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\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}
