%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Typesetting REDUCE output with TeX %%%%%
%%%%% by Werner Antweiler %%%%%
%%%%% University of Cologne Computer Center %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\input tridefs
\def\9{\TeX-REDUCE-Interface}
%
% LaTeX Version
%
\documentstyle[twocolumn]{article}
\begin{document}
%
% Titel
%
\title{\Huge A \9}
\author{Werner Antweiler\thanks{All three authors are with:
Rechenzentrum der Universit\"{a}t zu K\"{o}ln
(University of Cologne Computer Center),
Abt. Anwendungssoftware (Application Software
Department), Robert-Koch-Stra\ss e 10, 5000 K\"{o}ln 41,
West Germany.} \\
Andreas Strotmann\\
Volker Winkelmann\\
University of Cologne Computer Center, West Germany}
\date{Revised Version, \today}
\maketitle
%
\section{Introduction}
%
REDUCE is a well known computer algebra system invented by
Anthony C. Hearn. While every effort was made to improve
the system's algebraic capabilities, the readability of the
output remained poor by modern typesetting standards.
Although a pretty-printer is already incorporated in REDUCE,
the output is produced only in line-printer quality.
The simple idea to produce high quality output from REDUCE
is to link REDUCE with Donald E. Knuth's famous \TeX\ typesetting
language. This draft reviews our efforts in this direction.
Our major goals we pursue with TRI are:
\begin{itemize}
\item We want to produce REDUCE-output in typesetting quality.
\item The intermediate files (\TeX-input files) should be easy to edit.
The reason is that it is likely that the proposed
line-breaks are sub-optimal from the user's point of view.
\item We apply a \TeX-like algorithm which ``optimizes''
the line-breaking over the whole expression. This differs
fundamentally from the standard left-to-right,
one-line look-ahead
pretty-printers of REDUCE, LISP and the like.
\end{itemize}
We introduce a program written in RLISP\footnote{the REDUCE
implementation language, an extended Standard LISP with an
ALGOL-like syntax.} which typesets REDUCE
formulas with \TeX. Our \9 incorporates three
levels of \TeX\ output: without line breaking, with line breaking,
and with line breaking plus indentation. While speed
without line breaking is comparable to that achieved with
REDUCE's pretty-printer, line breaking consumes much more CPU time.
Nevertheless, we reckon with a cost increase due to line breaking
which is almost linear in the length of the expression to be broken.
Our line breaking algorithm mimics \TeX's line breaking in a
rather rudimentary fashion.
This paper deals with some of the ideas and algorithms we have
programmed and it summarizes some of the experiments we have made
with our program.
Furthermore, at the end of this paper we provide a small user's manual
which gives a short introduction to the use of our \9.
For simplicity's sake the name ``\9'' will be abbreviated to ``TRI''
in this paper.
%
\section{From REDUCE to \TeX: basic concepts}
%
The function {\tt TeXvarpri} is the interface to the algebraic
portion of REDUCE. It first calls a function named {\tt makeprefix}.
Its job is to change a REDUCE algebraic expression to a standard
prefix list while retaining the tree structure of the whole expression.
After this has been done, the
new intermediate expression is passed to the most important module
of TRI: the {\tt maketag/makefunc}-family of functions.
These functions recursively expand the structured list
(or operator tree) into a flat list, translating each REDUCE symbol or
expression into so-called \TeX-items on passing by.
For that reason, the resulting list is called the \TeX-item list.
If the simple \TeX-mode (without line breaking) was chosen, this list
is then printed immediately without further considerations.
Translation and printing this way is almost as fast as with
the standard REDUCE pretty printer.\par
When line-breaking has been enabled things get a bit more complicated.
The greatest effort with TRI was to implement the line-breaking
algorithm. More than half of the entire TRI code deals with this
task.
The ultimate goal is to add some ``break items'', i.e. \verb|\nl|-%
\TeX-commands\footnote{This is not a \TeX-primitive but a TRI-specific
\TeX-macro command which expands into a lot of stuff.},
marking --- in a certain way --- optimal line-breaks.
Additionally, these break items can be followed immediately by
``indentation items'', i.e. \verb|\OFF{...}| \TeX-commands\footnote{see
previous footnote}, specifying the amount of indentation
(in scaled points) applicable for the next new line.
The problem is to choose the right points where to insert these
special \TeX-items. Therefore, the \TeX-item list undergoes
three further transformation steps.\par
First, the \TeX-item list gets enlarged by so-called ``glue items''.
Glue items are two-element lists, where the first element is a width
info and the second element is a penalty info.
The width is determined according to the class of the two \TeX-items
to the left and to the right. Each \TeX-item is associated with
one out of eight classes. Following Knuth's \TeX book [Knu86],
these classes are ORD (for ordinary items),
LOP (for large operators such as square roots),
BIN and REL (for binary and relational operators), OPN and CLO (for
opening and closing parentheses), PCT (for punctuation) and finally
INN (for ``inner'' items such as \TeX-brackets and -macros).
The data for the width information is stored in a matrix.
Therefore, for each pair of classes it is easy to determine how much
(if any) space should be inserted between two consecutive \TeX-items.
\par
The ``penalty'' is
a value in the range of $-10000\ldots+10000$ indicating a mark-up on a
potential break-point, thus determining if this point is a
fairly good (if negative) or bad (if positive) choice.
The amount of penalty depends heuristically
(a) on the kind of \TeX-items surrounding the glue item,
(b) on the bracket nesting, and finally
(c) on special characteristics.\footnote{For example, the plus- and
the difference operator have special impact on the amount of
penalty. These operators are considered as extremely good places
for breaking up a line.}\par
During the second level, the \TeX-item list is transformed
into a so-called breaklist consisting of {\em active} and
{\em passive} nodes.
A passive node is simply a width info giving the total width of
\TeX-items not interspersed by glue items. On the other hand,
active nodes are glue items enlarged by a third element, the
offset info, indicating an indentation level which is used later
for computing the actual amount of indentation.
Active nodes are used as potential breakpoints.
Moreover, while creating the breaklist, the \TeX-item list will be
modified if necessary according to the length of fractions and square
roots which cannot be broken if retained in their ``classical''
form. Hence fractions look like {\tt (...)/(...)} if they don't fit
into a single line, especially in the case of large
polynomial fractions.\par
The third and most important level is the line-breaking algorithm
itself. The idea how to break lines is based
on the article by Knuth/Plass(1981). Line-breaking can occur at
active nodes only. So, you can loop through the breaklist considering
all potential break-points. But in order to find a suitable way
in a reasonable amount of time you have to limit the number of
potential breakpoints considered. This is performed by associating
a ``badness'' with each potential breakpoint, describing how
good looking or bad looking a line turns out.
If the badness is less than
a given amount of ``tolerance'' --- as set by the user ---
then an active node is considered to be feasible and becomes
a delta node. A delta node is simply an active node enlarged
by four further infos: an identification number for this node,
a pointer to the best feasible break-point (i.e. delta-node)
to come from,%
\footnote{If one were to break the formula at this delta-node, the
best place to start this line is given by this pointer. Note that
this pointer points backward through the list. When all delta nodes
have been inserted, you start at the very last delta node (which
is always at the end of breaklist) and skip backward from one
delta node to the next delta node pointed at. Thus, you find the
best way to break up the list.}
the total amount of demerits (a compound value derived from
badness and penalty) accumulated so far, and a value indicating
the amount of indentation applied to a new line beginning at this
node.
When the function dealing with line-breaking has stepped through
the list, the breakpoints will have been determined.
Afterwards all glue items (i.e. active nodes)
are deleted from the \TeX-item list while break- and indentation-%
items for those nodes marked as break-points are inserted.
\par
Finally the \TeX-item list is printed with regular ASCII-characters.
The readabiltiy of the intermediate output is low,
but it should be quite good enough for users to do some final
editing work.
Nevertheless, the nesting structure of the term is kept
visible when printed, so it will be easy to distinguish between
parenthesis levels simply by considering the amount of indentation.
% ----------------------------------------------------------------------
\section{Postprocessing with the module ``tridefs.tex''}
% ----------------------------------------------------------------------
When a \TeX-output file has been created with TRI it has to be
processed by \TeX\ itself. But before you run \TeX\ you should make sure
the file looks the way you want it. Sometimes you will
find it necessary to add some \TeX-code of your own or delete some
other \TeX-code. This is also the right time to
check the line breaks marked by \verb|\nl|-commands.
This job is up to you before you finally run \TeX.\par
During the \TeX-run the sizes of brackets are determined. This task is
not done by TRI. In order to produce proper sized brackets
we put some \verb|\left(| and \verb|\right)| \TeX-commands
where brackets are opened or closed. A new problem arises when
an expression has been broken up into several lines. Since, for every
line, the number of \verb|\left(| and \verb|\right)|
\TeX-commands must match
but bracketed expressions may start in one line and end in another,
we have to insert the required number of ``dummy'' parentheses
(i.e. \verb|\right.| and \verb|\left.| \TeX-commands)
at the end of the current line and the beginning of the following line.
This task is handled by the \verb|\nl|-\TeX-macro.\par
There is a caveat against this method.
Since opening and closing brackets
needn't lie in the same line, it is possible that the height of the
brackets can differ although they should correspond in height. That will
happen if the height of the text in the opening line has a height
different from the text in the closing line. We haven't found a
way of tackling this problem yet, but we think it is possible to program
a small \TeX-macro for this task.\par
There is at least one more line of \TeX-code you have to
insert by hand into the \TeX-input file produced by TRI. This line runs
\begin{verbatim}
\input tridefs
\end{verbatim}
and inputs the module {\tt tridefs.tex} into the file. This is necessary
because otherwise \TeX\ won't know how to deal with our macro calls.
If you use the \TeX-input file as a ``stand-alone'' file, don't
forget a final \verb|\bye| at the end of the text. If you use
code produced by TRI as part of a larger text then simply put
the input-line just at the beginning of your text.
%
\section{Experiments}
%
We have tested TRI on a Micro-VAX operating under ULTRIX,
with no other users working during this phase in order to minimize
interference with other processes, e.g., caused by paging.
The TRI code has been compiled with the PSL~3.4 compiler.
The following table presents results obtained with a small number of
different terms. All data were measured in CPU-seconds
as displayed by LISP's TIME-facility.
For expressions where special packages such as {\tt solve} and {\tt int}
were involved we have taken only effective output-time, i.e. the time
consumption caused by producing the output and not by calculating
the algebraic result.\footnote{That means we assigned the result of an
evaluation to an itermediate variable, and then we printed this
intermediate variable. Thus we could eliminate the time overhead
produced by ``pure'' evaluation. Nevertheless, in terms of effective
interactive answering time, the sum of evaluation and printing time
might be much more interesting than the ``pure'' printing time.
In such a context the percentage overhead caused by printing is
the critical point. But since we talk about printing we decided
to document the ``pure'' printing time.} First we display
the six expressions we have tested.\par
$$(x+y)^{12}\eqno(1)$$
$$(x+y)^{24}\eqno(2)$$
$$(x+y)^{36}\eqno(3)$$
$$(x+y)^{16}/(v-w)^{16}\eqno(4)$$
$$solve((1+\xi)x^2-2\xi x+\xi,x)\eqno(5)$$
$$solve(x^3+x^2\mu+\nu,x)\eqno(6)$$
The following table shows the results:
%
\medskip
\begin{center}
{\tt
\begin{tabular}{|c|r|r|r|r|} \hline
{\small\rm No.}&{\small\rm normal}&{\small\rm TeX}&
{\small\rm Break}&{\small\rm Indent}\\
\hline
1& 0.65& 0.75& 3.30& 3.37\\
2& 1.38& 1.79&11.81&12.10\\
3& 2.31& 3.11&19.33&19.77\\
4& 1.87& 2.26&11.78& 9.64\\
5& 0.46& 0.75& 0.90& 0.88\\
6& 4.52&21.52&31.34&29.78\\
\hline
\end{tabular}
}
\medskip
\end{center}
%
This short table should give you an impression of the
performance of TRI. It goes without saying that on other machines
results may turn out which are quite different from our results. But our
intention is to show the relative and not the absolute performance.
Note that printing times are a function of expression complexity,
as shown by rows three and six.
% ----------------------------------------------------------------------
\section{User's Guide to the TRI}
% ----------------------------------------------------------------------
If you intend to use TRI you are required to load the compiled code.
This can be performed with the command
\begin{verbatim}
load!-package 'tri;
\end{verbatim}
During the load, some default initializations are performed. The
default page width is set to 15 centimeters, the tolerance for
line breaking is set to 20 by default. Moreover, TRI is enabled
to translate greek names, e.g. TAU or PSI, into equivalent \TeX\
symbols, e.g. $\tau$ or $\psi$, respectively. Letters are printed
lowercase as defined through assertion of the set LOWERCASE. The whole
operation produces the following lines of output
\begin{verbatim}
% TeX-REDUCE-Interface 0.50
% set GREEK asserted
% set LOWERCASE asserted
% \hsize=150mm
% \tolerance 20
\end{verbatim}
Make sure you have at least version 0.50 installed at your site.
Now you can switch the three TRI modes on and off as you like.
You can use the switches alternatively and incrementally.
That means you have to switch on TeX for receiving standard
\TeX-output, or TeXBreak to receive broken \TeX-output, or
TeXIndent to receive broken \TeX-output plus indentation.
More specifically, if you switch off {\tt TeXBreak} you implicitly quit
{\tt TeXIndent}, too, or, if you switch off {\tt TeX}, you
implicitly quit {\tt TeXBreak} and, consequently, {\tt TeXIndent}.\par
The most crucial point in defining how TRI breaks multiple lines
of \TeX-code is your choice of the page width and the tolerance.
As mentioned earlier, ``tolerance'' is related to \TeX's famous
line-breaking algorithm. The value of ``tolerance'' determines which
potential breakpoints are considered feasible and which are not. The
higher the tolerance, the more breakpoints become feasible as determined
by the value of ``badness'' associated with each breakpoint. Breakpoints
are considered feasible if the badness is less than the tolerance.
You can easily change the tolerance using
\begin{verbatim}
TeXtolerance(tolerance);
\end{verbatim}
where the {\em tolerance} is a positive integer in the closed interval
$[0,10000]$.
A tolerance of 0 means that actually no breakpoint will be considered
feasible (except those carrying a negative penalty), while a value
of 10000 allows any breakpoint to be considered feasible.
Obviously, the choice of a tolerance has a great impact on the time
consumption of our line-breaking algorithm since time consumption
increases in proportion to the number of feasible breakpoints.
So, the question is what values to choose. For line-breaking without
indentation, suitable values for the tolerance lie between 10 and 100.
As a rule of thumb, you should use higher values the deeper the term
is nested --- if you can estimate. If you use indentation, you have to
use much higher tolerance values. This is necessary because badness
is worsened by indentation. Accordingly, TRI has to try harder to find
suitable places where to break. Reasonable values for tolerance
here lie between 700 and 1500. A value of 1000 should be your first
guess. That will work for most expressions in a reasonable amount of
time.\par
The page width of the \TeX\ output page,
measured in millimeters\footnote{You can
also specify page width in scaled points (sp).
Note: 1~pt = 65536~sp = 1/72.27~inch. The function automatically chooses
the appropiate dimension according to the size: all values greater than
400 are considered to be scaled points.}, can be changed by using
\begin{verbatim}
TeXpagewidth(page-width);
\end{verbatim}
You should choose a page width according to your purposes,
but allow a few centimeters for errors in TRI's attempt to
emulate \TeX's metric.
For example, specify 140 millimeters for an effective page width of
150 or 160 millimeters. That way you have a certain safety-margin to
the borders of the page.\par
Sometimes you want to add your own translations for REDUCE-symbols
to be mapped to \TeX-items.
For such a task, TRI provides a function named
{\tt TeXlet} which binds any REDUCE-symbol to one of the predefined
\TeX-items. A call to this function has the following syntax:
\begin{verbatim}
TeXlet(REDUCE-symbol,TeX-item)
\end{verbatim}
Three examples show how to do it right:
\begin{verbatim}
TeXlet(velocity,'!v);
TeXlet(gamma,"\Gamma ");
TeXlet(acceleration,"\vartheta ");
\end{verbatim}
Besides this method of single assertions you can choose to assert
one of (currently) two standard sets providing substitutions
for lowercase and greek letters. These sets are loaded by default.
You can switch these sets on or off using the functions
\begin{verbatim}
TeXassertset setname;
TeXretractset setname;
\end{verbatim}
where the setnames {\tt GREEK} and {\tt LOWERCASE}
are currently defined and available.
So far you have learned only how to connect REDUCE-atoms with predefined
\TeX-items but not how to create new \TeX-items itself. We provide a
way for adding standard \TeX-items of any class {\tt ORD, BIN, REL,
OPN, CLO, PCT} and {\tt LOP} except for class {\tt INN} which is
reserved for internal use by TRI only. You can call the function
\begin{verbatim}
TeXitem(item,class,list-of-widths)
\end{verbatim}
e.g. together with a binding
\begin{verbatim}
TeXitem("\nabla ",ORD,
{546135,437818,377748});
TeXlet(NABLA,"\nabla ");
\end{verbatim}
where {\em item} is a legal \TeX-code name\footnote{Please note that
any \TeX-name ending with a letter must be followed by a blank
to prevent interference with letters of following \TeX-items.
Note also that you can legalize a name by defining it as a \TeX-macro
and declaring its width.},
{\em class} is one of seven classes (see above) and {\em list-of-widths}
is a non-empty list of elements, each one representing the width of
the item in successive super-/subscript depth levels.
That means that the first entry is the width in display mode,
the second stands for scriptstyle and the third stands for
scriptscriptstyle in \TeX-terminology.
Starting with version 0.50, all arguments can be supplied without
quotation marks, i.e., LISP notation is no longer required
but still possible.\par
But how can you retrieve the width information required?
For this purpose we provide a small interactive \TeX\ facility
called {\tt redwidth.tex}.
It repeatedly prompts you for the \TeX-items for which you want
to retrieve the width information.\par
Finally, another command is supplied which displays all
information stored about a specific \TeX-item.
If, for example, you call
\begin{verbatim}
TeXdisplay(NABLA);
\end{verbatim}
TRI will respond with
\begin{verbatim}
% TeX item \nabla is of class ORD and
has following widths:
% 8.333358pt 6.680572pt 5.763977pt
\end{verbatim}
%
\section{Examples}
%
Some examples shall demonstrate the capabilities of TRI.
For each example we state (a) the REDUCE command (i.e.
the input), (b) the tolerance if it differs from the default, and
(c) the output as produced in a \TeX\ run. The examples are
displayed at the end of this article.\par
%
\section{Caveats}
%
Techniques for printing mathematical expressions are available
everywhere. TRI adds only a highly specialized version for
most REDUCE output. The emphasis is on the word {\em most}.
One major caveat is that we cannot print SYMBOLIC-mode output
from REDUCE. This could be done best in a WEB-like programming-plus-%
documentation style. Nevertheless, as Knuth's WEB is allready available
for PASCAL and C, hopefully someone will write a LISP-WEB or a
REDUCE-WEB as well.\par
\LaTeX\ users will be disappointed that we have not mentioned yet if and
how TRI co-operates with \LaTeX. Nevertheless, TRI can be used
together with \LaTeX, as this document proves. But there are some
important restrictions. First, read in the module ``tridefs.tex''
at the very beginning of a \LaTeX-run, before you apply any
\LaTeX-specific command. Furthermore, the \TeX-macros used
in this module {\em do} interfere with \LaTeX's own macros.
You cannot use \LaTeX's \verb|\(|, \verb|\)|, \verb|\[| and \verb|\]|
commands while you are using ``tridefs.tex''. If you avoid to
use them you should be able to run TRI with \LaTeX\ smoothly.
\par
Whenever you discover a bug in our program please let us
know. Send us a short report accompanied by an output listing.%
\footnote{You can reach us via electronic mail at the following
address: reduce@rrz.uni-koeln.de} We will try to fix the error.
%
\section{Distribution}
%
The whole TRI package consists of following files:
\begin{itemize}
\item {\tt tri.latex}: This text as a \TeX-input file.
\item {\tt tri.tex}: A long version of this report, where we explain
our breaking algorithm thoroughly.
\item {\tt tri.red}: This is the REDUCE-LISP source code
for TRI (approximately 58 KBytes of code).
\item {\tt tridefs.tex}: The \TeX-input file to be used together with
output from TRI.
\item {\tt redwidth.tex}: This is the \TeX-input file for interactive
determination of \TeX-item widths.
\item {\tt tritest.red}: Run this REDUCE file to check if TRI works
correctly.
\item {\tt tritest.tex}: When you have run the file {\tt tritest.red},
just make a \TeX\ run with this file to see all the nice things
TRI is able to produce.
\end{itemize}
You can obtain TRI package from a network library at The
RAND Corporation, Santa Monica, Ca./USA.
Simply send a note {\tt send index} to the network address
{\tt reduce-netlib@rand.org} and you will receive a description
on how to use the REDUCE netlib followed by a directory.
Specific files may be requested with the command
{\tt send }{\em library file}, where {\em library} is one the
catalogues containing the different REDUCE packages, and {\em file}
is the name of a particular file. For example, you can use
the message {\tt send tex tri.red} to obtain the RLISP-code of
the \9.
% ----------------------------------------------------------------------
\begin{thebibliography}{Knu88}
\bibitem[Ant86]{an:be}Antweiler, W.; Strotmann, A.; Pfenning, Th.;
Winkelmann, V.: {\em Zwischenbericht \"{u}ber den Status der
Arbeiten am REDUCE-\TeX-Anschlu\ss.}
Internal Paper, Rechenzentrum an der Universit\"{a}t
zu K\"{o}ln, November 1986.
\bibitem[Ant89]{an:Ab}Antweiler, W.: {\em A \TeX{}-REDUCE-Interface.}
Arbeits\-bericht RRZK 8901 des Regionalen Rechen\-zentrums an der
Uni\-versi\-t\"{a}t zu K\"{o}ln, Februar 1989.
\bibitem[Fat87]{fa:Te}Fateman, Richard J.:
{\em\TeX\ Output from MACSYMA-like Systems.}
ACM SIGSAM Bulletin, Vol. 21, No. 4, Issue \#82, pp. 1--5,
November 1987.
\bibitem[Knu81]{kn:br}Knuth, Donald E.; Plass, Michael F.:
{\em Breaking Paragraphs into Lines.}
Software---Practice and Experience, Vol. 11,
pp. 1119--1184, 1981.
\bibitem[Knu86]{kn:Te}Knuth, Donald E.: {\em The \TeX\-book.}
Addison-Wesley, Readings/Ma. Sixth printing, 1986.
\bibitem[Hea87]{he:RE}Hearn, Anthony C.: {\em REDUCE User's Manual,
Version 3.3.} The RAND Corporation, Santa Monica, Ca.,
July 1987.
\end{thebibliography}
%
%
\onecolumn
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% %%%%%
%%%%% Examples for the TRI %%%%%
%%%%% %%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\TRIexa{Standard}{TeXindent}{250}{\verb|(x+y)**16/(v-w)**16|}
$$\displaylines{\qdd
\(x^{16}
+16\cdot x^{15}\cdot y
+120\cdot x^{14}\cdot y^{2}
+560\cdot x^{13}\cdot y^{3}\nl
\OFF{327680}
+1820\cdot x^{12}\cdot y^{4}
+4368\cdot x^{11}\cdot y^{5}
+8008\cdot x^{10}\cdot y^{6}\nl
\OFF{327680}
+11440\cdot x^{9}\cdot y^{7}
+12870\cdot x^{8}\cdot y^{8}
+11440\cdot x^{7}\cdot y^{9}\nl
\OFF{327680}
+8008\cdot x^{6}\cdot y^{10}
+4368\cdot x^{5}\cdot y^{11}
+1820\cdot x^{4}\cdot y^{12}\nl
\OFF{327680}
+560\cdot x^{3}\cdot y^{13}
+120\cdot x^{2}\cdot y^{14}
+16\cdot x\cdot y^{15}
+y^{16}
\)
/\nl
\(v^{16}
-16\cdot v^{15}\cdot w
+120\cdot v^{14}\cdot w^{2}
-560\cdot v^{13}\cdot w^{3}\nl
\OFF{327680}
+1820\cdot v^{12}\cdot w^{4}
-4368\cdot v^{11}\cdot w^{5}
+8008\cdot v^{10}\cdot w^{6}
-11440\cdot v^{9}\cdot w^{7}\nl
\OFF{327680}
+12870\cdot v^{8}\cdot w^{8}
-11440\cdot v^{7}\cdot w^{9}
+8008\cdot v^{6}\cdot w^{10}
-4368\cdot v^{5}\cdot w^{11}\nl
\OFF{327680}
+1820\cdot v^{4}\cdot w^{12}
-560\cdot v^{3}\cdot w^{13}
+120\cdot v^{2}\cdot w^{14}
-16\cdot v\cdot w^{15}
+w^{16}
\)
\Nl}$$
\TRIexa{Integration}{TeX}{-}{\verb|int(1/(x**3+2),x)|}
$$
-
\(\frac{2^{\frac{1}{
3}}\cdot
\(2\cdot
\sqrt{3}\cdot
\arctan
\(\frac{2^{\frac{1}{
3}}
-2\cdot x}{
2^{\frac{1}{
3}}\cdot
\sqrt{3}}
\)
+
\ln
\(2^{\frac{2}{
3}}
-2^{\frac{1}{
3}}\cdot x
+x^{2}
\)
-2\cdot
\ln
\(2^{\frac{1}{
3}}
+x
\)
\)
}{
12}
\)
$$
\TRIexa{Integration}{TeXindent}{1000}%
{\verb|int(1/(x{*}{*}4+3*x{*}{*}2-1,x)|}
$$\displaylines{\qdd
\(\sqrt{2}\cdot
\(3\cdot
\sqrt{
\sqrt{13}
-3}\cdot
\sqrt{13}\cdot
\ln
\(-
\(\sqrt{
\sqrt{13}
-3}\cdot
\sqrt{2}
\)
+2\cdot x
\)
\nl
\OFF{2260991}
-3\cdot
\sqrt{
\sqrt{13}
-3}\cdot
\sqrt{13}\cdot
\ln
\(\sqrt{
\sqrt{13}
-3}\cdot
\sqrt{2}
+2\cdot x
\)
\nl
\OFF{2260991}
+13\cdot
\sqrt{
\sqrt{13}
-3}\cdot
\ln
\(-
\(\sqrt{
\sqrt{13}
-3}\cdot
\sqrt{2}
\)
+2\cdot x
\)
\nl
\OFF{2260991}
-13\cdot
\sqrt{
\sqrt{13}
-3}\cdot
\ln
\(\sqrt{
\sqrt{13}
-3}\cdot
\sqrt{2}
+2\cdot x
\)
\nl
\OFF{2260991}
+6\cdot
\sqrt{
\sqrt{13}
+3}\cdot
\sqrt{13}\cdot
\arctan
\(\frac{2\cdot x}{
\sqrt{
\sqrt{13}
+3}\cdot
\sqrt{2}}
\)
\nl
\OFF{2260991}
-26\cdot
\sqrt{
\sqrt{13}
+3}\cdot
\arctan
\(\frac{2\cdot x}{
\sqrt{
\sqrt{13}
+3}\cdot
\sqrt{2}}
\)
\)
\)
/104
\Nl}$$
\TRIexa{Solving Equations}{TeXindent}{1000}%
{\verb|solve(x**3+x**2*mu+nu=0,x)|}
$$\displaylines{\qdd
\{x=
\[-
\(\(\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{2}{
3}}\cdot
\sqrt{3}\cdot i\nl
\OFF{3675021}
+
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{2}{
3}}\nl
\OFF{3675021}
+2\cdot
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{1}{
3}}\cdot \nl
\OFF{3675021}
2^{\frac{1}{
3}}\cdot 3^{\frac{1}{
6}}\cdot
\mu
-2^{\frac{2}{
3}}\cdot
\sqrt{3}\cdot 3^{\frac{1}{
3}}\cdot
i\cdot \mu ^{2}
+2^{\frac{2}{
3}}\cdot 3^{\frac{1}{
3}}\cdot
\mu ^{2}
\)
/\nl
\OFF{3347341}
\(6\cdot
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{1}{
3}}\cdot 2^{\frac{1}{
3}}\cdot
3^{\frac{1}{
6}}
\)
\)
\]
\CO\nl
\OFF{327680}
x=
\[\(\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{2}{
3}}\cdot
\sqrt{3}\cdot i\nl
\OFF{2837617}
-
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{2}{
3}}\nl
\OFF{2837617}
-2\cdot
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{1}{
3}}\cdot \nl
\OFF{2837617}
2^{\frac{1}{
3}}\cdot 3^{\frac{1}{
6}}\cdot \mu
-2^{\frac{2}{
3}}\cdot
\sqrt{3}\cdot 3^{\frac{1}{
3}}\cdot i\cdot
\mu ^{2}
-2^{\frac{2}{
3}}\cdot 3^{\frac{1}{
3}}\cdot
\mu ^{2}
\)
/\nl
\OFF{2509937}
\(6\cdot
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{1}{
3}}\cdot 2^{\frac{1}{
3}}\cdot 3^{
\frac{1}{
6}}
\)
\]
\CO\nl
\OFF{327680}
x=
\[\(\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{2}{
3}}\nl
\OFF{2837617}
-
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot \nl
\OFF{3675021}
\sqrt{3}\cdot \nu
\)
^{\frac{1}{
3}}\cdot 2^{\frac{1}{
3}}\cdot 3^{
\frac{1}{
6}}\cdot \mu
+2^{\frac{2}{
3}}\cdot 3^{\frac{1}{
3}}\cdot
\mu ^{2}
\)
/\nl
\OFF{2509937}
\(3\cdot
\(9\cdot
\sqrt{4\cdot \mu ^{3}\cdot \nu
+27\cdot \nu ^{2}}
-2\cdot
\sqrt{3}\cdot \mu ^{3}
-27\cdot
\sqrt{3}\cdot \nu
\)
^{\frac{1}{
3}}\cdot 2^{\frac{1}{
3}}\cdot 3^{
\frac{1}{
6}}
\)
\]
\}
\Nl}$$
\TRIexa{Matrix Printing}{TeX}{--}{%
\verb|mat((1,a-b,1/(c-d)),(a**2-b**2,1,sqrt(c)),((a+b)/(c-d),sqrt(d),1))|
}
$$
\pmatrix{1&a
-b&
\frac{1}{
c
-d}\cr
a^{2}
-b^{2}&1&
\sqrt{c}\cr
\frac{a
+b}{
c
-d}&
\sqrt{d}&1\cr
}
$$
%
%
\end{document}