style
-case, i.e. enter an empty value. (Alternatively, use the star option
-of the \verb`\eval` command.)
-
-Another minor wrinkle occurs if you want to change parentheses around
-the vv-list to square brackets. Because the settings option is a square-bracket
-delimited argument, the square brackets in the specification will,
-like commas, need to be hidden in braces, although you can get away
-with braces around the whole spec.:
-\begin{lyxcode}
-vvi=\{,\textbackslash mskip~12mu~plus~6mu~minus~9mu~{[}vv{]}\}
-\end{lyxcode}
+vv-list are suppressed (wrapped in braces). In that case nothing is
+displayed. Compare the last example with
+\begin{centred}
+\verb`\eval[vvi=?!]{$ \pi $}[{\pi}=3]` $\Longrightarrow$ \eval[vvi=?!]{$ \pi $}[{\pi}=3]
+\end{centred}
+and
+\begin{centred}
+\verb`\eval[vvi=?!]{$ \pi $}` $\Longrightarrow$ \eval[vvi=?!]{$ \pi $}
+\end{centred}
+See also the punctuation setting below, §\ref{subsec:settingsPunctuation}.
\subsection{Displaying the vv-list on a new line }
@@ -2652,9 +2943,9 @@ how to display a following punctuation mark? For an inline display
we can simply follow the \verb`\eval` command with the appropriate
punctuation, for instance: \verb`\eval{$ 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 \verb`\eval` command a trailing fullstop will slide 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
@@ -2664,17 +2955,18 @@ 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).
+and slow evaluation.
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.
+to place the {\ttfamily\verb`<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}
@@ -2682,6 +2974,12 @@ Nor is one limited to a single punctuation mark:
\eval[p=\ \text{(but no 8!)}]{\[ \frac{1}{81} \]}[9]
\end{centred}
+\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,
+§\ref{subsec:supplReuseEvalSetting}.
+
\section{Infinite sums and products}
\label{sec:settingsInfiniteSumsProds}There are ways of tweaking various
@@ -2689,7 +2987,31 @@ 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}
+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. 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 term taking unit value.
+Sometimes the initial terms of series or products are `irregular'
+and take these `stopping' values meaning sum or product would stop
+after only one or two additions/multiplications and far from any limit.
+
+\begin{table}[t]
\centering
\noindent \centering{}\caption{Settings for infinite sums \& products}\label{tab:settingsSumsProducts}
\noindent \begin{center}
@@ -2706,31 +3028,6 @@ key & type & meaning & default\tabularnewline
\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.
@@ -2775,7 +3072,7 @@ by increasing the additional rounding:
\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
+argument \verb`prod`; see §\ref{sec:supplInfo}. The product now
produces the correct three-figure value, but it takes $350$ factors
to do so.
@@ -2797,60 +3094,42 @@ value of $\sinh1$ beforehand, noting the number of factors required
would make us justly cautious about accepting the result of the infinite
product calculation.
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
One way to gain confidence in a result is to choose a possibly unrealistic
-rounding value – say, the default $6$ for the infinite product \-\-
+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},
+rounding values $s$ of $6+(-5)=1$, one decimal place, $6+(-4)=2$,
+two decimal places, and so on, but the result is always presented
+to $6$ decimal places. You 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 the infinite product
+for $\sinh1$ the number of factors $N_{s}$ required at a stopping
+rounding value $s$ increases in geometric proportion with a scale
+factor of about $3$: $N_{s}\approx\text{const}\times3^{s}$. This
+rapidly becomes large ($3^{4}=81,3^{5}=243\dots$). For the exponential
+series on the other hand $N_{s}=4+s$, the number of terms increases
+only slowly, in direct proportion to the stopping rounding value.
+Similar experiments with the sums of inverse fourth, third and second
+powers of the integers using \verb`\nmcInfo` to find how many terms
+are required at each stopping rounding value, show that at least over
+the rounding value range $1$ to $8$, for inverse fourth powers $N_{s}\approx\text{const}\times1.7^{s}$,
+for inverse third powers $N_{s}\approx\text{const}\times2^{s}$ and
+for inverse squares $N_{s}\approx\text{const}\times3^{s}$. 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 (e.g. $1.7^6\approx\eval{1.7^6}[0]$). 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.
+\eval[p=.]{\[ \sum_{k=1}^\infty \frac1{k^4} \]}
+\end{centred}
+\end{shaded}%
+\end{minipage}
\subsection{Premature ending of infinite sums}
@@ -2862,12 +3141,12 @@ 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 $\frac{4}{n^{2}\pi^{2}}(1-\cos n\pi)\cos2\pi nt$
+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
@@ -2885,11 +3164,11 @@ $\Longrightarrow$ \eval[p]{\[
(1-\cos n\pi)\cos2\pi nt
\]}[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}:
+\noindent Only one term? Of course – since for 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}
@@ -2903,22 +3182,12 @@ $\Longrightarrow$ \eval[p,S?=1]{\[
(1-\cos n\pi)\cos2\pi nt
\]}[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}
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+\begin{wraptable}{o}{0.35\columnwidth}%
+\centering{}\vspace{-5ex}
\caption{Finite sums}\label{tab:settingsFinite-sums}
-\setlength{\abovetopsep}{1ex}%
+\abovetopsep=2ex %
\begin{tabular}{cc}
\toprule
$N$ & $\Sigma$\tabularnewline
@@ -2930,12 +3199,21 @@ $66$ & $0.5999$\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 $0.6$.
+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 the specified rounding value is $2$ 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. 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. \end{shaded}%
+\end{minipage}
For a final example, consider the error function
\[
@@ -2986,26 +3264,29 @@ meaningful rounding value. For example, when $z=7$ the sum apparently
\subsection{Double sums or products}
-Sums or products can be iterated. For instance, the exponential function
+Sums or products can be iterate d. 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}
+\begin{verbatim}
+ \eval[p]
+ {\[ \sum_{k=0}^{\infty}
+ \prod_{m=1}^{k}\frac{x}{m} \]}[x=2]
+\end{verbatim}
+$\Longrightarrow$ \eval[p]
+ {\[ \sum_{k=0}^{\infty}
+ \prod_{m=1}^{k}\frac{x}{m} \]}[x=2]
+
\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
+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]
+ \eval[p]
+ {\[ \sum_{n=0}^{\infty}
+ \frac{(-1)^{n}}{n!} \]}[3] \info{sum}.
\end{verbatim}
-$\Longrightarrow$ \eval[p={,}\quad \mbox{\nmcInfo{sum}}.]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+$\Longrightarrow$ \eval[p]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3] \info{sum}.
-(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
+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}}
@@ -3032,7 +3313,15 @@ 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]
+The way to do that is to create a \emph{configuration file} named
+\texttt{numerica.cfg} in a text editor. Its entries 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.
+
+\begin{table}[t]
\centering
\noindent \centering{}\caption{Default values, \texttt{\textbackslash eval} command}\label{tab:settingsDefaults}
\noindent \begin{center}
@@ -3042,38 +3331,33 @@ of settings.
\midrule
{\small rounding} & {\small\texttt{6}}\tabularnewline
{\small pad} & {\small\texttt{0}}\tabularnewline
-{\small output-sci-notation } & {\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-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 multitoken-variables} & {\small\texttt{1}}\tabularnewline
+{\small use-degrees} & {\small\texttt{0}}\tabularnewline
+{\small 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 vv-inline} & {\small\texttt{\{,\}\textbackslash mskip 12mu 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
+\% & \tabularnewline
+{\small intify-rounding} & {\small\texttt{14}}\tabularnewline
+{\small eval-reuse} & {\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`.
+zeros), \verb`output-sci-notation`, \verb`input-sci-notation`, (check
+for) \verb`multitoken-variables`, and \verb`use-degrees` (for trig.
+functions).
The table is divided into four parts.
\begin{itemize}
@@ -3091,42 +3375,43 @@ 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.
+see §\ref{sec:settingsOption}. The key names are more expansive here
+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).
+\item 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}.
+\item The third block of 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}.
+\item The last block is for `left-overs': specifying at what rounding
+value a floating point should be considered an integer (see §\ref{subsec:defaultsIntifyingRounding}
+below), and specifying what kind of result is saved to file when the
+\verb`\nmcReuse` command is used (see §\ref{subsec:supplReuseEvalSetting}).
\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{ $d.d_{1}d_{2}d_{3}d_{4}\times10^{n}$},
+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}.
+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
+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}
@@ -3162,15 +3447,16 @@ 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.
+\label{subsec:settingsPersonal-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
@@ -3193,7 +3479,7 @@ 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
+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.
@@ -3213,11 +3499,11 @@ then add a line to \texttt{numerica.cfg} like
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.
+{\ttfamily\verb`<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}
@@ -3231,7 +3517,7 @@ arguments of functions, or inserting explicit multiplication signs
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
+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,
@@ -3242,7 +3528,7 @@ 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}
+meanings: to adhere or cling to, and to split apart or separate. \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});
@@ -3268,16 +3554,19 @@ $\Longrightarrow$ \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]`
+\begin{verbatim}
+ \eval[p]{\[
+ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T)
+ \]}[T=2,t=1,n=3]
+\end{verbatim}
$\Longrightarrow$ \eval[p]{\[ \cos\frac{2\pi}{T}\q n\q(t+\tfrac12T) \]}[T=2,t=1,n=3]
-\end{centred}
-which is $\cos(\pi\times3\times2)$.
+
+\noindent 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.
+handling arguments that contain parentheses (and fractions). This
+is discussed in §\ref{subsec:parseTrigFns} below.
For the \verb`\Q` command which splits an argument we have, without
it,
@@ -3289,21 +3578,22 @@ 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`.
+which is one half of $e$, although it is unlikely to be read that
+way. If one half of $e$ is intended then parenthesize the $1/2$
+or present 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.
+in this document, think of the letter \textsf{q} as a circle \emph{cleaving
+to} a vertical descender; think of the letter \textsf{Q} as a circle
+\emph{cleaved apart} 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}
+group will not be that used for 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
@@ -3314,23 +3604,25 @@ 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}
+\begin{table}
+\centering
+
+\caption{ Parsing groups}\label{tab:settingsParsing-groups}
{\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{II}} & {\small\textrm{unary functions (direct trig. functions default), /}}\tabularnewline
+{\small\textrm{III}} & {\small\textrm{direct trig. functions with special setting}}\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}%
+
+\end{table}
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
@@ -3343,7 +3635,7 @@ 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).
+and even \emph{third }positions).
For counting position, we need to allow for formatting elements and
multi-token numbers – in both decimal and scientific formats. Formatting
@@ -3405,15 +3697,14 @@ is performed first.
\verb`\eval{$ \surd 3^4 $}` $\Longrightarrow$ \eval{$ \surd 3^4 $}.
\end{centred}
-\subsubsection{Parsing group II: unary functions, slash fractions}
+\subsubsection{Parsing group II}
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}.
+exponential functions, the signum function \verb`\sgn`, and the denominators
+of slash fractions \verb`/`. 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}
@@ -3467,7 +3758,8 @@ or double factorial sign, and ends there; otherwise
\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.
+and \verb`\tfrac`-s, as instanced with $\sin2\tfrac{p}{q}\pi x$
+which exhibits all elements.
Illustrating 1, the exponent is included in the argument but not the
following variable:
@@ -3478,15 +3770,10 @@ 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 $\log_{10}5000.$
+be considered part of the argument of the logarithm. If that is the
+case, inserting a \verb`\q` command before \verb`n` would achieve
+this, but that would still be confusing for the reader of the pdf.
+Inserting parentheses is the only sensible thing to do.
Illustrating 2(a)ii, again the exponent is included in the argument
but not the following variable:
@@ -3495,20 +3782,24 @@ but not the following variable:
\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:
+the intention was for the $n$ to be included in the argument of the
+logarithm, then again the \verb`\q` command could be used or, better
+in this case, the $n$ could 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 $}[m=10,n=5].
+\verb`\eval{$ \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.
+the logarithm of $5000$, or better still, $m^{3}n$ could (and should)
+be parenthesized for the sake of the reader.
+
+Why the difference between $nm^{3}$ where $n$ and $m^{3}$ are included
+in the argument, and $m^{3}n$ where $n$ 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 in the pdf.
It is simple and easy to remember if it is understood that anything
-that breaks the visual appearance of juxtaposed numbers, variables,
+that breaks the `visual flow' of juxtaposed numbers, variables,
constants and \verb`\tfrac`-s ends the argument. An exponent does
-just that.
+just that. If you feel there is ambiguity, parenthesize to clarify.
Illustrating 2(b)ii, the argument stops with the \verb`\dfrac` and
its arguments and does not extend to the following constant:
@@ -3517,7 +3808,8 @@ its arguments and does not extend to the following constant:
\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.
+be used since the \verb`\dfrac` breaks the `visual flow' of the
+argument.
\begin{description}
\item [{Fractions}]~
@@ -3526,10 +3818,10 @@ 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}).
+to different results depending on whether it lay between \verb`$ $`
+or \verb`\[ \]` delimiters, which ruled it out. Because \verb`\frac`
+sometimes displays as \verb`\dfrac`, it 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
@@ -3538,14 +3830,13 @@ 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.
+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 (in the author's view) is
+acceptable since they display differently and are not instinctively
+read in the same way.
\end{description}
\subsubsection{Parsing group III}
@@ -3573,23 +3864,37 @@ 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
+()~=~<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.
+where \texttt{<integer>} is one of \texttt{0, 1, 2}.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+I remain unsure whether to persist with the \texttt{()} setting. A
+principal aim of \texttt{numerica} is to avoid having to modify a
+formula to bring it into a form that can be evaluated. The \texttt{()}
+setting arose in that context, but it complicates the code and may
+well confuse the user. Inserting cleave functions, \texttt{\textbackslash q}
+or \texttt{\textbackslash Q}, to achieve the same effects does mean
+modifying formulas, but is straightforward and easier to understand.
+(And \texttt{\textbackslash q} and \texttt{\textbackslash Q} have
+no effect on the visual appearance of formulas.)\end{shaded}%
+\end{minipage}
+
+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
+\item [{\texttt{()=0}}] is the \emph{default} setting, parsing group II
+behaviour; b/Brackets are included in the argument only if
\begin{itemize}
\item the left b/Bracket is in first position;
\begin{itemize}
@@ -3599,6 +3904,11 @@ which ends there, otherwise
\item the argument ends with the right b/Bracket.
\end{itemize}
\end{itemize}
+\end{description}
+The default setting allows things like $\sin\tfrac{1}{2}a$, $\cos2\pi nt$
+and $\tan(A+B)$. It does \emph{not} encompass examples like $\tan\tfrac{1}{2}(A+B)$
+or $\cos2(n+\tfrac{1}{2})\pi$. For that we need the setting \verb`()=1`:
+\begin{description}
\item [{\texttt{()=1}}] includes a b/Bracketed expression in the argument,
provided
\begin{itemize}
@@ -3628,6 +3938,37 @@ at which point the argument ends, or
\end{itemize}
\end{itemize}
\end{itemize}
+\end{description}
+With the \verb`()=1` setting, the arguments of $\tan\tfrac{1}{2}(A+B)$,
+$\sec\pi(\tfrac{1}{4}+\tfrac{1}{2}az)$, $\cos(2m+p)z$, $\sin(2n+1)v$
+are all accommodated, as is $\sin\tfrac{1}{2}(m+n)\pi$ with items
+on both sides of the parentheses. But, note, there must be at most
+\emph{one} item before the left parenthesis:
+\begin{centred}
+\verb`\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2]`. $\Longrightarrow$
+\eval[()=1]{$ \sin\tfrac16(m+n)\pi $}[m=1,n=2],
+\end{centred}
+whereas, with two items before the left parenthesis,
+\begin{centred}
+\verb`\eval[()=1]{$ \sin2\tfrac1{12}(m+n)\pi $}[m=1,n=2]`. $\Longrightarrow$
+
+\eval[()=1]{$ \sin2\tfrac1{12}(m+n)\pi $}[m=1,n=2].
+\end{centred}
+Whatever the \verb`()` setting, \texttt{numerica} does not check
+what is included between the parentheses (or brackets generally) –
+it could be anything. However inserting \verb`\left`, \verb`\right`
+or other modifiers before the parentheses restricts the argument of
+the sine in this example, despite the \verb`()=1` setting, to the
+\verb`\tfrac`:
+\begin{centred}
+\verb`\eval[()=1]{$ \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}
+Although \verb`()=1` serves well for the kinds of expressions and
+identities involved in trigonometry, perusal of any text on Fourier
+series will show it does not cover the kinds of expressions met there.
+For that we need
+\begin{description}
\item [{\texttt{()=2}}] includes a b/Bracketed expression in the argument
provided
\begin{itemize}
@@ -3655,28 +3996,11 @@ or \verb`\tfrac`, at which point the argument ends, or
\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.
+{\ttfamily\verb`()=2`} draws no distinction between brackets
+and Brackets. It allows all \verb`()=1` possibilities but also \emph{two
+}items (of a suitable kind) before a left b/Bracket; it also treats
+\verb`\dfrac`-s and \verb`\frac`-s like \verb`\tfrac`-s for determining
+the scope of arguments.
The following examples are taken from different texts on Fourier series.
The first shows a \verb`\frac` being included in the argument, the
@@ -3695,12 +4019,12 @@ instance
\[ \sin(N+\tfrac12)\frac{2\pi\tau}T \]
}[N=1,\tau=2,T=3]
\end{verbatim}
-$\Longrightarrow$ \eval[p,()=2]
+$\Longrightarrow$ \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.
+ }[N=1,\tau=2,T=3]which is the sine of $2\pi=(\tfrac{3}{2})\times(\tfrac{4}{3}\pi)$
+where a \verb`\frac` trailing the parentheses has been included in
+the argument, and \emph{not }$(\sin\tfrac{3}{2})(\tfrac{4}{3}\pi)$.
Or consider
\begin{verbatim}
\eval[p,()=2]
@@ -3713,11 +4037,10 @@ $\Longrightarrow$ \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.
+ \]}[x=1,\lambda=2,t=3,T=4]which is the sine of $-\tfrac{1}{2}\pi=2\pi\times(-\tfrac{1}{4})$
+where there are two items before the parentheses and \verb`\left`
+and \verb`\right` modifiers, and \emph{not} $\sin2\pi$ times the
+parenthesised expression.
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
@@ -3814,14 +4137,14 @@ For parsing group VI
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.
+often do write formulas. It is \emph{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
\[
-\sum\left(\mathtt{<stuff>}\right)\mathtt{<more-stuff>}
+\sum\left(\mathtt{<stuff>}\right)\mathtt{<more\ stuff>}
\]
does not necessarily end the summand at the right parenthesis: it
ends at the first naked $+$ or $-$ sign, or \verb`\Q` command,
@@ -3829,142 +4152,65 @@ 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
+even if it fits within the parsing rules, is to be deprecated. The
+\emph{intent} is that \verb`\eval` can parse unambiguous expressions
+correctly.
+
+\chapter{Supplementary commands}
+
+\label{chap:Supplementary-commands}This chapter introduces four commands,
+\verb`\nmcInfo` (which we have already met), \verb`\nmcMacros`,
+\verb`\nmcConstants` and \verb`\nmcReuse`, supplementary to the
+principal command \verb`\nmcEvaluate`. They use the same machinery
+as \verb`\nmcEvaluate` and so have the same syntax. If all arguments
+are present it is
+\begin{centred}
+\noindent \verb`\nmc<cmd>*[settings]{main arg}[vv-list][rounding]`
+\end{centred}
+where \verb`<cmd>` is one of \verb`Info`, \verb`Macros`, \verb`Constants`
+and \verb`Reuse`. All four commands have short-name forms: \verb`\info`,
+\verb`\macros`, \verb`\constants`, \verb`\reuse`.
+
+Generally the final two optional arguments will not be used. The user
+should be aware of this if following a command with a square bracketed
+expression – the expression will be absorbed without trace unless
+it is preceded by, for example, an empty brace pair.
+
+Because the commands share the machinery of \verb`\nmcEvaluate`,
+the settings discussed earlier (Chapter~\ref{chap:Settings}) for
+the \verb`\eval` command are also available for these commands, although
+they will, in the main, be irrelevant. The `debug' code has been
+used by the \verb`view` setting of some of these supplementary commands
+to produce its effects.
+
+The starred form of command is available in all four cases and in
+all cases produces a pure number. If both star and \verb`view` are
+used at the same time, the \verb`view` setting prevails over starring.
+
+\section{Feedback on \textquoteleft infinite\textquoteright{} processes:\texttt{ \textbackslash nmcInfo}}
+
+\label{sec:supplInfo}Used after the evaluation of an `infinite'
+process, the \verb`\nmcInfo` command, or its short-name form \verb`\info`
+will tell you how many terms or factors or other operations\footnote{It also applies to the commands \texttt{\textbackslash nmcIterate
+}and \texttt{\textbackslash nmcSolve} from the \texttt{numerica-plus
+}package and to derivatives and integrals from the \texttt{numerica-calculus}
+package.} were needed to arrive at the result.The main argument contains an
+identifier for the `infinite' process:
\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 $\sinh x=x\prod_{k=1}^{\infty}\left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)$.
+(or \verb`\info{<arg>}`) where, at this stage, \verb`<arg>` is either
+\verb`sum` or \verb`prod`. The display, as we have seen in earlier
+examples, is a number followed by a space then a descriptor. For \verb`sum`
+and \verb`prod` the descriptors are \verb`terms` and \verb`factors`.
+Starring \verb`\nmcInfo` – \verb`\nmcInfo*{arg}` or \verb`\info*{arg}`
+– suppresses the descriptor and leaves only the number. This allows
+the starred form to be nested in an \verb`\eval` command, which might
+sometimes be convenient.
+
+As an example, let's test `the hard way' a standard identity, $\cosh^{2}x-\sinh^{2}x=1$.
+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{\[
@@ -3974,48 +4220,48 @@ The difference of their squares should be $1$:
\left[x\prod_{k=1}^{\infty}
\left(1+\frac{x^{2}}{k^{2}\pi^{2}}\right)
\right]^2
- \]}[x=1][3] \info{sum}\quad \info{prod}
+ \]}[x=1][3] \info{sum},\quad \info{prod}
\end{verbatim}
-$\Longrightarrow$ \eval{\[
+$\Longrightarrow$ \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}.
+ \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.
+in the one evaluation. One does not exclude the other.
+
+\subsection{Suppressing the descriptor: \texttt{\textbackslash nmcInfo{*}}}
-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:
+The starred form of the \verb`\info` command suppresses the descriptor
+(`terms', `factors') and gives a purely numerical result:
\begin{verbatim}
- \eval[p=\mbox{,\quad\nmcInfo{sum}.}]
- {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+ \eval{$
+ \sum_{k=0}^{\infty}\binom \alpha k x^k
+ $}[x=1/2,\alpha=3],
+ requiring $ \info*{sum}-1 $ additions.
\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.
+$\Longrightarrow$ \eval{$
+ \sum_{k=0}^{\infty}\binom \alpha k x^k
+ $}[x=1/2,\alpha=3],
+requiring $ \info*{sum}-1 $ additions. (Four terms added, therefore $3$ additions.)
-\section{Errors}
+\subsection{Errors}
-Should the wrong argument be used in the \verb`\nmcInfo` command,
+Should the \emph{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 $}[x=1/2,\alpha=3],\ \info{prod}.
+$\Longrightarrow$ \eval{$ \sum_{k=0}^{\infty}\binom \alpha k x^k $}[x=1/2,\alpha=3],\ \info{prod}.\medskip{}
$119$ \emph{factors}? The information command is remembering a previous
result, the last time \verb`prod` was used as its argument. Changing
@@ -4030,147 +4276,897 @@ Should a non-existent argument be used, an error message is generated:
\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
+\subsection{\texttt{view} setting}
+
+As noted at the start of this chapter, \verb`\nmcInfo` uses the `machinery'
+of \verb`\nmcEvaluate`. Most of the settings available for \verb`\eval`
+are also available for \verb`\info` but of these only one seems relevant:
+the \verb`dbg` setting. However, rather than use the obscure \verb`dbg=<integer>`
+(which is possible), it suffices to enter \verb`view` in this argument:
+\begin{centred}
+\verb`\info[view]{}` $\Longrightarrow$ \info[view] {}
+\end{centred}
+The result is a display of all the current values of all the `infinite'
+processes available. All such values are initialized to $0$. (Further
+processes \verb`iter` and \verb`solve` become relevant if the \verb`numerica-plus`
+package is used; \verb`deriv` and \verb`integ` become relevant if
+the \verb`numerica-calculus` package, currently under development,
+is used.)
+
+\section{User-defined macros: \texttt{\textbackslash nmcMacros}}
+
+\label{sec:supplMacros}The \verb`\nmcMacros` and \verb`\nmcConstants`
+commands were prompted by a question on \TeX{} Stack Exchange.\cprotect\footnote{A question from Giacomo Petrillo on \TeX{} Stack Exchange and a response
+by `egreg' prompted the introduction of the \cprotect\texttt{\textbackslash nmcMacros}
+and \cprotect\texttt{\textbackslash nmcConstants} commands. See \url{https://tex.stackexchange.com/questions/602993/use-macros-in-numerica-vv-list/602998#602998}} Some time later the maintainer of the \verb`mandi` package\footnote{The maintainer is Joe Heafner, who explains that `mandi' is an abbreviation
+of `matter and interactions' after a physics textbook of that name
+(by different authors). Among other things, the package defines a
+long list of macros, each containing the value of a physical constant.} approached me with a similar problem. Suppose one has defined a macro
+to contain a value, say
+\begin{itemize}
+\item \verb`\def\myvalue{0.35}`, or
+\item \verb`\newcommand\myvalue{0.35}`, or
+\item \verb`\NewDocumentCommand\myvalue{}{0.35}`, if you're using \verb`xparse`.
+\end{itemize}
+(If you're using the document processor \LyX{} then there is good reason
+to prefer \verb`\gdef` to define your macro, \verb`\gdef\myvalue{0.35}`;
+see Chapter~\ref{chap:LyX}). After one of these commands, \verb`\myvalue`
+is now known to \LaTeX , but it is not known to \verb`numerica`.
+The quantities \verb`numerica` \emph{does }know about are variables
+in the vv-list of an \verb`\eval` command, and those \LaTeX{} (and
+\verb`amsmath` and \verb`mathtools`) commands used for writing mathematical
+expressions. These quantities are stored in \verb`numerica` in structures
+called property lists. Since \verb`\myvalue` is not recorded in these
+lists yet, putting \verb`x=\myvalue` in the formula or vv-list of
+an \verb`\eval` command will produce an `Unknown token' error message:
+\begin{verbatim}
+ \NewDocumentCommand \myvalue {} {0.35}
+ \eval{ \myvalue }
+\end{verbatim}
+$\Longrightarrow$ \NewDocumentCommand \myvalue {} {0.35}
+ \eval{ \myvalue }
+
+With version 2 of \verb`numerica` a command is now available, \verb`\nmcMacros`,
+to register macros and their values with the property lists used internally
+by \verb`numerica`. (This command was not available in version 1.)
+The macro must have been defined earlier in the document or in a supporting
+package.
+
+The basic usage is simple. If you have a list of macros you wish to
+make available to \verb`\nmcEvaluate`, enter them in a comma list
+in the mandatory argument of \verb`\nmcMacros`:
\begin{lyxcode}
-{\small\textbackslash nmcReuse{*}{[}csname{]}}{\small\par}
+\textbackslash nmcMacros\{~\textbackslash macro1,~\textbackslash macro2,~\ldots ~\}
\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.
+There is an equivalent short-name form of the command, \verb`\macros`.
+
+Multiple \verb`\nmcMacros` commands can be used in a document. If
+the command is placed in the preamble (\emph{after} the definition
+of the macros)\emph{ }then the user-defined macros and their values
+are available throughout the document, otherwise they are available
+from the position of the \verb`\macros` statement. However, macros
+do not need to be defined in your current document provided they are
+defined and accessible from elsewhere – for example from a loaded
+\LaTeX{} package. But always an \verb`\nmcMacros` command is required
+to `register' them with \verb`numerica` for use in an \verb`\eval`
+command.
+
+\subsection{What can be stored in a macro?}
+
+Generally a user-defined macro will store a number. This macro might
+well be defined in an external package – for example the \verb`mandi`
+package defines a large number of macros containing the values of
+physical constants, some fundamental like the speed of light, others
+contingent like the earth–moon distance. If the \verb`mandi` package
+is loaded then writing, for instance,
+\begin{verbatim}
+ \macros{ \electronmassprecisevalue,
+ \protonmassprecisevalue }
+\end{verbatim}
+will make these two macros available for use in \verb`numerica`.
+One could then write in the vv-list of an \verb`\eval` command
+\begin{verbatim}
+m_e=\electronmassprecisevalue,m_p=\protonmassprecisevalue
+\end{verbatim}
+which would allow (among other things) calculation of the mass ratio
+$m_{p}/m_{e}$ of proton to electron. (The length of name of some
+of the macros in the \verb`mandi` package has a pedagogical purpose,
+but makes them unwieldy for direct use in mathematical expressions.)
+
+\subsubsection{Macros containing formulas }
+
+Numbers are not the only quantities that can be stored in a macro
+for use in \verb`numerica`. In fact any mathematical expression that
+can be \verb`\eval`-uated can be stored in a macro:
+\begin{verbatim}
+ \NewDocumentCommand \mysumC {}
+ { \sum_{n=1}^{100}1/n - \ln 100 }
+ \macros{ \mysumC }
+ \eval{$ \mysumC $}[4]
+\end{verbatim}
+$\Longrightarrow$ \NewDocumentCommand \mysumC {}
+ { \sum_{n=1}^{100}1/n - \ln 100 }
+ \macros{ \mysumC }
+ \eval{$ \mysumC $}[4], \medskip{}
+
+\noindent (to be compared with Euler's constant \eval{$ \gamma $}[4]
+– obviously many more terms are needed). The \verb`\eval` command
+wraps around math delimiters in the example. Hence the result is presented
+in the form \emph{formula=result}. In that presentation, note how
+\verb`\mysumC` displays as the formula it contains.
+
+\paragraph{The essential space: }
+
+But the critical thing to notice in the example is \emph{the space
+preceding }\verb`\sum`\emph{ in the definition of }\verb`\mysumC`.
+When a formula starts with an expandable token, \emph{this space is
+essential}. For macros to register successfully with \verb`numerica`,
+the first character in their definition must be \emph{un}expandable.
+Thus a digit is fine: storing a number in a macro is straightforward
+and you don't need to fuss about such niceties. But a control sequence
+like \verb`\sum` does expand (to $\sum$ ). If it is the initial
+token of the formula, then it will cause a possibly obscure error
+– see §\ref{subsec:supplMacrosErrors} – unless preceded by an unexpandable
+token. Hence the space before \verb`\sum` in the \verb`\NewDocumentCommand`
+statement. (On the other hand the spacing in the \verb`\macros` statement
+is purely aesthetic.)
+
+When using macros from another package, this is a matter to be aware
+of. If the macros contain only numbers, there should be no problem,
+but if they contain more complicated expressions, the absence of an
+initial space could make them unusable in \verb`numerica`.
+
+\subsubsection{Vv-list}
+
+In the example it would be nice to be able to vary the number of terms
+summed. This is easily done by using a vv-list in the \verb`\macros`
+statement:
+\begin{verbatim}
+ \NewDocumentCommand \mysumN {}
+ { \sum_{n=1}^{N}1/n - \ln N }
+ \macros{ \mysumN }[N=150]
+ \eval{$ \mysumN $}
+\end{verbatim}
+$\Longrightarrow$ \NewDocumentCommand \mysumN {}
+ { \sum_{n=1}^{N}1/n - \ln N }
+ \macros{ \mysumN }[N=150]
+ \eval{$ \mysumN $}.\medskip{}
+
+\noindent \verb`numerica` needs a definite value to store; it does
+not store the formula as such. To give \verb`\mysumN` a definite
+value, give the variable \verb`N` a value. This is done in the vv-list
+added to the \verb`\macros` statement: \verb`N=150`. In this way
+a definite value is stored in \verb`numerica` against the macro \verb`\mysumN`.
+The definition of the macro is unaffected. If a new value is given
+to \verb`N` in the \verb`\macros` statement (which is the point
+of using a variable), the old value is overwritten and the new value
+is used in subsequent calculations.
+
+\subsection{Seeing what macros are available}
+
+Perhaps your document has a number of \verb`\nmcMacros` statements
+scattered through it and you want to remind yourself of what exactly
+has been stored. \verb`\nmcMacros` has the \verb`view` setting for
+this purpose. Writing
+\begin{centred}
+\verb`\macros[view]{}` $\Longrightarrow$ \macros[view]{}
+\end{centred}
+produces a list of all macros registered with \verb`numerica` and
+their values, as you can see.
+
+If the braced argument is not empty, the display is slightly modified:
+\begin{verbatim}
+ \def\mydef{ \sin(m\pi/n) }
+ \newcommand\mynewcmd{ \cos(m\pi/n) }
+ \macros[view]{ \mydef,\mynewcmd }[m=3,n=18]
+\end{verbatim}
+$\Longrightarrow$ \def\mydef{ \sin(m\pi/n) }
+ \newcommand\mynewcmd{ \cos(m\pi/n) }
+ \macros[view]{ \mydef,\mynewcmd }[m=3,n=18]
+
+\noindent \verb`\mydef` and \verb`\mynewcmd` have been added to
+those available for use in \verb`numerica`.
+
+\subsubsection{Freeing macros from storage}
+
+Rather than cluttering \verb`numerica`'s property lists with no-longer-needed
+macros, it is possible to remove them from there with the \verb`free`
+setting. This has no effect on the \LaTeX{} definition of the macro.
+It merely `de-registers' the macro with \verb`numerica`.
+\begin{centred}
+\verb`\macros[free,view]{ \mysumC }` $\Longrightarrow$ \macros[free,view]{ \mysumC }
+\end{centred}
+If you want to free \emph{all} macros registered with \verb`numerica`
+use an empty main argument with the \verb`free` setting. For an example,
+see just below.
+
+\subsubsection{Counting how many macros are available}
+
+You can count how many macros are currently registered with \verb`numerica`
+by starring the \verb`\nmcMacros`~command:
+\begin{centred}
+\verb`\macros*{}` $\Longrightarrow$ \macros*{}.
+\end{centred}
+If the braced argument is not empty, the list of macros it contains
+will be added to those registered with \verb`numerica` and included
+in the overall count.
+
+Note that the \verb`view` setting prevails over starring if both
+are used.
+
+The star can also be used with the \verb`free` setting. As mentioned
+above, if the main argument is empty, then \emph{all} macros are freed:
+\begin{centred}
+\verb`\macros*[free]{}` $\Longrightarrow$ \macros*[free]{ }
+\end{centred}
+
+\subsection{Errors}
+
+\label{subsec:supplMacrosErrors}If a macro is used in a \verb`\macros`
+statement and the macro has not been defined in the document or a
+supporting package it will cause an error:
+\begin{verbatim}
+ \macros{ \mymacro }
+\end{verbatim}
+$\Longrightarrow$ \macros{\mymacro }
+
+\noindent As noted in the introduction to this section, an undefined
+macro used in an \verb`\eval`-uation will cause an `Unknown token'
+message in \verb`numerica`. The solution in this and the preceding
+case is (obviously) to define the macro.
+
+If a macro contains a formula which begins with an expandable token
+and a preceding space is omitted (see above), then entering that macro
+in a \verb`\macros` statement to register it with \verb`numerica`
+will generally cause a puzzling error:
+\begin{verbatim}
+ \newcommand\mysin{\sin(\pi/7)}
+ \macros{ \mysin }
+\end{verbatim}
+$\Longrightarrow$ \newcommand\mysin{\sin(\pi/7)}
+ \macros{ \mysin }
+
+\noindent The \verb`\protect` seems to be plucked from nowhere. In
+fact it comes from the expansion of \verb`\sin`. If \verb`\sum`
+had been the first token in the macro definition, again with no preceding
+space, then \verb`\protect` would have been replaced by the even
+more puzzling \verb`\DOTSB`. The solution is to insert a space as
+the first token in the macro definition.
+
+If a macro is defined but the \verb`\macros` statement is overlooked,
+and the macro is used in an \verb`\eval`-uation, it will generate
+an `Unknown token' message.
+
+If your macro stores a formula with variables, and you forget to give
+those variables values in the \verb`\macros` statement that will
+produce a message:
+\begin{verbatim}
+ \def\mysumk{ \sum_{n=1}^k n }
+ \macros{ \mysumk }
+\end{verbatim}
+$\Longrightarrow$ \def\mysumk{ \sum_{n=1}^k n }
+ \macros{ \mysumk }
+
+\noindent The `where' part of the message is specific in this case,
+but is generally `\verb`\nmcMacros` command'.
+
+And of course there can be `all the usual suspects' discussed at
+§\ref{sec:evalErrors} in the evaluation of the vv-list or the formula.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+
+\subsubsection{Display of macros}
+
+\label{subsec:supplMacrosDisplay}As shown in earlier examples, macros
+display as their content. Thus \verb`\mysumC` displayed as $\sum_{n=1}^{100}1/n-\ln100$.
+But once a macro is known to \LaTeX{} (not necessarily to \verb`numerica`)
+it can be used as a variable name. This has the same potential for
+abuse as noted earler for multi-token variables (§\ref{subsec:evalDon't-do-this!}).
+In the following example note that there is no \verb`\macros` statement.
+It suffices for the macro to be known to \LaTeX .
+\begin{verbatim}
+ \def\mymac{1}
+ \eval[vvi=\,???]{$ \mymac+\mymac $}[\mymac=2]
+\end{verbatim}
+$\Longrightarrow$ \def\mymac{1}
+ \eval[vvi=\,???]{$ \mymac+\mymac $}[\mymac=2]
+
+The value assigned to a variable name – in this case \verb`\mymac`
+– by \verb`numerica` for \cprotect\emph{calculational} purposes and
+how that variable name \cprotect\emph{displays} in \LaTeX{} are two
+separate things. One relies on the user not to do something deliberately
+deceptive.\end{shaded}%
+\end{minipage}
+
+\subsection{Rounding value}
+
+\label{subsec:supplMacrosRounding}Values are stored to $16$ significant
+figures (if available). In most cases appending a rounding value to
+a \verb`\macros` statement has no effect on the value stored. In
+the following example note the \verb`o` setting, meaning the sine
+reads angles in degrees:
+\begin{verbatim}
+ \NewDocumentCommand\testi{}{ \sin 60 }
+ \NewDocumentCommand\testii{}{ \sin 60 }
+ \macros[o]{ \testi }[10]
+ \macros[o]{ \testii }[3]
+ \macros[view]{}
+\end{verbatim}
+$\Longrightarrow$ \NewDocumentCommand\testi{}{ \sin 60 }
+ \NewDocumentCommand\testii{}{ \sin 60 }
+ \macros[o]{ \testi }[10]
+ \macros[o]{ \testii }[3]
+ \macros[view]{}
+
+\noindent Despite the different rounding values the same $16$ figures
+are stored in both \verb`\testi` and \verb`\testii`.
+
+For the \verb`\eval` command, rounding values specify how results
+are \emph{displayed}. The rounding value matters only \emph{after},
+not during, the calculation. Only for infinite sums or products is
+this otherwise. There the rounding value is used to determine when
+to stop adding further terms or factors. The same is true of the \verb`\macros`
+command. Only if a macro contains an infinite sum or product does
+the rounding value become relevant. Sixteen figures are still stored,
+but most of them will be `wrong' since the infinite sum or product
+has stopped early, after only a finite number of terms or factors.
+Exactly how many of the first few figures are correct depends on the
+rounding value. An example may clarify the matter.
+\begin{verbatim}
+ \macros[free]{}
+ \def\zetaiii{ \sum_{n=1}^\infty 1/n^3 }
+ \macros[view]{ \zetaiii }[3]
+ \info{sum}
+ \macros[view]{ \zetaiii }[6]
+ \info{sum}
+\end{verbatim}
+$\Longrightarrow$ \macros[free]{}
+ \def\zetaiii{ \sum_{n=1}^\infty 1/n^3 }
+ \macros[view]{ \zetaiii }[3]
+ \info{sum}
+ \macros[view]{ \zetaiii }[6]
+ \info{sum}
+
+\emph{HMF }Table 23.3 tells me that $\zeta(3)=1.202056903159594\dots$
+The different rounding numbers have restricted the infinite sums to
+the very finite $47$ and $468$ terms respectively. Although $16$
+figures are stored, only the first few are correct. Just how many
+depends on the number of terms summed which depends on when the stopping
+criterion is met which depends on the rounding value.
+
+\section{User-defined constants:\texttt{ \textbackslash nmcConstants}}
+
+\label{sec:supplConstants}As noted much earlier in this document
+(§\ref{subsec:Built-in-Constants}), there are five built-in constants:
+\verb`\pi`, \verb`e`, \verb`\phi`, \verb`\gamma` and \verb`\deg`,
+but a user may well want to define their own constant or constants.
+There are contexts where it would make sense to permanently record
+fundamental constants like the speed of light or Planck's constant,
+or more down-to-earth constants like the acceleration due to gravity
+or the viscosity of water, rather than having to enter them in the
+vv-list for each calculation. Or a parameter might be held constant
+for a particular problem or class of problems where other variables
+change – for example triangles of constant perimeter but varying sides.
+This is the purpose of the \verb`\nmcConstants` command.
+
+The symbols used to denote constants are subject to exactly the same
+constraints and freedoms as the symbols used to denote variables.
+They might be single latin letters like \verb`c` (e.g. $c=3\times10^{8}$),
+or greek letters like \verb`\alpha` (e.g. $\alpha=1/137)$, or multitoken
+combinations like the Rydberg constants \verb`R_\infty` or \verb`R_{\mathrm{H}}`
+from atomic physics, or \verb`\mu_0` and \verb`\epsilon_0` used
+to denote the permeability and permittivity of free space, or personal
+constants like \verb`total` of no wider significance. \verb`numerica`
+handles all these different forms of constant with the command \verb`\nmcConstants`:
+\begin{verbatim}
+ \nmcConstants{ const-n=value-n, ... ,
+ const2=value2, const1=value1 }
+\end{verbatim}
+This is the simplest use – each constant is assigned a (numerical)
+value. But it is easy to envisage situations where it would be convenient
+to have a constant with value $1/\sqrt{2\pi}$ say, or another with
+value $e^{\tfrac{\pi}{2}}$, and so on. That is easy: simply put the
+expession for the value on the right:
+\begin{verbatim}
+ \constants{ a=1/\sqrt{2\pi},b=e^{\tfrac\pi2} }
+\end{verbatim}
+Or the values could be expressions depending on parameters:
+\begin{verbatim}
+ \constants{ s=\tfrac12(a+b+c) }[a=3,b=5,c=7]
+\end{verbatim}
+Some constants might depend on earlier constants in the list:
+\begin{verbatim}
+ \constants{ A=\sqrt{s(s-a)(s-b)(s-c),
+ s=\tfrac12(a+b+c) }[a=3,b=5,c=7]
+\end{verbatim}
+Or the values could involve an `infinite' process, requiring a rounding
+number:
+\begin{verbatim}
+ \constants{ \zeta=\sum_{n=1}^\infty(1/n^k) }[k=4][5]
+\end{verbatim}
+In this, although $16$ figures will be stored, only the first few
+will be accurate, the precise number depending on the value of \verb`k`
+and the rounding number (\verb`5` in the example); see the discussion
+on this issue for user-defined macros, §\ref{subsec:supplMacrosRounding}.
+
+\subsection{New list replaces old by default}
+
+A particular group of constants may be relevant only to a particular
+part of a document. Another part of the document may use other constants.
+By default, a second list of constants \emph{replaces }the first list.
+Thus each of the \verb`\constants` statements above would replace
+the previous one.
+
+There is a technical reason why replacing rather than appending is
+the default. For each calculation all \emph{multi-token} constants
+(e.g. \verb`R_\infty`, \verb`N_0`, \ldots ) are added internally
+to the start of the vv-list of the \verb`\eval` command. Even if
+the vv-list is empty, this is still the case since the formula might
+well use constants. Like multi-token variables and for the same reason
+(see §\ref{subsec:evalVariableNames}), multi-token constants are
+mapped internally to single tokens. This occurs afresh for each calculation.
+If there are a lot of multi-token constants then each calculation
+is going to involve not only this mapping from multi- to single tokens
+but the evaluation of a long vv-list. In that case it seems better
+to make the default behaviour replacement of one constant list by
+another, rather than appending them.
+
+\subsection{Adding constants to a list}
+
+Despite the default behaviour, there will be occasions when you want
+to add a new constant or constants to the current list. This is easily
+done with the \verb`add` setting. For instance,
+\begin{verbatim}
+ \nmcConstants[add]{ \sigma=5.67\times10^{-8},
+ k_B = 1.381\times10^{-23} }
+\end{verbatim}
+would add \verb`\sigma` and \verb`k_B` to the current list. The
+presence of the \verb`add` setting triggers appending rather than
+replacement.
+
+\subsection{Examples of use}
+
+\subsubsection{Example 1: atomic constants}
+
+In the following example, the values of various atomic constants are
+taken from the \verb`mandi` package. I use two \verb`\constants`
+statements in order to show the use of the \verb`add` setting. I've
+also included a \verb`view` setting in the second \verb`\constants`
+statement.
+
+The constants are used to calculate the fine-structure constant \verb`\alpha`
+in the vv-list of the \verb`\eval` command, and its well-known reciprocal
+(close to $137)$ in the main argument. Note that the constants do
+not need to be entered in the vv-list of the \verb`\eval` command.
+Their values are available from the \verb`\constants` statements.
+\begin{verbatim}
+ \constants{ c=2.99792458\times10^{8},
+ h=6.62607015\times10^{-34},
+ e=1.602176634\times10^{-19} }
+ \constants[view,add]
+ { \epsilon_0=8.854187817\times10^{-12} }
+ \eval{$ 1/\alpha $}[\alpha=e^2/2\epsilon_0hc]
+\end{verbatim}
+$\Longrightarrow$ \constants{ c=2.99792458\times10^{8},
+ h=6.62607015\times10^{-34},
+ e=1.602176634\times10^{-19} }
+ \constants[view,add]{
+ \epsilon_0=8.854187817\times10^{-12} }
+ \eval{$ 1/\alpha $}[\alpha=e^2/2\epsilon_0hc].
+
+The \verb`view` setting produces a now familiar kind of display.
+It shows that the three-token \verb`\epsilon_0` (the control sequence
+\verb`\epsilon`, the underscore \verb`_` and the digit \verb`0`)
+has been replaced by \verb`\nmc_q` – which may look as if it is also
+three tokens but is in fact a single control sequence.
+
+\subsubsection{Example 2: local constants}
+
+Long ago, when there were such creatures as reference librarians,
+I was asked about a school physics project along these lines.
+
+A\emph{ car is travelling at 50 km/hr when it hits a lamppost. The
+bonnet crumples 1 metre and the car comes to an immediate halt. Although
+she herself is wearing a seat-belt, a woman in the passenger cabin
+is holding her 5 kg baby. Does the baby survive?}
+
+The enquirer was familiar with the equations describing constant acceleration,
+\[
+x=ut+\tfrac{1}{2}at^{2},\quad\text{and}\quad v^{2}-u^{2}=2ax,
+\]
+and Newton's second law, $F=ma$, force equals mass times acceleration.
+The question was really about understanding these laws and how to
+think with them. Here, $s$ is the distance travelled in time $t,$
+with initial speed $u$ at $t=0$, speed $v$ at time $t$, and constant
+acceleration $a$ – a deceleration in this case.
+
+The given data provide our constants: distance $x=1$ metre, initial
+speed $u=1000*50/(60*60)=(10/36)*50$ metres per second, final speed
+$v=0$. To estimate whether the woman can hold on to her baby, we
+will need to make a comparison with forces we have personally experienced.
+Most of us have tried lifting someone else, so let's use a characteristic
+human weight as our test mass. Thus, we have the (baby's) mass $m=5$
+kilograms, and a test mass, $M$ say, which we will leave as a variable.
+But dealing with weight, we will need the acceleration due to gravity.
+For the kind of rough estimating we are doing, $g=10$ metres per
+second per second will be an adequate approximation.
+\begin{verbatim}
+ \constants{ x=1,v=0,u=(10/36)50,m=5,g=10 }
+\end{verbatim}
+The deceleration experienced by the woman is found from the second
+equation of constant acceleration, $a=(v^{2}-u^{2})/2x$. Even if
+the deceleration isn't constant this will give an estimate of its
+magnitude. (If some of the deceleration is less than this $a$, some
+must be greater.) This is also the deceleration experienced by the
+baby as long as the woman holds onto her. Hence the magnitude of the
+force exerted by the baby on the woman's arms is $ma=m(v^{2}-u^{2})/2x=-mu^{2}/2x$
+which we want to compare with our test force, say that required to
+lift $M=70$ kilograms, which was once considered the mass of an average
+western adult male (but is doubtless a considerable underestimate
+now). Hence the test force is $Mg$. Let's do the calculations. (I
+have altered the \verb`\constants` statement to allow for a later
+comparison with the effect of a small increase in speed.)
+\begin{verbatim}
+ \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=50]
+ \eval{$ mu^2/2x $}[0], \par
+ \eval{$ Mg $}[M=70].
+\end{verbatim}
+$\Longrightarrow$ \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=50]
+ \eval{$ mu^2/2x $}[0], \par
+ \eval{$ Mg $}[M=70].
+
+The force required to hold on to the baby is noticeably less than
+that required to lift a $70$~kg person – in fact about the same
+as that needed to lift a $50$~kg person. But we have ignored the
+force experienced by the mothers forearms – perhaps doubling $m$
+(baby plus forearms) would give a better estimate of the force she
+experiences. In that case $mu^{2}/2x$ obviously doubles and the total
+force required by the woman to retain her baby – now $964$ newtons
+– is significantly more than that required to lift a $70$~kg person.
+I think it almost certain that the baby is torn from her arms.
+
+What difference does increasing the speed to 60 km/hr make?
+\begin{verbatim}
+ \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=60]
+ \eval{$ mu^2/2x $}[1], \par
+ \eval{$ Mg $}[M=70].
+\end{verbatim}
+$\Longrightarrow$ \constants{ x=1,u=(10/36)U,m=5,g=10 }[U=60]
+ \eval{$ mu^2/2x $}[0], \par
+ \eval{$ Mg $}[M=70].
+
+Now the force of baby alone is comparable to that required to lift
+a $70$ kg person. Including the woman's forearms in $m$, doubling
+$m$ say, will result in a force twice as great – like that required
+to lift two $70$~kg people or one $140$~kg person. There is no
+chance of the woman holding on to her baby. The force is too great.
+
+\subsubsection{Example 3: macros and constants}
+
+Constants can depend on previously defined and registered user macros.
+Suppose I have defined two macros
+\begin{verbatim}
+ \NewDocumentCommand\electronmassprecisevalue {}
+ {9.1093837015\times10^{-31}}
+ \NewDocumentCommand\protonmassprecisevalue {}
+ {1.672621898\times10^{-27}}
+\end{verbatim}
+(I have taken both the names and the values from the \verb`mandi`
+package.) The long explicit names of the macros has a pedagogic purpose,
+but they are too cumbersome to use in calculations. For that purpose
+we need, first, a \verb`\macros` statement registering the two macros
+with \verb`numerica`, and then a \verb`\constants` statement like
+\begin{verbatim}
+ \nmcConstants{ m_e=\electronmassprecisevalue,
+ m_p=\protonmassprecisevalue }
+\end{verbatim}
+With that \verb`m_e` and \verb`m_p` could be entered in formulas,
+taking the values contained in the macros. Let's do it:
+\begin{verbatim}
+ \NewDocumentCommand\electronmassprecisevalue {}
+ {9.1093837015\times10^{-31}}
+ \NewDocumentCommand\protonmassprecisevalue {}
+ {1.672621898\times10^{-27}}
+ \nmcMacros{ \electronmassprecisevalue,
+ \protonmassprecisevalue }
+ \nmcConstants{ m_e=\electronmassprecisevalue,
+ m_p=\protonmassprecisevalue }
+ \eval{$ m_p/m_e $}
+\end{verbatim}
+$\Longrightarrow$ \NewDocumentCommand\electronmassprecisevalue {}
+ {9.1093837015\times10^{-31}}
+\NewDocumentCommand\protonmassprecisevalue {}
+ {1.672621898\times10^{-27}}
+\nmcMacros{ \electronmassprecisevalue,
+ \protonmassprecisevalue }
+\nmcConstants{ m_e=\electronmassprecisevalue,
+ m_p=\protonmassprecisevalue }
+\eval{$ m_p/m_e $},
+
+\noindent the familiar mass ratio of proton and electron.
+
+\subsection{Viewing, counting constants}
+
+To see all constants currently `in play', use the \verb`view` setting
+in the \verb`\constants` command. The main argument can be empty,
+\begin{centred}
+\verb`\constants[view]{}` $\Longrightarrow$ \constants[view]{}
+\end{centred}
+or contain a list of constants. In the latter case, the display is
+of the above form but featuring the constants of the new list or,
+if the \verb`add` setting is used, featuring the joined lists, old
+and new:
+\begin{centred}
+\verb`\constants[view,add]{X=42}` $\Longrightarrow$ \constants[view,add]{X=42}
+\end{centred}
+To count how many constants are currently in play, star the \verb`\constants`
+command. The number will depend on whether the main argument is empty
+or not, and whether the \verb`add` setting is active:
+\begin{centred}
+\verb`\constants*{}` $\Longrightarrow$ \constants*{}.
+\end{centred}
+If the \verb`view` setting is being used at the same time as the
+star, the \verb`view` prevails.
+
+\subsection{Errors}
+
+When contemplating error messages from \verb`numerica` it needs to
+be remembered that \emph{multi-token} constants are added to the vv-list
+for every calculation. Hence an error may not be in the vv-list as
+indicated in the message but in the \verb`\constants` statement,
+specifically, the multi-token constants.
+
+\section{Saving and reusing results: \texttt{\textbackslash nmcReuse}}
+
+\label{sec:supplReuse}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` (short-name form, \verb`\reuse`)
+which saves a result to a control sequence that can then be used elsewhere
+in the document, expanding to the saved result. The control sequence
+and its content are also saved to file, allowing the possibility of
+using the result in other documents.
+
+\noindent\begin{minipage}[t]{1\columnwidth}%
+\begin{shaded}%
+The \texttt{\textbackslash nmcReuse} command in version 2 of \texttt{numerica
+}has been completely rewritten. Its use is not compatible with how
+the command was used in version 1. I found that I could bring \texttt{\textbackslash nmcReuse}
+along with \texttt{\textbackslash nmcMacros} and \texttt{\textbackslash nmcConstants}
+into the coding scheme used for \texttt{\textbackslash nmcEvaluate}
+and the reasons for doing so were too compelling. \end{shaded}%
+\end{minipage}
+
+\subsection{Use of \texttt{\textbackslash nmcReuse}}
+
+As noted, all the supplementary commands share the syntax of the \verb`\eval`
+command, so that \verb`\nmcReuse` has an optional settings argument
+preceding a mandatory main argument, followed by two trailing optional
+arguments. \verb`\nmcReuse` does not use the last two. The command
+is used mainly in two ways:
+\begin{enumerate}
+\item {\small\verb`\nmcReuse{}`}{\small , which loads the saved control
+sequences from file, if not already loaded; and}{\small\par}
+\item {\small\verb`\nmcReuse{csname}`}{\small , which loads the saved
+control sequences from file, if not already loaded, assigns the latest
+result from }{\small\verb`\eval`}{\small{} to the control sequence
+}{\small\verb`\csname`}{\small , and saves }{\small\verb`\csname`}{\small{}
+to file.}{\small\par}
+\end{enumerate}
+You may wish to put \verb`\nmcReuse{}` in the preamble of your document
+(\emph{after} \verb`\usepackage{numerica}` of course). In that way,
+saved control sequences are available from the start. Indeed, control
+sequences saved from later in the document can be used in earlier
+sections in a later \LaTeX{} run.
+
+Note that only the \emph{name}, \verb`csname`, of the control sequence
+is supplied to \verb`\reuse`, not the control sequence (\verb`\csname`).
+The name should be composed of letters only. If the name has already
+been defined in \LaTeX{} a \verb`numerica` error is produced, see
+below §\ref{subsec:reuseDeletingOverwriting:}, although if you want
+to save a \emph{new} value in a previously saved control sequence,
+that can be done without invoking a message; see §\ref{subsec:reuseDeletingOverwriting:}.
+
+Once defined {\small with a }{\small\verb`\nmcReuse{csname}`}
+command, \verb`\csname` becomes available for use elsewhere in the
+document.
+
+\subsubsection{What is saved?}
+
+What is saved is the most recent result of an \verb`\eval`-uation.
+This is the \emph{full} result. It may include the vv-list; it may
+include formatting elements; it may include math delimiters. Thus,
+using \verb`\csname` in your document (after the command \verb`\nmcReuse{csname}`)
+may not be straightforward – simply writing \verb`\csname` where
+you want the value it expands to, may produce a \LaTeX{} error and
+halt compilation. You may have to write \verb`$ \csname $` or provide
+some other math environment in order for the control sequence to display
+correctly.
+
+It can be helpful to see \emph{exactly} what has been saved; to do
+that see §\ref{subsec:reuseSeeingSaved}.
+
+\paragraph{Use of \texttt{\textbackslash eval{*}}}
+
+Users will make life simpler for themselves if they make a habit of
+using the starred form \verb`\eval*` to produce the results to save.
+\verb`\eval*` produces solely a number with no formatting or delimiters;
+even a negative result uses a hyphen for the minus sign, just as one
+would type it. In this case \verb`\csname` can be used freely in
+both text and math environments.
+
+\subsubsection{The \texttt{.nmc} file}
+
+\label{subsec:suppleReuse.nmc-file}The file that control sequences
+are saved to has a filename composed of the document name with the
+extension \verb`.nmc`. If your document is \verb`mydoc.tex` (so
+that the \LaTeX{} command {\small\verb`\jobname`} expands to
+\verb`mydoc`) then the file to which results are saved is \verb`mydoc.nmc`,
+located in the document directory.
+
+\verb`mydoc.nmc` is a comma list of pairs of the form \verb`\csname {value}`.
+Thus, the contents of \verb`mydoc.nmc` might be \texttt{\textbackslash csname1
+\{value1\},\textbackslash csname2 \{value2\},..., \textbackslash csname$n$
+\{value$n$\}}. If \verb`mydoc.nmc` does not already exist then it
+is created in the document directory, and \verb`\csname {value}`
+becomes its first element.
+
+\paragraph{Editing the \texttt{.nmc} file externally}
+
+The \verb`.nmc` file is a text file and can be edited in a text editor.
+Thus it is possible to externally add control sequences and values
+to it provided the structure of the file is strictly adhered to. It
+is also possible to delete items from it or rename control sequences
+or edit values by the same mechanism. Editing the file externally
+like this, or renaming it, or transferring items from one \verb`.nmc`
+file to another, provides a way of using saved values in multiple
+documents.
+
+\subsubsection{Messages}
+
+If a control sequence \verb`\csname` is already known to \LaTeX ,
+then writing \verb`\reuse` \verb`{csname}` will produce a \verb`numerica`
+message and the result of the latest \verb`\eval`-uation will \emph{not}
+be saved:
+\begin{verbatim}
+ \eval*{\sum_{n=1}^{10}n}\par
+ \reuse{sigma}
+\end{verbatim}
+$\Longrightarrow$ \eval*{\sum_{n=1}^{10}n} \par \reuse{sigma}
+
+If there is no result to save – perhaps an \verb`\eval`-uation produces
+an error message instead – then another message is generated:
+\begin{verbatim}
+ \eval*{1/0}\par
+ \reuse{oops}
+\end{verbatim}
+$\Longrightarrow$ \eval*{1/0}\par
+ \reuse{oops}
+
+\subsubsection{Deleting and renewing}
+
+\label{subsec:reuseDeletingOverwriting:}There may be occasions when
+you wish to change a previously saved value and yet, irritatingly,
+the control sequence name will now be known to \LaTeX{} and so will
+generate an `already known' message. If you choose a different name
+for the control sequence to save the new value to, do you want the
+old name cluttering the \verb`.nmc` file? Deleting and renewing the
+values of saved control sequences are controlled by the settings \verb`delete`
+and \verb`renew`.
+
+Entering \verb`delete` in the settings option \emph{deletes} a control
+sequence and its value from the \verb`.nmc` file and undefines it
+in \LaTeX{} terms. Thus \verb`\reuse[delete]` \verb`{csname}` would
+delete \verb`\csname` and its value from the \verb`.nmc` file and
+undefine \verb`\csname`. If \verb`\csname` is not present in the
+file, nothing happens. Entering \verb`renew` replaces the value of
+a saved control sequence with a new value. If there is no such \emph{saved}
+control sequence but the control sequence is otherwise known to \LaTeX{}
+the `already defined' message will still be generated. This prevents
+giving control sequences like \verb`\sin`~or \verb`\frac` new meanings
+with the \verb`renew` setting.
+\begin{itemize}
+\item \verb`\reuse[delete]{csname}` deletes \verb`\csname` and its value
+from the \verb`.nmc` file and from memory if present; otherwise has
+no effect;
+\item \verb`\reuse{csname}` (the default) saves the result of the latest
+\verb`\eval` command to \verb`\csname`, provided \verb`\csname`
+is not already defined; in that case a warning message is presented
+and the result is not saved;
+\item \verb`\reuse[renew]{csname}` behaves like the default mode unless
+\verb`\csname` is already a saved control sequence in the \verb`.nmc`
+file, in which case its previous value is replaced by the result of
+the latest \verb`\eval` command;
+\item if \verb`delete` and \verb`renew` are used together, whichever occurs
+second prevails.
+\end{itemize}
+In the following example, the first \verb`\reuse` deletes \verb`\suma`
+should it be present in the \verb`.nmc` file, the second saves the
+result, $55$, of the latest \verb`\eval`-uation (in fact an \verb`\eval*`-uation)
+and the third overwrites that saved value with the new value, $210$.
+\begin{verbatim}
+ \reuse[delete]{suma}
+ \eval*{\sum_{n=1}^{10}n} \par
+ \reuse{suma}
+ \eval*{\sum_{n=1}^{20}n} \par
+ \reuse[renew]{suma}
+\end{verbatim}
+$\Longrightarrow$ \reuse[delete]{suma}
+\eval*{\sum_{n=1}^{10}n} \par
+\reuse{suma}
+\eval*{\sum_{n=1}^{20}n} \par
+\reuse[renew]{suma}
+
+\subsubsection{Viewing what has been saved}
+
+\label{subsec:reuseSeeingSaved}It would be good in this example to
+see that the new value $210$ has in fact been saved. That is easy.
+Simply enter \verb`view` in the settings option of \verb`\nmcReuse`
+(I've removed the now unnecessary \verb`\par` tokens from the example.)
+\begin{verbatim}
+ \reuse[delete]{suma}
+ \eval*{\sum_{n=1}^{10}n}
+ \reuse[view]{suma}
+ \eval*{\sum_{n=1}^{20}n}
+ \reuse[renew,view]{suma}
+\end{verbatim}
+$\Longrightarrow$ \reuse[delete]{suma}
+ \eval*{\sum_{n=1}^{10}n}
+ \reuse[view]{suma}
+ \eval*{\sum_{n=1}^{20}n}
+ \reuse[renew,view]{suma}
+
+\noindent First the original value $55$ was saved to \verb`\suma`
+but then the value was overwritten by the new value $210$.
+
+The \verb`view` setting allows us to see how formatting is stored
+if the \emph{un}starred form of the \verb`\eval` command is used.
+In the following example, \verb`\eval` wraps around math delimiters:
+\begin{centred}
+\verb`\eval{$ 1+1 $} \reuse[view,renew]{two}` $\Longrightarrow$
+\eval{$ 1+1 $} \reuse[view,renew]{two}
+\end{centred}
+The full \emph{formula=result} display has been captured in \verb`\two`
+along with the math delimiters. If a vv-list is also involved, things
+become messy (but informative):
+\begin{verbatim}
+ \eval{$ x+y $}[x=1,y=2]
+ \reuse[view,renew]{three}
+\end{verbatim}
+$\Longrightarrow$ \eval{$ x+y $}[x=1,y=2]
+ \reuse[view,renew]{three}
+
+You may want to see \emph{all} saved control sequences. In that case
+use an \emph{empty} main argument: \verb`\nmcReuse[view]{}`. We now
+have enough saved control sequences to make this worthwhile:
+\begin{verbatim}
+ \reuse[view]{}
+\end{verbatim}
+$\Longrightarrow$ \reuse [view]{}
+
+\noindent (The \verb`\seven` that appears here is defined shortly.
+Its appearance \emph{before} definition is presumably due to \LaTeX{}
+making a number of passes when compiling this document.)
+
+\subsubsection{Counting saved control sequences: \texttt{\textbackslash nmcReuse{*}}}
+
+Because \verb`\nmcReuse` uses the same machinery as \verb`\nmcEvaluate`,
+it has a starred form, \verb`\nmcReuse*`, which produces a purely
+numerical result (just like \verb`\eval*`, \verb`info*`, \verb`\macros*`
+and \verb`\constants*`). In this case, the number is the count of
+how many control sequences have been saved:
+\begin{centred}
+\verb`\reuse*{}` $\Longrightarrow$ \reuse*{ }.
+\end{centred}
+
+\subsection{\texttt{reuse} setting of \texttt{\textbackslash eval} command}
+
+\label{subsec:supplReuseEvalSetting}Using \verb`\eval*` for a calculation
+ensures a purely numerical result, with no vv-list or formatting in
+the display of the result. But sometimes we might want the full display
+yet wish to save only the numerical result. This is the point of the
+\verb`reuse` setting of the \verb`\eval` command.
For the \emph{starred} form of the \verb`\eval` command it is always
\emph{only the numerical result} that is saved, whatever the value
@@ -4186,44 +5182,278 @@ reuse~=~<integer>
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;
+including the vv-list if there is one and possibly a formatting component
+(like math delimiters). Note that if the vv-list is empty, a formatting
+component (math delimiters) may still be 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).
+numerical result with no other elements of the display (no vv-list,
+no formatting component, no math delimiters).
\end{itemize}
-Thus, with the default setting (\texttt{reuse=0}) the full content
-of the display is saved:
+As we saw earlier, saving the result from \verb`\eval{$ x+y $}[x=1,y=2]`,
+corresponding to \verb`reuse=0`, means the full display is saved.
+Check by writing \verb`\three` $\Longrightarrow$ \three. The full
+display was saved (including math delimiters).
+
+On the other hand, with {\ttfamily\verb`reuse=1`} only
+the numerical value is saved:
+\begin{centred}
+\verb`\eval[reuse=1]{$ x + y $}[x=3,y=4] \reuse[renew]{seven}` $\Longrightarrow$
+\eval[reuse=1]{$ x + y $}[x=3,y=4] \reuse[renew]{seven}.
+\end{centred}
+The numerical result only of the calculation should be saved, although
+the formula and vv-list are displayed as the result of the \verb`\eval`-uation.
+We can easily check: \verb`\seven` $\Longrightarrow$ \seven. Indeed,
+only the numerical result was saved.
+
+\chapter{Nesting commands}
+
+\label{chap:Nesting} The \verb`\eval` command and the supplementary
+commands of the previous chapter can be \emph{nested} –\emph{ }used
+within other \verb`\eval` or supplementary commands. Nesting may
+occur in the main argument, or the vv-list, or the settings option,
+or some combination of all three. With the commands currently introduced,
+nesting is unlikely to be a major concern, but it becomes significant
+for the commands defined in the associated package \texttt{numerica-plus}
+(see §\ref{subsec:Related-packages}). Since those additional commands
+are not available for this document, the examples below use the commands
+introduced earlier: \verb`\eval`, \verb`\info`, \verb`\macros`,
+\verb`\constants` and \verb`\reuse`.
+
+\section{Nesting in the formula}
+
+Consider a statement like \verb`\eval{...\eval...}`. There is an
+inner \verb`\eval` and an outer \verb`\eval`. The inner \verb`\eval`
+`digests' \emph{its} \LaTeX{} formula to produce an \verb`l3fp`-readable
+expression which is fed to \verb`l3fp` to evaluate. The result is
+then returned to (the inner) \verb`\eval` to display. In version
+1 of \verb`numerica` that meant the inner command \emph{had} to be
+starred, \verb`\eval*`, so that no display formatting was fed to
+the outer command to try to digest (and cause an error).\emph{ }In
+version 2 of \verb`numerica` this is no longer the case. \verb`numerica`
+detects whether a command is inner or outer, and if inner, suppresses
+all display formatting, producing only a number, as if the command
+had been starred:
+\begin{centred}
+\verb`\eval{$ \sin(\eval{\sin x}[x=\pi/6]\pi) + 1 $}` $\Longrightarrow$
+\eval{$ \sin(\eval{\sin x}[x=\pi/6]\pi) + 1$}.
+\end{centred}
+In the presentation of the overall result, the inner \verb`\eval`
+command is evaluated, displaying as a number.
+
+In this example, the \verb`x=\pi/6` could be removed from the inner
+\verb`\eval` and placed in the vv-list of the outer command since
+outer variables are available to the inner command:
+\begin{centred}
+\verb`\eval{$ \sin(\eval{\sin x}\pi) + 1 $}[x=\pi/6]` $\Longrightarrow$
+\eval{$ \sin(\eval{\sin x}\pi) + 1$}[x=\pi/6].
+\end{centred}
+Just to show that it is possible, the next example shows \verb`\eval`
+being used in a \verb`\constants` command. The \verb`o` setting
+in the \verb`\constants` command pervades its argument; hence it
+needs to be explicitly turned off for the \verb`\eval` if \verb`\sin(\pi/6)`
+is to evaluate as expected.
+\begin{verbatim}
+ \constants[o]{ y=\sin 30,x=\eval[o=0]{\sin(\pi/6)} }
+ \eval{$ x+y $}
+\end{verbatim}
+$\Longrightarrow$ \constants[o]{ y=\sin 30,x=\eval[o=0]{\sin(\pi/6)} }
+ \eval{$ x+y $}.
+
+\subsection{Math delimiters and double evaluations}
+
+Any math delimiters in the inner \verb`\eval` are ignored. (This
+also differs from version 1 of \verb`numerica` where they caused
+an error.) Obviously it is simpler to omit them as I have done in
+the examples.
+
+However, math delimiters in the \emph{outer} \verb`\eval` command
+still have their normal effect and produce a \emph{formula = result,
+(vv-list)} display. One consequence of such a display is that the
+formula in the \emph{inner} \verb`\eval` command is evaluated \emph{twice}
+– once when the overall result is being calculated (i.e. the formula
+of the \emph{outer} \verb`\eval`) and later when the overall display
+of the result is created. In the \emph{formula} part of the \emph{formula
+= result, {[}vv-list{]}} display, the tokens in the \emph{formula}
+are expanded to their display form. For example, \verb`\sin` is expanded
+to $\sin$, \verb`\pi` is expanded to $\pi$ – and the inner \verb`\eval`
+is expanded to the numerical result of its evaluation – a second evaluation.
+If the inner formula is simple, this will be of little moment, but
+should the inner formula contain, say, a slowly converging infinite
+series, then evaluating it twice is a bad idea and it would be better
+to remove the delimiters from the outer \verb`\eval`. That prevents
+the second evaluation.
+
+The problem does not arise if the outer \verb`\eval` lies within
+a math environment (e.g. \verb`$ \eval{...} $`) since that produces
+a display of the form \emph{result, {[}vv-list{]}.} The formula is
+not displayed and so the second evaluation does not occur. The inner
+\verb`\eval` is evaluated once only to calculate the result.
+
+\section{Nesting in the vv-list}
+
+The inner \verb`\eval` can be placed in the vv-list of the outer
+command. If the vv-list of the inner \verb`\eval` contains a comma
+(meaning there are at least two variables), then the entire inner
+\verb`\eval` and its \LaTeX{} arguments needs to be wrapped in braces
+to hide the comma or commas of its vv-list from the outer \verb`\eval`.
+To show the effect of not doing so, I have slightly complicated the
+previous example by adding a second (unnecessary) variable. The first
+example is with braces, the second without:
\begin{centred}
-\verb`\eval{$ x + y $}[x=1,y=1] \reuse[iandi]` $\Longrightarrow$
-\eval{$ x + y $}[x=1,y=1] \reuse[iandi].
+\verb`\eval{$ \sin k\pi + 1 $}[k={\eval{y\sin x}[x=\pi/6,y=1]}]`
+$\Longrightarrow$ \eval{$ \sin k\pi + 1 $} [k={\eval{y\sin x}[x=\pi/6,y=1]}].
+
+\verb`\eval{$ \sin k\pi + 1 $}[k=\eval{y\sin x}[x=\pi/6,y=1]]` $\Longrightarrow$
+\eval{$ \sin k\pi + 1 $}[k=\eval{y\sin x}[x=\pi/6,y=1]].
\end{centred}
-To check that this is the case, \verb`\reuse \iandi` $\Longrightarrow$
-\reuse \iandi .
+The vv-list of the outer \verb`\eval` is parsed as containing two
+entries, \verb`k=\eval` \verb`{y\sin x}[x=\pi/6` and \verb`y=1]`.
+Both will cause errors but since the vv-list is evaluated from the
+right, it is \verb`y=1]` which actually does so.
+
+\section{Nesting in the settings option}
-On the other hand, with \texttt{reuse=1} only the numerical value
-is saved:
+This will be rare, but commands can occur in the settings option of
+the outer command. The \verb`\info` command provides a good example.
+I have included it in the punctuation setting of an \verb`\eval`-uation.
+\begin{verbatim}
+ \eval[p=\mbox{,\qquad\info{sum} terms.}]
+ {\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+\end{verbatim}
+$\Longrightarrow$ \eval[p=\mbox{,\qquad\info{sum} terms.}]{\[ \sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!} \]}[3]
+
+Because of the \verb`\[ \]` math delimiters, if the \verb`\info`
+command had been placed \emph{after} the \texttt{\textbackslash eval}
+command, it would have slid down to the next line. Used in the settings,
+as here, the display is \emph{inside} the \verb`\[ \]` delimiters,
+on the same line as the expression. This may be significant for adjusting
+vertical spacing of later parts of the document – widow and orphan
+control for instance.
+
+A point to note is the explicit writing of the `terms' descriptor.
+Normally \verb`\info{sum}` would automatically supply the descriptor,
+but as noted earlier, nesting of one command in another suppresses
+all elements of display of the inner command beyond the numerical
+result. It is as if the inner command is starred. Because the \verb`\info`
+command is nested in the \verb`\eval` command, the `terms' descriptor
+is suppressed and has had to be explicitly supplied by hand.
+
+\section{Rounding and display}
+
+In the display of the overall result, it is the result of the inner
+command which is shown, not the formula that the inner command acts
+on. How that number is displayed is determined by the number-format
+specification of the \emph{inner} command. Note however that this
+specification affects only how the result of the inner command is
+shown. Always 16 figures are passed from the inner command to the
+outer, as you can see in this example:
+\begin{verbatim}
+ \eval{$ \pi - \eval{ \pi }[4] $}[15]
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \pi - \eval{ \pi }[4] $}[15].
+
+\noindent The outer result would not be zero to $15$ places of decimals
+if the inner result were restricted to $4$ decimal places. It is
+only the \emph{display} of the inner result which is so restricted.
+
+For infinite sums and products (and for \verb`\nmcIterate` and \verb`\nmcSolve`
+of the \verb`numerica-plus` package), the rounding value is not just
+for display purposes but is also used to determine the result. This
+may require judicious use of the extra rounding setting to get a sensible
+display. In the first instance below, the second sum stops at an effective
+rounding value of $5+2=7$, since the default extra rounding is $+2$,
+and the first sum (the inner one) also stops at $7=4+3$. No surprise
+then that the overall result is $0$. In the second instance, the
+inner sum stops at a rounding value of $4+2=6$. Although the left-hand
+side of the display is unaltered, the result is no longer $0$.
+\begin{verbatim}
+ \eval{$ \eval[S+=3]{\sum_{n=1}^\infty 1/n^3}[4*]
+ - \sum_{n=1}^\infty 1/n^3$}[5]
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \eval[S+=3]{\sum_{n=1}^\infty 1/n^3}[4*]
+ - \sum_{n=1}^\infty 1/n^3$}[5]
+
+\noindent whereas
+\begin{verbatim}
+ \eval{$ \eval[S+=2]{\sum_{n=1}^\infty 1/n^3}[4*]
+ - \sum_{n=1}^\infty 1/n^3$}[5]
+\end{verbatim}
+$\Longrightarrow$ \eval{$ \eval[S+=2]{\sum_{n=1}^\infty 1/n^3}[4*]
+ - \sum_{n=1}^\infty 1/n^3$}[5].
+
+\section{Error messages}
+
+Errors in an inner command create a small change in error message
+display.
+\begin{centred}
+\verb`\eval{ 1 + \eval{ 1 + \eval{ k } } }` $\Longrightarrow$ \eval{ 1 + \eval{ 1 + \eval{ k } } }
+
+\verb`\eval{ x + \eval{ k }[k=\arcsin 2] }[x=1]` $\Longrightarrow$
+\eval{ x + \eval{ k }[k=\arcsin 2] }[x=1]
+\end{centred}
+An integer is added to the `where' part of the error message. The
+integer indicates the \emph{level of nesting} where the error occurs.
+
+If there is no nesting where the error occurs, the integer is suppressed,
+even though there may be nesting elsewhere in the overall expression.
+This is in the interests of straightforwardness when nesting is absent,
+which will be overwhelmingly the most common situation.
\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],
+\verb`\eval{ k + \eval{ x }[x=1] }[k=\arcsin 2]` $\Longrightarrow$
+\eval{ k + \eval{ x }[x=1] }[k=\arcsin 2]
\end{centred}
-which we can check here: \verb`\reuse\iandi` $\Longrightarrow$ \reuse\iandi .
-\subsubsection{\texttt{\textbackslash reuse} in the preamble}
+\section{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=1]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1 $}` $\Longrightarrow$
+\eval[dbg=210]{$ \sin \eval*{\sin x}[x=\pi/6]\pi + 1$}
+\end{centred}
+There is no vv-list for the outer command whence the two empty slots
+in the display but when the inner \verb`\eval` is in the vv-list,
+they are filled:
+\begin{centred}
+\verb`\eval[dbg=1]{$ \sin k\pi + 1 $}[k=\eval*{\sin x}[x=\pi/6]]`
+$\Longrightarrow$ \eval[dbg=1]{$ \sin k\pi + 1$} [k=\eval*{\sin x}[x=\pi/6]]
+\end{centred}
+For the inner \verb`\eval` command debugging may still work but in
+an idiosyncratic way. To clarify exactly what is going on I have added
+a \verb`\left( \right)` pair around the entire inner \verb`\eval`
+command. Note that I have also used a \emph{negative} \texttt{dbg}
+value. With a positive value, the right parenthesis is pressed toward
+the right margin of the page. The negative value limits the display
+to the text width and gives the much neater result shown.
+\begin{verbatim}
+ \eval[()=2]{$
+ \sin\left(
+ \eval*[dbg=-1]{ \sin x }[x=\pi/6]
+ \right)\pi + 1 $}
+\end{verbatim}
+$\Longrightarrow$ \eval[()=2]{$
+ \sin\left(
+ \eval*[dbg=-1]{ \sin x }[x=\pi/6]
+ \right)\pi + 1 $}
-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}`).
+\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 despite the potential for some
+rather odd displays, since there can be no confusion about which \verb`\eval`
+command is being `debugged'. 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{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
+enables snippets from a 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{}
@@ -4231,49 +5461,119 @@ 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`,
+then click \textsf{OK}. You may wish to follow the above line in the
+preamble with \verb`\nmcReuse{}`:
\begin{lyxcode}
\textbackslash usepackage{[}lyx{]}\{numerica\}
-\textbackslash nmcReuse
+\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.
+In that case, type the extra line 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.
+The instant preview facility of \LyX{} performs 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 mini-\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
+\& Feel \lyxarrow{} Display}, ensuring that the \textsf{Display graphics}
+checkbox is checked, and against \textsf{Instant preview} selecting
\textsf{On}, then clicking \textsf{OK}.
-\subsection{Conflict with hyperref support?}
+\subsection{Document location}
+
+It also matters where your document is located. You may have your
+own local or personal texmf tree (see §\ref{subsec:settingsPersonal-texmf-tree}).
+If your document is located there, perhaps in the \verb`doc` folder,
+then not all features of preview will work as expected. Presumably
+this is because both \LyX{} and your \LaTeX{} distribution (e.g. \TeX Live
+or MiK\TeX ) are interacting with the location and interfere. Move
+your document to another location which your \LaTeX{} distribution
+has no interest in, and open it in \LyX{} there.
+
+\subsection{Global vs local previewing}
+
+\label{subsec:LyXGlobal-vs-local}Compilation of previews occurs in
+two distinct modes.
+
+\paragraph{Global preview generation:}
+
+When a document is opened (and preview is \emph{on}), all previews
+in the document are formed in sequence in the one \LaTeX{} run. This
+is the global mode. The mini-\LaTeX{} run may well be substantial.
+It compiles a \verb`.tex` file that begins with the document's preamble
+with some additions then comes \verb`\begin{document}`. That is followed
+by a sequence of preview environments,
+\begin{lyxcode}
+\textbackslash begin\{preview\}
-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.
+<stuff>
+
+\textbackslash end\{preview\}
+\end{lyxcode}
+one for each preview in the document. Finally there is an \verb`\end{document}`
+statement. The critical point is that all previews are between the
+same \verb`\begin{document}`, \verb`\end{document}` statements,
+and so earlier previews in the sequence can communicate with later
+ones.
+
+\paragraph{Local preview generation:}
+
+The other mode in which preview operates is local. Suppose you have
+your document open and want to add to it, for instance with a simple
+evaluation, \verb`\eval{x+y}[x=1,y=2]` in an ERT inset in a preview
+inset. The resulting mini-\LaTeX{} run is of the form
+\begin{lyxcode}
+<preamble>
+
+\textbackslash begin\{document\}
+
+\textbackslash begin\{preview\}
+
+\textbackslash eval\{x+y\}{[}x=1,y=2{]}
+
+\textbackslash end\{preview\}
+
+\textbackslash end\{document\}
+\end{lyxcode}
+The preamble is as before but there is only \emph{one} preview between
+the \verb`\begin{document}`, \verb`\end{document}` statements. That
+preview is isolated from all other, previous previews and will be
+isolated from all other, later previews.
+
+This has implications for the supplementary commands of the previous
+chapter and means that if you want to transfer information (a macro,
+a constant, a result) from one preview to another, you need to do
+it through the preamble or by means of an external file or, in some
+cases, by forcing a global preview run in which all previews are recompiled
+between the same \verb`\begin{document}`, \verb`\end{document}`
+statements.
+
+\subsubsection{Forcing a global preview run}
+
+Closing then opening a document is one way to force a global preview
+compilation. Another is to change the zoom level. This causes \LyX{}
+to recompile all previews at the new zoom level. But you may not want
+to work at the new zoom level. Going back to the old zoom level will
+force a second recompilation of all previews. For a large document
+\emph{two} recompilations is too heavy a burden. The secret is to
+combine a zoom in and a zoom out into one command and attach it to
+a shortcut.
+
+If you go to \textsf{Tools \lyxarrow{} Preferences \lyxarrow{} Editing
+\lyxarrow{} Shortcuts}, click on the \textsf{New} button and enter
+\begin{lyxcode}
+command-sequence~buffer-zoom-in;~buffer-zoom-out
+\end{lyxcode}
+then assign a shortcut to it (\verb`Alt+Z` for zoom?) you will gain
+a simple means of forcing a global recompilation of previews.
\section{Mathed}
@@ -4295,22 +5595,23 @@ The content of mathed is immediately displayed and can be edited.
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.
+by contrast, \LyX{} supplies these automatically in the form of blue-outlined
+boxes.) 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{ \{}.
+typing a backslash \textsf{\textbackslash{} }then a left brace\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}
+\section{Preview insets}
There are problems with using mathed for calculations.
\begin{itemize}
@@ -4318,13 +5619,14 @@ There are problems with using mathed for calculations.
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.
+In mathed the only way to effect such a display 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.
+settings option content and the vv-list, often extending well beyond
+the edge of the screen.
\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
@@ -4334,14 +5636,15 @@ 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'.
+which means holding the \textsf{Ctrl} key down and tapping two diagonally
+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
@@ -4355,18 +5658,22 @@ 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}
+Instant preview will display error messages generated by \verb`numerica`
+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.
+
+\subsection{Temporary directory of \protect\LyX}
+
+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
@@ -4379,12 +5686,13 @@ 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.
+\texttt{.dvi}, \texttt{.tex}, and – depending on the number of previews
+in your document – a number, perhaps a lot, of image files with the
+extension \texttt{.png}, each one of which is 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
@@ -4394,14 +5702,14 @@ 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}
+It is possible when a preview stalls 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
@@ -4409,46 +5717,142 @@ for the relevant name in the list, select it, and end the process
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
+\section{Hyperref support vs speed}
+
+If you want the pdf produced from your document to support hyperref
+links and show an outline window in your pdf viewer (generally placed
+on the left in the viewer) then you need to ensure the checkbox at
+\textsf{Document Settings \lyxarrow{} PDF Properties \lyxarrow{} Use
+Hyperref Support} is indeed checked. But you don't need to do this
+until the final compilation of the document. The advantage of leaving
+this until the last is that in a large document with many previews
+the time for preview generation is essentially halved. If hyperref
+support is enabled, preview generation not only creates all the individual
+image files that are the previews (files of extension \verb`.png`)
+but also requires the compilation of a single pdf document showing
+all the previews in sequence. (Like the previews, the pdf document
+`hides' in the termporary directory where \LyX{} does its work.)
+In other words, \emph{two} images are created for each preview, the
+\verb`.png` image which is the one \LyX{} displays, and another image
+buried inside the pdf of all images. That second step does not occur
+if hyperref support is disabled. In a small document, this is not
+going to matter; in a large document it becomes significant. It is
+well worth temporarily turning off hyperref support and then, when
+the time for final compilation comes, turning it back on.
+
+\section{Supplementary commands in \protect\LyX}
+
+There are some difficulties using the supplementary commands successfully
+with instant preview.
+
+\subsection{Reuse of earlier previews}
+
+One is that whenever \LyX{} has generated a preview image for a particular
+\LaTeX{} expression, it will use that same image whenever it meets
+that same \LaTeX{} expression later. That means that a statement like
+\verb`\macros[view]{}` and the same statement later will display
+the same image, even though there may have been macros defined or
+freed in between. The same goes for all the other supplementary functions,
+including, for example, \verb`\info{sum}`. A second instance of \verb`\info{sum}`
+will display the image generated by the first instance even though
+further infinite sums may have been evaluated between the \verb`\info`
+statements.
+
+The remedy is to make some small but insignificant difference to the
+\LaTeX{} expression in the second instance – generally a change in
+white space will do. For example: first time \verb`\macros[view]{}`,
+second time \verb`\macros[view]{ }` where a space has been inserted
+between the braces; or: first time \verb`\info{sum}`, second time
+\verb`\info{ sum}` where a space has been inserted before \verb`sum`.
+This will ensure \LyX{} doesn't fall back on the previously generated
+image.
+
+\subsection{\textquoteleft Stalled\textquoteright{} previews}
+
+It is possible to put content into an ERT inset inside a preview inset
+(\textsf{Ctrl+P Ctrl+L}) and for nothing to happen. The preview has
+apparently stalled. Certainly this can be the case if there is an
+error in the input (e.g. a missing brace) but it also occurs if there
+is no output to display. For instance \verb`\constants{ c=300000000 }`
+does not produce any visual output. There is nothing for the preview
+to display and so the preview inset sits there, apparently stalled.
+This is a security measure for previews in \LyX{} to provide at least
+some guard against malicious code being run in the preview. If the
+preview resolved, it would disappear completely from view in the \LyX{}
+window.
+
+If you find the visual appearance of such apparently stalled previews
+distracting, the addition of some displayable content to the preview
+will result in it resolving to that content; the content could be
+as small as a full stop.
+
+\subsection{Using \texttt{\textbackslash nmcMacros}}
+
+As noted earlier, previews are mini-\LaTeX{} runs, either local or
+global. Each local preview is of the form<preamble>
\begin{lyxcode}
-\textbackslash usepackage{[}lyx{]}\{numerica\}
+\textbackslash begin\{document\}
+
+\textbackslash begin\{preview\}
-\textbackslash nmcReuse
+<stuff>
+
+\textbackslash end\{preview\}
+
+\textbackslash end\{document\}
\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.
+Whatever goes into or comes out of the preview is isolated from any
+other local preview, unless it is through the preamble or an external
+file. Sometimes a global preview run can overcome this problem for
+then all the previews lie between the same \verb`\begin{document}`,
+\verb`end{document}` statements. However, this does not help with
+macro definitions. \verb`\def`, \verb`\newcommand`, \verb`\NewDocumentCommand`
+all provide \emph{local} definitions which remain trapped within their
+own \verb`\begin{preview}`, \verb`\end{preview}`) statements. Another
+preview, say containing an \verb`\eval` command, between a different
+pair of \verb`\begin{preview}`, \verb`\end{preview}`) statements,
+will not know about the macro definition.
+
+There are (at least) three ways out:
+\begin{enumerate}
+\item Confine everything to the same preview inset: the definition of a
+macro, the \verb`\macros` statement, and the use of the macro in
+an \verb`\eval` command.
+\item Confine macro definitions to the preamble (\textsf{Document \lyxarrow{}
+Settings \lyxarrow{} \LaTeX{} Preamble}).
+\item Within previews use \verb`\gdef` (or \verb`\global\def`) exclusively
+for making your macro definitions; this makes the macro available
+to all later previews.
+\end{enumerate}
+
+\subsection{Using \texttt{\textbackslash nmcConstants}}
+
+Because \verb`\nmcConstants` doesn't use \verb`\def` or \verb`\newcommand`
+or \verb`\NewDocumentCommand` it is not subject to the same localisation
+problem as \verb`\nmcMacros`, but the reach of a \verb`\constants`
+command will still be confined to its own preview unless a \emph{global}
+preview run is forced; see above §\ref{subsec:LyXGlobal-vs-local}.
+
+\subsection{Using \texttt{\textbackslash nmcReuse}}
+
+As noted earlier, \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
+do so as described earlier (§\ref{sec:supplReuse}), but the file
+\texttt{mydoc.nmc} containing the saved results will be located in
+the temporary directory. When \LyX{} is closed the file will be deleted
+along with all the other contents of that directory.
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:
+the temporary directory and into the document 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}
@@ -4458,7 +5862,7 @@ 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).
+with \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
@@ -4494,7 +5898,7 @@ 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?}
+\subsubsection{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
@@ -4511,33 +5915,38 @@ 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.
+\LyX{} 2.4.0}.\footnote{Due for release in 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.
+
+Note that in the other direction, the \verb`.nmc` file in your document
+directory is \emph{automatically} copied to the temporary directory
+when needed. Nothing needs to be done by you, the user.
-\section{Using \protect\LyX{} notes}
+\subsubsection{Use of \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.
+the result saved using \verb`\nmcReuse` within the same note. 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)
+\item \texttt{\textbackslash nmcInfo, \textbackslash info}
+\item \texttt{\textbackslash nmcMacros, \textbackslash macros}
+\item \texttt{\textbackslash nmcConstants, \textbackslash constants}
+\item \texttt{\textbackslash nmcReuse, \textbackslash reuse}
\end{enumerate}
Provided they have not already been defined when \texttt{numerica}
is loaded, the following commands are defined in \texttt{numerica}
@@ -4578,7 +5987,7 @@ list)
\item constants
\begin{enumerate}
\item \texttt{e, \textbackslash pi, \textbackslash gamma, \textbackslash phi,
-\textbackslash deg, \textbackslash infty }(sometimes)
+\textbackslash deg, \textbackslash infty}
\end{enumerate}
\item arithmetic operators
\begin{enumerate}
@@ -4661,11 +6070,17 @@ arccsc, \textbackslash arcsec, \textbackslash arccot }
\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 begin\{\}, \textbackslash end\{\}, \$, \textbackslash{[},
+\textbackslash{]}}
+\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 thinspace, \textbackslash medspace, \textbackslash thickspace,}
+\item \textbackslash\texttt{negthinspace, \textbackslash negmedspace,
+\textbackslash negthickspace,}
+\item \textbackslash\texttt{hspace{*}\{\}, \textbackslash mspace\{\},}
+\item \texttt{\textbackslash quad, \textbackslash qquad , \textbackslash hfill,
+\textbackslash hfil}
\item \texttt{\textbackslash phantom\{\}, \textbackslash vphantom\{\},
\textbackslash hphantom\{\}}
\item \texttt{\textbackslash xmathstrut{[}{]}\{\}} \texttt{, \textbackslash splitfrac\{\}\{\},
@@ -4673,15 +6088,16 @@ arccsc, \textbackslash arcsec, \textbackslash arccot }
\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\{\}}
+\textbackslash text\{\}, \textbackslash mbox\{\}, \textbackslash smash\{\}}
+\item \texttt{\textbackslash color{[}{]}\{\}, \textbackslash textcolor{[}{]}\{\}\{\}}
\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\{\}}
+\textbackslash mathsf\{\}, \textbackslash mathfrak\{\}, \textbackslash mathscr\{\}}
+\item \texttt{\textbackslash mathnormal\{\}, \textbackslash boldsymbol\{\}}
+\item \texttt{\textbackslash textrm, \textbackslash textsf, \textbackslash texttt}
\end{enumerate}
\end{enumerate}
@@ -4689,17 +6105,19 @@ arccsc, \textbackslash arcsec, \textbackslash arccot }
\subsection{Available \texttt{\textbackslash nmcEvaluate} settings}
\begin{center}
-\begin{tabular}{ll>{\raggedright}p{4cm}>{\raggedright}p{4cm}}
+\begin{tabular}[t]{>{\raggedright}p{1.5cm}l>{\raggedright}m{4cm}>{\raggedright}m{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{view}} & & {\small equivalent to }{\small\texttt{dbg=1}} & \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{o}} & {\small int (0/1)} & {\small degree switch for trig. funcions} & {\small\texttt{1}}\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{vv@}} & {\small int (0/1)} & {\small vv-list calculation mode} & {\small\texttt{0}}\tabularnewline
+{\small\texttt{vvmode}} & {\small int (0/1)} & {\small equivalent to }{\small\texttt{vv@}} & {\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{}
@@ -4714,6 +6132,28 @@ in }{\small\texttt{vvd}}{\small )} & \tabularnewline
\end{tabular}
\par\end{center}
+\subsection{Available settings for supplementary commands}
+
+All settings for \verb`\nmcEvaluate`, the \verb`view` setting in
+particular (although most will be irrelevant), plus for
+\begin{itemize}
+\item \verb`\nmcMacros`
+\begin{itemize}
+\item \verb`free` `deregister' a macro from \verb`numerica`
+\end{itemize}
+\item \verb`\nmcConstants`
+\begin{itemize}
+\item \verb`add` add the new list of constants to the current one
+\end{itemize}
+\item \verb`\nmcReuse`
+\begin{itemize}
+\item \verb`delete` remove the listed control sequences from the \verb`.nmc`
+file
+\item \verb`renew` change the value of a control sequence in the \verb`.nmc`
+file
+\end{itemize}
+\end{itemize}
+
\subsection{Available configuration file settings}
\begin{center}
\bigskip{}
@@ -4728,16 +6168,17 @@ key & default\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{use-degrees}} & {\small\texttt{0}}\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{vv-inline}} & {\small\texttt{\{,\}\textbackslash mskip 12mu 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}
+\end{tabular}
\par\end{center}
\end{document}