[ginac.git] / doc / tutorial / ginac.texi

\input texinfo  @c -*-texinfo-*-
@c %**start of header
@setfilename ginac.info
@settitle GiNaC, an open framework for symbolic computation within the C++ programming language
@setchapternewpage on
@afourpaper
@c For `info' only.
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@iftex
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@c %**end of header

@include version.texi

@dircategory Mathematics
@direntry
* ginac: (ginac).                   C++ library for symbolic computation.
@end direntry

@ifinfo
This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.

Copyright (C) 1999-2022 Johannes Gutenberg University Mainz, Germany

Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
are preserved on all copies.

@ignore
Permission is granted to process this file through TeX and print the
results, provided the printed document carries copying permission
notice identical to this one except for the removal of this paragraph

@end ignore
Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the entire
resulting derived work is distributed under the terms of a permission
notice identical to this one.
@end ifinfo

@finalout
@c finalout prevents ugly black rectangles on overfull hbox lines
@titlepage
@title GiNaC @value{VERSION}
@subtitle An open framework for symbolic computation within the C++ programming language
@subtitle @value{UPDATED}
@author @uref{https://www.ginac.de}

@page
@vskip 0pt plus 1filll
Copyright @copyright{} 1999-2022 Johannes Gutenberg University Mainz, Germany
@sp 2
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
are preserved on all copies.

Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the entire
resulting derived work is distributed under the terms of a permission
notice identical to this one.
@end titlepage

@page
@contents

@page


@node Top, Introduction, (dir), (dir)
@c    node-name, next, previous, up
@top GiNaC

This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.

@menu
* Introduction::                 GiNaC's purpose.
* A tour of GiNaC::              A quick tour of the library.
* Installation::                 How to install the package.
* Basic concepts::               Description of fundamental classes.
* Methods and functions::        Algorithms for symbolic manipulations.
* Extending GiNaC::              How to extend the library.
* A comparison with other CAS::  Compares GiNaC to traditional CAS.
* Internal structures::          Description of some internal structures.
* Package tools::                Configuring packages to work with GiNaC.
* Bibliography::
* Concept index::
@end menu


@node Introduction, A tour of GiNaC, Top, Top
@c    node-name, next, previous, up
@chapter Introduction
@cindex history of GiNaC

The motivation behind GiNaC derives from the observation that most
present day computer algebra systems (CAS) are linguistically and
semantically impoverished.  Although they are quite powerful tools for
learning math and solving particular problems they lack modern
linguistic structures that allow for the creation of large-scale
projects.  GiNaC is an attempt to overcome this situation by extending a
well established and standardized computer language (C++) by some
fundamental symbolic capabilities, thus allowing for integrated systems
that embed symbolic manipulations together with more established areas
of computer science (like computation-intense numeric applications,
graphical interfaces, etc.) under one roof.

The particular problem that led to the writing of the GiNaC framework is
still a very active field of research, namely the calculation of higher
order corrections to elementary particle interactions.  There,
theoretical physicists are interested in matching present day theories
against experiments taking place at particle accelerators.  The
computations involved are so complex they call for a combined symbolical
and numerical approach.  This turned out to be quite difficult to
accomplish with the present day CAS we have worked with so far and so we
tried to fill the gap by writing GiNaC.  But of course its applications
are in no way restricted to theoretical physics.

This tutorial is intended for the novice user who is new to GiNaC but
already has some background in C++ programming.  However, since a
hand-made documentation like this one is difficult to keep in sync with
the development, the actual documentation is inside the sources in the
form of comments.  That documentation may be parsed by one of the many
Javadoc-like documentation systems.  If you fail at generating it you
may access it from @uref{https://www.ginac.de/reference/, the GiNaC home
page}.  It is an invaluable resource not only for the advanced user who
wishes to extend the system (or chase bugs) but for everybody who wants
to comprehend the inner workings of GiNaC.  This little tutorial on the
other hand only covers the basic things that are unlikely to change in
the near future.

@section License
The GiNaC framework for symbolic computation within the C++ programming
language is Copyright @copyright{} 1999-2021 Johannes Gutenberg
University Mainz, Germany.

This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License as
published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; see the file COPYING.  If not, write to the
Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
MA 02110-1301, USA.


@node A tour of GiNaC, How to use it from within C++, Introduction, Top
@c    node-name, next, previous, up
@chapter A Tour of GiNaC

This quick tour of GiNaC wants to arise your interest in the
subsequent chapters by showing off a bit.  Please excuse us if it
leaves many open questions.

@menu
* How to use it from within C++::  Two simple examples.
* What it can do for you::         A Tour of GiNaC's features.
@end menu


@node How to use it from within C++, What it can do for you, A tour of GiNaC, A tour of GiNaC
@c    node-name, next, previous, up
@section How to use it from within C++

The GiNaC open framework for symbolic computation within the C++ programming
language does not try to define a language of its own as conventional
CAS do.  Instead, it extends the capabilities of C++ by symbolic
manipulations.  Here is how to generate and print a simple (and rather
pointless) bivariate polynomial with some large coefficients:

@example
#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

int main()
@{
    symbol x("x"), y("y");
    ex poly;

    for (int i=0; i<3; ++i)
        poly += factorial(i+16)*pow(x,i)*pow(y,2-i);

    cout << poly << endl;
    return 0;
@}
@end example

Assuming the file is called @file{hello.cc}, on our system we can compile
and run it like this:

@example
$ c++ hello.cc -o hello -lginac -lcln
$ ./hello
355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
@end example

(@xref{Package tools}, for tools that help you when creating a software
package that uses GiNaC.)

@cindex Hermite polynomial
Next, there is a more meaningful C++ program that calls a function which
generates Hermite polynomials in a specified free variable.

@example
#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

ex HermitePoly(const symbol & x, int n)
@{
    ex HKer=exp(-pow(x, 2));
    // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
    return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
@}

int main()
@{
    symbol z("z");

    for (int i=0; i<6; ++i)
        cout << "H_" << i << "(z) == " << HermitePoly(z,i) << endl;

    return 0;
@}
@end example

When run, this will type out

@example
H_0(z) == 1
H_1(z) == 2*z
H_2(z) == 4*z^2-2
H_3(z) == -12*z+8*z^3
H_4(z) == -48*z^2+16*z^4+12
H_5(z) == 120*z-160*z^3+32*z^5
@end example

This method of generating the coefficients is of course far from optimal
for production purposes.

In order to show some more examples of what GiNaC can do we will now use
the @command{ginsh}, a simple GiNaC interactive shell that provides a
convenient window into GiNaC's capabilities.


@node What it can do for you, Installation, How to use it from within C++, A tour of GiNaC
@c    node-name, next, previous, up
@section What it can do for you

@cindex @command{ginsh}
After invoking @command{ginsh} one can test and experiment with GiNaC's
features much like in other Computer Algebra Systems except that it does
not provide programming constructs like loops or conditionals.  For a
concise description of the @command{ginsh} syntax we refer to its
accompanied man page. Suffice to say that assignments and comparisons in
@command{ginsh} are written as they are in C, i.e. @code{=} assigns and
@code{==} compares.

It can manipulate arbitrary precision integers in a very fast way.
Rational numbers are automatically converted to fractions of coprime
integers:

@example
> x=3^150;
369988485035126972924700782451696644186473100389722973815184405301748249
> y=3^149;
123329495011708990974900260817232214728824366796574324605061468433916083
> x/y;
3
> y/x;
1/3
@end example

Exact numbers are always retained as exact numbers and only evaluated as
floating point numbers if requested.  For instance, with numeric
radicals is dealt pretty much as with symbols.  Products of sums of them
can be expanded:

@example
> expand((1+a^(1/5)-a^(2/5))^3);
1+3*a+3*a^(1/5)-5*a^(3/5)-a^(6/5)
> expand((1+3^(1/5)-3^(2/5))^3);
10-5*3^(3/5)
> evalf((1+3^(1/5)-3^(2/5))^3);
0.33408977534118624228
@end example

The function @code{evalf} that was used above converts any number in
GiNaC's expressions into floating point numbers.  This can be done to
arbitrary predefined accuracy:

@example
> evalf(1/7);
0.14285714285714285714
> Digits=150;
150
> evalf(1/7);
0.1428571428571428571428571428571428571428571428571428571428571428571428
5714285714285714285714285714285714285
@end example

Exact numbers other than rationals that can be manipulated in GiNaC
include predefined constants like Archimedes' @code{Pi}.  They can both
be used in symbolic manipulations (as an exact number) as well as in
numeric expressions (as an inexact number):

@example
> a=Pi^2+x;
x+Pi^2
> evalf(a);
9.869604401089358619+x
> x=2;
2
> evalf(a);
11.869604401089358619
@end example

Built-in functions evaluate immediately to exact numbers if
this is possible.  Conversions that can be safely performed are done
immediately; conversions that are not generally valid are not done:

@example
> cos(42*Pi);
1
> cos(acos(x));
x
> acos(cos(x));
acos(cos(x))
@end example

(Note that converting the last input to @code{x} would allow one to
conclude that @code{42*Pi} is equal to @code{0}.)

Linear equation systems can be solved along with basic linear
algebra manipulations over symbolic expressions.  In C++ GiNaC offers
a matrix class for this purpose but we can see what it can do using
@command{ginsh}'s bracket notation to type them in:

@example
> lsolve(a+x*y==z,x);
y^(-1)*(z-a);
> lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
@{x==19/8,y==-1/40@}
> M = [ [1, 3], [-3, 2] ];
[[1,3],[-3,2]]
> determinant(M);
11
> charpoly(M,lambda);
lambda^2-3*lambda+11
> A = [ [1, 1], [2, -1] ];
[[1,1],[2,-1]]
> A+2*M;
[[1,1],[2,-1]]+2*[[1,3],[-3,2]]
> evalm(%);
[[3,7],[-4,3]]
> B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
> evalm(B^(2^12345));
[[1,0,0],[0,1,0],[0,0,1]]
@end example

Multivariate polynomials and rational functions may be expanded,
collected, factorized, and normalized (i.e. converted to a ratio of
two coprime polynomials):

@example
> a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
> b = x^2 + 4*x*y - y^2;
4*x*y-y^2+x^2
> expand(a*b);
8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
> factor(%);
(4*x*y+x^2-y^2)^2*(x^2+3*y^2)
> collect(a+b,x);
4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
> collect(a+b,y);
12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
> normal(a/b);
3*y^2+x^2
@end example

Here we have made use of the @command{ginsh}-command @code{%} to pop the
previously evaluated element from @command{ginsh}'s internal stack.

You can differentiate functions and expand them as Taylor or Laurent
series in a very natural syntax (the second argument of @code{series} is
a relation defining the evaluation point, the third specifies the
order):

@cindex Zeta function
@example
> diff(tan(x),x);
tan(x)^2+1
> series(sin(x),x==0,4);
x-1/6*x^3+Order(x^4)
> series(1/tan(x),x==0,4);
x^(-1)-1/3*x+Order(x^2)
> series(tgamma(x),x==0,3);
x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
(-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
> evalf(%);
x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
-(0.90747907608088628905)*x^2+Order(x^3)
> series(tgamma(2*sin(x)-2),x==Pi/2,6);
-(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
-Euler-1/12+Order((x-1/2*Pi)^3)
@end example

Often, functions don't have roots in closed form.  Nevertheless, it's
quite easy to compute a solution numerically, to arbitrary precision:

@cindex fsolve
@example
> Digits=50:
> fsolve(cos(x)==x,x,0,2);
0.7390851332151606416553120876738734040134117589007574649658
> f=exp(sin(x))-x:
> X=fsolve(f,x,-10,10);
2.2191071489137460325957851882042901681753665565320678854155
> subs(f,x==X);
-6.372367644529809108115521591070847222364418220770475144296E-58
@end example

Notice how the final result above differs slightly from zero by about
@math{6*10^(-58)}.  This is because with 50 decimal digits precision the
root cannot be represented more accurately than @code{X}.  Such
inaccuracies are to be expected when computing with finite floating
point values.

If you ever wanted to convert units in C or C++ and found this is
cumbersome, here is the solution.  Symbolic types can always be used as
tags for different types of objects.  Converting from wrong units to the
metric system is now easy:

@example
> in=.0254*m;
0.0254*m
> lb=.45359237*kg;
0.45359237*kg
> 200*lb/in^2;
140613.91592783185568*kg*m^(-2)
@end example


@node Installation, Prerequisites, What it can do for you, Top
@c    node-name, next, previous, up
@chapter Installation

@cindex CLN
GiNaC's installation follows the spirit of most GNU software. It is
easily installed on your system by three steps: configuration, build,
installation.

@menu
* Prerequisites::                Packages upon which GiNaC depends.
* Configuration::                How to configure GiNaC.
* Building GiNaC::               How to compile GiNaC.
* Installing GiNaC::             How to install GiNaC on your system.
@end menu


@node Prerequisites, Configuration, Installation, Installation
@c    node-name, next, previous, up
@section Prerequisites

In order to install GiNaC on your system, some prerequisites need to be
met.  First of all, you need to have a C++-compiler adhering to the
ISO standard @cite{ISO/IEC 14882:2011(E)}.  We used GCC for development
so if you have a different compiler you are on your own.  For the
configuration to succeed you need a Posix compliant shell installed in
@file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
required for the configuration, it can be downloaded from
@uref{http://pkg-config.freedesktop.org}.
Last but not least, the CLN library
is used extensively and needs to be installed on your system.
Please get it from @uref{https://www.ginac.de/CLN/} (it is licensed under
the GPL) and install it prior to trying to install GiNaC.  The configure
script checks if it can find it and if it cannot, it will refuse to
continue.


@node Configuration, Building GiNaC, Prerequisites, Installation
@c    node-name, next, previous, up
@section Configuration
@cindex configuration
@cindex Autoconf

To configure GiNaC means to prepare the source distribution for
building.  It is done via a shell script called @command{configure} that
is shipped with the sources and was originally generated by GNU
Autoconf.  Since a configure script generated by GNU Autoconf never
prompts, all customization must be done either via command line
parameters or environment variables.  It accepts a list of parameters,
the complete set of which can be listed by calling it with the
@option{--help} option.  The most important ones will be shortly
described in what follows:

@itemize @bullet

@item
@option{--disable-shared}: When given, this option switches off the
build of a shared library, i.e. a @file{.so} file.  This may be convenient
when developing because it considerably speeds up compilation.

@item
@option{--prefix=@var{PREFIX}}: The directory where the compiled library
and headers are installed. It defaults to @file{/usr/local} which means
that the library is installed in the directory @file{/usr/local/lib},
the header files in @file{/usr/local/include/ginac} and the documentation
(like this one) into @file{/usr/local/share/doc/GiNaC}.

@item
@option{--libdir=@var{LIBDIR}}: Use this option in case you want to have
the library installed in some other directory than
@file{@var{PREFIX}/lib/}.

@item
@option{--includedir=@var{INCLUDEDIR}}: Use this option in case you want
to have the header files installed in some other directory than
@file{@var{PREFIX}/include/ginac/}. For instance, if you specify
@option{--includedir=/usr/include} you will end up with the header files
sitting in the directory @file{/usr/include/ginac/}. Note that the
subdirectory @file{ginac} is enforced by this process in order to
keep the header files separated from others.  This avoids some
clashes and allows for an easier deinstallation of GiNaC. This ought
to be considered A Good Thing (tm).

@item
@option{--datadir=@var{DATADIR}}: This option may be given in case you
want to have the documentation installed in some other directory than
@file{@var{PREFIX}/share/doc/GiNaC/}.

@end itemize

In addition, you may specify some environment variables.  @env{CXX}
holds the path and the name of the C++ compiler in case you want to
override the default in your path.  (The @command{configure} script
searches your path for @command{c++}, @command{g++}, @command{gcc},
@command{CC}, @command{cxx} and @command{cc++} in that order.)  It may
be very useful to define some compiler flags with the @env{CXXFLAGS}
environment variable, like optimization, debugging information and
warning levels.  If omitted, it defaults to @option{-g
-O2}.@footnote{The @command{configure} script is itself generated from
the file @file{configure.ac}.  It is only distributed in packaged
releases of GiNaC.  If you got the naked sources, e.g. from git, you
must generate @command{configure} along with the various
@file{Makefile.in} by using the @command{autoreconf} utility.  This will
require a fair amount of support from your local toolchain, though.}

The whole process is illustrated in the following two
examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
@command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
your login shell.)

Here is a simple configuration for a site-wide GiNaC library assuming
everything is in default paths:

@example
$ export CXXFLAGS="-Wall -O2"
$ ./configure
@end example

And here is a configuration for a private static GiNaC library with
several components sitting in custom places (site-wide GCC and private
CLN).  The compiler is persuaded to be picky and full assertions and
debugging information are switched on:

@example
$ export CXX=/usr/local/gnu/bin/c++
$ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
$ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
$ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
$ ./configure --disable-shared --prefix=$(HOME)
@end example


@node Building GiNaC, Installing GiNaC, Configuration, Installation
@c    node-name, next, previous, up
@section Building GiNaC
@cindex building GiNaC

After proper configuration you should just build the whole
library by typing
@example
$ make
@end example
at the command prompt and go for a cup of coffee.  The exact time it
takes to compile GiNaC depends not only on the speed of your machines
but also on other parameters, for instance what value for @env{CXXFLAGS}
you entered.  Optimization may be very time-consuming.

Just to make sure GiNaC works properly you may run a collection of
regression tests by typing

@example
$ make check
@end example

This will compile some sample programs, run them and check the output
for correctness.  The regression tests fall in three categories.  First,
the so called @emph{exams} are performed, simple tests where some
predefined input is evaluated (like a pupils' exam).  Second, the
@emph{checks} test the coherence of results among each other with
possible random input.  Third, some @emph{timings} are performed, which
benchmark some predefined problems with different sizes and display the
CPU time used in seconds.  Each individual test should return a message
@samp{passed}.  This is mostly intended to be a QA-check if something
was broken during development, not a sanity check of your system.  Some
of the tests in sections @emph{checks} and @emph{timings} may require
insane amounts of memory and CPU time.  Feel free to kill them if your
machine catches fire.  Another quite important intent is to allow people
to fiddle around with optimization.

By default, the only documentation that will be built is this tutorial
in @file{.info} format. To build the GiNaC tutorial and reference manual
in HTML, DVI, PostScript, or PDF formats, use one of

@example
$ make html
$ make dvi
$ make ps
$ make pdf
@end example

Generally, the top-level Makefile runs recursively to the
subdirectories.  It is therefore safe to go into any subdirectory
(@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
@var{target} there in case something went wrong.


@node Installing GiNaC, Basic concepts, Building GiNaC, Installation
@c    node-name, next, previous, up
@section Installing GiNaC
@cindex installation

To install GiNaC on your system, simply type

@example
$ make install
@end example

As described in the section about configuration the files will be
installed in the following directories (the directories will be created
if they don't already exist):

@itemize @bullet

@item
@file{libginac.a} will go into @file{@var{PREFIX}/lib/} (or
@file{@var{LIBDIR}}) which defaults to @file{/usr/local/lib/}.
So will @file{libginac.so} unless the configure script was
given the option @option{--disable-shared}.  The proper symlinks
will be established as well.

@item
All the header files will be installed into @file{@var{PREFIX}/include/ginac/}
(or @file{@var{INCLUDEDIR}/ginac/}, if specified).

@item
All documentation (info) will be stuffed into
@file{@var{PREFIX}/share/doc/GiNaC/} (or
@file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).

@end itemize

For the sake of completeness we will list some other useful make
targets: @command{make clean} deletes all files generated by
@command{make}, i.e. all the object files.  In addition @command{make
distclean} removes all files generated by the configuration and
@command{make maintainer-clean} goes one step further and deletes files
that may require special tools to rebuild (like the @command{libtool}
for instance).  Finally @command{make uninstall} removes the installed
library, header files and documentation@footnote{Uninstallation does not
work after you have called @command{make distclean} since the
@file{Makefile} is itself generated by the configuration from
@file{Makefile.in} and hence deleted by @command{make distclean}.  There
are two obvious ways out of this dilemma.  First, you can run the
configuration again with the same @var{PREFIX} thus creating a
@file{Makefile} with a working @samp{uninstall} target.  Second, you can
do it by hand since you now know where all the files went during
installation.}.


@node Basic concepts, Expressions, Installing GiNaC, Top
@c    node-name, next, previous, up
@chapter Basic concepts

This chapter will describe the different fundamental objects that can be
handled by GiNaC.  But before doing so, it is worthwhile introducing you
to the more commonly used class of expressions, representing a flexible
meta-class for storing all mathematical objects.

@menu
* Expressions::                  The fundamental GiNaC class.
* Automatic evaluation::         Evaluation and canonicalization.
* Error handling::               How the library reports errors.
* The class hierarchy::          Overview of GiNaC's classes.
* Symbols::                      Symbolic objects.
* Numbers::                      Numerical objects.
* Constants::                    Pre-defined constants.
* Fundamental containers::       Sums, products and powers.
* Lists::                        Lists of expressions.
* Mathematical functions::       Mathematical functions.
* Relations::                    Equality, Inequality and all that.
* Integrals::                    Symbolic integrals.
* Matrices::                     Matrices.
* Indexed objects::              Handling indexed quantities.
* Non-commutative objects::      Algebras with non-commutative products.
@end menu


@node Expressions, Automatic evaluation, Basic concepts, Basic concepts
@c    node-name, next, previous, up
@section Expressions
@cindex expression (class @code{ex})
@cindex @code{has()}

The most common class of objects a user deals with is the expression
@code{ex}, representing a mathematical object like a variable, number,
function, sum, product, etc@dots{}  Expressions may be put together to form
new expressions, passed as arguments to functions, and so on.  Here is a
little collection of valid expressions:

@example
ex MyEx1 = 5;                       // simple number
ex MyEx2 = x + 2*y;                 // polynomial in x and y
ex MyEx3 = (x + 1)/(x - 1);         // rational expression
ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
ex MyEx5 = MyEx4 + 1;               // similar to above
@end example

Expressions are handles to other more fundamental objects, that often
contain other expressions thus creating a tree of expressions
(@xref{Internal structures}, for particular examples).  Most methods on
@code{ex} therefore run top-down through such an expression tree.  For
example, the method @code{has()} scans recursively for occurrences of
something inside an expression.  Thus, if you have declared @code{MyEx4}
as in the example above @code{MyEx4.has(y)} will find @code{y} inside
the argument of @code{sin} and hence return @code{true}.

The next sections will outline the general picture of GiNaC's class
hierarchy and describe the classes of objects that are handled by
@code{ex}.

@subsection Note: Expressions and STL containers

GiNaC expressions (@code{ex} objects) have value semantics (they can be
assigned, reassigned and copied like integral types) but the operator
@code{<} doesn't provide a well-defined ordering on them. In STL-speak,
expressions are @samp{Assignable} but not @samp{LessThanComparable}.

This implies that in order to use expressions in sorted containers such as
@code{std::map<>} and @code{std::set<>} you have to supply a suitable
comparison predicate. GiNaC provides such a predicate, called
@code{ex_is_less}. For example, a set of expressions should be defined
as @code{std::set<ex, ex_is_less>}.

Unsorted containers such as @code{std::vector<>} and @code{std::list<>}
don't pose a problem. A @code{std::vector<ex>} works as expected.

@xref{Information about expressions}, for more about comparing and ordering
expressions.


@node Automatic evaluation, Error handling, Expressions, Basic concepts
@c    node-name, next, previous, up
@section Automatic evaluation and canonicalization of expressions
@cindex evaluation

GiNaC performs some automatic transformations on expressions, to simplify
them and put them into a canonical form. Some examples:

@example
ex MyEx1 = 2*x - 1 + x;  // 3*x-1
ex MyEx2 = x - x;        // 0
ex MyEx3 = cos(2*Pi);    // 1
ex MyEx4 = x*y/x;        // y
@end example

This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
evaluation}. GiNaC only performs transformations that are

@itemize @bullet
@item
at most of complexity
@tex
$O(n\log n)$
@end tex
@ifnottex
@math{O(n log n)}
@end ifnottex
@item
algebraically correct, possibly except for a set of measure zero (e.g.
@math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
@end itemize

There are two types of automatic transformations in GiNaC that may not
behave in an entirely obvious way at first glance:

@itemize
@item
The terms of sums and products (and some other things like the arguments of
symmetric functions, the indices of symmetric tensors etc.) are re-ordered
into a canonical form that is deterministic, but not lexicographical or in
any other way easy to guess (it almost always depends on the number and
order of the symbols you define). However, constructing the same expression
twice, either implicitly or explicitly, will always result in the same
canonical form.
@item
Expressions of the form 'number times sum' are automatically expanded (this
has to do with GiNaC's internal representation of sums and products). For
example
@example
ex MyEx5 = 2*(x + y);   // 2*x+2*y
ex MyEx6 = z*(x + y);   // z*(x+y)
@end example
@end itemize

The general rule is that when you construct expressions, GiNaC automatically
creates them in canonical form, which might differ from the form you typed in
your program. This may create some awkward looking output (@samp{-y+x} instead
of @samp{x-y}) but allows for more efficient operation and usually yields
some immediate simplifications.

@cindex @code{eval()}
Internally, the anonymous evaluator in GiNaC is implemented by the methods

@example
ex ex::eval() const;
ex basic::eval() const;
@end example

but unless you are extending GiNaC with your own classes or functions, there
should never be any reason to call them explicitly. All GiNaC methods that
transform expressions, like @code{subs()} or @code{normal()}, automatically
re-evaluate their results.


@node Error handling, The class hierarchy, Automatic evaluation, Basic concepts
@c    node-name, next, previous, up
@section Error handling
@cindex exceptions
@cindex @code{pole_error} (class)

GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
generated by GiNaC are subclassed from the standard @code{exception} class
defined in the @file{<stdexcept>} header. In addition to the predefined
@code{logic_error}, @code{domain_error}, @code{out_of_range},
@code{invalid_argument}, @code{runtime_error}, @code{range_error} and
@code{overflow_error} types, GiNaC also defines a @code{pole_error}
exception that gets thrown when trying to evaluate a mathematical function
at a singularity.

The @code{pole_error} class has a member function

@example
int pole_error::degree() const;
@end example

that returns the order of the singularity (or 0 when the pole is
logarithmic or the order is undefined).

When using GiNaC it is useful to arrange for exceptions to be caught in
the main program even if you don't want to do any special error handling.
Otherwise whenever an error occurs in GiNaC, it will be delegated to the
default exception handler of your C++ compiler's run-time system which
usually only aborts the program without giving any information what went
wrong.

Here is an example for a @code{main()} function that catches and prints
exceptions generated by GiNaC:

@example
#include <iostream>
#include <stdexcept>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

int main()
@{
    try @{
        ...
        // code using GiNaC
        ...
    @} catch (exception &p) @{
        cerr << p.what() << endl;
        return 1;
    @}
    return 0;
@}
@end example


@node The class hierarchy, Symbols, Error handling, Basic concepts
@c    node-name, next, previous, up
@section The class hierarchy

GiNaC's class hierarchy consists of several classes representing
mathematical objects, all of which (except for @code{ex} and some
helpers) are internally derived from one abstract base class called
@code{basic}.  You do not have to deal with objects of class
@code{basic}, instead you'll be dealing with symbols, numbers,
containers of expressions and so on.

@cindex container
@cindex atom
To get an idea about what kinds of symbolic composites may be built we
have a look at the most important classes in the class hierarchy and
some of the relations among the classes:

@ifnotinfo
@image{classhierarchy}
@end ifnotinfo
@ifinfo
<PICTURE MISSING>
@end ifinfo

The abstract classes shown here (the ones without drop-shadow) are of no
interest for the user.  They are used internally in order to avoid code
duplication if two or more classes derived from them share certain
features.  An example is @code{expairseq}, a container for a sequence of
pairs each consisting of one expression and a number (@code{numeric}).
What @emph{is} visible to the user are the derived classes @code{add}
and @code{mul}, representing sums and products.  @xref{Internal
structures}, where these two classes are described in more detail.  The
following table shortly summarizes what kinds of mathematical objects
are stored in the different classes:

@cartouche
@multitable @columnfractions .22 .78
@item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
@item @code{constant} @tab Constants like 
@tex
$\pi$
@end tex
@ifnottex
@math{Pi}
@end ifnottex
@item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
@item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
@item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
@item @code{ncmul} @tab Products of non-commutative objects
@item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b}, 
@tex
$\sqrt{2}$
@end tex
@ifnottex
@code{sqrt(}@math{2}@code{)}
@end ifnottex
@dots{}
@item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
@item @code{function} @tab A symbolic function like
@tex
$\sin 2x$
@end tex
@ifnottex
@math{sin(2*x)}
@end ifnottex
@item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
@item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
@item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
@item @code{indexed} @tab Indexed object like @math{A_ij}
@item @code{tensor} @tab Special tensor like the delta and metric tensors
@item @code{idx} @tab Index of an indexed object
@item @code{varidx} @tab Index with variance
@item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
@item @code{wildcard} @tab Wildcard for pattern matching
@item @code{structure} @tab Template for user-defined classes
@end multitable
@end cartouche


@node Symbols, Numbers, The class hierarchy, Basic concepts
@c    node-name, next, previous, up
@section Symbols
@cindex @code{symbol} (class)
@cindex hierarchy of classes

@cindex atom
Symbolic indeterminates, or @dfn{symbols} for short, are for symbolic
manipulation what atoms are for chemistry.

A typical symbol definition looks like this:
@example
symbol x("x");
@end example

This definition actually contains three very different things:
@itemize
@item a C++ variable named @code{x}
@item a @code{symbol} object stored in this C++ variable; this object
  represents the symbol in a GiNaC expression
@item the string @code{"x"} which is the name of the symbol, used (almost)
  exclusively for printing expressions holding the symbol
@end itemize

Symbols have an explicit name, supplied as a string during construction,
because in C++, variable names can't be used as values, and the C++ compiler
throws them away during compilation.

It is possible to omit the symbol name in the definition:
@example
symbol x;
@end example

In this case, GiNaC will assign the symbol an internal, unique name of the
form @code{symbolNNN}. This won't affect the usability of the symbol but
the output of your calculations will become more readable if you give your
symbols sensible names (for intermediate expressions that are only used
internally such anonymous symbols can be quite useful, however).

Now, here is one important property of GiNaC that differentiates it from
other computer algebra programs you may have used: GiNaC does @emph{not} use
the names of symbols to tell them apart, but a (hidden) serial number that
is unique for each newly created @code{symbol} object. If you want to use
one and the same symbol in different places in your program, you must only
create one @code{symbol} object and pass that around. If you create another
symbol, even if it has the same name, GiNaC will treat it as a different
indeterminate.

Observe:
@example
ex f(int n)
@{
    symbol x("x");
    return pow(x, n);
@}

int main()
@{
    symbol x("x");
    ex e = f(6);

    cout << e << endl;
     // prints "x^6" which looks right, but...

    cout << e.degree(x) << endl;
     // ...this doesn't work. The symbol "x" here is different from the one
     // in f() and in the expression returned by f(). Consequently, it
     // prints "0".
@}
@end example

One possibility to ensure that @code{f()} and @code{main()} use the same
symbol is to pass the symbol as an argument to @code{f()}:
@example
ex f(int n, const ex & x)
@{
    return pow(x, n);
@}

int main()
@{
    symbol x("x");

    // Now, f() uses the same symbol.
    ex e = f(6, x);

    cout << e.degree(x) << endl;
     // prints "6", as expected
@}
@end example

Another possibility would be to define a global symbol @code{x} that is used
by both @code{f()} and @code{main()}. If you are using global symbols and
multiple compilation units you must take special care, however. Suppose
that you have a header file @file{globals.h} in your program that defines
a @code{symbol x("x");}. In this case, every unit that includes
@file{globals.h} would also get its own definition of @code{x} (because
header files are just inlined into the source code by the C++ preprocessor),
and hence you would again end up with multiple equally-named, but different,
symbols. Instead, the @file{globals.h} header should only contain a
@emph{declaration} like @code{extern symbol x;}, with the definition of
@code{x} moved into a C++ source file such as @file{globals.cpp}.

A different approach to ensuring that symbols used in different parts of
your program are identical is to create them with a @emph{factory} function
like this one:
@example
const symbol & get_symbol(const string & s)
@{
    static map<string, symbol> directory;
    map<string, symbol>::iterator i = directory.find(s);
    if (i != directory.end())
        return i->second;
    else
        return directory.insert(make_pair(s, symbol(s))).first->second;
@}
@end example

This function returns one newly constructed symbol for each name that is
passed in, and it returns the same symbol when called multiple times with
the same name. Using this symbol factory, we can rewrite our example like
this:
@example
ex f(int n)
@{
    return pow(get_symbol("x"), n);
@}

int main()
@{
    ex e = f(6);

    // Both calls of get_symbol("x") yield the same symbol.
    cout << e.degree(get_symbol("x")) << endl;
     // prints "6"
@}
@end example

Instead of creating symbols from strings we could also have
@code{get_symbol()} take, for example, an integer number as its argument.
In this case, we would probably want to give the generated symbols names
that include this number, which can be accomplished with the help of an
@code{ostringstream}.

In general, if you're getting weird results from GiNaC such as an expression
@samp{x-x} that is not simplified to zero, you should check your symbol
definitions.

As we said, the names of symbols primarily serve for purposes of expression
output. But there are actually two instances where GiNaC uses the names for
identifying symbols: When constructing an expression from a string, and when
recreating an expression from an archive (@pxref{Input/output}).

In addition to its name, a symbol may contain a special string that is used
in LaTeX output:
@example
symbol x("x", "\\Box");
@end example

This creates a symbol that is printed as "@code{x}" in normal output, but
as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
information about the different output formats of expressions in GiNaC).
GiNaC automatically creates proper LaTeX code for symbols having names of
greek letters (@samp{alpha}, @samp{mu}, etc.). You can retrieve the name
and the LaTeX name of a symbol using the respective methods:
@cindex @code{get_name()}
@cindex @code{get_TeX_name()}
@example
symbol::get_name() const;
symbol::get_TeX_name() const;
@end example

@cindex @code{subs()}
Symbols in GiNaC can't be assigned values. If you need to store results of
calculations and give them a name, use C++ variables of type @code{ex}.
If you want to replace a symbol in an expression with something else, you
can invoke the expression's @code{.subs()} method
(@pxref{Substituting expressions}).

@cindex @code{realsymbol()}
By default, symbols are expected to stand in for complex values, i.e. they live
in the complex domain.  As a consequence, operations like complex conjugation,
for example (@pxref{Complex expressions}), do @emph{not} evaluate if applied
to such symbols. Likewise @code{log(exp(x))} does not evaluate to @code{x},
because of the unknown imaginary part of @code{x}.
On the other hand, if you are sure that your symbols will hold only real
values, you would like to have such functions evaluated. Therefore GiNaC
allows you to specify
the domain of the symbol. Instead of @code{symbol x("x");} you can write
@code{realsymbol x("x");} to tell GiNaC that @code{x} stands in for real values.

@cindex @code{possymbol()}
Furthermore, it is also possible to declare a symbol as positive. This will,
for instance, enable the automatic simplification of @code{abs(x)} into 
@code{x}. This is done by declaring the symbol as @code{possymbol x("x");}.


@node Numbers, Constants, Symbols, Basic concepts
@c    node-name, next, previous, up
@section Numbers
@cindex @code{numeric} (class)

@cindex GMP
@cindex CLN
@cindex rational
@cindex fraction
For storing numerical things, GiNaC uses Bruno Haible's library CLN.
The classes therein serve as foundation classes for GiNaC.  CLN stands
for Class Library for Numbers or alternatively for Common Lisp Numbers.
In order to find out more about CLN's internals, the reader is referred to
the documentation of that library.  @xref{Top,,, cln, The CLN Manual}, for
more information. Suffice to say that it is by itself build on top of
another library, the GNU Multiple Precision library GMP, which is an
extremely fast library for arbitrary long integers and rationals as well
as arbitrary precision floating point numbers.  It is very commonly used
by several popular cryptographic applications.  CLN extends GMP by
several useful things: First, it introduces the complex number field
over either reals (i.e. floating point numbers with arbitrary precision)
or rationals.  Second, it automatically converts rationals to integers
if the denominator is unity and complex numbers to real numbers if the
imaginary part vanishes and also correctly treats algebraic functions.
Third it provides good implementations of state-of-the-art algorithms
for all trigonometric and hyperbolic functions as well as for
calculation of some useful constants.

The user can construct an object of class @code{numeric} in several
ways.  The following example shows the four most important constructors.
It uses construction from C-integer, construction of fractions from two
integers, construction from C-float and construction from a string:

@example
#include <iostream>
#include <ginac/ginac.h>
using namespace GiNaC;

int main()
@{
    numeric two = 2;                      // exact integer 2
    numeric r(2,3);                       // exact fraction 2/3
    numeric e(2.71828);                   // floating point number
    numeric p = "3.14159265358979323846"; // constructor from string
    // Trott's constant in scientific notation:
    numeric trott("1.0841015122311136151E-2");
    
    std::cout << two*p << std::endl;  // floating point 6.283...
    ...
@end example

@cindex @code{I}
@cindex complex numbers
The imaginary unit in GiNaC is a predefined @code{numeric} object with the
name @code{I}:

@example
    ...
    numeric z1 = 2-3*I;                    // exact complex number 2-3i
    numeric z2 = 5.9+1.6*I;                // complex floating point number
@}
@end example

It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
This would, however, call C's built-in operator @code{/} for integers
first and result in a numeric holding a plain integer 1.  @strong{Never
use the operator @code{/} on integers} unless you know exactly what you
are doing!  Use the constructor from two integers instead, as shown in
the example above.  Writing @code{numeric(1)/2} may look funny but works
also.

@cindex @code{Digits}
@cindex accuracy
We have seen now the distinction between exact numbers and floating
point numbers.  Clearly, the user should never have to worry about
dynamically created exact numbers, since their `exactness' always
determines how they ought to be handled, i.e. how `long' they are.  The
situation is different for floating point numbers.  Their accuracy is
controlled by one @emph{global} variable, called @code{Digits}.  (For
those readers who know about Maple: it behaves very much like Maple's
@code{Digits}).  All objects of class numeric that are constructed from
then on will be stored with a precision matching that number of decimal
digits:

@example
#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

void foo()
@{
    numeric three(3.0), one(1.0);
    numeric x = one/three;

    cout << "in " << Digits << " digits:" << endl;
    cout << x << endl;
    cout << Pi.evalf() << endl;
@}

int main()
@{
    foo();
    Digits = 60;
    foo();
    return 0;
@}
@end example

The above example prints the following output to screen:

@example
in 17 digits:
0.33333333333333333334
3.1415926535897932385
in 60 digits:
0.33333333333333333333333333333333333333333333333333333333333333333334
3.1415926535897932384626433832795028841971693993751058209749445923078
@end example

@cindex rounding
Note that the last number is not necessarily rounded as you would
naively expect it to be rounded in the decimal system.  But note also,
that in both cases you got a couple of extra digits.  This is because
numbers are internally stored by CLN as chunks of binary digits in order
to match your machine's word size and to not waste precision.  Thus, on
architectures with different word size, the above output might even
differ with regard to actually computed digits.

It should be clear that objects of class @code{numeric} should be used
for constructing numbers or for doing arithmetic with them.  The objects
one deals with most of the time are the polymorphic expressions @code{ex}.

@subsection Tests on numbers

Once you have declared some numbers, assigned them to expressions and
done some arithmetic with them it is frequently desired to retrieve some
kind of information from them like asking whether that number is
integer, rational, real or complex.  For those cases GiNaC provides
several useful methods.  (Internally, they fall back to invocations of
certain CLN functions.)

As an example, let's construct some rational number, multiply it with
some multiple of its denominator and test what comes out:

@example
#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

// some very important constants:
const numeric twentyone(21);
const numeric ten(10);
const numeric five(5);

int main()
@{
    numeric answer = twentyone;

    answer /= five;
    cout << answer.is_integer() << endl;  // false, it's 21/5
    answer *= ten;
    cout << answer.is_integer() << endl;  // true, it's 42 now!
@}
@end example

Note that the variable @code{answer} is constructed here as an integer
by @code{numeric}'s copy constructor, but in an intermediate step it
holds a rational number represented as integer numerator and integer
denominator.  When multiplied by 10, the denominator becomes unity and
the result is automatically converted to a pure integer again.
Internally, the underlying CLN is responsible for this behavior and we
refer the reader to CLN's documentation.  Suffice to say that
the same behavior applies to complex numbers as well as return values of
certain functions.  Complex numbers are automatically converted to real
numbers if the imaginary part becomes zero.  The full set of tests that
can be applied is listed in the following table.

@cartouche
@multitable @columnfractions .30 .70
@item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
@item @code{.is_zero()}
@tab @dots{}equal to zero
@item @code{.is_positive()}
@tab @dots{}not complex and greater than 0
@item @code{.is_negative()}
@tab @dots{}not complex and smaller than 0
@item @code{.is_integer()}
@tab @dots{}a (non-complex) integer
@item @code{.is_pos_integer()}
@tab @dots{}an integer and greater than 0
@item @code{.is_nonneg_integer()}
@tab @dots{}an integer and greater equal 0
@item @code{.is_even()}
@tab @dots{}an even integer
@item @code{.is_odd()}
@tab @dots{}an odd integer
@item @code{.is_prime()}
@tab @dots{}a prime integer (probabilistic primality test)
@item @code{.is_rational()}
@tab @dots{}an exact rational number (integers are rational, too)
@item @code{.is_real()}
@tab @dots{}a real integer, rational or float (i.e. is not complex)
@item @code{.is_cinteger()}
@tab @dots{}a (complex) integer (such as @math{2-3*I})
@item @code{.is_crational()}
@tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
@end multitable
@end cartouche

@page

@subsection Numeric functions

The following functions can be applied to @code{numeric} objects and will be
evaluated immediately:

@cartouche
@multitable @columnfractions .30 .70
@item @strong{Name} @tab @strong{Function}
@item @code{inverse(z)}
@tab returns @math{1/z}
@cindex @code{inverse()} (numeric)
@item @code{pow(a, b)}
@tab exponentiation @math{a^b}
@item @code{abs(z)}
@tab absolute value
@item @code{real(z)}
@tab real part
@cindex @code{real()}
@item @code{imag(z)}
@tab imaginary part
@cindex @code{imag()}
@item @code{csgn(z)}
@tab complex sign (returns an @code{int})
@item @code{step(x)}
@tab step function (returns an @code{numeric})
@item @code{numer(z)}
@tab numerator of rational or complex rational number
@item @code{denom(z)}
@tab denominator of rational or complex rational number
@item @code{sqrt(z)}
@tab square root
@item @code{isqrt(n)}
@tab integer square root
@cindex @code{isqrt()}
@item @code{sin(z)}
@tab sine
@item @code{cos(z)}
@tab cosine
@item @code{tan(z)}
@tab tangent
@item @code{asin(z)}
@tab inverse sine
@item @code{acos(z)}
@tab inverse cosine
@item @code{atan(z)}
@tab inverse tangent
@item @code{atan(y, x)}
@tab inverse tangent with two arguments
@item @code{sinh(z)}
@tab hyperbolic sine
@item @code{cosh(z)}
@tab hyperbolic cosine
@item @code{tanh(z)}
@tab hyperbolic tangent
@item @code{asinh(z)}
@tab inverse hyperbolic sine
@item @code{acosh(z)}
@tab inverse hyperbolic cosine
@item @code{atanh(z)}
@tab inverse hyperbolic tangent
@item @code{exp(z)}
@tab exponential function
@item @code{log(z)}
@tab natural logarithm
@item @code{Li2(z)}
@tab dilogarithm
@item @code{zeta(z)}
@tab Riemann's zeta function
@item @code{tgamma(z)}
@tab gamma function
@item @code{lgamma(z)}
@tab logarithm of gamma function
@item @code{psi(z)}
@tab psi (digamma) function
@item @code{psi(n, z)}
@tab derivatives of psi function (polygamma functions)
@item @code{factorial(n)}
@tab factorial function @math{n!}
@item @code{doublefactorial(n)}
@tab double factorial function @math{n!!}
@cindex @code{doublefactorial()}
@item @code{binomial(n, k)}
@tab binomial coefficients
@item @code{bernoulli(n)}
@tab Bernoulli numbers
@cindex @code{bernoulli()}
@item @code{fibonacci(n)}
@tab Fibonacci numbers
@cindex @code{fibonacci()}
@item @code{mod(a, b)}
@tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
@cindex @code{mod()}
@item @code{smod(a, b)}
@tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
@cindex @code{smod()}
@item @code{irem(a, b)}
@tab integer remainder (has the sign of @math{a}, or is zero)
@cindex @code{irem()}
@item @code{irem(a, b, q)}
@tab integer remainder and quotient, @code{irem(a, b, q) == a-q*b}
@item @code{iquo(a, b)}
@tab integer quotient
@cindex @code{iquo()}
@item @code{iquo(a, b, r)}
@tab integer quotient and remainder, @code{r == a-iquo(a, b)*b}
@item @code{gcd(a, b)}
@tab greatest common divisor
@item @code{lcm(a, b)}
@tab least common multiple
@end multitable
@end cartouche

Most of these functions are also available as symbolic functions that can be
used in expressions (@pxref{Mathematical functions}) or, like @code{gcd()},
as polynomial algorithms.

@subsection Converting numbers

Sometimes it is desirable to convert a @code{numeric} object back to a
built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
class provides a couple of methods for this purpose:

@cindex @code{to_int()}
@cindex @code{to_long()}
@cindex @code{to_double()}
@cindex @code{to_cl_N()}
@example
int numeric::to_int() const;
long numeric::to_long() const;
double numeric::to_double() const;
cln::cl_N numeric::to_cl_N() const;
@end example

@code{to_int()} and @code{to_long()} only work when the number they are
applied on is an exact integer. Otherwise the program will halt with a
message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
rational number will return a floating-point approximation. Both
@code{to_int()/to_long()} and @code{to_double()} discard the imaginary
part of complex numbers.

Note the signature of the above methods, you may need to apply a type
conversion and call @code{evalf()} as shown in the following example:
@example
    ...
    ex e1 = 1, e2 = sin(Pi/5);
    cout << ex_to<numeric>(e1).to_int() << endl
         << ex_to<numeric>(e2.evalf()).to_double() << endl;
    ...
@end example

@node Constants, Fundamental containers, Numbers, Basic concepts
@c    node-name, next, previous, up
@section Constants
@cindex @code{constant} (class)

@cindex @code{Pi}
@cindex @code{Catalan}
@cindex @code{Euler}
@cindex @code{evalf()}
Constants behave pretty much like symbols except that they return some
specific number when the method @code{.evalf()} is called.

The predefined known constants are:

@cartouche
@multitable @columnfractions .14 .32 .54
@item @strong{Name} @tab @strong{Common Name} @tab @strong{Numerical Value (to 35 digits)}
@item @code{Pi}
@tab Archimedes' constant
@tab 3.14159265358979323846264338327950288
@item @code{Catalan}
@tab Catalan's constant
@tab 0.91596559417721901505460351493238411
@item @code{Euler}
@tab Euler's (or Euler-Mascheroni) constant
@tab 0.57721566490153286060651209008240243
@end multitable
@end cartouche


@node Fundamental containers, Lists, Constants, Basic concepts
@c    node-name, next, previous, up
@section Sums, products and powers
@cindex polynomial
@cindex @code{add}
@cindex @code{mul}
@cindex @code{power}

Simple rational expressions are written down in GiNaC pretty much like
in other CAS or like expressions involving numerical variables in C.
The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
been overloaded to achieve this goal.  When you run the following
code snippet, the constructor for an object of type @code{mul} is
automatically called to hold the product of @code{a} and @code{b} and
then the constructor for an object of type @code{add} is called to hold
the sum of that @code{mul} object and the number one:

@example
    ...
    symbol a("a"), b("b");
    ex MyTerm = 1+a*b;
    ...
@end example

@cindex @code{pow()}
For exponentiation, you have already seen the somewhat clumsy (though C-ish)
statement @code{pow(x,2);} to represent @code{x} squared.  This direct
construction is necessary since we cannot safely overload the constructor
@code{^} in C++ to construct a @code{power} object.  If we did, it would
have several counterintuitive and undesired effects:

@itemize @bullet
@item
Due to C's operator precedence, @code{2*x^2} would be parsed as @code{(2*x)^2}.
@item
Due to the binding of the operator @code{^}, @code{x^a^b} would result in
@code{(x^a)^b}. This would be confusing since most (though not all) other CAS
interpret this as @code{x^(a^b)}.
@item
Also, expressions involving integer exponents are very frequently used,
which makes it even more dangerous to overload @code{^} since it is then
hard to distinguish between the semantics as exponentiation and the one
for exclusive or.  (It would be embarrassing to return @code{1} where one
has requested @code{2^3}.)
@end itemize

@cindex @command{ginsh}
All effects are contrary to mathematical notation and differ from the
way most other CAS handle exponentiation, therefore overloading @code{^}
is ruled out for GiNaC's C++ part.  The situation is different in
@command{ginsh}, there the exponentiation-@code{^} exists.  (Also note
that the other frequently used exponentiation operator @code{**} does
not exist at all in C++).

To be somewhat more precise, objects of the three classes described
here, are all containers for other expressions.  An object of class
@code{power} is best viewed as a container with two slots, one for the
basis, one for the exponent.  All valid GiNaC expressions can be
inserted.  However, basic transformations like simplifying
@code{pow(pow(x,2),3)} to @code{x^6} automatically are only performed
when this is mathematically possible.  If we replace the outer exponent
three in the example by some symbols @code{a}, the simplification is not
safe and will not be performed, since @code{a} might be @code{1/2} and
@code{x} negative.

Objects of type @code{add} and @code{mul} are containers with an
arbitrary number of slots for expressions to be inserted.  Again, simple
and safe simplifications are carried out like transforming
@code{3*x+4-x} to @code{2*x+4}.


@node Lists, Mathematical functions, Fundamental containers, Basic concepts
@c    node-name, next, previous, up
@section Lists of expressions
@cindex @code{lst} (class)
@cindex lists
@cindex @code{nops()}
@cindex @code{op()}
@cindex @code{append()}
@cindex @code{prepend()}
@cindex @code{remove_first()}
@cindex @code{remove_last()}
@cindex @code{remove_all()}

The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
expressions. They are not as ubiquitous as in many other computer algebra
packages, but are sometimes used to supply a variable number of arguments of
the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
constructors, so you should have a basic understanding of them.

Lists can be constructed from an initializer list of expressions:

@example
@{
    symbol x("x"), y("y");
    lst l = @{x, 2, y, x+y@};
    // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
    // in that order
    ...
@end example

Use the @code{nops()} method to determine the size (number of expressions) of
a list and the @code{op()} method or the @code{[]} operator to access
individual elements:

@example
    ...
    cout << l.nops() << endl;                // prints '4'
    cout << l.op(2) << " " << l[0] << endl;  // prints 'y x'
    ...
@end example

As with the standard @code{list<T>} container, accessing random elements of a
@code{lst} is generally an operation of order @math{O(N)}. Faster read-only
sequential access to the elements of a list is possible with the
iterator types provided by the @code{lst} class:

@example
typedef ... lst::const_iterator;
typedef ... lst::const_reverse_iterator;
lst::const_iterator lst::begin() const;
lst::const_iterator lst::end() const;
lst::const_reverse_iterator lst::rbegin() const;
lst::const_reverse_iterator lst::rend() const;
@end example

For example, to print the elements of a list individually you can use:

@example
    ...
    // O(N)
    for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
        cout << *i << endl;
    ...
@end example

which is one order faster than

@example
    ...
    // O(N^2)
    for (size_t i = 0; i < l.nops(); ++i)
        cout << l.op(i) << endl;
    ...
@end example

These iterators also allow you to use some of the algorithms provided by
the C++ standard library:

@example
    ...
    // print the elements of the list (requires #include <iterator>)
    std::copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));

    // sum up the elements of the list (requires #include <numeric>)
    ex sum = std::accumulate(l.begin(), l.end(), ex(0));
    cout << sum << endl;  // prints '2+2*x+2*y'
    ...
@end example

@code{lst} is one of the few GiNaC classes that allow in-place modifications
(the only other one is @code{matrix}). You can modify single elements:

@example
    ...
    l[1] = 42;       // l is now @{x, 42, y, x+y@}
    l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
    ...
@end example

You can append or prepend an expression to a list with the @code{append()}
and @code{prepend()} methods:

@example
    ...
    l.append(4*x);   // l is now @{x, 7, y, x+y, 4*x@}
    l.prepend(0);    // l is now @{0, x, 7, y, x+y, 4*x@}
    ...
@end example

You can remove the first or last element of a list with @code{remove_first()}
and @code{remove_last()}:

@example
    ...
    l.remove_first();   // l is now @{x, 7, y, x+y, 4*x@}
    l.remove_last();    // l is now @{x, 7, y, x+y@}
    ...
@end example

You can remove all the elements of a list with @code{remove_all()}:

@example
    ...
    l.remove_all();     // l is now empty
    ...
@end example

You can bring the elements of a list into a canonical order with @code{sort()}:

@example
    ...
    lst l1 = @{x, 2, y, x+y@};
    lst l2 = @{2, x+y, x, y@};
    l1.sort();
    l2.sort();
    // l1 and l2 are now equal
    ...
@end example

Finally, you can remove all but the first element of consecutive groups of
elements with @code{unique()}:

@example
    ...
    lst l3 = @{x, 2, 2, 2, y, x+y, y+x@};
    l3.unique();        // l3 is now @{x, 2, y, x+y@}
@}
@end example


@node Mathematical functions, Relations, Lists, Basic concepts
@c    node-name, next, previous, up
@section Mathematical functions
@cindex @code{function} (class)
@cindex trigonometric function
@cindex hyperbolic function

There are quite a number of useful functions hard-wired into GiNaC.  For
instance, all trigonometric and hyperbolic functions are implemented
(@xref{Built-in functions}, for a complete list).

These functions (better called @emph{pseudofunctions}) are all objects
of class @code{function}.  They accept one or more expressions as
arguments and return one expression.  If the arguments are not
numerical, the evaluation of the function may be halted, as it does in
the next example, showing how a function returns itself twice and
finally an expression that may be really useful:

@cindex Gamma function
@cindex @code{subs()}
@example
    ...
    symbol x("x"), y("y");    
    ex foo = x+y/2;
    cout << tgamma(foo) << endl;
     // -> tgamma(x+(1/2)*y)
    ex bar = foo.subs(y==1);
    cout << tgamma(bar) << endl;
     // -> tgamma(x+1/2)
    ex foobar = bar.subs(x==7);
    cout << tgamma(foobar) << endl;
     // -> (135135/128)*Pi^(1/2)
    ...
@end example

Besides evaluation most of these functions allow differentiation, series
expansion and so on.  Read the next chapter in order to learn more about
this.

It must be noted that these pseudofunctions are created by inline
functions, where the argument list is templated.  This means that
whenever you call @code{GiNaC::sin(1)} it is equivalent to
@code{sin(ex(1))} and will therefore not result in a floating point
number.  Unless of course the function prototype is explicitly
overridden -- which is the case for arguments of type @code{numeric}
(not wrapped inside an @code{ex}).  Hence, in order to obtain a floating
point number of class @code{numeric} you should call
@code{sin(numeric(1))}.  This is almost the same as calling
@code{sin(1).evalf()} except that the latter will return a numeric
wrapped inside an @code{ex}.


@node Relations, Integrals, Mathematical functions, Basic concepts
@c    node-name, next, previous, up
@section Relations
@cindex @code{relational} (class)

Sometimes, a relation holding between two expressions must be stored
somehow.  The class @code{relational} is a convenient container for such
purposes.  A relation is by definition a container for two @code{ex} and
a relation between them that signals equality, inequality and so on.
They are created by simply using the C++ operators @code{==}, @code{!=},
@code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.

@xref{Mathematical functions}, for examples where various applications
of the @code{.subs()} method show how objects of class relational are
used as arguments.  There they provide an intuitive syntax for
substitutions.  They are also used as arguments to the @code{ex::series}
method, where the left hand side of the relation specifies the variable
to expand in and the right hand side the expansion point.  They can also
be used for creating systems of equations that are to be solved for
unknown variables.

But the most common usage of objects of this class
is rather inconspicuous in statements of the form @code{if
(expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}.  Here, an implicit
conversion from @code{relational} to @code{bool} takes place.  Note,
however, that @code{==} here does not perform any simplifications, hence
@code{expand()} must be called explicitly.

Simplifications of
relationals may be more efficient if preceded by a call to
@example
ex relational::canonical() const
@end example
which returns an equivalent relation with the zero
right-hand side. For example:
@example
possymbol p("p");
relational rel = (p >= (p*p-1)/p);
if (ex_to<relational>(rel.canonical().normal()))
        cout << "correct inequality" << endl;
@end example
However, a user shall not expect that any inequality can be fully
resolved by GiNaC.

@node Integrals, Matrices, Relations, Basic concepts
@c    node-name, next, previous, up
@section Integrals
@cindex @code{integral} (class)

An object of class @dfn{integral} can be used to hold a symbolic integral.
If you want to symbolically represent the integral of @code{x*x} from 0 to
1, you would write this as
@example
integral(x, 0, 1, x*x)
@end example
The first argument is the integration variable. It should be noted that
GiNaC is not very good (yet?) at symbolically evaluating integrals. In
fact, it can only integrate polynomials. An expression containing integrals
can be evaluated symbolically by calling the
@example
.eval_integ()
@end example
method on it. Numerical evaluation is available by calling the
@example
.evalf()
@end example
method on an expression containing the integral. This will only evaluate
integrals into a number if @code{subs}ing the integration variable by a
number in the fourth argument of an integral and then @code{evalf}ing the
result always results in a number. Of course, also the boundaries of the
integration domain must @code{evalf} into numbers. It should be noted that
trying to @code{evalf} a function with discontinuities in the integration
domain is not recommended. The accuracy of the numeric evaluation of
integrals is determined by the static member variable
@example
ex integral::relative_integration_error
@end example
of the class @code{integral}. The default value of this is 10^-8.
The integration works by halving the interval of integration, until numeric
stability of the answer indicates that the requested accuracy has been
reached. The maximum depth of the halving can be set via the static member
variable
@example
int integral::max_integration_level
@end example
The default value is 15. If this depth is exceeded, @code{evalf} will simply
return the integral unevaluated. The function that performs the numerical
evaluation, is also available as
@example
ex adaptivesimpson(const ex & x, const ex & a, const ex & b, const ex & f,
                   const ex & error)
@end example
This function will throw an exception if the maximum depth is exceeded. The
last parameter of the function is optional and defaults to the
@code{relative_integration_error}. To make sure that we do not do too
much work if an expression contains the same integral multiple times,
a lookup table is used.

If you know that an expression holds an integral, you can get the
integration variable, the left boundary, right boundary and integrand by
respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
@code{.op(3)}. Differentiating integrals with respect to variables works
as expected. Note that it makes no sense to differentiate an integral
with respect to the integration variable.

@node Matrices, Indexed objects, Integrals, Basic concepts
@c    node-name, next, previous, up
@section Matrices
@cindex @code{matrix} (class)

A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
matrix with @math{m} rows and @math{n} columns are accessed with two
@code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
second one in the range 0@dots{}@math{n-1}.

There are a couple of ways to construct matrices, with or without preset
elements. The constructor

@example
matrix::matrix(unsigned r, unsigned c);
@end example

creates a matrix with @samp{r} rows and @samp{c} columns with all elements
set to zero.

The easiest way to create a matrix is using an initializer list of
initializer lists, all of the same size:

@example
@{
    matrix m = @{@{1, -a@},
                @{a,  1@}@};
@}
@end example

You can also specify the elements as a (flat) list with

@example
matrix::matrix(unsigned r, unsigned c, const lst & l);
@end example

The function

@cindex @code{lst_to_matrix()}
@example
ex lst_to_matrix(const lst & l);
@end example

constructs a matrix from a list of lists, each list representing a matrix row.

There is also a set of functions for creating some special types of
matrices:

@cindex @code{diag_matrix()}
@cindex @code{unit_matrix()}
@cindex @code{symbolic_matrix()}
@example
ex diag_matrix(const lst & l);
ex diag_matrix(initializer_list<ex> l);
ex unit_matrix(unsigned x);
ex unit_matrix(unsigned r, unsigned c);
ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
                   const string & tex_base_name);
@end example

@code{diag_matrix()} constructs a square diagonal matrix given the diagonal
elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
matrix filled with newly generated symbols made of the specified base name
and the position of each element in the matrix.

Matrices often arise by omitting elements of another matrix. For
instance, the submatrix @code{S} of a matrix @code{M} takes a
rectangular block from @code{M}. The reduced matrix @code{R} is defined
by removing one row and one column from a matrix @code{M}. (The
determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
can be used for computing the inverse using Cramer's rule.)

@cindex @code{sub_matrix()}
@cindex @code{reduced_matrix()}
@example
ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
@end example

The function @code{sub_matrix()} takes a row offset @code{r} and a
column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
columns. The function @code{reduced_matrix()} has two integer arguments
that specify which row and column to remove:

@example
@{
    matrix m = @{@{11, 12, 13@},
                @{21, 22, 23@},
                @{31, 32, 33@}@};
    cout << reduced_matrix(m, 1, 1) << endl;
    // -> [[11,13],[31,33]]
    cout << sub_matrix(m, 1, 2, 1, 2) << endl;
    // -> [[22,23],[32,33]]
@}
@end example

Matrix elements can be accessed and set using the parenthesis (function call)
operator:

@example
const ex & matrix::operator()(unsigned r, unsigned c) const;
ex & matrix::operator()(unsigned r, unsigned c);
@end example

It is also possible to access the matrix elements in a linear fashion with
the @code{op()} method. But C++-style subscripting with square brackets
@samp{[]} is not available.

Here are a couple of examples for constructing matrices:

@example
@{
    symbol a("a"), b("b");

    matrix M = @{@{a, 0@},
                @{0, b@}@};
    cout << M << endl;
     // -> [[a,0],[0,b]]

    matrix M2(2, 2);
    M2(0, 0) = a;
    M2(1, 1) = b;
    cout << M2 << endl;
     // -> [[a,0],[0,b]]

    cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
     // -> [[a,0],[0,b]]

    cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
     // -> [[a,0],[0,b]]

    cout << diag_matrix(lst@{a, b@}) << endl;
     // -> [[a,0],[0,b]]

    cout << unit_matrix(3) << endl;
     // -> [[1,0,0],[0,1,0],[0,0,1]]

    cout << symbolic_matrix(2, 3, "x") << endl;
     // -> [[x00,x01,x02],[x10,x11,x12]]
@}
@end example

@cindex @code{is_zero_matrix()} 
The method @code{matrix::is_zero_matrix()} returns @code{true} only if
all entries of the matrix are zeros. There is also method
@code{ex::is_zero_matrix()} which returns @code{true} only if the
expression is zero or a zero matrix.

@cindex @code{transpose()}
There are three ways to do arithmetic with matrices. The first (and most
direct one) is to use the methods provided by the @code{matrix} class:

@example
matrix matrix::add(const matrix & other) const;
matrix matrix::sub(const matrix & other) const;
matrix matrix::mul(const matrix & other) const;
matrix matrix::mul_scalar(const ex & other) const;
matrix matrix::pow(const ex & expn) const;
matrix matrix::transpose() const;
@end example

All of these methods return the result as a new matrix object. Here is an
example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
and @math{C}:

@example
@{
    matrix A = @{@{ 1, 2@},
                @{ 3, 4@}@};
    matrix B = @{@{-1, 0@},
                @{ 2, 1@}@};
    matrix C = @{@{ 8, 4@},
                @{ 2, 1@}@};

    matrix result = A.mul(B).sub(C.mul_scalar(2));
    cout << result << endl;
     // -> [[-13,-6],[1,2]]
    ...
@}
@end example

@cindex @code{evalm()}
The second (and probably the most natural) way is to construct an expression
containing matrices with the usual arithmetic operators and @code{pow()}.
For efficiency reasons, expressions with sums, products and powers of
matrices are not automatically evaluated in GiNaC. You have to call the
method

@example
ex ex::evalm() const;
@end example

to obtain the result:

@example
@{
    ...
    ex e = A*B - 2*C;
    cout << e << endl;
     // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
    cout << e.evalm() << endl;
     // -> [[-13,-6],[1,2]]
    ...
@}
@end example

The non-commutativity of the product @code{A*B} in this example is
automatically recognized by GiNaC. There is no need to use a special
operator here. @xref{Non-commutative objects}, for more information about
dealing with non-commutative expressions.

Finally, you can work with indexed matrices and call @code{simplify_indexed()}
to perform the arithmetic:

@example
@{
    ...
    idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
    e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
    cout << e << endl;
     // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
    cout << e.simplify_indexed() << endl;
     // -> [[-13,-6],[1,2]].i.j
@}
@end example

Using indices is most useful when working with rectangular matrices and
one-dimensional vectors because you don't have to worry about having to
transpose matrices before multiplying them. @xref{Indexed objects}, for
more information about using matrices with indices, and about indices in
general.

The @code{matrix} class provides a couple of additional methods for
computing determinants, traces, characteristic polynomials and ranks:

@cindex @code{determinant()}
@cindex @code{trace()}
@cindex @code{charpoly()}
@cindex @code{rank()}
@example
ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
ex matrix::trace() const;
ex matrix::charpoly(const ex & lambda) const;
unsigned matrix::rank(unsigned algo=solve_algo::automatic) const;
@end example

The optional @samp{algo} argument of @code{determinant()} and @code{rank()}
functions allows to select between different algorithms for calculating the
determinant and rank respectively. The asymptotic speed (as parametrized
by the matrix size) can greatly differ between those algorithms, depending
on the nature of the matrix' entries. The possible values are defined in
the @file{flags.h} header file. By default, GiNaC uses a heuristic to
automatically select an algorithm that is likely (but not guaranteed)
to give the result most quickly.

@cindex @code{solve()}
Linear systems can be solved with:

@example
matrix matrix::solve(const matrix & vars, const matrix & rhs,
                     unsigned algo=solve_algo::automatic) const;
@end example

Assuming the matrix object this method is applied on is an @code{m}
times @code{n} matrix, then @code{vars} must be a @code{n} times
@code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
times @code{p} matrix.  The returned matrix then has dimension @code{n}
times @code{p} and in the case of an underdetermined system will still
contain some of the indeterminates from @code{vars}.  If the system is
overdetermined, an exception is thrown.

@cindex @code{inverse()} (matrix)
To invert a matrix, use the method:

@example
matrix matrix::inverse(unsigned algo=solve_algo::automatic) const;
@end example

The @samp{algo} argument is optional.  If given, it must be one of
@code{solve_algo} defined in @file{flags.h}.

@node Indexed objects, Non-commutative objects, Matrices, Basic concepts
@c    node-name, next, previous, up
@section Indexed objects

GiNaC allows you to handle expressions containing general indexed objects in
arbitrary spaces. It is also able to canonicalize and simplify such
expressions and perform symbolic dummy index summations. There are a number
of predefined indexed objects provided, like delta and metric tensors.

There are few restrictions placed on indexed objects and their indices and
it is easy to construct nonsense expressions, but our intention is to
provide a general framework that allows you to implement algorithms with
indexed quantities, getting in the way as little as possible.

@cindex @code{idx} (class)
@cindex @code{indexed} (class)
@subsection Indexed quantities and their indices

Indexed expressions in GiNaC are constructed of two special types of objects,
@dfn{index objects} and @dfn{indexed objects}.

@itemize @bullet

@cindex contravariant
@cindex covariant
@cindex variance
@item Index objects are of class @code{idx} or a subclass. Every index has
a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
the index lives in) which can both be arbitrary expressions but are usually
a number or a simple symbol. In addition, indices of class @code{varidx} have
a @dfn{variance} (they can be co- or contravariant), and indices of class
@code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.

@item Indexed objects are of class @code{indexed} or a subclass. They
contain a @dfn{base expression} (which is the expression being indexed), and
one or more indices.

@end itemize

@strong{Please notice:} when printing expressions, covariant indices and indices
without variance are denoted @samp{.i} while contravariant indices are
denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
value. In the following, we are going to use that notation in the text so
instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
not visible in the output.

A simple example shall illustrate the concepts:

@example
#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

int main()
@{
    symbol i_sym("i"), j_sym("j");
    idx i(i_sym, 3), j(j_sym, 3);

    symbol A("A");
    cout << indexed(A, i, j) << endl;
     // -> A.i.j
    cout << index_dimensions << indexed(A, i, j) << endl;
     // -> A.i[3].j[3]
    cout << dflt; // reset cout to default output format (dimensions hidden)
    ...
@end example

The @code{idx} constructor takes two arguments, the index value and the
index dimension. First we define two index objects, @code{i} and @code{j},
both with the numeric dimension 3. The value of the index @code{i} is the
symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
@code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
construct an expression containing one indexed object, @samp{A.i.j}. It has
the symbol @code{A} as its base expression and the two indices @code{i} and
@code{j}.

The dimensions of indices are normally not visible in the output, but one
can request them to be printed with the @code{index_dimensions} manipulator,
as shown above.

Note the difference between the indices @code{i} and @code{j} which are of
class @code{idx}, and the index values which are the symbols @code{i_sym}
and @code{j_sym}. The indices of indexed objects cannot directly be symbols
or numbers but must be index objects. For example, the following is not
correct and will raise an exception:

@example
symbol i("i"), j("j");
e = indexed(A, i, j); // ERROR: indices must be of type idx
@end example

You can have multiple indexed objects in an expression, index values can
be numeric, and index dimensions symbolic:

@example
    ...
    symbol B("B"), dim("dim");
    cout << 4 * indexed(A, i)
          + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
     // -> B.j.2.i+4*A.i
    ...
@end example

@code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
index of value 2, and a symbolic index @samp{i} with the symbolic dimension
@samp{dim}. Note that GiNaC doesn't automatically notify you that the free
indices of @samp{A} and @samp{B} in the sum don't match (you have to call
@code{simplify_indexed()} for that, see below).

In fact, base expressions, index values and index dimensions can be
arbitrary expressions:

@example
    ...
    cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
     // -> (B+A).(1+2*i)
    ...
@end example

It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
get an error message from this but you will probably not be able to do
anything useful with it.

@cindex @code{get_value()}
@cindex @code{get_dim()}
The methods

@example
ex idx::get_value();
ex idx::get_dim();
@end example

return the value and dimension of an @code{idx} object. If you have an index
in an expression, such as returned by calling @code{.op()} on an indexed
object, you can get a reference to the @code{idx} object with the function
@code{ex_to<idx>()} on the expression.

There are also the methods

@example
bool idx::is_numeric();
bool idx::is_symbolic();
bool idx::is_dim_numeric();
bool idx::is_dim_symbolic();
@end example

for checking whether the value and dimension are numeric or symbolic
(non-numeric). Using the @code{info()} method of an index (see @ref{Information
about expressions}) returns information about the index value.

@cindex @code{varidx} (class)
If you need co- and contravariant indices, use the @code{varidx} class:

@example
    ...
    symbol mu_sym("mu"), nu_sym("nu");
    varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
    varidx mu_co(mu_sym, 4, true);       // covariant index .mu

    cout << indexed(A, mu, nu) << endl;
     // -> A~mu~nu
    cout << indexed(A, mu_co, nu) << endl;
     // -> A.mu~nu
    cout << indexed(A, mu.toggle_variance(), nu) << endl;
     // -> A.mu~nu
    ...
@end example

A @code{varidx} is an @code{idx} with an additional flag that marks it as
co- or contravariant. The default is a contravariant (upper) index, but
this can be overridden by supplying a third argument to the @code{varidx}
constructor. The two methods

@example
bool varidx::is_covariant();
bool varidx::is_contravariant();
@end example

allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
to get the object reference from an expression). There's also the very useful
method

@example
ex varidx::toggle_variance();
@end example

which makes a new index with the same value and dimension but the opposite
variance. By using it you only have to define the index once.

@cindex @code{spinidx} (class)
The @code{spinidx} class provides dotted and undotted variant indices, as
used in the Weyl-van-der-Waerden spinor formalism:

@example
    ...
    symbol K("K"), C_sym("C"), D_sym("D");
    spinidx C(C_sym, 2), D(D_sym);          // default is 2-dimensional,
                                            // contravariant, undotted
    spinidx C_co(C_sym, 2, true);           // covariant index
    spinidx D_dot(D_sym, 2, false, true);   // contravariant, dotted
    spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted

    cout << indexed(K, C, D) << endl;
     // -> K~C~D
    cout << indexed(K, C_co, D_dot) << endl;
     // -> K.C~*D
    cout << indexed(K, D_co_dot, D) << endl;
     // -> K.*D~D
    ...
@end example

A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
dotted or undotted. The default is undotted but this can be overridden by
supplying a fourth argument to the @code{spinidx} constructor. The two
methods

@example
bool spinidx::is_dotted();
bool spinidx::is_undotted();
@end example

allow you to check whether or not a @code{spinidx} object is dotted (use
@code{ex_to<spinidx>()} to get the object reference from an expression).
Finally, the two methods

@example
ex spinidx::toggle_dot();
ex spinidx::toggle_variance_dot();
@end example

create a new index with the same value and dimension but opposite dottedness
and the same or opposite variance.

@subsection Substituting indices

@cindex @code{subs()}
Sometimes you will want to substitute one symbolic index with another
symbolic or numeric index, for example when calculating one specific element
of a tensor expression. This is done with the @code{.subs()} method, as it
is done for symbols (see @ref{Substituting expressions}).

You have two possibilities here. You can either substitute the whole index
by another index or expression:

@example
    ...
    ex e = indexed(A, mu_co);
    cout << e << " becomes " << e.subs(mu_co == nu) << endl;
     // -> A.mu becomes A~nu
    cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
     // -> A.mu becomes A~0
    cout << e << " becomes " << e.subs(mu_co == 0) << endl;
     // -> A.mu becomes A.0
    ...
@end example

The third example shows that trying to replace an index with something that
is not an index will substitute the index value instead.

Alternatively, you can substitute the @emph{symbol} of a symbolic index by
another expression:

@example
    ...
    ex e = indexed(A, mu_co);
    cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
     // -> A.mu becomes A.nu
    cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
     // -> A.mu becomes A.0
    ...
@end example

As you see, with the second method only the value of the index will get
substituted. Its other properties, including its dimension, remain unchanged.
If you want to change the dimension of an index you have to substitute the
whole index by another one with the new dimension.

Finally, substituting the base expression of an indexed object works as
expected:

@example
    ...
    ex e = indexed(A, mu_co);
    cout << e << " becomes " << e.subs(A == A+B) << endl;
     // -> A.mu becomes (B+A).mu
    ...
@end example

@subsection Symmetries
@cindex @code{symmetry} (class)
@cindex @code{sy_none()}
@cindex @code{sy_symm()}
@cindex @code{sy_anti()}
@cindex @code{sy_cycl()}

Indexed objects can have certain symmetry properties with respect to their
indices. Symmetries are specified as a tree of objects of class @code{symmetry}
that is constructed with the helper functions

@example
symmetry sy_none(...);
symmetry sy_symm(...);
symmetry sy_anti(...);
symmetry sy_cycl(...);
@end example

@code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
represents a cyclic symmetry. Each of these functions accepts up to four
arguments which can be either symmetry objects themselves or unsigned integer
numbers that represent an index position (counting from 0). A symmetry
specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
or @code{sy_cycl()} with no arguments specifies the respective symmetry for
all indices.

Here are some examples of symmetry definitions:

@example
    ...
    // No symmetry:
    e = indexed(A, i, j);
    e = indexed(A, sy_none(), i, j);     // equivalent
    e = indexed(A, sy_none(0, 1), i, j); // equivalent

    // Symmetric in all three indices:
    e = indexed(A, sy_symm(), i, j, k);
    e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
    e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
                                               // different canonical order

    // Symmetric in the first two indices only:
    e = indexed(A, sy_symm(0, 1), i, j, k);
    e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent

    // Antisymmetric in the first and last index only (index ranges need not
    // be contiguous):
    e = indexed(A, sy_anti(0, 2), i, j, k);
    e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent

    // An example of a mixed symmetry: antisymmetric in the first two and
    // last two indices, symmetric when swapping the first and last index
    // pairs (like the Riemann curvature tensor):
    e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);

    // Cyclic symmetry in all three indices:
    e = indexed(A, sy_cycl(), i, j, k);
    e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent

    // The following examples are invalid constructions that will throw
    // an exception at run time.

    // An index may not appear multiple times:
    e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
    e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR

    // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
    // same number of indices:
    e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR

    // And of course, you cannot specify indices which are not there:
    e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
    ...
@end example

If you need to specify more than four indices, you have to use the
@code{.add()} method of the @code{symmetry} class. For example, to specify
full symmetry in the first six indices you would write
@code{sy_symm(0, 1, 2, 3).add(4).add(5)}.

If an indexed object has a symmetry, GiNaC will automatically bring the
indices into a canonical order which allows for some immediate simplifications:

@example
    ...
    cout << indexed(A, sy_symm(), i, j)
          + indexed(A, sy_symm(), j, i) << endl;
     // -> 2*A.j.i
    cout << indexed(B, sy_anti(), i, j)
          + indexed(B, sy_anti(), j, i) << endl;
     // -> 0
    cout << indexed(B, sy_anti(), i, j, k)
          - indexed(B, sy_anti(), j, k, i) << endl;
     // -> 0
    ...
@end example

@cindex @code{get_free_indices()}
@cindex dummy index
@subsection Dummy indices

GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
that a summation over the index range is implied. Symbolic indices which are
not dummy indices are called @dfn{free indices}. Numeric indices are neither
dummy nor free indices.

To be recognized as a dummy index pair, the two indices must be of the same
class and their value must be the same single symbol (an index like
@samp{2*n+1} is never a dummy index). If the indices are of class
@code{varidx} they must also be of opposite variance; if they are of class
@code{spinidx} they must be both dotted or both undotted.

The method @code{.get_free_indices()} returns a vector containing the free
indices of an expression. It also checks that the free indices of the terms
of a sum are consistent:

@example
@{
    symbol A("A"), B("B"), C("C");

    symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
    idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);

    ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
    cout << exprseq(e.get_free_indices()) << endl;
     // -> (.i,.k)
     // 'j' and 'l' are dummy indices

    symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
    varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);

    e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
      + indexed(C, mu, sigma, rho, sigma.toggle_variance());
    cout << exprseq(e.get_free_indices()) << endl;
     // -> (~mu,~rho)
     // 'nu' is a dummy index, but 'sigma' is not

    e = indexed(A, mu, mu);
    cout << exprseq(e.get_free_indices()) << endl;
     // -> (~mu)
     // 'mu' is not a dummy index because it appears twice with the same
     // variance

    e = indexed(A, mu, nu) + 42;
    cout << exprseq(e.get_free_indices()) << endl; // ERROR
     // this will throw an exception:
     // "add::get_free_indices: inconsistent indices in sum"
@}
@end example

@cindex @code{expand_dummy_sum()}
A dummy index summation like 
@tex
$ a_i b^i$
@end tex
@ifnottex
a.i b~i
@end ifnottex
can be expanded for indices with numeric
dimensions (e.g. 3)  into the explicit sum like
@tex
$a_1b^1+a_2b^2+a_3b^3 $.
@end tex
@ifnottex
a.1 b~1 + a.2 b~2 + a.3 b~3.
@end ifnottex
This is performed by the function

@example
    ex expand_dummy_sum(const ex & e, bool subs_idx = false);
@end example

which takes an expression @code{e} and returns the expanded sum for all
dummy indices with numeric dimensions. If the parameter @code{subs_idx}
is set to @code{true} then all substitutions are made by @code{idx} class
indices, i.e. without variance. In this case the above sum 
@tex
$ a_i b^i$
@end tex
@ifnottex
a.i b~i
@end ifnottex
will be expanded to
@tex
$a_1b_1+a_2b_2+a_3b_3 $.
@end tex
@ifnottex
a.1 b.1 + a.2 b.2 + a.3 b.3.
@end ifnottex


@cindex @code{simplify_indexed()}
@subsection Simplifying indexed expressions

In addition to the few automatic simplifications that GiNaC performs on
indexed expressions (such as re-ordering the indices of symmetric tensors
and calculating traces and convolutions of matrices and predefined tensors)
there is the method

@example
ex ex::simplify_indexed();
ex ex::simplify_indexed(const scalar_products & sp);
@end example

that performs some more expensive operations:

@itemize @bullet
@item it checks the consistency of free indices in sums in the same way
  @code{get_free_indices()} does
@item it tries to give dummy indices that appear in different terms of a sum
  the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
@item it (symbolically) calculates all possible dummy index summations/contractions
  with the predefined tensors (this will be explained in more detail in the
  next section)
@item it detects contractions that vanish for symmetry reasons, for example
  the contraction of a symmetric and a totally antisymmetric tensor
@item as a special case of dummy index summation, it can replace scalar products
  of two tensors with a user-defined value
@end itemize

The last point is done with the help of the @code{scalar_products} class
which is used to store scalar products with known values (this is not an
arithmetic class, you just pass it to @code{simplify_indexed()}):

@example
@{
    symbol A("A"), B("B"), C("C"), i_sym("i");
    idx i(i_sym, 3);

    scalar_products sp;
    sp.add(A, B, 0); // A and B are orthogonal
    sp.add(A, C, 0); // A and C are orthogonal
    sp.add(A, A, 4); // A^2 = 4 (A has length 2)

    e = indexed(A + B, i) * indexed(A + C, i);
    cout << e << endl;
     // -> (B+A).i*(A+C).i

    cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
         << endl;
     // -> 4+C.i*B.i
@}
@end example

The @code{scalar_products} object @code{sp} acts as a storage for the
scalar products added to it with the @code{.add()} method. This method
takes three arguments: the two expressions of which the scalar product is
taken, and the expression to replace it with.

@cindex @code{expand()}
The example above also illustrates a feature of the @code{expand()} method:
if passed the @code{expand_indexed} option it will distribute indices
over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.

@cindex @code{tensor} (class)
@subsection Predefined tensors

Some frequently used special tensors such as the delta, epsilon and metric
tensors are predefined in GiNaC. They have special properties when
contracted with other tensor expressions and some of them have constant
matrix representations (they will evaluate to a number when numeric
indices are specified).

@cindex @code{delta_tensor()}
@subsubsection Delta tensor

The delta tensor takes two indices, is symmetric and has the matrix
representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
@code{delta_tensor()}:

@example
@{
    symbol A("A"), B("B");

    idx i(symbol("i"), 3), j(symbol("j"), 3),
        k(symbol("k"), 3), l(symbol("l"), 3);

    ex e = indexed(A, i, j) * indexed(B, k, l)
         * delta_tensor(i, k) * delta_tensor(j, l);
    cout << e.simplify_indexed() << endl;
     // -> B.i.j*A.i.j

    cout << delta_tensor(i, i) << endl;
     // -> 3
@}
@end example

@cindex @code{metric_tensor()}
@subsubsection General metric tensor

The function @code{metric_tensor()} creates a general symmetric metric
tensor with two indices that can be used to raise/lower tensor indices. The
metric tensor is denoted as @samp{g} in the output and if its indices are of
mixed variance it is automatically replaced by a delta tensor:

@example
@{
    symbol A("A");

    varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);

    ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
    cout << e.simplify_indexed() << endl;
     // -> A~mu~rho

    e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
    cout << e.simplify_indexed() << endl;
     // -> g~mu~rho

    e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
      * metric_tensor(nu, rho);
    cout << e.simplify_indexed() << endl;
     // -> delta.mu~rho

    e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
      * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
        + indexed(A, mu.toggle_variance(), rho));
    cout << e.simplify_indexed() << endl;
     // -> 4+A.rho~rho
@}
@end example

@cindex @code{lorentz_g()}
@subsubsection Minkowski metric tensor

The Minkowski metric tensor is a special metric tensor with a constant
matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
It is created with the function @code{lorentz_g()} (although it is output as
@samp{eta}):

@example
@{
    varidx mu(symbol("mu"), 4);

    e = delta_tensor(varidx(0, 4), mu.toggle_variance())
      * lorentz_g(mu, varidx(0, 4));       // negative signature
    cout << e.simplify_indexed() << endl;
     // -> 1

    e = delta_tensor(varidx(0, 4), mu.toggle_variance())
      * lorentz_g(mu, varidx(0, 4), true); // positive signature
    cout << e.simplify_indexed() << endl;
     // -> -1
@}
@end example

@cindex @code{spinor_metric()}
@subsubsection Spinor metric tensor

The function @code{spinor_metric()} creates an antisymmetric tensor with
two indices that is used to raise/lower indices of 2-component spinors.
It is output as @samp{eps}:

@example
@{
    symbol psi("psi");

    spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
    ex A_co = A.toggle_variance(), B_co = B.toggle_variance();

    e = spinor_metric(A, B) * indexed(psi, B_co);
    cout << e.simplify_indexed() << endl;
     // -> psi~A

    e = spinor_metric(A, B) * indexed(psi, A_co);
    cout << e.simplify_indexed() << endl;
     // -> -psi~B

    e = spinor_metric(A_co, B_co) * indexed(psi, B);
    cout << e.simplify_indexed() << endl;
     // -> -psi.A

    e = spinor_metric(A_co, B_co) * indexed(psi, A);
    cout << e.simplify_indexed() << endl;
     // -> psi.B

    e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
    cout << e.simplify_indexed() << endl;
     // -> 2

    e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
    cout << e.simplify_indexed() << endl;
     // -> -delta.A~C
@}
@end example

The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.

@cindex @code{epsilon_tensor()}
@cindex @code{lorentz_eps()}
@subsubsection Epsilon tensor

The epsilon tensor is totally antisymmetric, its number of indices is equal
to the dimension of the index space (the indices must all be of the same
numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
defined to be 1. Its behavior with indices that have a variance also
depends on the signature of the metric. Epsilon tensors are output as
@samp{eps}.

There are three functions defined to create epsilon tensors in 2, 3 and 4
dimensions:

@example
ex epsilon_tensor(const ex & i1, const ex & i2);
ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
               bool pos_sig = false);
@end example

The first two functions create an epsilon tensor in 2 or 3 Euclidean
dimensions, the last function creates an epsilon tensor in a 4-dimensional
Minkowski space (the last @code{bool} argument specifies whether the metric
has negative or positive signature, as in the case of the Minkowski metric
tensor):

@example
@{
    varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
           sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
    e = lorentz_eps(mu, nu, rho, sig) *
        lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
    cout << simplify_indexed(e) << endl;
     // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam

    idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
    symbol A("A"), B("B");
    e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
    cout << simplify_indexed(e) << endl;
     // -> -B.k*A.j*eps.i.k.j
    e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
    cout << simplify_indexed(e) << endl;
     // -> 0
@}
@end example

@subsection Linear algebra

The @code{matrix} class can be used with indices to do some simple linear
algebra (linear combinations and products of vectors and matrices, traces
and scalar products):

@example
@{
    idx i(symbol("i"), 2), j(symbol("j"), 2);
    symbol x("x"), y("y");

    // A is a 2x2 matrix, X is a 2x1 vector
    matrix A = @{@{1, 2@},
                @{3, 4@}@};
    matrix X = @{@{x, y@}@};

    cout << indexed(A, i, i) << endl;
     // -> 5

    ex e = indexed(A, i, j) * indexed(X, j);
    cout << e.simplify_indexed() << endl;
     // -> [[2*y+x],[4*y+3*x]].i

    e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
    cout << e.simplify_indexed() << endl;
     // -> [[3*y+3*x,6*y+2*x]].j
@}
@end example

You can of course obtain the same results with the @code{matrix::add()},
@code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
but with indices you don't have to worry about transposing matrices.

Matrix indices always start at 0 and their dimension must match the number
of rows/columns of the matrix. Matrices with one row or one column are
vectors and can have one or two indices (it doesn't matter whether it's a
row or a column vector). Other matrices must have two indices.

You should be careful when using indices with variance on matrices. GiNaC
doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
@samp{F.mu.nu} are different matrices. In this case you should use only
one form for @samp{F} and explicitly multiply it with a matrix representation
of the metric tensor.


@node Non-commutative objects, Methods and functions, Indexed objects, Basic concepts
@c    node-name, next, previous, up
@section Non-commutative objects

GiNaC is equipped to handle certain non-commutative algebras. Three classes of
non-commutative objects are built-in which are mostly of use in high energy
physics:

@itemize
@item Clifford (Dirac) algebra (class @code{clifford})
@item su(3) Lie algebra (class @code{color})
@item Matrices (unindexed) (class @code{matrix})
@end itemize

The @code{clifford} and @code{color} classes are subclasses of
@code{indexed} because the elements of these algebras usually carry
indices. The @code{matrix} class is described in more detail in
@ref{Matrices}.

Unlike most computer algebra systems, GiNaC does not primarily provide an
operator (often denoted @samp{&*}) for representing inert products of
arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
classes of objects involved, and non-commutative products are formed with
the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
figuring out by itself which objects commutate and will group the factors
by their class. Consider this example:

@example
    ...
    varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
    idx a(symbol("a"), 8), b(symbol("b"), 8);
    ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
    cout << e << endl;
     // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
    ...
@end example

As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
groups the non-commutative factors (the gammas and the su(3) generators)
together while preserving the order of factors within each class (because
Clifford objects commutate with color objects). The resulting expression is a
@emph{commutative} product with two factors that are themselves non-commutative
products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
parentheses are placed around the non-commutative products in the output.

@cindex @code{ncmul} (class)
Non-commutative products are internally represented by objects of the class
@code{ncmul}, as opposed to commutative products which are handled by the
@code{mul} class. You will normally not have to worry about this distinction,
though.

The advantage of this approach is that you never have to worry about using
(or forgetting to use) a special operator when constructing non-commutative
expressions. Also, non-commutative products in GiNaC are more intelligent
than in other computer algebra systems; they can, for example, automatically
canonicalize themselves according to rules specified in the implementation
of the non-commutative classes. The drawback is that to work with other than
the built-in algebras you have to implement new classes yourself. Both
symbols and user-defined functions can be specified as being non-commutative.
For symbols, this is done by subclassing class symbol; for functions,
by explicitly setting the return type (@pxref{Symbolic functions}).

@cindex @code{return_type()}
@cindex @code{return_type_tinfo()}
Information about the commutativity of an object or expression can be
obtained with the two member functions

@example
unsigned      ex::return_type() const;
return_type_t ex::return_type_tinfo() const;
@end example

The @code{return_type()} function returns one of three values (defined in
the header file @file{flags.h}), corresponding to three categories of
expressions in GiNaC:

@itemize @bullet
@item @code{return_types::commutative}: Commutates with everything. Most GiNaC
  classes are of this kind.
@item @code{return_types::noncommutative}: Non-commutative, belonging to a
  certain class of non-commutative objects which can be determined with the
  @code{return_type_tinfo()} method. Expressions of this category commutate
  with everything except @code{noncommutative} expressions of the same
  class.
@item @code{return_types::noncommutative_composite}: Non-commutative, composed
  of non-commutative objects of different classes. Expressions of this
  category don't commutate with any other @code{noncommutative} or
  @code{noncommutative_composite} expressions.
@end itemize

The @code{return_type_tinfo()} method returns an object of type
@code{return_type_t} that contains information about the type of the expression
and, if given, its representation label (see section on dirac gamma matrices for
more details).  The objects of type @code{return_type_t} can be tested for
equality to test whether two expressions belong to the same category and
therefore may not commute.

Here are a couple of examples:

@cartouche
@multitable @columnfractions .6 .4
@item @strong{Expression} @tab @strong{@code{return_type()}}
@item @code{42} @tab @code{commutative}
@item @code{2*x-y} @tab @code{commutative}
@item @code{dirac_ONE()} @tab @code{noncommutative}
@item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
@item @code{2*color_T(a)} @tab @code{noncommutative}
@item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
@end multitable
@end cartouche

A last note: With the exception of matrices, positive integer powers of
non-commutative objects are automatically expanded in GiNaC. For example,
@code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
non-commutative expressions).


@cindex @code{clifford} (class)
@subsection Clifford algebra


Clifford algebras are supported in two flavours: Dirac gamma
matrices (more physical) and generic Clifford algebras (more
mathematical). 

@cindex @code{dirac_gamma()}
@subsubsection Dirac gamma matrices
Dirac gamma matrices (note that GiNaC doesn't treat them
as matrices) are designated as @samp{gamma~mu} and satisfy
@samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where
@samp{eta~mu~nu} is the Minkowski metric tensor. Dirac gammas are
constructed by the function

@example
ex dirac_gamma(const ex & mu, unsigned char rl = 0);
@end example

which takes two arguments: the index and a @dfn{representation label} in the
range 0 to 255 which is used to distinguish elements of different Clifford
algebras (this is also called a @dfn{spin line index}). Gammas with different
labels commutate with each other. The dimension of the index can be 4 or (in
the framework of dimensional regularization) any symbolic value. Spinor
indices on Dirac gammas are not supported in GiNaC.

@cindex @code{dirac_ONE()}
The unity element of a Clifford algebra is constructed by

@example
ex dirac_ONE(unsigned char rl = 0);
@end example

@strong{Please notice:} You must always use @code{dirac_ONE()} when referring to
multiples of the unity element, even though it's customary to omit it.
E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
GiNaC will complain and/or produce incorrect results.

@cindex @code{dirac_gamma5()}
There is a special element @samp{gamma5} that commutates with all other
gammas, has a unit square, and in 4 dimensions equals
@samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by

@example
ex dirac_gamma5(unsigned char rl = 0);
@end example

@cindex @code{dirac_gammaL()}
@cindex @code{dirac_gammaR()}
The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
objects, constructed by

@example
ex dirac_gammaL(unsigned char rl = 0);
ex dirac_gammaR(unsigned char rl = 0);
@end example

They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
and @samp{gammaL gammaR = gammaR gammaL = 0}.

@cindex @code{dirac_slash()}
Finally, the function

@example
ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
@end example

creates a term that represents a contraction of @samp{e} with the Dirac
Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
with a unique index whose dimension is given by the @code{dim} argument).
Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.

In products of dirac gammas, superfluous unity elements are automatically
removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
and @samp{gammaR} are moved to the front.

The @code{simplify_indexed()} function performs contractions in gamma strings,
for example

@example
@{
    ...
    symbol a("a"), b("b"), D("D");
    varidx mu(symbol("mu"), D);
    ex e = dirac_gamma(mu) * dirac_slash(a, D)
         * dirac_gamma(mu.toggle_variance());
    cout << e << endl;
     // -> gamma~mu*a\*gamma.mu
    e = e.simplify_indexed();
    cout << e << endl;
     // -> -D*a\+2*a\
    cout << e.subs(D == 4) << endl;
     // -> -2*a\
    ...
@}
@end example

@cindex @code{dirac_trace()}
To calculate the trace of an expression containing strings of Dirac gammas
you use one of the functions

@example
ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
               const ex & trONE = 4);
ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
@end example

These functions take the trace over all gammas in the specified set @code{rls}
or list @code{rll} of representation labels, or the single label @code{rl};
gammas with other labels are left standing. The last argument to
@code{dirac_trace()} is the value to be returned for the trace of the unity
element, which defaults to 4.

The @code{dirac_trace()} function is a linear functional that is equal to the
ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
functional is not cyclic in
@tex $D \ne 4$
@end tex
@ifnottex
@math{D != 4}
@end ifnottex
dimensions when acting on
expressions containing @samp{gamma5}, so it's not a proper trace. This
@samp{gamma5} scheme is described in greater detail in the article
@cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).

The value of the trace itself is also usually different in 4 and in
@tex $D \ne 4$
@end tex
@ifnottex
@math{D != 4}
@end ifnottex
dimensions:

@example
@{
    // 4 dimensions
    varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
    ex e = dirac_gamma(mu) * dirac_gamma(nu) *
           dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
    cout << dirac_trace(e).simplify_indexed() << endl;
     // -> -8*eta~rho~nu
@}
...
@{
    // D dimensions
    symbol D("D");
    varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
    ex e = dirac_gamma(mu) * dirac_gamma(nu) *
           dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
    cout << dirac_trace(e).simplify_indexed() << endl;
     // -> 8*eta~rho~nu-4*eta~rho~nu*D
@}
@end example

Here is an example for using @code{dirac_trace()} to compute a value that
appears in the calculation of the one-loop vacuum polarization amplitude in
QED:

@example
@{
    symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
    varidx mu(symbol("mu"), D), nu(symbol("nu"), D);

    scalar_products sp;
    sp.add(l, l, pow(l, 2));
    sp.add(l, q, ldotq);

    ex e = dirac_gamma(mu) *
           (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *    
           dirac_gamma(mu.toggle_variance()) *
           (dirac_slash(l, D) + m * dirac_ONE());   
    e = dirac_trace(e).simplify_indexed(sp);
    e = e.collect(lst@{l, ldotq, m@});
    cout << e << endl;
     // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
@}
@end example

The @code{canonicalize_clifford()} function reorders all gamma products that
appear in an expression to a canonical (but not necessarily simple) form.
You can use this to compare two expressions or for further simplifications:

@example
@{
    varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
    ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
    cout << e << endl;
     // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu

    e = canonicalize_clifford(e);
    cout << e << endl;
     // -> 2*ONE*eta~mu~nu
@}
@end example

@cindex @code{clifford_unit()}
@subsubsection A generic Clifford algebra

A generic Clifford algebra, i.e. a
@tex $2^n$
@end tex
@ifnottex
2^n
@end ifnottex
dimensional algebra with
generators 
@tex $e_k$
@end tex 
@ifnottex
e_k
@end ifnottex
satisfying the identities 
@tex
$e_i e_j + e_j e_i = M(i, j) + M(j, i)$
@end tex
@ifnottex
e~i e~j + e~j e~i = M(i, j) + M(j, i) 
@end ifnottex
for some bilinear form (@code{metric})
@math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180) 
and contain symbolic entries. Such generators are created by the
function 

@example
    ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);    
@end example

where @code{mu} should be a @code{idx} (or descendant) class object
indexing the generators.
Parameter @code{metr} defines the metric @math{M(i, j)} and can be
represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
object. In fact, any expression either with two free indices or without
indices at all is admitted as @code{metr}. In the later case an @code{indexed}
object with two newly created indices with @code{metr} as its
@code{op(0)} will be used.
Optional parameter @code{rl} allows to distinguish different
Clifford algebras, which will commute with each other. 

Note that the call @code{clifford_unit(mu, minkmetric())} creates
something very close to @code{dirac_gamma(mu)}, although
@code{dirac_gamma} have more efficient simplification mechanism. 
@cindex @code{get_metric()}
Also, the object created by @code{clifford_unit(mu, minkmetric())} is
not aware about the symmetry of its metric, see the start of the previous
paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
specifies as follows:

@example
    clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
@end example

The method @code{clifford::get_metric()} returns a metric defining this
Clifford number.

If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
the Clifford algebra units with a call like that

@example
    ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
@end example

since this may yield some further automatic simplifications. Again, for a
metric defined through a @code{matrix} such a symmetry is detected
automatically. 

Individual generators of a Clifford algebra can be accessed in several
ways. For example 

@example
@{
    ... 
    idx i(symbol("i"), 4);
    realsymbol s("s");
    ex M = diag_matrix(lst@{1, -1, 0, s@});
    ex e = clifford_unit(i, M);
    ex e0 = e.subs(i == 0);
    ex e1 = e.subs(i == 1);
    ex e2 = e.subs(i == 2);
    ex e3 = e.subs(i == 3);
    ...
@}
@end example

will produce four anti-commuting generators of a Clifford algebra with properties
@tex
$e_0^2=1 $, $e_1^2=-1$,  $e_2^2=0$ and $e_3^2=s$.
@end tex
@ifnottex
@code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
@code{pow(e3, 2) = s}.
@end ifnottex

@cindex @code{lst_to_clifford()}
A similar effect can be achieved from the function

@example
    ex lst_to_clifford(const ex & v, const ex & mu,  const ex & metr,
                       unsigned char rl = 0);
    ex lst_to_clifford(const ex & v, const ex & e);
@end example

which converts a list or vector 
@tex
$v = (v^0, v^1, ..., v^n)$
@end tex
@ifnottex
@samp{v = (v~0, v~1, ..., v~n)} 
@end ifnottex
into the
Clifford number 
@tex
$v^0 e_0 + v^1 e_1 + ... + v^n e_n$
@end tex
@ifnottex
@samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
@end ifnottex
with @samp{e.k}
directly supplied in the second form of the procedure. In the first form
the Clifford unit @samp{e.k} is generated by the call of
@code{clifford_unit(mu, metr, rl)}. 
@cindex pseudo-vector
If the number of components supplied
by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
1 then function @code{lst_to_clifford()} uses the following
pseudo-vector representation: 
@tex
$v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
@end tex
@ifnottex
@samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
@end ifnottex

The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows

@example
@{
    ...
    idx i(symbol("i"), 4);
    realsymbol s("s");
    ex M = diag_matrix(@{1, -1, 0, s@});
    ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
    ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
    ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
    ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
  ...
@}
@end example

@cindex @code{clifford_to_lst()}
There is the inverse function 

@example
    lst clifford_to_lst(const ex & e, const ex & c, bool algebraic = true);
@end example

which takes an expression @code{e} and tries to find a list
@tex
$v = (v^0, v^1, ..., v^n)$
@end tex
@ifnottex
@samp{v = (v~0, v~1, ..., v~n)} 
@end ifnottex
such that the expression is either vector 
@tex
$e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
@end tex
@ifnottex
@samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
@end ifnottex
or pseudo-vector 
@tex
$v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
@end tex
@ifnottex
@samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
@end ifnottex
with respect to the given Clifford units @code{c}. Here none of the
@samp{v~k} should contain Clifford units @code{c} (of course, this
may be impossible). This function can use an @code{algebraic} method
(default) or a symbolic one. With the @code{algebraic} method the
@samp{v~k} are calculated as 
@tex
$(e c_k + c_k e)/c_k^2$. If $c_k^2$
@end tex
@ifnottex
@samp{(e c.k + c.k e)/pow(c.k, 2)}.   If @samp{pow(c.k, 2)} 
@end ifnottex
is zero or is not @code{numeric} for some @samp{k}
then the method will be automatically changed to symbolic. The same effect
is obtained by the assignment (@code{algebraic = false}) in the procedure call.

@cindex @code{clifford_prime()}
@cindex @code{clifford_star()}
@cindex @code{clifford_bar()}
There are several functions for (anti-)automorphisms of Clifford algebras:

@example
    ex clifford_prime(const ex & e)
    inline ex clifford_star(const ex & e)
    inline ex clifford_bar(const ex & e)
@end example

The automorphism of a Clifford algebra @code{clifford_prime()} simply
changes signs of all Clifford units in the expression. The reversion
of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
units in any product. Finally the main anti-automorphism
of a Clifford algebra @code{clifford_bar()} is the composition of the
previous two, i.e. it makes the reversion and changes signs of all Clifford units
in a product. These functions correspond to the notations
@math{e'},
@tex
$e^*$
@end tex
@ifnottex
e*
@end ifnottex
and
@tex
$\overline{e}$
@end tex
@ifnottex
@code{\bar@{e@}}
@end ifnottex
used in Clifford algebra textbooks.

@cindex @code{clifford_norm()}
The function

@example
    ex clifford_norm(const ex & e);
@end example

@cindex @code{clifford_inverse()}
calculates the norm of a Clifford number from the expression
@tex
$||e||^2 = e\overline{e}$.
@end tex
@ifnottex
@code{||e||^2 = e \bar@{e@}}
@end ifnottex
 The inverse of a Clifford expression is returned by the function

@example
    ex clifford_inverse(const ex & e);
@end example

which calculates it as 
@tex
$e^{-1} = \overline{e}/||e||^2$.
@end tex
@ifnottex
@math{e^@{-1@} = \bar@{e@}/||e||^2}
@end ifnottex
 If
@tex
$||e|| = 0$
@end tex
@ifnottex
@math{||e||=0}
@end ifnottex
then an exception is raised.

@cindex @code{remove_dirac_ONE()}
If a Clifford number happens to be a factor of
@code{dirac_ONE()} then we can convert it to a ``real'' (non-Clifford)
expression by the function

@example
    ex remove_dirac_ONE(const ex & e);
@end example

@cindex @code{canonicalize_clifford()}
The function @code{canonicalize_clifford()} works for a
generic Clifford algebra in a similar way as for Dirac gammas.

The next provided function is

@cindex @code{clifford_moebius_map()}
@example
    ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
                            const ex & d, const ex & v, const ex & G,
                            unsigned char rl = 0);
    ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
                            unsigned char rl = 0);
@end example 

It takes a list or vector @code{v} and makes the Moebius (conformal or
linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
the metric of the surrounding (pseudo-)Euclidean space. This can be an
indexed object, tensormetric, matrix or a Clifford unit, in the later
case the optional parameter @code{rl} is ignored even if supplied.
Depending from the type of @code{v} the returned value of this function
is either a vector or a list holding vector's components.

@cindex @code{clifford_max_label()}
Finally the function

@example
char clifford_max_label(const ex & e, bool ignore_ONE = false);
@end example

can detect a presence of Clifford objects in the expression @code{e}: if
such objects are found it returns the maximal
@code{representation_label} of them, otherwise @code{-1}. The optional
parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
be ignored during the search.
 
LaTeX output for Clifford units looks like
@code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
@code{representation_label} and @code{\nu} is the index of the
corresponding unit. This provides a flexible typesetting with a suitable
definition of the @code{\clifford} command. For example, the definition
@example
    \newcommand@{\clifford@}[1][]@{@}
@end example
typesets all Clifford units identically, while the alternative definition
@example
    \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
@end example
prints units with @code{representation_label=0} as 
@tex
$e$,
@end tex
@ifnottex
@code{e},
@end ifnottex
with @code{representation_label=1} as 
@tex
$\tilde{e}$
@end tex
@ifnottex
@code{\tilde@{e@}}
@end ifnottex
 and with @code{representation_label=2} as 
@tex
$\breve{e}$.
@end tex
@ifnottex
@code{\breve@{e@}}.
@end ifnottex

@cindex @code{color} (class)
@subsection Color algebra

@cindex @code{color_T()}
For computations in quantum chromodynamics, GiNaC implements the base elements
and structure constants of the su(3) Lie algebra (color algebra). The base
elements @math{T_a} are constructed by the function

@example
ex color_T(const ex & a, unsigned char rl = 0);
@end example

which takes two arguments: the index and a @dfn{representation label} in the
range 0 to 255 which is used to distinguish elements of different color
algebras. Objects with different labels commutate with each other. The
dimension of the index must be exactly 8 and it should be of class @code{idx},
not @code{varidx}.

@cindex @code{color_ONE()}
The unity element of a color algebra is constructed by

@example
ex color_ONE(unsigned char rl = 0);
@end example

@strong{Please notice:} You must always use @code{color_ONE()} when referring to
multiples of the unity element, even though it's customary to omit it.
E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
GiNaC may produce incorrect results.

@cindex @code{color_d()}
@cindex @code{color_f()}
The functions

@example
ex color_d(const ex & a, const ex & b, const ex & c);
ex color_f(const ex & a, const ex & b, const ex & c);
@end example

create the symmetric and antisymmetric structure constants @math{d_abc} and
@math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
and @math{[T_a, T_b] = i f_abc T_c}.

These functions evaluate to their numerical values,
if you supply numeric indices to them. The index values should be in
the range from 1 to 8, not from 0 to 7. This departure from usual conventions
goes along better with the notations used in physical literature.

@cindex @code{color_h()}
There's an additional function

@example
ex color_h(const ex & a, const ex & b, const ex & c);
@end example

which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.

The function @code{simplify_indexed()} performs some simplifications on
expressions containing color objects:

@example
@{
    ...
    idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
        k(symbol("k"), 8), l(symbol("l"), 8);

    e = color_d(a, b, l) * color_f(a, b, k);
    cout << e.simplify_indexed() << endl;
     // -> 0

    e = color_d(a, b, l) * color_d(a, b, k);
    cout << e.simplify_indexed() << endl;
     // -> 5/3*delta.k.l

    e = color_f(l, a, b) * color_f(a, b, k);
    cout << e.simplify_indexed() << endl;
     // -> 3*delta.k.l

    e = color_h(a, b, c) * color_h(a, b, c);
    cout << e.simplify_indexed() << endl;
     // -> -32/3

    e = color_h(a, b, c) * color_T(b) * color_T(c);
    cout << e.simplify_indexed() << endl;
     // -> -2/3*T.a

    e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
    cout << e.simplify_indexed() << endl;
     // -> -8/9*ONE

    e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
    cout << e.simplify_indexed() << endl;
     // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
    ...
@end example

@cindex @code{color_trace()}
To calculate the trace of an expression containing color objects you use one
of the functions

@example
ex color_trace(const ex & e, const std::set<unsigned char> & rls);
ex color_trace(const ex & e, const lst & rll);
ex color_trace(const ex & e, unsigned char rl = 0);
@end example

These functions take the trace over all color @samp{T} objects in the
specified set @code{rls} or list @code{rll} of representation labels, or the
single label @code{rl}; @samp{T}s with other labels are left standing. For
example:

@example
    ...
    e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
    cout << e << endl;
     // -> -I*f.a.c.b+d.a.c.b
@}
@end example


@node Methods and functions, Information about expressions, Non-commutative objects, Top
@c    node-name, next, previous, up
@chapter Methods and functions
@cindex polynomial

In this chapter the most important algorithms provided by GiNaC will be
described.  Some of them are implemented as functions on expressions,
others are implemented as methods provided by expression objects.  If
they are methods, there exists a wrapper function around it, so you can
alternatively call it in a functional way as shown in the simple
example:

@example
    ...
    cout << "As method:   " << sin(1).evalf() << endl;
    cout << "As function: " << evalf(sin(1)) << endl;
    ...
@end example

@cindex @code{subs()}
The general rule is that wherever methods accept one or more parameters
(@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
wrapper accepts is the same but preceded by the object to act on
(@var{object}, @var{arg1}, @var{arg2}, @dots{}).  This approach is the
most natural one in an OO model but it may lead to confusion for MapleV
users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
@code{A} and @code{x}).  On the other hand, since MapleV returns 3 on
@code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
here.  Also, users of MuPAD will in most cases feel more comfortable
with GiNaC's convention.  All function wrappers are implemented
as simple inline functions which just call the corresponding method and
are only provided for users uncomfortable with OO who are dead set to
avoid method invocations.  Generally, nested function wrappers are much
harder to read than a sequence of methods and should therefore be
avoided if possible.  On the other hand, not everything in GiNaC is a
method on class @code{ex} and sometimes calling a function cannot be
avoided.

@menu
* Information about expressions::
* Numerical evaluation::
* Substituting expressions::
* Pattern matching and advanced substitutions::
* Applying a function on subexpressions::
* Visitors and tree traversal::
* Polynomial arithmetic::           Working with polynomials.
* Rational expressions::            Working with rational functions.
* Symbolic differentiation::
* Series expansion::                Taylor and Laurent expansion.
* Symmetrization::
* Built-in functions::              List of predefined mathematical functions.
* Multiple polylogarithms::
* Iterated integrals::
* Complex expressions::
* Solving linear systems of equations::
* Input/output::                    Input and output of expressions.
@end menu


@node Information about expressions, Numerical evaluation, Methods and functions, Methods and functions
@c    node-name, next, previous, up
@section Getting information about expressions

@subsection Checking expression types
@cindex @code{is_a<@dots{}>()}
@cindex @code{is_exactly_a<@dots{}>()}
@cindex @code{ex_to<@dots{}>()}
@cindex Converting @code{ex} to other classes
@cindex @code{info()}
@cindex @code{return_type()}
@cindex @code{return_type_tinfo()}

Sometimes it's useful to check whether a given expression is a plain number,
a sum, a polynomial with integer coefficients, or of some other specific type.
GiNaC provides a couple of functions for this:

@example
bool is_a<T>(const ex & e);
bool is_exactly_a<T>(const ex & e);
bool ex::info(unsigned flag);
unsigned ex::return_type() const;
return_type_t ex::return_type_tinfo() const;
@end example

When the test made by @code{is_a<T>()} returns true, it is safe to call
one of the functions @code{ex_to<T>()}, where @code{T} is one of the
class names (@xref{The class hierarchy}, for a list of all classes). For
example, assuming @code{e} is an @code{ex}:

@example
@{
    @dots{}
    if (is_a<numeric>(e))
        numeric n = ex_to<numeric>(e);
    @dots{}
@}
@end example

@code{is_a<T>(e)} allows you to check whether the top-level object of
an expression @samp{e} is an instance of the GiNaC class @samp{T}
(@xref{The class hierarchy}, for a list of all classes). This is most useful,
e.g., for checking whether an expression is a number, a sum, or a product:

@example
@{
    symbol x("x");
    ex e1 = 42;
    ex e2 = 4*x - 3;
    is_a<numeric>(e1);  // true
    is_a<numeric>(e2);  // false
    is_a<add>(e1);      // false
    is_a<add>(e2);      // true
    is_a<mul>(e1);      // false
    is_a<mul>(e2);      // false
@}
@end example

In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
top-level object of an expression @samp{e} is an instance of the GiNaC
class @samp{T}, not including parent classes.

The @code{info()} method is used for checking certain attributes of
expressions. The possible values for the @code{flag} argument are defined
in @file{ginac/flags.h}, the most important being explained in the following
table:

@cartouche
@multitable @columnfractions .30 .70
@item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
@item @code{numeric}
@tab @dots{}a number (same as @code{is_a<numeric>(...)})
@item @code{real}
@tab @dots{}a real number, symbol or constant (i.e. is not complex)
@item @code{rational}
@tab @dots{}an exact rational number (integers are rational, too)
@item @code{integer}
@tab @dots{}a (non-complex) integer
@item @code{crational}
@tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
@item @code{cinteger}
@tab @dots{}a (complex) integer (such as @math{2-3*I})
@item @code{positive}
@tab @dots{}not complex and greater than 0
@item @code{negative}
@tab @dots{}not complex and less than 0
@item @code{nonnegative}
@tab @dots{}not complex and greater than or equal to 0
@item @code{posint}
@tab @dots{}an integer greater than 0
@item @code{negint}
@tab @dots{}an integer less than 0
@item @code{nonnegint}
@tab @dots{}an integer greater than or equal to 0
@item @code{even}
@tab @dots{}an even integer
@item @code{odd}
@tab @dots{}an odd integer
@item @code{prime}
@tab @dots{}a prime integer (probabilistic primality test)
@item @code{relation}
@tab @dots{}a relation (same as @code{is_a<relational>(...)})
@item @code{relation_equal}
@tab @dots{}a @code{==} relation
@item @code{relation_not_equal}
@tab @dots{}a @code{!=} relation
@item @code{relation_less}
@tab @dots{}a @code{<} relation
@item @code{relation_less_or_equal}
@tab @dots{}a @code{<=} relation
@item @code{relation_greater}
@tab @dots{}a @code{>} relation
@item @code{relation_greater_or_equal}
@tab @dots{}a @code{>=} relation
@item @code{symbol}
@tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
@item @code{list}
@tab @dots{}a list (same as @code{is_a<lst>(...)})
@item @code{polynomial}
@tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
@item @code{integer_polynomial}
@tab @dots{}a polynomial with (non-complex) integer coefficients
@item @code{cinteger_polynomial}
@tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
@item @code{rational_polynomial}
@tab @dots{}a polynomial with (non-complex) rational coefficients
@item @code{crational_polynomial}
@tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
@item @code{rational_function}
@tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
@end multitable
@end cartouche

To determine whether an expression is commutative or non-commutative and if
so, with which other expressions it would commutate, you use the methods
@code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
for an explanation of these.


@subsection Accessing subexpressions
@cindex container

Many GiNaC classes, like @code{add}, @code{mul}, @code{lst}, and
@code{function}, act as containers for subexpressions. For example, the
subexpressions of a sum (an @code{add} object) are the individual terms,
and the subexpressions of a @code{function} are the function's arguments.

@cindex @code{nops()}
@cindex @code{op()}
GiNaC provides several ways of accessing subexpressions. The first way is to
use the two methods

@example
size_t ex::nops();
ex ex::op(size_t i);
@end example

@code{nops()} determines the number of subexpressions (operands) contained
in the expression, while @code{op(i)} returns the @code{i}-th
(0..@code{nops()-1}) subexpression. In the case of a @code{power} object,
@code{op(0)} will return the basis and @code{op(1)} the exponent. For
@code{indexed} objects, @code{op(0)} is the base expression and @code{op(i)},
@math{i>0} are the indices.

@cindex iterators
@cindex @code{const_iterator}
The second way to access subexpressions is via the STL-style random-access
iterator class @code{const_iterator} and the methods

@example
const_iterator ex::begin();
const_iterator ex::end();
@end example

@code{begin()} returns an iterator referring to the first subexpression;
@code{end()} returns an iterator which is one-past the last subexpression.
If the expression has no subexpressions, then @code{begin() == end()}. These
iterators can also be used in conjunction with non-modifying STL algorithms.

Here is an example that (non-recursively) prints the subexpressions of a
given expression in three different ways:

@example
@{
    ex e = ...

    // with nops()/op()
    for (size_t i = 0; i != e.nops(); ++i)
        cout << e.op(i) << endl;

    // with iterators
    for (const_iterator i = e.begin(); i != e.end(); ++i)
        cout << *i << endl;

    // with iterators and STL copy()
    std::copy(e.begin(), e.end(), std::ostream_iterator<ex>(cout, "\n"));
@}
@end example

@cindex @code{const_preorder_iterator}
@cindex @code{const_postorder_iterator}
@code{op()}/@code{nops()} and @code{const_iterator} only access an
expression's immediate children. GiNaC provides two additional iterator
classes, @code{const_preorder_iterator} and @code{const_postorder_iterator},
that iterate over all objects in an expression tree, in preorder or postorder,
respectively. They are STL-style forward iterators, and are created with the
methods

@example
const_preorder_iterator ex::preorder_begin();
const_preorder_iterator ex::preorder_end();
const_postorder_iterator ex::postorder_begin();
const_postorder_iterator ex::postorder_end();
@end example

The following example illustrates the differences between
@code{const_iterator}, @code{const_preorder_iterator}, and
@code{const_postorder_iterator}:

@example
@{
    symbol A("A"), B("B"), C("C");
    ex e = lst@{lst@{A, B@}, C@};

    std::copy(e.begin(), e.end(),
              std::ostream_iterator<ex>(cout, "\n"));
    // @{A,B@}
    // C

    std::copy(e.preorder_begin(), e.preorder_end(),
              std::ostream_iterator<ex>(cout, "\n"));
    // @{@{A,B@},C@}
    // @{A,B@}
    // A
    // B
    // C

    std::copy(e.postorder_begin(), e.postorder_end(),
              std::ostream_iterator<ex>(cout, "\n"));
    // A
    // B
    // @{A,B@}
    // C
    // @{@{A,B@},C@}
@}
@end example

@cindex @code{relational} (class)
Finally, the left-hand side and right-hand side expressions of objects of
class @code{relational} (and only of these) can also be accessed with the
methods

@example
ex ex::lhs();
ex ex::rhs();
@end example


@subsection Comparing expressions
@cindex @code{is_equal()}
@cindex @code{is_zero()}

Expressions can be compared with the usual C++ relational operators like
@code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
the result is usually not determinable and the result will be @code{false},
except in the case of the @code{!=} operator. You should also be aware that
GiNaC will only do the most trivial test for equality (subtracting both
expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
@code{false}.

Actually, if you construct an expression like @code{a == b}, this will be
represented by an object of the @code{relational} class (@pxref{Relations})
which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.

There are also two methods

@example
bool ex::is_equal(const ex & other);
bool ex::is_zero();
@end example

for checking whether one expression is equal to another, or equal to zero,
respectively. See also the method @code{ex::is_zero_matrix()}, 
@pxref{Matrices}. 


@subsection Ordering expressions
@cindex @code{ex_is_less} (class)
@cindex @code{ex_is_equal} (class)
@cindex @code{compare()}

Sometimes it is necessary to establish a mathematically well-defined ordering
on a set of arbitrary expressions, for example to use expressions as keys
in a @code{std::map<>} container, or to bring a vector of expressions into
a canonical order (which is done internally by GiNaC for sums and products).

The operators @code{<}, @code{>} etc. described in the last section cannot
be used for this, as they don't implement an ordering relation in the
mathematical sense. In particular, they are not guaranteed to be
antisymmetric: if @samp{a} and @samp{b} are different expressions, and
@code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
yield @code{true}.

By default, STL classes and algorithms use the @code{<} and @code{==}
operators to compare objects, which are unsuitable for expressions, but GiNaC
provides two functors that can be supplied as proper binary comparison
predicates to the STL:

@example
class ex_is_less @{
public:
    bool operator()(const ex &lh, const ex &rh) const;
@};

class ex_is_equal @{
public:
    bool operator()(const ex &lh, const ex &rh) const;
@};
@end example

For example, to define a @code{map} that maps expressions to strings you
have to use

@example
std::map<ex, std::string, ex_is_less> myMap;
@end example

Omitting the @code{ex_is_less} template parameter will introduce spurious
bugs because the map operates improperly.

Other examples for the use of the functors:

@example
std::vector<ex> v;
// fill vector
...

// sort vector
std::sort(v.begin(), v.end(), ex_is_less());

// count the number of expressions equal to '1'
unsigned num_ones = std::count_if(v.begin(), v.end(),
                                  [](const ex& e) @{ return ex_is_equal()(e, 1); @});
@end example

The implementation of @code{ex_is_less} uses the member function

@example
int ex::compare(const ex & other) const;
@end example

which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
after @code{other}.


@node Numerical evaluation, Substituting expressions, Information about expressions, Methods and functions
@c    node-name, next, previous, up
@section Numerical evaluation
@cindex @code{evalf()}

GiNaC keeps algebraic expressions, numbers and constants in their exact form.
To evaluate them using floating-point arithmetic you need to call

@example
ex ex::evalf() const;
@end example

@cindex @code{Digits}
The accuracy of the evaluation is controlled by the global object @code{Digits}
which can be assigned an integer value. The default value of @code{Digits}
is 17. @xref{Numbers}, for more information and examples.

To evaluate an expression to a @code{double} floating-point number you can
call @code{evalf()} followed by @code{numeric::to_double()}, like this:

@example
@{
    // Approximate sin(x/Pi)
    symbol x("x");
    ex e = series(sin(x/Pi), x == 0, 6);

    // Evaluate numerically at x=0.1
    ex f = evalf(e.subs(x == 0.1));

    // ex_to<numeric> is an unsafe cast, so check the type first
    if (is_a<numeric>(f)) @{
        double d = ex_to<numeric>(f).to_double();
        cout << d << endl;
         // -> 0.0318256
    @} else
        // error
@}
@end example


@node Substituting expressions, Pattern matching and advanced substitutions, Numerical evaluation, Methods and functions
@c    node-name, next, previous, up
@section Substituting expressions
@cindex @code{subs()}

Algebraic objects inside expressions can be replaced with arbitrary
expressions via the @code{.subs()} method:

@example
ex ex::subs(const ex & e, unsigned options = 0);
ex ex::subs(const exmap & m, unsigned options = 0);
ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
@end example

In the first form, @code{subs()} accepts a relational of the form
@samp{object == expression} or a @code{lst} of such relationals:

@example
@{
    symbol x("x"), y("y");

    ex e1 = 2*x*x-4*x+3;
    cout << "e1(7) = " << e1.subs(x == 7) << endl;
     // -> 73

    ex e2 = x*y + x;
    cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
     // -> -10
@}
@end example

If you specify multiple substitutions, they are performed in parallel, so e.g.
@code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.

The second form of @code{subs()} takes an @code{exmap} object which is a
pair associative container that maps expressions to expressions (currently
implemented as a @code{std::map}). This is the most efficient one of the
three @code{subs()} forms and should be used when the number of objects to
be substituted is large or unknown.

Using this form, the second example from above would look like this:

@example
@{
    symbol x("x"), y("y");
    ex e2 = x*y + x;

    exmap m;
    m[x] = -2;
    m[y] = 4;
    cout << "e2(-2, 4) = " << e2.subs(m) << endl;
@}
@end example

The third form of @code{subs()} takes two lists, one for the objects to be
replaced and one for the expressions to be substituted (both lists must
contain the same number of elements). Using this form, you would write

@example
@{
    symbol x("x"), y("y");
    ex e2 = x*y + x;

    cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
@}
@end example

The optional last argument to @code{subs()} is a combination of
@code{subs_options} flags. There are three options available:
@code{subs_options::no_pattern} disables pattern matching, which makes
large @code{subs()} operations significantly faster if you are not using
patterns. The second option, @code{subs_options::algebraic} enables
algebraic substitutions in products and powers.
@xref{Pattern matching and advanced substitutions}, for more information
about patterns and algebraic substitutions. The third option,
@code{subs_options::no_index_renaming} disables the feature that dummy
indices are renamed if the substitution could give a result in which a
dummy index occurs more than two times. This is sometimes necessary if
you want to use @code{subs()} to rename your dummy indices.

@code{subs()} performs syntactic substitution of any complete algebraic
object; it does not try to match sub-expressions as is demonstrated by the
following example:

@example
@{
    symbol x("x"), y("y"), z("z");

    ex e1 = pow(x+y, 2);
    cout << e1.subs(x+y == 4) << endl;
     // -> 16

    ex e2 = sin(x)*sin(y)*cos(x);
    cout << e2.subs(sin(x) == cos(x)) << endl;
     // -> cos(x)^2*sin(y)

    ex e3 = x+y+z;
    cout << e3.subs(x+y == 4) << endl;
     // -> x+y+z
     // (and not 4+z as one might expect)
@}
@end example

A more powerful form of substitution using wildcards is described in the
next section.


@node Pattern matching and advanced substitutions, Applying a function on subexpressions, Substituting expressions, Methods and functions
@c    node-name, next, previous, up
@section Pattern matching and advanced substitutions
@cindex @code{wildcard} (class)
@cindex Pattern matching

GiNaC allows the use of patterns for checking whether an expression is of a
certain form or contains subexpressions of a certain form, and for
substituting expressions in a more general way.

A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
represents an arbitrary expression. Every wildcard has a @dfn{label} which is
an unsigned integer number to allow having multiple different wildcards in a
pattern. Wildcards are printed as @samp{$label} (this is also the way they
are specified in @command{ginsh}). In C++ code, wildcard objects are created
with the call

@example
ex wild(unsigned label = 0);
@end example

which is simply a wrapper for the @code{wildcard()} constructor with a shorter
name.

Some examples for patterns:

@multitable @columnfractions .5 .5
@item @strong{Constructed as} @tab @strong{Output as}
@item @code{wild()} @tab @samp{$0}
@item @code{pow(x,wild())} @tab @samp{x^$0}
@item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
@item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
@end multitable

Notes:

@itemize @bullet
@item Wildcards behave like symbols and are subject to the same algebraic
  rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
@item As shown in the last example, to use wildcards for indices you have to
  use them as the value of an @code{idx} object. This is because indices must
  always be of class @code{idx} (or a subclass).
@item Wildcards only represent expressions or subexpressions. It is not
  possible to use them as placeholders for other properties like index
  dimension or variance, representation labels, symmetry of indexed objects
  etc.
@item Because wildcards are commutative, it is not possible to use wildcards
  as part of noncommutative products.
@item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
  are also valid patterns.
@end itemize

@subsection Matching expressions
@cindex @code{match()}
The most basic application of patterns is to check whether an expression
matches a given pattern. This is done by the function

@example
bool ex::match(const ex & pattern);
bool ex::match(const ex & pattern, exmap& repls);
@end example

This function returns @code{true} when the expression matches the pattern
and @code{false} if it doesn't. If used in the second form, the actual
subexpressions matched by the wildcards get returned in the associative
array @code{repls} with @samp{wildcard} as a key. If @code{match()}
returns false,  @code{repls} remains unmodified.

The matching algorithm works as follows:

@itemize
@item A single wildcard matches any expression. If one wildcard appears
  multiple times in a pattern, it must match the same expression in all
  places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
  @samp{x*(x+1)} but not @samp{x*(y+1)}).
@item If the expression is not of the same class as the pattern, the match
  fails (i.e. a sum only matches a sum, a function only matches a function,
  etc.).
@item If the pattern is a function, it only matches the same function
  (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
@item Except for sums and products, the match fails if the number of
  subexpressions (@code{nops()}) is not equal to the number of subexpressions
  of the pattern.
@item If there are no subexpressions, the expressions and the pattern must
  be equal (in the sense of @code{is_equal()}).
@item Except for sums and products, each subexpression (@code{op()}) must
  match the corresponding subexpression of the pattern.
@end itemize

Sums (@code{add}) and products (@code{mul}) are treated in a special way to
account for their commutativity and associativity:

@itemize
@item If the pattern contains a term or factor that is a single wildcard,
  this one is used as the @dfn{global wildcard}. If there is more than one
  such wildcard, one of them is chosen as the global wildcard in a random
  way.
@item Every term/factor of the pattern, except the global wildcard, is
  matched against every term of the expression in sequence. If no match is
  found, the whole match fails. Terms that did match are not considered in
  further matches.
@item If there are no unmatched terms left, the match succeeds. Otherwise
  the match fails unless there is a globa