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\usepackage{numerica}
\usepackage{numerica-tables}
\usepackage{numerica-plus}
\newcommand\rel{\,\varrho\;}
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\DeclareMathOperator{\gd}{gd}
\usepackage{upquote}
\usepackage{xfrac}
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initsep=10.25pt plus 5pt minus 3pt]{idxlayout}
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\begin{document}
\title{\texttt{numerica-tables}~\\
{\large version 3.1.0}}
\author{Andrew Parsloe\\
(\url{ajparsloe@gmail.com})}
\maketitle
\begin{abstract}
The \texttt{numerica-tables} package defines a command \verb`\nmcTabulate`
which enables the creation of multi-column tables of mathematical
function values. Key\textendash value assignments allow presentation
in a wide variety of table styles within the `formal table' framework
of the \texttt{booktabs} package. \texttt{numerica-tables} requires
the \texttt{numerica} package to be loaded.\medskip{}
\end{abstract}
\noindent{}%
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%
\begin{itemize}
\item This document applies to version 3.1.0 of \texttt{numerica-tables}.
\item Version 3 of \texttt{numerica} needs to be loaded before \texttt{numerica-tables};
(\texttt{numerica} requires \texttt{amsmath}, \texttt{mathtools} and
a reasonably recent \LaTeX{} system).
\item The \texttt{booktabs} package is required.
\item I refer many times in this document to \emph{Handbook of Mathematical
Functions}, edited by Milton Abramowitz and Irene A. Stegun, Dover,
1965, abbreviated to \emph{HMF}, and often followed by a reference,
like Table 1.2, to a specific table.
\item Version 3.1.0 of \texttt{numerica-tables}
\begin{itemize}
\item adds an index to the documentation;
\item adds the ability to round table entries to different values depending
on row or column (or both) in the table;
\item fixes two minor presentation bugs.
\end{itemize}
\item Version 3.0.0 of \texttt{numerica-tables}
\begin{itemize}
\item adds the ability to use numbers or expressions from
\begin{itemize}
\item a comma list for row variable values, either as values or verbatim;
\item a macro containing a comma list for row variable values, either as
values or verbatim;
\item a file for row variable values, either as values or verbatim;
\end{itemize}
\item adds the ability to use functions of a stepped variable to generate
varyingly stepped row variable values;
\item adds the ability to suppress the header row;
\item is compatible with the additional features of \texttt{numerica} version
3.0.0,
\begin{itemize}
\item including the decimal comma if the \verb`comma` package option is
used with \texttt{numerica}; and
\item fraction-form row variables, or fraction-form output if the \verb`/`
or \verb`//` `triggers' are used in the number-format option;
\end{itemize}
\item amends and adds to documentation.
\end{itemize}
\end{itemize}
\end{shaded}%
\end{minipage}

\tableofcontents{}

\chapter{Introduction}

Entering 
\begin{verbatim}
    \usepackage[<options>]{numerica}
    \usepackage{numerica-tables}
\end{verbatim}
\noindent in the preamble of a document gives access to a command
for creating tables of function values in a wide variety of styles.
Contrary to previous practice, for version 3.0.0 of \texttt{numerica-tables},
the \texttt{numerica} package is not loaded automatically but must
be loaded explicitly (as above), with options if desired, \emph{before}
\texttt{numerica-tables}. It is \emph{essential} that the version
of \texttt{numerica} loaded is version 3. 

All tables are `formal tables' in the sense of the \texttt{booktabs}\index{booktabs@\texttt{booktabs}}
package, which is loaded automatically. Such tables have no vertical
rules\index{rules} and few horizontal rules.

\section{Table structure}

I\index{table structure} take as my source of models of mathematical
tables those presented in \emph{Handbook of Mathematical Functions},
edited by Milton Abramowitz and Irene A. Stegun, Dover, 1965, not
because the typesetting is elegant (it often isn't) but because \emph{HMF}
displays a wide variety of table styles. The editors of that volume
were faced with a host of different problems requiring a host of different
solutions. The \verb`\nmcTabulate` command of \texttt{numerica-tables}
aims to reproduce most of those different solutions, within \texttt{booktabs}
\index{booktabs@\texttt{booktabs}} elegance.

To create a table we need to specify a function or functions to tabulate.
The values a function takes will generally depend on a primary parameter
and, possibly, a number of secondary parameters (which is where much
of the complexity comes from). Mathematical tables are structured
in \emph{columns}. We (nearly always) read \emph{down} a column as
the primary parameter is incremented, generally in regular steps.
We need to decide on the range of values the primary parameter will
take and how fine-grained the tabulation will be \textendash{} what
the step size of its increments will be. Assigning different values
to a second parameter generates a second, third,\ldots{} column. Sometimes
rather than a second parameter, a second, third, \ldots{} function
of the first parameter is tabulated in the successive columns. 

In this document the first parameter is called the \emph{row variable}\index{row variable}
\textendash{} its value determines which row we are in; the second
parameter, if present, is called the \emph{column variable\index{column variable}}
\emph{\textendash{} }its value determines which column we are in.
A table generally (but not always) presents the values of the row
variable in the first column, the \emph{row variable column}, sometimes
in distinctive type (e.g. bolded). The values of the column variable
are presented in a \emph{header row} above the table body of function
values. Above the header row there may be a \emph{title row} and perhaps
a \emph{subtitle row} where other explanatory material can be displayed.
Sometimes there is a \emph{footer row} beneath the table body. Vertical
rules are absent, horizontal rules used sparingly \textendash{} for
example, at the top and bottom of the table, or under the header row,
but not in the body of the table.

\section{Shared syntax}

The \verb`\nmcTabulate`\index{nmcTabulate@\texttt{\textbackslash nmcTabulate}}
command (short-name form \verb`\tabulate`\index{tabulate@\texttt{\textbackslash tabulate}|see{\texttt{\textbackslash nmcTabulate}}})
shares the syntax of \verb`\nmcEvaluate` (see \texttt{numerica.pdf}).
When all options are used the command looks like 
\begin{verbatim}
    \nmcTabulate*[settings]{expr.}[vv-list][num. format]
\end{verbatim}
\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; see §\ref{sec:Star-option};
\item \verb`[settings]` comma-separated list of \emph{key=value }settings;
this option is at the heart of creating a table of function values;
see Chapter~\ref{chap:=00005CnmcTabulate-settings};
\item \verb`{expr.}` mandatory argument specifying the mathematical expression
or expressions in \LaTeX{} form to be tabulated;
\item \verb`[vv-list]` comma-separated list (or semicolon-separated list
if the \verb`comma` package option is used with \texttt{numerica})
of \emph{variable=value }items, in particular containing the initial
value of the row variable, and column variable if one is used; 
\item \verb`[num. format]` optional format specification for presentation
of the numerical results (rounding, padding with zeros, scientific
notation, fraction-form output); see §\ref{subsec:Trailing-optional-argument}.\index{rounding value}
\end{enumerate}
Unlike \verb`\nmcEvaluate` (the main command in\texttt{ numerica}),
for \verb`\nmcTabulate` the two apparently optional arguments straddling
the main argument (\verb`settings` and \verb`vv-list`) are \emph{essential}.
Although both are delimited by square brackets, that is in order to
draw on the {\ttfamily\verb`numerica`} code. Each argument
contains items \emph{necessary} for the construction of any table
of function values.

Should \texttt{numerica} be loaded with the \verb`comma` package
option, numbers in tables will be displayed with a decimal comma\index{decimal comma}.
In this case items in the vv-list\index{vv-list!and decimal comma}
\emph{must} be separated by a semicolon\index{semicolon and decimal comma}
since the decimal comma is likely to feature there. Similarly, $n$-ary
functions \textendash{} \verb`\max`, \verb`\min` and \verb`\gcd`\index{n-ary@$n$-ary functions}
\textendash{} must use the semicolon as their argument separator.

Math environments are significant for \verb`\nmcEvaluate`. They are
essentially irrelevant for \verb`\nmcTabulate`\index{nmcTabulate@\texttt{\textbackslash nmcTabulate}}\index{environments},
although an external environment may determine how a table is positioned
on the page\index{table placement!on the page} (\verb`\[ <table> \]`
for instance will centre the table between the margins). Environments
specified in the main argument (the formula) should be avoided. \verb`$ $`,
\verb`\( \)` or \verb`\[ \]` will not cause error, but a \verb`\begin`\textendash \verb`\end`
environment there will.

Just as the syntax is shared, so the settings available to the command
\verb`\nmcEvaluate` in \texttt{numerica} are also available to \verb`\nmcTabulate`,
although not all will be relevant. See the associated document \texttt{numerica.pdf}
for a list of these and associated discussion. I will point out instances
of their use in the following examples.

\chapter{\texttt{\textbackslash nmcTabulate} settings}

\label{chap:=00005CnmcTabulate-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. For the main discussion of row variable settings, see §\ref{sec:Row-variable-settings};
for column variable settings see §\ref{sec:Column-variable-settings};
for whole-of-table formatting see §\ref{sec:Whole-of-table-formatting};
for formatting the function values in table cells see §\ref{sec:Function-value-formatting}.

\section{Row-variable settings}

\label{sec:Row-variable-settings}

\subsection{Row-variable specification: uniform case}

\label{subsec:Row-var-spec-uniform}
\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}{{\small use only one of rstop or rows}}\tabularnewline
{\small\texttt{rows}} & {\small int} & {\small number of rows} & \tabularnewline
{\small\texttt{rspec}} & {\small comma list} & {\small\texttt{\{rvar}}{\small , }{\small\texttt{step}}{\small , }{\small\texttt{rows\}}} & {\small short form spec.}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}
Deciding\index{row variable!start, step, stop values|(} on a function
to tabulate (entered in the main or mandatory argument of \verb`\nmcTabulate`)
will inevitably also mean deciding on the tabulation variable, the
\emph{row} variable, \verb`rvar`\index{rvar@\texttt{rvar}}, what
value to start tabulating from \textendash{} \emph{that} is specified
in the vv-list\index{vv-list!row/col. variable initial values}\index{row variable!initial value in vv-list}
and so does not need a specific key \textendash{} what value to tabulate
to, \verb`rstop`\index{rstop@\texttt{rstop}}, and how fine-grained
the tabulation is to be, the step size \verb`rstep`\index{rstep@\texttt{rstep}}.
In the uniform case (which makes up the overwhelming majority of cases
in \emph{HMF} for instance) the step size is constant. It does not
change as the value of the row variable changes. (The non-uniform
case, available from version 3.0.0 of \texttt{numerica-tables}, is
discussed in §\ref{subsec:Row-var-spec-nonuniform} below. Quite different
keys are required.)

The two tables in the example below tabulate $\sin x$ and $\cos x$
between $0$ and $1$ in increments of $0.2$. Note that the start
value of the tabulation variable is entered in the vv-list. The reason
for placing it there is that for more complicated functions other
parameters in the function and therefore in the vv-list may depend
on the row variable. Although it will often be the first entry in
the vv-list, it does not need to be. The initial value of the row
variable may depend on other quantities which must necessarily precede
it \textendash{} lie to the right of it \textendash{} in the list.

In the vv-list, the start value of the row variable may be a \LaTeX{}
expression. Both \verb`rstep` and \verb`rstop` may also be \LaTeX{}
expressions. They are evaluated \emph{after} the vv-list is evaluated
and so may depend on the values of variables in the vv-list, including
the \emph{initial} value of the row variable. Thus setting \verb`rstep=1/x`,
where \verb`x` is the row variable, will give a \emph{constant} step
size equal to the reciprocal of the \emph{initial }value of the row
variable.

The difference in appearance of the two tables results from padding
with zeros\index{function value formatting!padding with zeros}\index{padding with zeros}
in the second (the asterisk in the trailing optional argument has
the same effect in \verb`\nmcTabulate` as in \verb`\nmcEvaluate`).
As you can see, padding applies not only to the values of the function
but also to the values of the row variable, although that has been
padded to only $1$ decimal place rather than the $6$ of the function
values. Padding makes an obvious improvement to the appearance. (How
many digits to pad to in the row variable column is discussed in §\ref{subsec:Row-var-col-formatting}.)
\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, \verb`rows`\index{rows@\texttt{rows}}, to tabulate
rather than a stop value. Only one of \verb`rows` and \verb`rstop`
should be given, but if both (inadvertently) are present, it is the
value of \verb`rows` that prevails. The first of the following three
tables shows an example where \verb`rows` is specified.

The second and third tables use an abbreviated form of the row variable
specification, \verb`rspec`\index{rspec@\texttt{rspec}}. This is
a three-element comma list, \verb`{rvar,rstep,rows}`. The second
table gives a straightforward example. In the third table a \LaTeX{}
expression has been inserted for \verb`rows` in the \verb`rspec`
comma list. Like \verb`rstep` and \verb`rstop`, \verb`rows` can
be a \LaTeX{} expression, but it is evaluated \emph{before} the vv-list
and therefore, unlike \verb`rstep` and \verb`rstop`, cannot depend
on quantites specified there like the initial row variable value.
\begin{verbatim}
    \tabulate[rvar=x,rstep=0.2,rows=6]
      { \sin x/\cos x }[x=0][*] \qquad
    \tabulate[rspec={x,0.2,6}]
      { \tan x }[x=0][*] \qquad
    \tabulate[rspec={x,0.2,1+(1/0.2)}]
      { \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,0.2,6}]
      { \tan x }[x=0][*] \qquad
    \tabulate[rspec={x,0.2,1+(1/0.2)}]
      { \sqrt{\sec^2 x - 1} }[x=0][*]\index{row variable!start, step, stop values|)}\medskip{}


\subsection{Row variable specification: non-uniform case}

\label{subsec:Row-var-spec-nonuniform}
\begin{table}[b]
\centering{}\caption{Non-uniform 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{rdata}} & {\small comma list} & {\small list of row-var. values } & {\small comma list may be stored in a macro}\tabularnewline
{\small\texttt{rfile}} & {\small chars} & {\small filepath/name} & {\small file of comma separated values}\tabularnewline
{\small\texttt{rverb}} & {\small int (}{\small\texttt{0}}/\texttt{1}) & {\small display }{\small\texttt{rdata }}{\small or}{\small\texttt{ rfile}}{\small{}
values verbatim} & {\small initialized to }{\small\texttt{0}}\tabularnewline
{\small\texttt{rfunc}} & {\small token(s)} & {\small function of a stepped variable} & \tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}
Occasionally one wants to form a table in which the row variable does
not increase or decrease in regular steps; for examples, see \emph{HMF}
Tables 1.1 and 3.1. (Tables 9.7, 10.5, 10.10 use two step values and
could also be handled with these settings.) For instance, one might
want a table of values of simple functions of a list of constants,
or a table of function values at $\pi$, $\pi/2$, $\pi/3$, $\pi/4$,
$\dots$, or at $1$, $10$, $100$, $1000$, $\dots$, or a table
of function values at thoroughly irregular, perhaps experimentally
determined, values.

\texttt{numerica-tables} provides two means of specifying such row
variables, either by means of a row variable function (\verb`rfunc`\index{rfunc@\texttt{rfunc}}),
when the row variable values change in a non-uniform but formulaic
way, or by explicitly listing the row variable values in a comma list
(\verb`rdata`\index{rdata@\texttt{rdata}}, \verb`rfile`\index{rfile@\texttt{rfile}}).
In this latter case, a setting \verb`rverb`\index{rverb@\texttt{rverb}}
enables the row variable column to be displayed either as a sequence
of \emph{values} or verbatim as a sequence of \emph{expressions} \textendash{}
like fractions of $\pi$.

\subsubsection{\texttt{rfunc}}

\index{rfunc@\texttt{rfunc}|(}\index{row variable!variably stepped|(}Suppose
\textendash{} perhaps with an interest in the distribution of prime
numbers \textendash{} that we want to create a small table of values
of $n/\ln n$ for, say, $n=10$, $100$, 1000, $10000$, $\ldots$
The prospective row variable $n$ is not increasing in constant steps,
although clearly in a regular manner. We handle such cases with the
\verb`rfunc` setting; in the present instance \verb`rfunc=10^n`
where now \verb`n` does increment by a constant amount:
\begin{verbatim}
    \tabulate[rpos=1,rfunc=10^n,rvar=n,rstep=1,
               rows=7,rround=0] 
     { n/\ln n }[n=1][0]
\end{verbatim}
$\Longrightarrow$     \tabulate[rpos=1,rfunc=10^n,rvar=n,rstep=1,
               rows=7,rround=0] 
     { n/\ln n }[n=1][0]\medskip{}

The variable \verb`n` has two different meanings here. Initially
it is the variable of a simple step function incrementing by $1$
at each step. The function takes values \verb`10^n`. Once the table
is compiled however \verb`n` is used to denote these successive function
values, $10,100,\ldots,10000000$. To the \emph{reader} of the table,
only this latter meaning is evident. Yes, it would be possible to
add further keys to specify a distinct step function variable and
its start, step and stop values, but the list of keys to specify a
table is already large. The `double usage' implemented, perhaps
confusing initially, economizes on keys and is invisible to the reader
of the table. The initial value \verb`n=1`\index{vv-list!row/col. variable initial values}
in the vv-list applies to the row variable function \verb`10^n`,
not to the function being tabulated (so the error-producing expression
\verb`1/\ln 1` does not arise).\index{rfunc@\texttt{rfunc}|)}\index{row variable!variably stepped|)}

\subsubsection{\texttt{rdata}, \texttt{rfile}, \texttt{rverb} }

\label{subsec:rdata-rfile-rverb}One\index{row variable!values from list, macro or file|(}
source of difficulty in reading the previous table is working out
just how many zeros there are in the larger numbers in the left column.
It would be more readable `at a glance' if we could write those
in scientific notation. To do that we use the \verb`rdata`\index{rdata@\texttt{rdata}|(}
and \verb`rverb`\index{rverb@\texttt{rverb}|(} keys. In the first
of the examples below, \verb`rverb` is absent (corresponding to the
default \verb`rverb=0`); in the second \verb`rverb=1`, the effect
of which is to use verbatim the row variable values provided in the
\verb`rdata` comma list:\index{row variable!verbatim values|(}
\begin{verbatim}
    \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7},
        rvar=n,rround=0,ralign=l] 
     { n/\ln n }[0]\qquad
    \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7},
        rverb=1,rvar=n,rround=0,ralign=l] 
     { n/\ln n }[0]
\end{verbatim}
$\Longrightarrow$     \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7},
        rvar=n,rround=0] 
     { n/\ln n }[0]\qquad
    \tabulate[rdata={10,100,1000,10^4,10^5,10^6,10^7},
        rverb=1,rvar=n,rround=0] 
     { n/\ln n }[0]\medskip{}

\verb`rverb` is particularly useful if you want to make a table of
simple functions of constants. In the following example, with its
initial setting (\verb`rverb=0`), the constants would be listed in
the first column as numbers and repeated in the second column (perhaps
to a different number of decimal places), which would all be rather
pointless. With \verb`rverb=1`, the constants are listed verbatim
in the first column against their numerical values in the second.
In the example, the data has been stored in macros, \verb`\mydatai`
and \verb`\mydataii`, prior to calling the \verb`\tabulate` command.
\verb`\mydataii` uses the \verb`\sfrac` command from the \texttt{xfrac}
package. By setting \verb`rdata` equal to these macros, the \verb`\tabulate`
command gains access to the values stored in them. 
\begin{verbatim}
    \def\mydatai{\tfrac12\pi,\tfrac13\pi,\tfrac23\pi,
                 \tfrac14\pi,\tfrac34\pi}
    \renewcommand\arraystretch{1.2}
    \tabulate[rdata=\mydatai,rverb=1,rvar=k,headless=1] 
      { k }[6*]
    \renewcommand\arraystretch{1} \qquad
    \def\mydataii{\sfrac\pi2,\sfrac\pi3,\sfrac{2\pi}3,
                  \sfrac\pi4,\sfrac{3\pi}4}
    \tabulate[rdata=\mydataii,rverb=1,rvar=k,headless=1] 
       { k }[6*] 
\end{verbatim}
$\Longrightarrow$     \renewcommand\arraystretch{1.2}
    \def\mydatai{\tfrac12\pi,\tfrac13\pi,
    \tfrac23\pi,\tfrac14\pi,\tfrac34\pi}
    \tabulate[rdata=\mydatai,rverb=1,rvar=k,headless=1 ] 
     { k }[6*]
    \renewcommand\arraystretch{1} \qquad
    \def\mydataii{\sfrac\pi2,\sfrac\pi3,
    \sfrac{2\pi}3,\sfrac\pi4,\sfrac{3\pi}4}
    \tabulate[rdata=\mydataii,rverb=1,rvar=k,headless=1] 
     { k }[6*]\medskip{}

The setting \verb`headless=1`\index{headless@\texttt{headless}}
(see §\ref{subsec:Header-row-suppress})\index{header row|see{column variable header, \texttt{headless}, row variable header}}
means the tables have no header row. To accommodate the tower of \verb`\tfrac`-s
in the first column of the first table, \verb`\arraystretch`\index{arraystretch@\texttt{\textbackslash arraystretch}}
has been used to add more space between the rows (and re-set afterwards).
Through the virtues of \verb`\sfrac` \index{sfrac@\texttt{\textbackslash sfrac}}
it is unnecessary in the second table which is neater and more readable
to my eye. But even better is the example at §\ref{subsec:rfracFraction-form-rvar}.

Note that the row variable can be chosen arbitrarily \textendash{}
I earlier used \verb`n` and have used \verb`k` here but it could
be anything. Nor does the function tabulated need to be the identity
\textendash{} see the earlier $n/\ln n$ example or a later example
in a multi-column setting in §\ref{subsec:Headerstyle:-multi-column-case}.
The identity function was appropriate for tables showing the decimal
values of symbolic constants, as was the \verb`rverb=1` setting.
The \verb`rverb` setting applies \emph{only} to the \verb`rdata`
and \verb`rfile` keys. It has no effect otherwise.\index{row variable!verbatim values|)}\index{rverb@\texttt{rverb}|)}\index{rdata@\texttt{rdata}|)}

In addition to a comma list or macro, data for the row variable can
also be stored in a file of comma-separated values \textendash{} say
\verb`mydata.txt`. If \verb`mydata.txt` is placed in the directory
of the current document and \verb`rfile=mydata.txt`\index{rfile@\texttt{rfile}}
entered in the settings option of the \verb`\tabulate` command, the
file will be found and the values used for the row variable. Alternatively,
the file could be placed in your \verb`texmf` tree and your \TeX{}
distro alerted to its presence (by refreshing the filename database).
Again \verb`rfile=mydata.txt` in the settings option will ensure
the file is found and the contents used for the row variable values.
Or, the file could be stored elsewhere and the \verb`rfile` key equated
to the full path and filename \textendash{} something like \verb`rfile=e:/mydocs/mydatafiles/mydata.txt`.
This ensures the file will be found and the contents used for the
row variable values. Note that even in Windows systems (where file
paths use backslashes) the path requires that forward slashes only
be used. \index{row variable!values from list, macro or file|)}

\subsection{Formatting row variable values \& column}

\label{subsec:Row-var-col-formatting}The padding option ({*}) of
the trailing optional argument is one way of formatting the row variable
values, but to how many decimal places? Aligned left or right or centred?
Under what heading \textendash{} the example tables so far have simply
used the row variable for the header? And should the row variable
column be at the left of the table, or the right \textendash{} or
both? These and related questions are answered by assigning values
to the keys listed in Table~\ref{tab:Formatting-the-row-variable}.
\begin{table}[tb]
\centering{}\caption{\protect\label{tab:Formatting-the-row-variable}Formatting the row
variable column}
\begin{center}
\begin{tabular}{ll>{\raggedright}p{4cm}l}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline
\midrule
{\small\texttt{rround}} & {\small int} & {\small rounding} & {\small\texttt{1}}\tabularnewline
{\small\texttt{rfont}} & {\small chars} & {\small font (}{\small\verb`\math<chars>`}{\small )} & \tabularnewline
{\small\texttt{ralign}} & {\small char (}{\small\texttt{r/c/l}}{\small )} & {\small horizontal alignment} & {\small\texttt{r}}\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 placement} & {\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
{\small\texttt{rfrac}} & {\small int (}{\small\texttt{0}}{\small\ldots 5)} & {\small fraction form} & {\small\texttt{0}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}


\subsubsection{Rounding: \texttt{rround}}

After\index{rounding value!row variable|(}\index{rround@\texttt{rround}|(}\index{formatting!row variable values|(}
studying some of the previous tables, we might decide to adjust the
step size, say from $0.2$ to $0.25$. But changing \verb`rstep`
to the new value gives a disconcerting result. \texttt{numerica-tables}
uses a default rounding value of $1$ for the row variable and has
rounded $0.25$ down to $0.2$, then $0.2+0.25=0.45$ down to $0.4$,
then $0.4+0.25=0.65$ down to $0.6$, then $0.6+0.25=0.85$ down to
$0.8$, at which point it stops since $0.8+0.85>1$ which is the stopping
value. The rounded-down values are the values used for caclulating
the sines. The second table corrects matters by adjusting the row
variable rounding with \verb`rround=2`.
\begin{verbatim}
    \tabulate[rvar=x,rstep=0.25,rstop=1]
      { \sin x }[x=0][*] (Eh???) \quad
    \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][*](Eh???) \quad
    \tabulate[rvar=x,rstep=0.25,rstop=1,rround=2]
      { \sin x }[x=0][*]\index{rround@\texttt{rround}|)}\index{rounding value!row variable|)}

\subsubsection{Font: \texttt{rfont}}

In\index{rfont@\texttt{rfont}} the second table below bolding by
means of the setting \verb`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.\index{formatting!row variable values|)}

\subsubsection{Alignment: \texttt{ralign}}

By\index{row variable column!alignment|(}\index{ralign@\texttt{ralign}}
default, the alignment of all columns is to the right, as in 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  interfere with
neat output in left or centred alignments.\index{function value formatting!sign handling}
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 \verb`ralign=r`; \verb`ralign=l`
(lowercase L) aligns entries in the row variable column to the left,
and \verb`ralign=c` centres entries in the row variable column. The
tables of the next example use a \verb`c` alignment to centre the
row variable column header. The third of those tables shows how minus
signs spoil the effect.

\subsubsection{Row-variable header: \texttt{rhead}}

In\index{rhead@\texttt{rhead}|(}\index{row variable header|(}\index{formatting!row variable header|(}
the second and third tables, the header for the row variable column
has also been bolded. The default header is the row variable symbol.
That can be replaced by giving a value to the key \verb`rhead`. I
have used \verb`rhead=\boldsymbol{x}` (rather than \verb`\mathbf{x}`)
in order to get an italicized bold symbol. Note that you do not need
to include math delimiters in the specification. It is assumed that
\verb`rhead` will sit between \verb`$ $` delimiters which are inserted
automatically by \verb`numerica-tables`.
\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][*]\qquad
    \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][*]\index{rfont@\texttt{rfont}}\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 in the
row variable column 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.)\index{rhead@\texttt{rhead}}\index{formatting!row variable header|)}

\subsubsection{Nudging~the~header: \texttt{rhnudge}}

However\index{rhnudge@\texttt{rhnudge}|(}, you might prefer to avoid
inserting positioning commands into the actual row variable header,
obscuring its true content. You can avoid doing this by setting the
key \verb`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 \verb`rhnudge=9`. For positive
nudge values, nudging works in the \emph{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 \verb`numerica-tables`.

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.
By giving \verb`rhnudge` a \emph{negative} value (\verb`rhnudge=-12`
in the example) it is brought 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*]\index{row variable header|)}\index{rhnudge@\texttt{rhnudge}|)}

\subsubsection{Position in the table: \texttt{rpos}}

\label{subsec:Row-var-col-pos}\index{row variable column!position in table}\index{rpos@\texttt{rpos}}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 \verb`rpos=0`, suppressed (no row variable column); 
\item \verb`rpos=1`, first column (the default); 
\item \verb`rpos=2`, last column; 
\item \verb`rpos=3`, first and last columns;\texttt{ }
\item \verb`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 \verb`rpos=1`.
\end{itemize}
An example with \verb`rpos=3` is given shortly below, §\ref{sec:Multiple-function-tables}. 

\subsubsection{\texttt{rvar'}, \texttt{rhead'}, \texttt{rhnudge'}}

These settings become relevant only when \verb`rpos=4`; see §\ref{subsec:Second-row-var-col}.

\subsubsection{Fraction-form values: rfrac}

\label{subsec:rfracFraction-form-rvar}\index{row variable!fraction form|(}\index{fraction form!row variable values|(}
By giving the setting \verb`rfrac` an integer value between $1$
and $5$ inclusive it is possible to render the row variable values
as fractions with no more than \verb`rround` digits in the denominator.
Initially \verb`rfrac=0`, which gives decimal output.
\begin{itemize}
\item \verb`rfrac=1` produces a slash fraction like $2/3$;
\item \verb`rfrac=2` produces a scalable \verb`\frac`-tion like $\frac{2}{3}$
in textstyle and $\dfrac{2}{3}$ in displaystyle;
\item \verb`rfrac=3` produces a non-scalable \verb`\tfrac` like $\tfrac{2}{3}$;
\item \verb`rfrac=4` produces a non-scalable \verb`\dfrac` like $\dfrac{2}{3}$;
\item \verb`rfrac=5` produces a slash fraction like $\sfrac 23$ (by means
of the \verb`\sfrac`\index{sfrac@\texttt{\textbackslash sfrac}}
command) in text- and displaystyles if the \texttt{xfrac} package
is loaded and like $\scriptstyle 2/3$ if it isn't; when used as a
super- or sub-script, the fractions reduce in size like $e^{\sfrac23}$
or $e^{{\scriptscriptstyle 2/3}}$.
\end{itemize}
The following table repeats an earlier table from §\ref{subsec:rdata-rfile-rverb},
but more neatly, through using \verb`\sfrac` (with the \verb`rfrac=5`
call) instead of \verb`\tfrac`, and moving $\pi$ from the row variable
column into the formula:
\begin{verbatim}
    \def\mydataii{1/2,1/3,2/3,1/4,3/4}
    \tabulate[rdata=\mydataii,rpos=1,
              rvar=k,rfrac=5,rhnudge=6,chnudge=18] 
      { k\pi }[k=0][6*]
\end{verbatim}
$\Longrightarrow$     \def\mydataii{1/2,1/3,2/3,1/4,3/4}
    \tabulate[rdata=\mydataii,rpos=1, 
        rvar=k,rfrac=5,rhnudge=6,chnudge=18] 
     { k\pi }[k=0][6*]\index{fraction form!row variable values|)}\index{row variable!fraction form|)}

\subsection{Adjoined multi-function tables}

\label{subsec:Adjoining-tables}\index{adjoined tables}\index{multi-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. 

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\index{rpos@\texttt{rpos}} \verb`rpos`
settings (\verb`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 \verb`%` 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}]
      { \sin x }[x=0][*]%
    \tabulate[rpos=0,rspec={x,0.2,6}]
      { \cos x }[x=0][*]%
    \tabulate[rpos=2,rspec={x,0.2,6}]
      { \tan x }[x=0][*]
\end{verbatim}
$\Longrightarrow$     \tabulate
      [rspec={x,0.2,6}]
      { \sin x }[x=0][*]%
    \tabulate
      [rpos=0,rspec={x,0.2,6}]
      { \cos x }[x=0][*]%
    \tabulate
      [rpos=2,rspec={x,0.2,6}]
      { \tan x }[x=0][*]\medskip{}

\noindent However, tabulating more than one function at a time is
too common a need to have to resort to a fudge like adjoining tables.
\verb`numerica-tables` offers a systematic way of doing this; see
§\ref{sec:Multiple-function-tables}.

\section{Column-variable settings}

\label{sec:Column-variable-settings}
\begin{table}
\caption{Column-variable specification}

\centering{}\begin{center}
\begin{tabular}{ll>{\raggedright}p{4cm}l}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small comment}\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{cols\}}} & {\small short form spec.}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}
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, as we have seen,
is to create and adjoin separate tables, one per parameter value,
but that 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\index{column variable!start, step, stop values|(}. 

In \index{cvar@\texttt{cvar}|(} the following example \verb`cvar=k`
is the column variable. I have chosen a step size \verb`cstep=2`\index{cstep@\texttt{cstep}|(}
and a stop value \verb`cstop=9`\index{cstop@\texttt{cstop}|(}. As
with the row variable, the start value ({\ttfamily\verb`k=3`})
of the column variable is specified in the vv-list\index{vv-list!row/col. variable initial values}\index{column variable!initial value in vv-list}.
Although in the example these values are numbers, all three values
could be \LaTeX{} expressions that evaluate to numbers. In particular,
the expressions for step and stop values may include the row and column
variables (in the example $x$ and $k$) which are assigned their
initial vv-list values. Note also the setting for \verb`rhead`\index{rhead@\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},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},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{ }{\ttfamily\verb`cstop`}, you might prefer
to specify the number of columns, \verb`cols`\index{cols@\texttt{cols}|(},
explicitly. I could have replaced {\ttfamily\verb`cstop=9`}
with \verb`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.

It is worth pointing out explicitly that if \verb`cols` is specified,
then it is possible to have a \emph{zero} step size, \verb`cstep=0`.
(A similar comment applies to \verb`rows` and \verb`rstep`.)

And again, as with the row variable, it is possible to condense the
specification into a comma list with the key \verb`cspec`\index{cspec@\texttt{cspec}|(}.
This is a $3$-element comma list of the form \verb`{cvar,cstep,cols}`.
Thus, for the preceding table I could also have written
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},rhead=x\backslash k,
              cvar=k,cstep=2,cols=4]
      { \sin kx }[k=3,x=0][*]
\end{verbatim}
or more succinctly
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},rhead=x\backslash k,
              cspec={k,2,4}]
      { \sin kx }[k=3,x=0][*]
\end{verbatim}
and produced the same table. 

As with the row equivalents, \verb`cstep`, \verb`cstop` and \verb`cols`
can all be \LaTeX{} expressions. Again like the row equivalents, the
first two are evaluated \emph{after }the vv-list and so may depend
not only on numbers and constants but also the initial values of the
row and column variables, which are given those values in the vv-list.\index{vv-list!row/col. variable initial values}
\verb`cols` is evaluated \emph{before }the vv-list; it may be a \LaTeX{}
expression but cannot depend on the row or column variable.\index{cvar@\texttt{cvar}|)}\index{cstep@\texttt{cstep}|)}\index{cstop@\texttt{cstop}|)}\index{cols@\texttt{cols}|)}\index{cspec@\texttt{cspec}|)}\index{column variable!start, step, stop values|)}

\subsection{Column header formatting}

\label{subsec:Column-header-formatting}\index{formatting!column variable header|(}\index{column variable header|(}
\begin{table}[b]
\centering{}\caption{Formatting the column variable header}
\begin{center}
\begin{tabular}{ll>{\raggedright}p{4cm}l}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline
\midrule
{\small\texttt{chstyle}} & {\small int (0\ldots 4)} & {\small header style} & {\small\texttt{0}}\tabularnewline
{\small\texttt{ctitle}} & {\small tokens} & {\small single col. alt. 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{} rounding} & {\small\texttt{0}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\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 \verb`chead`. First I discuss the built-in styles.

\subsubsection{Single-column header}

When\index{column variable header!single column case} 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
\verb`ctitle=*`\index{ctitle@\texttt{ctitle}!* setting@\texttt{{*}} setting}
(see §\ref{subsec:Title:-ctitle-setting} below). You may want some
other header. Then give \verb`ctitle`\index{ctitle@\texttt{ctitle}}
some other value (although note that giving it the value \texttt{{*}{*}}\index{ctitle@\texttt{ctitle}!** setting@\texttt{{*}{*}} setting}
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 \emph{two} pairs
of braces, \verb`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 \verb`chead`\index{chead@\texttt{chead}}
a value. This key I discuss below in §\ref{subsec:chead}. \verb`chead`
is entirely up to the user to specify, including any math delimiters
and positioning (nudging) of elements. 

If both \verb`ctitle` and \verb`chead` are given, the \verb`chead`
value prevails.

\subsubsection{Multi-column header: \texttt{chstyle}}

\label{subsec:Headerstyle:-multi-column-case}\index{column variable header!multi-column case|(}\verb`chstyle=0`\index{chstyle@\texttt{chstyle}|(}\index{column variable header!multi-column case!chstyle@\texttt{chstyle}|(}
which is the default gives a header of the form displayed in the last
example, with only the column variable \emph{value} at the head of
each column. This style generally requires the row variable header
to indicate what the values denote, as in the example above where
\verb`rhead=x\backslash k`, the backslash separating row from column
variable. \emph{HMF} contains a multitude of instances of this style;
see Tables~9.7, 17.5, 21.1, 24.3, 27.4, etc. for examples.

\verb`chstyle=1` changes the header of the \emph{first} function
value column to the form \emph{variable=value} \textendash{} in the
example below, to $k=1$. This may be an appropriate choice when a
small rounding value is being used and the resulting columns are narrow.
I can find only one real instance in \emph{HMF}, Table~26.7. In the
example I have used the \verb`rdata` setting to collect an assortment
of nonsense values and for some weird reason wish to tabulate the
sines of multiples of these oddballs. Note that the row variable setting
\verb`rhead` (producing $X\backslash k$) is no longer needed since
the column variable is now explicitly indicated. (But the table is
lacking a title \textendash{} what on earth are we calculating with
this strange group of numbers?)
\begin{verbatim}
    \tabulate[rdata={-e^2,1.234e2,3.1416,\pi/\gamma,1/9},
              ^,rvar=X,rverb=1,cspec={k,1,3},chstyle=1]
      { \sin kX }[k=1][3*]
\end{verbatim}
$\Longrightarrow$    \tabulate[rdata={-e^2,1.234e2,3.1416,\pi/\gamma,1/9},
              ^,rvar=X,rverb=1,cspec={k,1,3},chstyle=1]
      { \sin kX }[k=1][3*]\medskip{}

\verb`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. Again, the row variable setting
\verb`rhead` no longer needs the \verb`\backslash k` part since
the column variable is now explicitly indicated.
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},
              cspec={k,2,3},chstyle=2]
      { \sin kx }[k=3,x=0][3*]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,0.2,6},
              cspec={k,2,3},chstyle=2]
      { \sin kx }[k=3,x=0][3*]\medskip{}

\verb`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.
Note that if the column variable value is \verb`1`, the \verb`1`
will be displayed:
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},
              cspec={k,2,3},chstyle=3]
      { \sin kx }[k=1,x=0][4*]
\end{verbatim}
$\Longrightarrow$     \tabulate
      [rspec={x,0.2,6},
         cspec={k,2,3},chstyle=3]
      { \sin kx }[k=1,x=0][4*]\medskip{}

In this last example you may not want the \verb`1` displayed. To
achieve that effect put \verb`chstyle=4`. This results in a header
as for \verb`chstyle=3` but if the column variable takes the value
\verb`1`, it has an empty replacement:
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},
              cspec={k,2,3},chstyle=4]
      { \sin kx }[k=1,x=0][4*]
\end{verbatim}
$\Longrightarrow$     \tabulate
      [rspec={x,0.2,6},
         cspec={k,2,3},chstyle=4]
      { \sin kx }[k=1,x=0][4*]\index{column variable header!multi-column case!chstyle@\texttt{chstyle}|)}\index{chstyle@\texttt{chstyle}|)}

\subsubsection{User-defined header: \texttt{chead}}

\label{subsec:chead}\index{chead@\texttt{chead}|(}If the function
in the last example were, for instance, $k+\sin kx$, then neither
replacing $k$ by \verb`1` nor an empty replacement would be appropriate.
In that case the only recourse is to use the \verb`chead` key. Users
can assign whatever value they like to \verb`chead`. The assignment
must contain the correct number of tab characters (\verb`&`) for
the \emph{column variable columns only}. It is a header only to the
function-value columns. The user will need to insert \verb`$` signs
or other math delimiters as appropriate. This differs from the practice
for \verb`rhead`, but \verb`chead` is potentially far more complicated.
Thus for $k+\sin kx$,
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},cspec={k,2,3},
              chead=$1+\sin x$&$3+\sin3x$&$5+\sin 5x$]
      { k+\sin kx }[k=1,x=0][4*]
\end{verbatim}
$\Longrightarrow$     \tabulate
      [rspec={x,0.2,6},
         cspec={k,2,3}, 
         chead=$1+\sin x$&$3+\sin3x$&$5+\sin 5x$]
      { k+\sin kx }[k=1,x=0][4*]\medskip{}

\noindent Non-empty content for the \verb`chead` key overrides any
\verb`chstyle` setting and, in the case of a table with only a single
function-value column, overrides any \verb`ctitle` setting.\index{chead@\texttt{chead}|)}

\subsubsection{Alignment: \texttt{calign}}

The\index{calign@\texttt{calign}|(} function-value columns are aligned
right, \verb`calign=r`, by default. Also available are \verb`calign=c`
for centred alignment and \verb`calign=l` (lowercase L) for left
alignment. Using centred alignment with {\ttfamily\verb`chstyle=2`}
in a previous example table gives
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},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},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}\index{calign@\texttt{calign}|)}.

\subsubsection{Nudging header entries: \texttt{chnudge}}

\label{subsec:Nudgingtheheaders:-chnudge}In\index{chnudge@\texttt{chnudge}|(}
left or right alignment it is possible to nudge the column headers
in the opposite direction by giving a numerical value to the the key
\verb`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. {\ttfamily\verb`numerica-tables`} provides
that. In the next example I have chosen \verb`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},ralign=c,
              cspec={k,2,3},chstyle=2,chnudge=12]
      { \sin kx }[k=3,x=0][*]
\end{verbatim}
$\Longrightarrow$     \tabulate
      [rspec={x,0.2,6},ralign=c,
         cspec={k,2,3},chstyle=2,chnudge=12]
      { \sin kx }[k=3,x=0][*]\medskip{}

The \verb`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 \verb`chnudge=-9`.
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},ralign=c,
              cspec={k,2,3},chstyle=2,chnudge=-9]
      { \sin kx }[k=100,x=0][3*]
\end{verbatim}
$\Longrightarrow$     \tabulate
      [rspec={x,0.2,6},ralign=c,
         cspec={k,2,3},chstyle=2,chnudge=-9]
      { \sin kx }[k=100,x=0][3*]\index{chnudge@\texttt{chnudge}|)}

\subsubsection{Rounding: \texttt{chround}}

In\index{chround@\texttt{chround}|(}\index{rounding value!header row|(}
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 {\ttfamily\verb`chround`}. For a step size of $0.25$
the appropriate setting is {\ttfamily\verb`chround=2`}.
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},ralign=c,
              cspec={k,0.25,3},chstyle=2,chround=2]
      { \sin kx }[k=3,x=0][*]
\end{verbatim}
$\Longrightarrow$     \tabulate
      [rspec={x,0.2,6},rhead=x,ralign=c,
         cspec={k,0.25,3},chstyle=2,chround=2]
      { \sin kx }[k=3,x=0][*]\index{column variable header|)}\index{rounding value!header row|)}\index{column variable header!multi-column case|)}\index{chround@\texttt{chround}|)}\index{formatting!column variable header}

\section{Multiple functions in a single table}

\label{sec:Multiple-function-tables}As\index{multi-function tables|(}
already noted in §\ref{subsec:Adjoining-tables}, tabulating more
than one function at a time is too common a need to have to resort
to a fudge like adjoining tables. The systematic way of handling this
task is to enter the functions in the main argument of a \verb`\tabulate`
command separated by a specified mark then alert \verb`\tabulate`
that this has happened with the \verb`ff`\index{ff@\texttt{ff}}
key in the settings option. 

By default the multi-function delimiter\index{multi-function tables!function delimiter}
is the comma if the decimal point is a dot (or period), or the semicolon\index{semicolon and decimal comma}
if the decimal point is a comma\index{decimal comma} (\texttt{numerica}
loaded with the \verb`comma` package option). If you are content
with the default delimiter then it suffices to enter \verb`ff` in
the settings option. If not, then enter \verb`ff=<mark>` there. For
example \verb`ff=|` would make the `pipe' character \verb`|` the
multi-formula delimiter. If the \verb`ff` key is overlooked then
multiple formulas in the main argument of \verb`\tabulate` will almost
certainly cause a \LaTeX{} error.

In the following example, using the default comma, note first the
\verb`ff` setting, and then the \verb`o` setting indicating that
the arguments of the trig functions are in degrees and (just to amuse
myself) I have put the row variable column on both sides with the
\verb`rpos=3`\index{rpos@\texttt{rpos}} setting: 
\begin{verbatim}
    \tabulate[ff,o,rpos=3,rround=0,
              rvar=\theta,rstep=10,rstop=90]
      { \sin \theta, \cos \theta }[\theta=0][*]
\end{verbatim}
$\Longrightarrow$     \tabulate[ff,o,rpos=3,rround=0,
              rvar=\theta,rstep=10,rstop=90]
      { \sin \theta, \cos \theta }[\theta=0][*]\medskip{}

The tables suggest a space saving possibility: since $\sin$ and $\cos$
are complementary functions\index{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 \verb`rpos=4`\index{rpos@\texttt{rpos}} setting (§\ref{subsec:Second-row-var-col})
which enables complementary functions to be tabulated in `half tables'
(\emph{HMF} Tables 4.10\textendash 4.12 are examples for the trigonometric
functions).

In the next example, the row variable column is again duplicated,
left and right, with the \verb`rpos=3` setting, and a centred alignment
is used for the function values. Because $\sin$ and $\cos$ are complementary,
I have stopped the table at \verb`rstop=45` since continuing to \verb`90`
would simply give a mirror reflection of the preceding values. To
accommodate the use of the comma in \verb`\max` and \verb`\min`\index{n-ary@$n$-ary functions}
I have stipulated \verb`ff=|`\index{ff@\texttt{ff}}. Visually, the
table has an unsatisfactory, sprawling appearance \textendash{} which
directs attention to how tables might be titled (§\ref{subsec:Title:-ctitle-setting})\index{rpos@\texttt{rpos}}:
\begin{verbatim}
    \tabulate[ff=|,o,rpos=3,rround=0,
        rvar=\theta,rstep=10,rstop=45,calign=c]
      { \max(\sin \theta,\cos \theta)|
             \min(\sin \theta,\cos \theta) }
      [\theta=0][*]
\end{verbatim}
$\Longrightarrow$     \tabulate[ff=|,o,rpos=3,rround=0,
        rvar=\theta,rstep=10,rstop=45,calign=c]
      { \max(\sin \theta,\cos \theta)|
             \min(\sin \theta,\cos \theta) }
      [\theta=0][*]   \index{multi-function tables|)}

\section{Whole-of-table formatting}

\label{sec:Whole-of-table-formatting}There\index{formatting!title, subtitle, footer|(}
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}
\centering{}\caption{\protect\label{tab:Table-formatting-settings}Table formatting}
\begin{center}
\begin{tabular}{ll>{\raggedright}p{4cm}l}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small initial}\tabularnewline
\midrule
{\small\texttt{ctitle}} & {\small token(s)} & {\small collective title for function-value columns} & \tabularnewline
 & {\small token(s)} & {\small subtitle row for} {\small function- value cols} & \tabularnewline
{\small\texttt{header}} & {\small int (}{\small\texttt{0/1}}{\small )} & {\small suppress/show header row} & {\small\texttt{1}}\tabularnewline
{\small\texttt{foot}} & {\small token(s)} & {\small table-wide footer row} & \tabularnewline
{\small\texttt{rules}} & {\small char(s)} & {\small horizontal rule spec.} & {\small\texttt{ThB}}\tabularnewline
{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row variable col. position(s)} & {\small\texttt{1}}\tabularnewline
{\small\texttt{rbloc}} & {\small integer comma list} & {\small row block specification} & \tabularnewline
{\small\texttt{valign}} & {\small char (}{\small\texttt{t}}/{\small\texttt{m}}/{\small\texttt{b}}{\small )} & {\small vertical alignment of table relative to text baseline} & {\small\texttt{m}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}


\subsection{Title for function-value columns: \texttt{ctitle}}

\label{subsec:Title:-ctitle-setting}\index{ctitle@\texttt{ctitle}|(}The\index{title row|(}
function-value columns have individual headers, formatted in the various
ways provided by the settings discussed in previous sections, but
it can also be helpful to have a collective title for these columns.
We saw the need in the last example. The need is met with the \verb`ctitle`
key. This can be set to whatever you like (e.g. \verb`ctitle=\text{Fred}`)
but, to more purpose, I shall use the setting to improve the look
of the table in the last example. Note that the content of the \verb`ctitle`
key is enclosed in braces. This is to shield the commas there from
being misinterpreted as item separators in the settings option. For
an analogous reason I have specified the semicolon (\verb`ff=;`)\index{ff@\texttt{ff}}
as the function separator in the main argument:\index{rpos@\texttt{rpos}}\index{n-ary@$n$-ary functions}
\begin{verbatim}
     \tabulate[ff=;,o,rpos=3,rround=0,
               rvar=\theta,rstep=10,rstop=45,
               ctitle={\max,\min(\sin\theta,\cos\theta)},
               chead=max\hphantom{00} & min\hphantom{00}]
       { \max(\sin \theta,\cos \theta);
             \min(\sin \theta,\cos \theta) } [\theta=0][*]
\end{verbatim}
$\Longrightarrow$      \tabulate[ff=;,o,rpos=3,rround=0,
               rvar=\theta,rstep=10,rstop=45,
               ctitle={\max,\min(\sin\theta,\cos\theta)},
               chead=max\hphantom{00} & min\hphantom{00}]
      { \max(\sin \theta,\cos \theta);
             \min(\sin \theta,\cos \theta) }
      [\theta=0][*]\medskip{}

\noindent Now it is clearer what is being tabulated and the wide separation
of the function-value columns is reduced. 

There are two built-in values for the \verb`ctitle` key: \verb`ctitle=*`,
which forms the title from the function being tabulated, and \verb`ctitle=**`
which uses the function and vv-list for the title. Obviously these,
particularly the latter, could easily become too long to be useful.
An example of \verb`ctitle=**` is presented below in §\ref{subsec:Rules:-rules-setting}.
In the following example, inclusion of the vv-list in the title is
pointless since the variables there form the row and column variables
of the table; \verb`ctitle=*` is all that is needed:
\begin{verbatim}
    \tabulate[rspec={n,1,5},rround=0,
              cspec={m,1,4},chstyle=2,ctitle=*]
      { \cos(m\pi/n) }[n=3,m=2][*4]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={n,1,5},rround=0,
              cspec={m,1,4},chstyle=2,ctitle=*]
      { \cos(m\pi/n) }[n=3,m=2][*4]

\subsection{Between header \& title: \texttt{csubttl}}

Some\index{csubttl@\texttt{csubttl}}\index{title row!subtitle row}
tables need to fit more header or title material into their rows than
can be comfortably accommodated in either row alone. For examples,
see \emph{HMF} Tables 7.9 (error function for complex arguments),
17.7 (Jacobian zeta function), 21.1 (eigenvalues of spheroidal wave
functions), and 26.7 (probability integrals). One way of handling
this problem is to resort to more complicated environments in header
and title rows. Another, more direct way, is to insert a row between
the header and title rows. 

I have chosen \verb`csubttl`, a contraction of `column variable
subtitle' for the key name. (In version 2 of \texttt{numerica-tables}
the name \verb`cmidrow`\index{cmidrow@\texttt{cmidrow }|see{\texttt{csubttl}}}
was used; that will still work but \verb`csubttl` gives a clearer
indication of what the key does.) The initial `c' emphasizes that
like \verb`chead` and \verb`ctitle` it is constrained to the span
of the column variable (or function-value) columns only. The content
of \verb`csubttl` is entirely the responsibility of the user, including
insertion of the necessary number of tab characters, \verb`&`, and
any math delimiters required. 
\begin{verbatim}
    \tabulate[ff,rspec={x,1,7},rround=0,chnudge=18,
      ctitle=\text{Hyperbolic functions},
      csubttl=\multicolumn{3}{c}{$\sinh x=\tfrac12
        (e^x-e^{-x}),\ \cosh x=\tfrac12(e^x+e^{-x})$}]
    { \tfrac12e^x, \sinh x, \cosh x }
      [{x}=0][8*]
\end{verbatim}
$\Longrightarrow$     \tabulate[ff,rspec={x,1,7},rround=0,chnudge=18,
      ctitle=\text{Hyperbolic functions},
      csubttl=\multicolumn{3}{c}{$\sinh x=\tfrac12
        (e^x-e^{-x}),\ \cosh x=\tfrac12(e^x+e^{-x})$}]
      { \tfrac12e^x, \sinh x, \cosh x }
      [{x}=0][8*]\index{ctitle@\texttt{ctitle}|)}\index{title row|)}

\subsection{Suppress/show header row}

\label{subsec:Header-row-suppress}Usually\index{headless@\texttt{headless}}
the header row in a table is shown. It carries essential information
as to the table's contents. However, there are occasions when it should
be suppressed. An example where this is appropriate is given in §\ref{subsec:rdata-rfile-rverb}
where a table listing fractions of $\pi$ and their values is shown.
As in that example, to suppress the header enter the setting \verb`headless=1`.
(Otherwise tables default to \verb`headless=0`.)

\subsection{Footer row: \texttt{foot} }

Some\index{foot@\texttt{foot}}\index{footer row} tables have a footer
row and \verb`numerica-tables` 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 usually 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`\multicolumn`. (\emph{HMF}
uses the footer mainly for cryptic descriptions of the accuracy and
needs of interpolation methods.) 

You can put into the footer what you wish with the setting \verb`foot=<tokens>`.

The one exception is when \verb`foot=*`\index{foot@\texttt{foot}}.
This will fill the footer with the header, but with the items of the
header presented in \emph{reversed} order \textendash{} the last item
first, and so on. 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}.

\noindent{}%
\noindent\begin{minipage}[t]{1\columnwidth}%
\begin{shaded}%

\subsubsection{Footer functions}

In\index{footer functions} previous versions of \texttt{numerica-tables}
it was possible to perform certain simple operations on columns \textendash{}
calculate the sum, the average and maximum and minimum values. This
is no longer so in version 3. Not only does it seem tangential to
the primary function of the \verb`\tabulate` command but it was also
acutely dependent on the format of the numbers being operated on.
A simple change in the number-format option could cause a \LaTeX{}
error.\end{shaded}%
\end{minipage}\index{formatting!title, subtitle, footer|)}

\subsection{Horizontal rules: \texttt{rules}}

\label{subsec:Rules:-rules-setting}\index{rules@\texttt{rules}|(}\index{rules|(}\index{formatting!rules|(}
\begin{table}[!b]
\centering{}\caption{\protect\label{tab:Rules}Rules. {\small (In the \textquoteleft span\textquoteright{}
column, \textquoteleft f-v\textquoteright =function-value; \textquoteleft r-v\textquoteright =row
variable; \textquoteleft$<$~table\textquoteright{} indicates that
the rule spans the table but is trimmed at each end.)}}
\begin{center}
\begin{tabular}{lll>{\raggedright}p{3cm}l}
\toprule 
{\small char} & {\small rule} & {\small position} & {\small span} & {\small default rule thickness}\tabularnewline
\midrule
{\small\texttt{T}} & {\small top} & {\small above table} & {\small table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline
{\small\texttt{t}} & {\small title} & {\small below title} & {\small f-v cols} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline
{\small\texttt{s}} & {\small subtitle} & {\small below subtitle} & {\small f-v cols (if $1$ r-v col.)}{\small\par}

{\small$<$ table (if $2$ r-v cols)} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline
{\small\texttt{h}} & {\small header} & {\small below header} & {\small table} & {\small\texttt{\textbackslash lightrulewidth=.05em}}\tabularnewline
{\small\texttt{f}} & {\small footer} & {\small above footer} & {\small < table} & {\small\texttt{\textbackslash cmidrulewidth =.03em}}\tabularnewline
{\small\texttt{B}} & {\small bottom} & {\small below table} & {\small table} & {\small\texttt{\textbackslash heavyrulewidth=.08em}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\end{table}
The \texttt{booktabs} \index{booktabs@\texttt{booktabs}} package
which \texttt{numerica} uses is most emphatic that one should `1.
Never, ever use vertical rules.\index{rules!edicts} 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). In the various examples in the present
document I have used horizontal rules because these too are tables
within text. Some form of delineation seems necessary. (Although many
of \emph{HMF}'s tables are inelegantly typeset, I have used it as
a valuable source for the variety of structures that one might need
for presenting a multitude of different kinds of numerical data.) 

The \verb`rules` key allows one to specify precisely which rules
are used. The content of the key is a `word' \textendash{} a sequence
of letters \textendash{} where the characters have the significance
and default thicknesses (from \texttt{booktabs}\index{booktabs@\texttt{booktabs}})
shown in Table~\ref{tab:Rules}. The default setting is \verb`rules=ThB`.
To insert a rule beneath the title, for example, change this to \verb`rules=TthB`.

If you are using a subtitle row between header and title rows and
want a rule beneath that too, then the spec. is \verb`rules=TtshB`.
(For legacy reasons, \verb`m` \textendash{} from `midrow' \textendash{}
can also be used instead of \verb`s`, as in version 2 of \texttt{numerica-tables}.)
To my eye rules beneath both title and subtitle don't work; a rule
beneath the subtitle alone gives a better result. The subtitle rule
changes its behaviour depending on whether there are two row variable
columns \textendash{} on the left and right of the table \textendash{}
or not. If there is only such column then, like the title rule, the
subtitle rule spans only the function-value columns. If there are
two row variable columns then the subtitle rule spans the table but
is trimmed by 0.5 em at each end. That degree of trim is the \texttt{booktabs}\index{booktabs@\texttt{booktabs}}
default but can be changed by giving a different value to \verb`\cmidrulekern`\index{cmidrulekern@\texttt{\textbackslash cmidrulekern}}
in the preamble, e.g. \verb`\cmidrulekern=1em`. Note that the changed
trim will also apply to the title rule.

If you are using a footer row\index{footer row} and want a rule above
it, then add \verb`f` to the specification, e.g. \verb`rules=TthfB`.
In version 3 of \texttt{numerica-tables} the rule is trimmed at each
end. Visually, having two table-spanning rules close together, the
\verb`f` and \verb`B` rules, doesn't work. The trimming makes a
difference. (For the \verb`T` and \verb`h` rules, the occurrence
of the table body beneath the \verb`h` rule seems to make a difference
to the visual impact of the rules.) But the question should always
be: is a rule necessary at all? Usually, less is more.

If you wish to change the thickness of a rule from its default, then
enter new values for any or all of \verb`\heavyrulewidth`\index{heavyrulewidth@\texttt{\textbackslash heavyrulewidth}},
\verb`\lightrulewidth`\index{lightrulewidth@\texttt{\textbackslash lightrulewidth}},
\verb`\cmidrulewidth`\index{cmidrulewidth@\texttt{\textbackslash cmidrulewidth}}
in the preamble. The values listed in Table~\ref{tab:Rules} are
the default values in the \texttt{booktabs}\index{booktabs@\texttt{booktabs}}
package (except for the midrow and footer rules, which \texttt{booktabs}\index{booktabs@\texttt{booktabs}}
does not cover; in \texttt{numerica-tables} these rules are assigned
a thickness of \verb`\cmidrulewidth`).

The order in which rules are placed in the specification doesn't matter.
I have entered them in their `natural' order simply because it feels
natural to do so, but it is their occurrence in the spec., not their
position, that matters.

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 \verb`ctitle=**`\index{ctitle@\texttt{ctitle}!** setting@\texttt{{*}{*}} setting}.
The formula 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 \verb`k` and \verb`x` in the vv-list
so that \emph{they} don't display).
\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][*]\index{rules|)}\index{rules@\texttt{rules}|)}\index{formatting!rules}

\subsection{Second row variable column: \texttt{rpos=4}}

\label{subsec:Second-row-var-col}In\index{row variable column!position in table}
\index{rpos@\texttt{rpos}|(}§\ref{subsec:Row-var-col-pos} I discussed
the settings \verb`rpos=0,1,2` and in §\ref{sec:Multiple-function-tables}
gave an example of using {\ttfamily\verb`rpos=3`} where
repeating the row variable column on the right is helpful. There is
another value available for this key, \verb`rpos=4`. Like {\ttfamily\verb`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 a function of those
in the left column ({\ttfamily\verb`rpos=3`} corresponds
to the function being the identity). The value given to the key \verb`rvar'`\index{rvar@\texttt{rvar'}|(}
determines the function used and the value given to the key \verb`rhead'`\index{rhead@\texttt{rhead'}|(}
determines the header for the right-hand row variable column. If \verb`rhead'`
is omitted it defaults to a blank header, unless the \verb`rvar'`
setting is also omitted, when \verb`rpos=4` behaves like \verb`rpos=3`.

For example, the sine and cosine are complementary functions\index{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.
In the table, $\theta'=90-\theta$ and \verb`rhead'=\theta'`. Simply
to illustrate the use of \verb`rhnudge'`\index{rhnudge@\texttt{rhnudge'}|(}
I have nudged the header in the second (right) row variable column
to sit above the tens digits of the row variable values. The example
also gives an illustration of the use of an  expression in the third
element of \verb`rspec`.\index{foot@\texttt{foot}}
\begin{verbatim}
    \tabulate[ff=;,o,rpos=4,
      rspec={\theta,5,1+45/5},rround=0,
        chnudge=14,rvar'=90-\theta,rhnudge'=4,
          rhead'=\theta',rules=ThfB,foot=*]
      { \sin\theta;\cos\theta }[\theta=0][*]
\end{verbatim}
$\Longrightarrow$     \tabulate[ff=;,o,rpos=4,
      rspec={\theta,5,1+45/5},rround=0,
        chnudge=14,rvar'=90-\theta,rhnudge'=4,
          rhead'=\theta',rules=ThfB,foot=*]
      { \sin\theta;\cos\theta }[\theta=0][*]\index{footer row!complementary functions}\index{rhnudge@\texttt{rhnudge'}|)}\index{rhead@\texttt{rhead'}|)}\index{ff@\texttt{ff}}\medskip{}

\noindent The values of sines from $0$ to $45$ degrees are read
downwards from the first column of function values, and from 45 to
90 degrees are read upwards from the second column of function values.
For cosines it is downwards from the second column and upwards from
the first column. The reversed footer line indicates the change of
columns to use. In the example note
\begin{itemize}
\item the setting of \verb`rvar'`\index{rvar@\texttt{rvar'}|)} to a function
\verb`90-\theta` of the row variable;
\item the footer setting \verb`foot=*` to obtain the reversed header in
the footer;
\item the rule \emph{above} the footer row specified by the \verb`f` added
to the \verb`rules` setting, \verb`rules=ThfB`.
\end{itemize}
Note also the degree setting \verb`o` in the settings option. 

Although there is a significant space saving with tables like this
(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 in the row variable column on the right (stemming
from $y!=\Gamma(x-1)$); \emph{HMF }Table 6.5 in effect uses $\langle1/x\rangle$
(the nearest integer to $1/x$) for the row variable on the right.\index{complementary functions|)}

\subsection{Separating blocks of rows: \texttt{rbloc}}

Readability\index{formatting!grouping rows|(}\index{row grouping|(}
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 \verb`rbloc`\index{rbloc@\texttt{rbloc}|(} key:
\begin{verbatim}
    rbloc = <comma list of positive integers>
\end{verbatim}
specifies how many rows belong to each block. For example, \verb`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 \verb`rbloc=5`
(strictly \verb`rbloc={5}` but the braces can be omitted in this
case since no comma is enclosed) divides a table into blocks of $5$
rows; \verb`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{}%
\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
mark to signal the end of the page or the table.\end{shaded}%
\end{minipage}

\medskip{}
In the next example I have divided the rows into blocks of $5$ by
means of the setting \verb`rbloc=5`.
\begin{verbatim}
    \tabulate[ff=;,o,rspec={\theta,10,1+90/10},
              rround=0,rbloc=5]
        { \sin\theta; \cos\theta }[\theta=0][*]
\end{verbatim}
$\Longrightarrow$     \tabulate[ff=;,o,rspec={\theta,10,1+90/10},
                          rround=0,rbloc=5]
		{ \sin \theta; \cos \theta}[\theta=0][*]\index{rbloc@\texttt{rbloc}|)}

\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
\verb`rblocsep=<length>`\index{rblocsep@\texttt{rblocsep}}. The
units need to be included in the specification. \index{formatting!grouping rows|)}\index{row grouping|)}

\subsection{Table placement}

\label{sec:Table-placement}Tables\index{table placement!on the page}
can be nudged vertically with the \LaTeX{} commands \verb`\bigskip`\index{bigskip@\texttt{\textbackslash bigskip}},
\verb`\medskip`\verb`\medskip`, \verb`\smallskip`\index{smallskip@\texttt{\textbackslash smallskip}},
usually about $1$, $\sfrac12$ and $\sfrac14$ line spaces (with
stretch and shrink). \texttt{booktabs}\index{booktabs@\texttt{booktabs}}
provides \verb`\abovetopsep`\index{abovetopsep@\texttt{\textbackslash abovetopsep}}
and \verb`\belowbottomsep`\index{belowbottomsep@\texttt{\textbackslash belowbottomsep}},
both set by default to \verb`0ex` 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). 

\subsubsection{Vertical~alignment}

\label{subsec:Verticalalignment}By\index{table placement!on the line}
writing\texttt{ }{\ttfamily\verb`valign=<char>`}\index{valign@\texttt{valign}}
where \verb`<char>` is one of \verb`t`, \verb`m` or \verb`b` the
vertical alignment of the table can be set relative to the text baseline.\texttt{ }\verb`valign=t`\texttt{
}aligns the top of the table with the text baseline, \verb`valign=b`
the bottom of the table with the text baseline, \verb`valign=m` aligns
the middle of the table with the text baseline. By default \verb`valign=m`
is set. 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][*] \quad
    B \tabulate[rspec={x,0.2,1+(1/0.2)}]
      { \tan x }[x=0][*] \quad
    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][*] \quad
    B \tabulate[rspec={x,0.2,1+(1/0.2)}]
      { \tan x }[x=0][*] \quad
    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 a top alignment is necessary (so that the header rows of
the tables align).

\section{Function value formatting}

\label{sec:Function-value-formatting}\index{formatting!function values|(}\index{function value formatting|(}
\begin{table}
\centering{}\caption{\protect\label{tab:Function-value-formatting}Formatting function
values}
\begin{center}
\begin{tabular}{cc>{\raggedright}p{5cm}c}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small initial}\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{round}} & {\small tokens} & {\small row or col. dependent rounding value} & \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}
\end{table}
In previous tables in this document, function values have generally
been limited to a fairly narrow range of values. What happens when
function values span orders of magnitude? Can we accommodate scientific
notation, expressly designed to cope with such orders of magnitude,
in a natural way? Can we round rows or columns to different rounding
values? Or, in a different direction, can we form tables of function
values in fraction form?

\subsection{Trailing optional argument}

\label{subsec:Trailing-optional-argument}The primary tool for function-value
formatting is the trailing optional argument of the \verb`\tabulate`
command where the rounding value\index{rounding value} is specified,
padding with zeros is set or not (generally \emph{set} in tables),
scientific notation is set or not, and fraction-form output can be
specified. 

\subsubsection{Fraction-form output}

\label{subsec:Fraction-form-output}In\index{fraction form!function values|(}\index{function value formatting!fraction form|(}
§\ref{subsec:rfracFraction-form-rvar} we saw how to display the row
variable in fraction form. Function values can also be presented in
that form. The problem is that such output requires far more computation
than other output since finding denominators at the specified accuracy
is an iterative process that needs to be performed for every function
value. However, it is feasible for small tables. In the tables below,
approximations to small positive and inverse powers of $\pi$ are
listed to $2$ and $4$ decimal places of accuracy. It is interesting
that all the powers listed can be approximated to $4$-place accuracy
by $3$-figure denominators (and $\pi^{2}$ by a $2$-figure denominator).\index{row variable!verbatim values|(}\index{sfrac@\texttt{\textbackslash sfrac}}
\begin{verbatim}
    \def\mydataiii{\pi,\pi^2,\pi^3,
        \pi^{\sfrac12},\pi^{\sfrac13}} 
    \tabulate[rdata=\mydataiii,rverb=1,rpos=1,
        rvar=k,ralign=l,chead={\small $2$ places}] 
      { k }[2/s] \qquad
    \tabulate[rdata=\mydataiii,rverb=1,rpos=1,
        rvar=k,ralign=l,chead={\small $4$ places}] 
      { k }[4/s]
\end{verbatim}
$\Longrightarrow$ \def\mydataiii{\pi,\pi^2,\pi^3,
    \pi^{\sfrac12},\pi^{\sfrac13}} 
    \tabulate[rdata=\mydataiii,rverb=1,rpos=1,
        rvar=k,ralign=l,chead={\small $2$ places}] 
     { k }[2/s]\qquad
\tabulate[rdata=\mydataiii,rverb=1,rpos=1,
        rvar=k,ralign=l,chead={\small $4$ places}] 
     { k }[4/s]\medskip{}

A second example shows that all four built-in constants to \texttt{numerica}
and their first few inverse powers can be approximated to $5$ decimal
places with $3$-figure denominators:\index{sfrac@\texttt{\textbackslash sfrac}}
\begin{verbatim}
    \tabulate[rdata={\pi ,e,\phi,\gamma},rverb=1,rvar=k,
        cspec={n,1,4},chstyle=3,chnudge=9,rules=TthB,/max=1000,
        ctitle=\lvert k^{\sfrac1n}-p/q \rvert<0.5\times10^{-5}] 
      { k^{\sfrac1n} }[n=1,k=1][/s5]
\end{verbatim}
$\Longrightarrow$     \tabulate[rdata={\pi ,e,\phi,\gamma},rverb=1,rvar=k,
        cspec={n,1,4},chstyle=3,chnudge=9,rules=TthB,/max=1000,
        ctitle=\lvert k^{\sfrac1n}-p/q \rvert<0.5\times10^{-5}] 
      { k^{\sfrac1n} }[n=1,k=1][/s5]\index{function value formatting!fraction form|)}\index{fraction form!function values|)}\index{row variable!verbatim values|)}

\subsubsection{Scientific notation}

Elegant\index{function value formatting!scientific notations|(} scientific
notation\index{scientific notation},\index{scientific notation|seealso{t-notation}}
set with an \verb`x` in the trailing optional argument, is generally
not appropriate for use in tables; see the first table below. Repeating
the \verb`x` \textendash{} \verb`xx` \textendash{} 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 distracting
and would be more so for a larger table. The \verb`x` specification
should be used in tables, if at all, only for \emph{small} tables
and special cases. The \verb`t` option is much preferred; see §\ref{subsec:t-option}
below.
\begin{verbatim}
    \tabulate[rspec={x,1,2*3+1},rround=0]
      { e^x}[x=-5][*x]\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,calign=l]
      { e^x}[x=-3][*xx]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,1,2*3+1},rround=0]
      { e^x}[x=-3][*x]\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,calign=l]
      { e^x}[x=-3][*xx]

\subsection{The \texttt{t} option}

\label{subsec:t-option}\emph{HMF}\index{t-notation|(} uses a special
notation\index{function value formatting|(seealso{t-notation}} for
coping with function values spanning orders of magnitude. This notation
can be invoked by inserting \verb`t` in the trailing optional argument.
Repeating the previous two tables, and adding a \verb`chnudge` value,
gives a more compact and visually appealing result:
\begin{verbatim}
    \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24]
      { e^x}[x=-3][*t]\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24]
      { e^x}[x=-3][*tt]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24]
      { e^x}[x=-3][*t] \qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,chnudge=24]
      { e^x}[x=-3][*tt]

\subsubsection{Padding the exponent: \texttt{(pad)}}

In\index{padding with phantoms|(} 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 \verb`numerica-tables` offers
the setting 
\begin{verbatim}
    (pad) = <integer>
\end{verbatim}
to achieve the effect. (The parentheses are part of the key \textendash{}
a reminder of the \verb`t`-form of scientific notation.) \verb`<integer>`
is the number of digits/characters to pad to. Repeating the last two
tables with the setting \verb`(pad)=2` produces the following results:
\begin{verbatim}
    \tabulate[rspec={x,1,2*3+1},rround=0,
                     chnudge=24,(pad)=2]
      { e^x}[x=-3][*t]\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,
                     chnudge=24,(pad)=2]
      { e^x}[x=-3][*tt]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,1,2*3+1},rround=0,
                       chnudge=24,(pad)=2]
      { e^x}[x=-3][*t]\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,
                       chnudge=24,(pad)=2]
      { e^x}[x=-3][*tt]\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.\index{padding with phantoms|)}

\paragraph{Accommodating signs: \texttt{signs}}

\label{subsec:Signs}Instead\index{t-notation!accommodating signs}\index{function value formatting!sign handling|(}
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 \verb`t`-notation the \emph{exponent} is the parenthesized integer
part of a number and the \emph{significand} the following decimal
figures. \verb`numerica-tables` offers the \verb`signs`\index{signs@\texttt{signs}|(}
key to align (or not) the exponents. The setting is
\begin{verbatim}
    signs = <integer>
\end{verbatim}
Besides the do-nothing default \verb`(signs=0)`, there are four effective
values for \verb`<integer>`:
\begin{itemize}
\item \verb`signs=2`\texttt{ }inserts a $+$ sign between exponent and
significand of every non-negative number;
\item \verb`signs=1`\texttt{ }inserts a $+$ sign between exponent and
significand of every non-negative number that immediately precedes
or follows a negative number;
\item \verb`signs=-1`\texttt{ }inserts a $+$ sign between exponent and
significand of any non-negative number that immediately precedes or
follows a negative number, and inserts a \emph{phantom} $+$ sign
between exponent and significand of every other non-negative number;
\item \verb`signs=-2`\texttt{ }inserts a \emph{phantom} $+$ sign between
exponent and significand of every non-negative number;
\end{itemize}
In the following examples, \verb`signs=-2`, \verb`signs=-1` and
\verb`signs=2`, all give acceptable results.
\begin{verbatim}
    \tabulate[rspec={x,1,2*3+1},rround=0,
              (pad)=2,signs=-2]
      { e^x-1}[x=-3][4*tt]\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,
              (pad)=2,signs=-1]
      { e^x-1}[x=-3][4*tt]\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,
              (pad)=2,signs=2]
      { e^x-1}[x=-3][4*tt]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,1,2*3+1},rround=0,
        (pad)=2,signs=-2]
      { e^x-1}[x=-3][4*t']\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,
        (pad)=2,signs=-1]
      { e^x-1}[x=-3][4*t']\qquad
    \tabulate[rspec={x,1,2*3+1},rround=0,
        (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`.

\verb`signs=1`, however, is an inappropriate setting for these function
values in the \verb`t`-notation:
\begin{verbatim}
    \tabulate[rspec={x,1,2*3+1},rround=0,
              (pad)=2,signs=1]
      { e^x-1}[x=-3][4*tt] \qquad
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,1,2*3+1},rround=0,
        (pad)=2,signs=1]
      { e^x-1}[x=-3][4*tt]\index{t-notation|)}\index{function value formatting!scientific notations|)}\index{padding with phantoms|)}\medskip{}


\subsection{Indicating signs outside the t-notation}

The \verb`signs` key is not limited to the \verb`t`-notation. In
the following tables where the notation is not used, positive values
for the key, including \verb`signs=1`\index{signs@\texttt{signs}},
give good results (I've included also the default setting \textendash{}
the third table):
\begin{verbatim}
    \tabulate[rspec={x,0.1,9},(pad)=2,signs=2]
      { 10\sin 5x}[x=-0.4][*4]\qquad
    \tabulate[rspec={x,0.1,9},(pad)=2,signs=1]
      { 10\sin 5x}[x=-0.4][*4]\qquad
    \tabulate[rspec={x,0.1,9},(pad)=2]
      { 10\sin 5x}[x=-0.4][*4]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,0.1,9},(pad)=2,signs=2]
      { 10\sin 5x}[x=-0.4][*4]\qquad
    \tabulate[rspec={x,0.1,9},(pad)=2,signs=1]
      { 10\sin 5x}[x=-0.4][*4]\qquad
    \tabulate[rspec={x,0.1,9},(pad)=2]
      { 10\sin 5x}[x=-0.4][*4]\medskip{}

\noindent\emph{HMF} seems to use \verb`signs=2` when the sign of
the function values changes every few entries and \verb`signs=1`
when there are runs of entries of the same sign. Over the range tabulated
here for $10\sin5x$, they would use the middle table of the three,
\verb`signs=1`.\index{function value formatting!sign handling|)}\index{signs@\texttt{signs}|)}

\subsection{Rounding to varying values}

\label{subsec:Rounding-varying} Above\index{rounding value!row/col-dependent|(},
in §\ref{subsec:Fraction-form-output}, we created two tables of fraction-form
approximations to simple power functions of $\pi$, one accurate to
two places of decimals, one to four places. Version 3.1 of \texttt{numerica-tables}
(as distinct from version 3.0) offers the means of producing tables
with rounding values depending on position in the table. This is effected
through the key \index{round@\texttt{round}|(}
\begin{verbatim}
    round = f(rvar,cvar)
\end{verbatim}
where \texttt{f(rvar,cvar)} denotes a function of row and column variables.
Usually this will mean dependence either on row variable or column
variable rather than both.  In the present instance we form a multi-function
table and with the \verb`round` key let the rounding value equal
the row variable value \verb`r` (\verb`round=r`) to obtain fractional
approximations to simple powers of $\pi$ at rounding values from
$1$ to $5$ (and discover that all these values can be approximated
to $5$ decimal places with $3$ figure denominators \textendash{}
$\pi^{2}$ only just).\index{multi-function tables}\index{fraction form!row/col-dependent rounding}
\begin{verbatim}
     \tabulate[ff,rspec={r,1,5},round=r,/max=999,chstyle=2,
            ctitle=\lvert\pi^k-\sfrac mn\rvert<0.5\times10^{-r}]
        { \pi,\pi^2,\pi^3,\pi^{1/2},\pi^{1/3}}[r=1][/s]
\end{verbatim}
$\Longrightarrow$      \tabulate[ff,rspec={r,1,5},round=r,/max=999,chstyle=2,
               ctitle=\lvert\pi^k-\sfrac mn\rvert<0.5\times10^{-r}]
       { \pi ,\pi^2,\pi^3,\pi^{1/2},\pi^{1/3}}[r=1][/s]\index{fraction form!function values}\index{function value formatting!fraction form}\medskip{}

Another place where a variable rounding value can be of value is when
a function being tabulated changes slowly for each step in the row
variable value; the value of the cosine for instance changes from
$1.0000$ to $\eval{\cos10\degree}[4]$ between $0\degree$ and $10\degree$.
\emph{Part} of a table of the cosine might be something like the following,
where values in the initial rows of the table are rounded to a higher
value than in later rows. \verb`round` is set to an expression involving
the row variable in boolean elements \verb`\theta<11` and \verb`\theta>10`
which evaluate to $0$ or $1$ so that \verb`round` takes the value
$6$ for the initial rows of the table and the value $4$ thereafter. 
\begin{verbatim}
    \tabulate[o,rspec={\theta,1,6},calign=l,chnudge=15,
              round=6(\theta<11)+4(\theta>10)] 
      { \cos\theta }[\theta=8]
\end{verbatim}
$\Longrightarrow$     \tabulate[o,rspec={\theta,1,6},calign=l,chnudge=15,
              round=6(\theta<11)+4(\theta>10)] 
      { \cos\theta }[\theta=8]\index{rounding value!row/col-dependent|)}\index{round@\texttt{round}|)}

\subsection{Differences: \texttt{diffs}}

In\index{differences|(} 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+(1.05-1)/0.01},rround=2,
      rhnudge=9,chnudge=21,diffs=3]
        { \sinh x }[x=1][*4]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},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.
With the default right alignment of the function-value columns, it
is easy to get this wrong. The evidence will be either 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 first table the first row of
function values is misaligned by one character. (\texttt{diffs=1}
would have produced a two-character misalignment.) In the second table
the unnecessary fourth digit for the differences results in pre-padding
with $0$. 

In the second table the function $-\sinh x$ is \emph{decreasing},
showing how it is the \emph{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+(1.05-1)/0.01},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=30,diffs=4]
        { -\sinh x }[x=1][*4]
\end{verbatim}
$\Longrightarrow$     \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},rround=2,
      rhnudge=9,chnudge=21,diffs=2]
        { \sinh x }[x=1][*4]\qquad
    \tabulate[rspec={x,0.01,1+(1.05-1)/0.01},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. (A left alignment of the function value column is another
way of tackling this issue.) 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. \index{differences|)}

\subsection{Formatting special values: \texttt{Q?} and \texttt{A!}}

You\index{function value formatting!highlighting special values|(}\index{Q?@\texttt{Q?}|(}\index{A@\texttt{A"!}|(}
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: \verb`Q?=<tokens>` and \verb`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', 0 for `false').
\emph{This is not a \LaTeX{} expression};\emph{ this is an }\verb`l3fp`
\emph{expression}.\footnote{Documentation about \texttt{l3fp }can be found in \texttt{interface3.pdf},
which is part of the \texttt{l3kernel} bundle.} For the user it should be enough to know that an expression formed
from decimal numbers (but only with the dot decimal point), parentheses
\verb`( )`, the familiar arithmetic symbols, \verb`+`,\texttt{ }\verb`-`,\texttt{
}\verb`*`,\texttt{ }\verb`/` and \verb`^`, relation symbols \verb`<`,\texttt{
}\verb`>`,\texttt{ }\verb`=` and combinations like \verb`!=` (for
$\ne$),\texttt{ }\verb`>=` (for $\ge$), and\texttt{ }\verb`<=`
(for $\le$) will be digested by \verb`l3fp`. In addition there are
\verb`||` for logical Or, \verb`&&` for logical And, and \verb`!`
for logical Not; \verb`exp(1)` for $e$ and \verb`pi` (no backslash)
for $\pi$. \texttt{numerica-tables} provides \verb`MAX`\index{MAX@\texttt{MAX}}
and \verb`MIN`\index{MIN@\texttt{MIN}} for the maximum and minimum
function values tabulated, and uses \verb`@` to denote the current
function value. 

So, a query might be \verb`Q?=@<0`, \emph{Is the current function
value negative?}, or \verb`Q?={@>=pi}`, \emph{Is the current function
value greater than or equal to $\pi$?} (The braces hide the equality
sign in the \emph{key=value} settings option.) \verb`Q?={@=MIN}`
(again note the braces) is the question: \emph{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 \verb`@` to denote the current function value. Thus \verb`A!=\mathbf{@}`
is a valid answer; so is \verb`A!=\color{red}{@}` (provided you have
\verb`\usepackage{color}` in the preamble); and so is \verb`A!=(@)`.
Another valid answer is \verb`A!= ` , meaning that function values
satisfying the \verb`Q?` question are omitted from the output. 

This can be useful to suppress `irrelevant' values in a particular
context. 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 `\verb`1e-14`'
in the query are there to prevent rounding errors confusing the result):
\begin{verbatim}
    \tabulate[rspec={n,1,1+(15-4)},rules=Tth,rround=0,
              rpos=2,cspec={m,1,1+(5-2)},chstyle=2,
              ctitle=*,Q?={@<-1e-14||@>0.5+1e-14},A!=]
      { \cos(m\pi/n) }[n=4,m=2][*4]
\end{verbatim}
$\Longrightarrow$      \tabulate
      [rspec={n,1,1+(15-4)},rround=0,rpos=2,rules=Tth,
         cspec={m,1,1+(5-2)},ctitle=*,chstyle=2,
           Q?={@<-1e-14||@>0.5+1e-14},A!=]
      { \cos(m\pi/n) }[n=4,m=2][*4]

\subsubsection{Star option: \texttt{\textbackslash nmcTabulate{*}}}

\label{sec:Star-option}If\index{nmcTabulate@\texttt{\textbackslash nmcTabulate}!star (*) option@star (\texttt{{*}}) option}
the \verb`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. Like other starred commands in the
\verb`numerica` suite (\verb`\eval*`, \verb`info*`, \verb`\macros*`,
\verb`\constants*`, \verb`\iter*`, \verb`\solve*` and \verb`\recur*`),
 \verb`\tabulate*` outputs a single number. Using the star means
you do not need an answering \verb`A!` to the query \verb`Q?` since
no formatting of table values is involved. 
\begin{verbatim}
    \tabulate*[rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)},
               Q?={@<-1e-14||@>0.5+1e-14}]
      { \cos(m\pi/n) }[n=4,m=2][*4]
\end{verbatim}
$\Longrightarrow$      \tabulate*
      [rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)},
           Q?={@<-1e-14||@>0.5+1e-14}]
      { \cos(m\pi/n) }[n=4,m=2][*4]. Indeed, if you omit the \verb`Q?` and \verb`A!` settings from
the previous table so that all function values are visible then this
is the value that follows $0.5000$ in the \verb`m=2` column \textendash{}
the first function value encountered outside the interval $[0,0.5]$.

If you want the \emph{maximum} value that has been tabulated then,
from version 3 of \texttt{numerica-tables}, you do not even need the
query: when \verb`\tabulate` is starred, \verb`Q?` is initialized
behind the scenes to \verb`@=MAX`\index{MAX@\texttt{MAX}}.\footnote{\noindent In the unlikely event that someone \emph{consistently} wanted
some other query to be asked \textendash{} the minimum value perhaps,
or first negative value or \ldots{} \textendash{} please let the author
know. It would be straightforward to use a package option to give
a choice in this matter.} Thus, repeating the example from §\ref{sec:Column-variable-settings},
\begin{verbatim}
    \tabulate[rspec={x,0.2,6},rhead=x\backslash k,
              cvar=k,cstep=2,cstop=9]
      { \sin kx }[k=3,x=0]
    \tabulate*[rspec={x,0.2,6},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},rhead=x\backslash k,
         cvar=k,cstep=2,cstop=9]
      { \sin kx }[k=3,x=0][*]   
\tabulate*
      [rspec={x,0.2,6},rhead=x\backslash k,
         cvar=k,cstep=2,cstop=9]
      { \sin kx }[k=3,x=0] [*]

\paragraph{Errors}

If\index{error messages} \emph{no} function value satisfies a query
then a message is generated:
\begin{verbatim}
    \tabulate*[rspec={n,1,1+(15-4)},
               cspec={m,1,1+(5-2)},Q?=@>1]
      { \cos(m\pi/n) }[n=4,m=2][*4]
\end{verbatim}
$\Longrightarrow$      \tabulate*
      [rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)},
           Q?=@>1]
      { \cos(m\pi/n) }[n=4,m=2][*4]\index{Q?@\texttt{Q?}|)}\index{A@\texttt{A"!}|)}

\paragraph{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
\verb`e`, but remember that if you enter the \verb`x` option you
will need to place the \verb`\tabulate*` command between math delimiters,
otherwise the \verb`\times` symbol resulting from the \verb`x` option
will generate a \LaTeX{} error (`Missing \$ inserted'):
\begin{verbatim}
    $ 
    \tabulate*[rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)},
               Q?={@<-1e-14||@>0.5+1e-14},A!=]
      { \cos(m\pi/n) }[n=4,m=2][*4x]
    $
\end{verbatim}
$\Longrightarrow$     $ 
    \tabulate*
      [rspec={n,1,1+(15-4)},cspec={m,1,1+(5-2)},
           Q?={@<-1e-14||@>0.5+1e-14},A!=]
      { \cos(m\pi/n) }[n=4,m=2][*4x]
    $.\index{formatting!function values|)}\index{function value formatting|)}\index{function value formatting!highlighting special values|)}

\section{Other matters}

Here I group items that do not fit naturally into the earlier categories.

\subsection{Nesting}

\label{subsec:Nesting}A\index{nmcTabulate@\texttt{\textbackslash nmcTabulate}!nesting}
\verb`\tabulate` command can be nested within other commands from
the \verb`numerica` suite, and those other commands can be nested
within a \verb`\tabulate` command.

Occasionally one might want to extract a value from a table to insert
in another command. This can be done by nesting a \verb`\tabulate*`
command with an appropriate \verb`Q?` setting within the other command.
In fact, from version 2 of \verb`numerica` on, the star is unnecessary.
All we require is that the \verb`Q?` setting is satisfied by at least
one tabulated function value. 
\begin{verbatim}
    \eval[env=$]{(\tabulate
      [rspec={n,1,15},cspec={m,1,5},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},cspec={m,1,5},
             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]. 

Forming the table
\begin{verbatim}
     \tabulate[rspec={n,1,15},rround=0,rpos=2,rules=Tth,
               cspec={m,1,5},ctitle=*,chstyle=2]
       { \cos(m\pi/n) }[n=4,m=2][*4]
\end{verbatim}
for the cosine and the table
\begin{verbatim}
    \tabulate[rspec={n,1,15},rround=0,rpos=2,rules=Tth,
              cspec={m,1,5},ctitle=*,chstyle=2]
      { \sin(m\pi/n) }[n=4,m=2][*4]
\end{verbatim}
for the sine and checking the entries shows that indeed the maximum
and minimum values are $0.9397$ and $-1.0000$ respectively.

If the \verb`Q?` setting is not satisfied by any function value a
familiar error message is shown \textendash{} with a tweak:
\begin{verbatim}
    \eval{$ (\tabulate
      [rspec={n,1,15},cspec={m,1,5},Q?=@>2]
         { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t 
          $}[t=2][4]
\end{verbatim}
$\Longrightarrow$     \eval{$ 
      (\tabulate
        [rspec={n,1,15},cspec={m,1,5},
          Q?=@>2] 
            { \cos(m\pi/n) }[n=4,m=2][*4])\sinh t 
          $}[t=2][4]

\noindent Here, the \verb`(2)` tells us that the message refers to
a command at the second level, a \emph{nested} command.

Perhaps a more likely situation is to want to nest other commands
within a \verb`\tabulate` command. I give an example in the documentation
to the associated package \verb`numerica-plus` around the timing
of signals between points fixed on a rotating disk. 

\subsection{Saving tables to file}

\label{sec:reuse-setting}In\index{tables!saving to file}\index{saving tables to file}
earlier versions of \texttt{numerica-tables} it was possible to save
a table to file, or a row or a column or a particular value from a
table, by giving a \emph{setting} \verb`reuse` a value. From version
3.0.0, in the interests of simplifying use (and avoiding code complications)
the \verb`reuse` \emph{setting} has been discontinued. The \verb`\reuse`
(or \verb`\nmcReuse`) \emph{command} remains (as part of the \texttt{numerica}
package) and can be used to save the most recent table to file. 

In the following example, a table is created and then saved to file
and to the macro \verb`\mytable` by the subsequent \verb`\reuse`
command:
\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]
      { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] 
    \reuse{mytable}
\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]
      { a\sin kx }[a=2/\pi,{k}=3,{x}=0][*] 
    \reuse[renew]{mytable}\medskip{}

\noindent Now test the content of the control sequence \medskip{}

\noindent\verb`\mytable` $\Longrightarrow$ \mytable \medskip{}

\noindent Certainly \verb`\mytable` contains the table. 

If we use the \verb`view` setting with \verb`\reuse` we can see
that \verb`\mytable` and its contents have also been saved to file:
\begin{verbatim}
    \reuse[view]{}
\end{verbatim}
$\Longrightarrow$ \reuse[view]{}

The file that \verb`\mytable` is saved to is the \texttt{.nmc} file
of the current document, hence \texttt{numerica-tables.nmc} in the
present instance. The contents of this file can be edited in a text
editor, or some limited file operations can be effected with the \verb`\reuse`
command. These have been described in the associated document \texttt{numerica.pdf}.

\subsection{Viewing the \protect\LaTeX{} form}

In\index{tables!viewing LaTeX form@viewing \LaTeX{} form} previous
versions of \texttt{numerica-tables} the \verb`dbg` and \verb`view`\index{view@\texttt{view}}
settings were disabled. In version 3, they have been enabled to the
extent that the \LaTeX{} form of a table can be viewed by entering
either \verb`dbg=11` or, less nerdishly, \verb`view` into the settings
option of \verb`\nmcTabulate`. In the example I first create the
table and then use the \verb`view` setting:
\begin{verbatim}
    
    \tabulate[view,rvar=x,rstep=0.2,rstop=1]
      { \sin x }[x=0]
\end{verbatim}
$\Longrightarrow$    \tabulate [view,rvar=x,rstep=0.2,rstop=1] 
      { \sin x }[x=0]

\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: uniform case §\ref{subsec:Row-var-spec-uniform}}
\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{rows\}}} & {\small short form spec.}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\par\end{center}

\subsubsection*{Row-variable specification: non-uniform case §\ref{subsec:Row-var-spec-nonuniform}}
\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{rdata}} & {\small comma list} & {\small list of row-var. values} & {\small may be stored in a macro}\tabularnewline
{\small\texttt{rfile}} & {\small chars} & {\small file of row-var. values} & {\small file path/name}\tabularnewline
{\small\texttt{rverb}} & {\small int (}{\small\texttt{0}}{\small /}{\small\texttt{1}}{\small )} & {\small display }{\small\texttt{rdata}}{\small{} or }{\small\texttt{rfile}}{\small{}
values verbatim} & {\small default }{\small\texttt{0}}\tabularnewline
{\small\texttt{rfunc}} & {\small token(s)} & {\small step function specifying row-var. values} & \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 initial}\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`rhnudge`}}{\small{}
mu} & {\small\texttt{0}}\tabularnewline
{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} &  & {\small\texttt{1}}\tabularnewline
{\small\texttt{rvar'}} & {\small token(s)} & {\small 2nd row variable col. spec.} & {\small\texttt{rvar}}\tabularnewline
{\small\texttt{rhead'}} & {\small token(s)} & {\small header of 2nd r-v col. (if it exists)} & {\small\texttt{rvar'}}\tabularnewline
{\small\texttt{rhnudge'}} & int & {\small nudge 2nd r-v col. header }{\small{\small\verb`rhnudge'`}}{\small{}
mu} & {\small\texttt{0}}\tabularnewline
{\small\texttt{rfrac}} & {\small int (}{\small\texttt{0}}{\small\ldots 5)} & {\small fraction form} & {\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{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 (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small header style} & {\small\texttt{0}}\tabularnewline
{\small\texttt{ctitle}} & {\small token(s)} & {\small single col. alternative header} & \tabularnewline
{\small\texttt{chead}} & {\small token(s)} & {\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`chnudge`}}{\small{}
mu} & {\small\texttt{0}}\tabularnewline
{\small\texttt{chround}} & {\small int} & {\small{} rounding} & {\small\texttt{0}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}\newpage{}
\par\end{center}

\subsubsection*{Function-value formatting §\ref{sec:Function-value-formatting}.}
\begin{center}
\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 token(s)} & {\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}.}
\begin{center}
\begin{center}
\begin{tabular}{ll>{\raggedright}p{4cm}l}
\toprule 
{\small key} & {\small type} & {\small meaning} & {\small default}\tabularnewline
\midrule
{\small\texttt{ctitle}} & {\small token(s)} & {\small collective title for function-value columns} & \tabularnewline
 & {\small token(s)} & {\small inter-header/title row for} {\small function-value columns} & \tabularnewline
{\small\texttt{header}} & {\small int (}{\small\texttt{0/1}}{\small )}  & {\small suppress/show header row}  & {\small\texttt{1}}\tabularnewline
{\small\texttt{rules}} & {\small char(s)} & {\small horizontal rules template} & {\small\texttt{ThB}}\tabularnewline
{\small\texttt{foot}} & {\small token(s)} & {\small content of footer line} & \tabularnewline
{\small\texttt{rpos}} & {\small int (}{\small\texttt{0}}{\small\ldots}{\small\texttt{4}}{\small )} & {\small row variable } & {\small\texttt{1}}\tabularnewline
{\small\texttt{rbloc}} & {\small comma list} & {\small division of rows into blocks} & \tabularnewline
{\small\texttt{valign}} & {\small char (}{\small\texttt{t}}/{\small\texttt{m}}/{\small\texttt{b}}{\small )} & {\small vertical alignment of table relative to text baseline} & {\small\texttt{m}}\tabularnewline
\bottomrule
\end{tabular}
\par\end{center}
\par\end{center}

\subsubsection*{Miscellaneous settings}

\verb`view`, equivalent to \verb`dbg=11`: show the \LaTeX{} expression
for the table.

{\small\printindex}{\small\par}
\end{document}