@c %**start of header
@setfilename ginac.info
@settitle GiNaC, an open framework for symbolic computation within the C++ programming language
-@setchapternewpage off
+@setchapternewpage on
@afourpaper
+@c For `info' only.
+@paragraphindent 0
+@c For TeX only.
+@iftex
+@c I hate putting "@noindent" in front of every paragraph.
+@parindent=0pt
+@end iftex
@c %**end of header
@include version.texi
@end direntry
@ifinfo
-This file documents GiNaC @value{VERSION}, an open framework for symbolic
-computation within the C++ programming language.
+This is a tutorial that documents GiNaC @value{VERSION}, an open
+framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2001 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
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 The GiNaC Group:
-@author Christian Bauer, Alexander Frink, Richard B. Kreckel
+@author Christian Bauer, Alexander Frink, Richard Kreckel
@page
@vskip 0pt plus 1filll
-Copyright @copyright{} 1999 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2001 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
@c node-name, next, previous, up
@top GiNaC
-This file documents GiNaC @value{VERSION}, an open framework for symbolic
-computation within the C++ programming language.
+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.
-* Important Algorithms:: Algorithms for symbolic manipulations.
+* 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. It is an attempt to overcome the current
-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.
-
-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{http://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
+semantically impoverished. Although they are quite powerful tools for
+learning math and solving particular problems they lack modern
+linguistical 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{http://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 Johannes Gutenberg University Mainz,
-Germany.
+language is Copyright @copyright{} 1999-2001 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
@c node-name, next, previous, up
@chapter A Tour of GiNaC
-This quick tour of GiNaC wants to rise your interest in the
+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.
@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 it's own as conventional
+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
+manipulations. Here is how to generate and print a simple (and rather
pointless) bivariate polynomial with some large coefficients:
-@subheading My first GiNaC program (a bivariate polynomial)
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
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.
-@subheading My second GiNaC program (Hermite polynomials)
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
-ex HermitePoly(symbol x, int deg)
+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,deg) * diff(HKer, x, deg) / HKer);
+ 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()
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.
+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 @command{ginsh}, a simple GiNaC interactive
-shell that provides a convenient window into GiNaC's capabilities.
+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
@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:
+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;
1/3
@end example
-All numbers occuring in GiNaC's expressions can be converted into floating
-point numbers with the @code{evalf} method, to arbitrary accuracy:
+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);
> a=Pi^2+x;
x+Pi^2
> evalf(a);
-x+9.869604401089358619L0
+9.869604401089358619+x
> x=2;
2
> evalf(a);
-11.869604401089358619L0
+11.869604401089358619
@end example
Built-in functions evaluate immediately to exact numbers if
@example
> lsolve(a+x*y==z,x);
y^(-1)*(z-a);
-lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
+> 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]] ]]
@example
> a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
--3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
+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;
--y^2+x^2+4*x*y
+4*x*y-y^2+x^2
> expand(a*b);
-3*y^6+x^6-24*x*y^5+43*x^2*y^4+16*x^3*y^3+17*x^4*y^2+8*x^5*y
-> collect(a*b,x);
-3*y^6+48*x*y^4+2*x^2*y^2+x^4*(-y^2+x^2+4*x*y)+4*x^3*y*(-y^2+x^2+4*x*y)
+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
+> 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
-You can differentiate functions and expand them as Taylor or
-Laurent series (the third argument of series is the evaluation point,
-the fourth defines the order):
+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);
+> series(sin(x),x==0,4);
x-1/6*x^3+Order(x^4)
-> series(1/tan(x),x,0,4);
+> 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
-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 therefore easy:
+Here we have made use of the @command{ginsh}-command @code{"} to pop the
+previously evaluated element from @command{ginsh}'s internal stack.
+
+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;
@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.
@c node-name, next, previous, up
@section Prerequisites
-In order to install GiNaC on your system, some prerequistes need
-to be met. First of all, you need to have a C++-compiler adhering to
-the ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
+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
+ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{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. Perl is needed
-by the built process as well, since some of the source files are automatically
-generated by Perl scripts. Last but not least, Bruno Haible's library
-@acronym{CLN} is extensively used and needs to be installed on your system.
-Please get it from @uref{ftp://ftp.santafe.edu/pub/gnu/} or from
-@uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP site}
-(it is covered by GPL) and install it prior to trying to install GiNaC.
-The configure script checks if it can find it and if it cannot
+by the built process as well, since some of the source files are
+automatically generated by Perl scripts. Last but not least, Bruno
+Haible's library @acronym{CLN} is extensively used and needs to be
+installed on your system. Please get it either from
+@uref{ftp://ftp.santafe.edu/pub/gnu/}, from
+@uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
+from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
+site} (it is covered by 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. (Actually, this script is by itself
-created with GNU Autoconf from the files @file{configure.in} and
-@file{aclocal.m4}.) 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:
+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
@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
+the header files in @file{/usr/local/include/ginac} and the documentation
(like this one) into @file{/usr/local/share/doc/GiNaC}.
@item
@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
+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).
@command{export @var{VARIABLE}=@var{value}} if the Berkeley C shell is
your login shell.)
-@subheading Sample sessions of how to call the configure script
-
-Simple configuration for a site-wide GiNaC library assuming everything
-is in default paths:
+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
-Configuration for a private static GiNaC library with several components
-sitting in custom places (site-wide @acronym{GCC} and private @acronym{CLN}),
-the compiler pursueded to be picky and full assertions switched on:
+And here is a configuration for a private static GiNaC library with
+several components sitting in custom places (site-wide @acronym{GCC} and
+private @acronym{CLN}). The compiler is pursuaded 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="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic -O2"
+$ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -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.
-at the command prompt and go for a cup of coffee.
-
-Just to make sure GiNaC works properly you may run a simple test
-suite by typing
+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 compare the output
-to reference output. Each of the checks should return a message @samp{passed}
-together with the CPU time used for that particular test. If it does
-not, something went wrong. This is mostly intended to be a QA-check
-if something was broken during the development, but not a sanity check
-of your system. Another intent is to allow people to fiddle around
-with optimization. If @acronym{CLN} was installed all right
-this step is unlikely to return any errors.
+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.
+
+Generally, the top-level Makefile runs recursively to the
+subdirectories. It is therfore safe to go into any subdirectory
+(@code{doc/}, @code{ginsh/}, ...) 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
$ 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):
+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
All documentation (HTML and Postscript) will be stuffed into
-@file{@var{PREFIX}/share/doc/GiNaC/} (or @file{@var{DATADIR}/doc/GiNaC/}, if
-specified).
+@file{@var{PREFIX}/share/doc/GiNaC/} (or
+@file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
@end itemize
-Just for the record 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 finally
-@command{make uninstall} removes the installed library and header
-files@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.}.
+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 with 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.
+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.
* Symbols:: Symbolic objects.
* Numbers:: Numerical objects.
* Constants:: Pre-defined constants.
-* Fundamental operations:: The power, add and mul classes
-* Built-in functions:: Mathematical functions.
+* Fundamental containers:: The power, add and mul classes.
+* Lists:: Lists of expressions.
+* Mathematical functions:: Mathematical functions.
+* Relations:: Equality, Inequality and all that.
+* Indexed objects:: Handling indexed quantities.
+* Non-commutative objects:: Algebras with non-commutative products.
@end menu
@node Expressions, The Class Hierarchy, 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... 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:
+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... 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:
-@subheading Examples of expressions
@example
ex MyEx1 = 5; // simple number
ex MyEx2 = x + 2*y; // polynomial in x and y
ex MyEx5 = MyEx4 + 1; // similar to above
@end example
-Before describing the more fundamental objects that form the building
-blocks of expressions we'll have a quick look under the hood by
-describing how expressions are internally managed.
-
-@unnumberedsubsec Digression: Expressions are reference counted
-
-An expression is extremely light-weight since internally it
-works like a handle to the actual representation and really holds
-nothing more than a pointer to some other object. What this means in
-practice is that whenever you create two @code{ex} and set
-the second equal to the first no copying process is involved. Instead,
-the copying takes place as soon as you try to change the second.
-Consider the simple sequence of code:
-
-@subheading Simple copy-on-write semantics
-@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- symbol x("x"), y("y"), z("z");
- ex e1, e2;
-
- e1 = sin(x + 2*y) + 3*z + 41;
- e2 = e1; // e2 points to same object as e1
- cout << e2 << endl; // prints sin(x+2*y)+3*z+41
- e2 += 1; // e2 is copied into a new object
- cout << e2 << endl; // prints sin(x+2*y)+3*z+42
- // ...
-@}
-@end example
-
-The line @code{e2 = e1;} creates a second expression pointing to the
-object held already by @code{e1}. The time involved for this operation
-is therefore constant, no matter how large @code{e1} was. Actual
-copying, however, must take place in the line @code{e2 += 1;} because
-@code{e1} and @code{e2} are not handles for the same object any more.
-This concept is called @dfn{copy-on-write semantics}. It increases
-performance considerably whenever one object occurs multiple times and
-represents a simple garbage collection scheme because when an @code{ex}
-runs out of scope its destructor checks whether other expressions handle
-the object it points to too and deletes the object from memory if that
-turns out not to be the case. A slightly less trivial example of
-differentiation using the chain-rule should make clear how powerful this
-can be.
-
-@subheading Advanced copy-on-write semantics
-@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- symbol x("x"), y("y");
-
- ex e1 = x + 3*y;
- ex e2 = pow(e1, 3);
- ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
- cout << e1 << endl // prints x+3*y
- << e2 << endl // prints (x+3*y)^3
- << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
- // ...
-@}
-@end example
-
-Here, @code{e1} will actually be referenced three times while @code{e2}
-will be referenced two times. When the power of an expression is built,
-that expression needs not be copied. Likewise, since the derivative of a
-power of an expression can be easily expressed in terms of that expression,
-no copying of @code{e1} is involved when @code{e3} is constructed. So,
-when @code{e3} is constructed it will print as
-@code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
-holds a reference to @code{e2} and the factor in front is just @code{3*e1^2}.
-
-As a user of GiNaC, you cannot see this mechanism of
-copy-on-write semantics. When you insert an expression into a second
-expression, the result behaves exactly as if the contents of the first
-expression were inserted. But it may be useful to remember that this
-is not what happens. Knowing this will enable you to write much more
-efficient code. If you still have an uncertain feeling with
-copy-on-write semantics, we recommend you have a look at the
-@uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
-Marshall Cline. Chapter 16 covers this issue and presents an
-implementation which is pretty close to the one in GiNaC.
+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}.
-So much for expressions. But what exactly are these expressions
-handles of? This will be answered in the following sections.
+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}.
@node The Class Hierarchy, Symbols, Expressions, Basic Concepts
@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 and functions of symbols. You'll soon learn in this chapter
-how many of the functions on symbols are really classes. This is
-because simple symbolic arithmetic is not supported by languages like
-C++ so in a certain way GiNaC has to implement its own arithmetic.
-
-To give an idea about what kinds of symbolic composits may be
-built we have a look at the most important classes in the class
-hierarchy. The dashed line symbolizes a "points to" or "handles"
-relationship while the solid lines stand for "inherits from"
-relationships.
-
-@subheading The GiNaC class hierarchy
-@image{classhierarchy}
-
-Some of the classes shown here (the ones sitting in white boxes) are
-abstract base classes that are of no interest at all 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 would be @code{expairseq}, which is 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 of terms and products,
-respectively. We'll come back later to some more details about these
-two classes and motivate the use of pairs in sums and products here.
-
-@subsection Digression: Internal representation of products and sums
-
-Although it should be completely transparent for the user of
-GiNaC a short discussion of this topic helps to understand the sources
-and also explain performance to a large degree. Consider the symbolic
-expression @math{2*d^3*(4*a+5*b-3)} which could naively be represented
-by a tree of linear containers for addition and multiplication, one
-container for exponentiation with base and exponent and some atomic
-leaves of symbols and numbers in this fashion:
-
-@subheading Naive internal representation-tree for @math{2*d^3*(4*a+5*b-3)}
-@image{repnaive}
-
-However, doing so results in a rather deeply nested tree which will
-quickly become inefficient to manipulate. If we represent the sum
-instead as a sequence of terms, each having a purely numeric
-multiplicative coefficient and the multiplication as a sequence of
-terms, each having a numeric exponent, the tree becomes much more
-flat.
-
-@subheading Pair-wise internal representation-tree for @math{2*d^3*(4*a+5*b-3)}
-@image{reppair}
+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 composits may be built we
+have a look at the most important classes in the class hierarchy and
+some of the relations among the classes:
-The number @code{3} above the symbol @code{d} shows that @code{mul}
-objects are treated similarly where the coefficients are interpreted as
-@emph{exponents} now. Addition of sums of terms or multiplication of
-products with numerical exponents can be coded to be very efficient with
-such a pair-representation. Internally, this handling is done by many CAS in
-this way. It typically speeds up manipulations by an order of
-magnitude. The overall multiplicative factor @code{2} and
-the additive term @code{-3} look somewhat cumbersome in
-this representation, however, since they are still carrying a trivial
-exponent and multiplicative factor @code{1} respectively.
-Within GiNaC, this is avoided by adding a field that carries overall
-numeric coefficient.
-
-@subheading Realistic picture of GiNaC's representation-tree for @math{2*d^3*(4*a+5*b-3)}
-@image{repreal}
+@image{classhierarchy}
-This also allows for a better handling of numeric radicals, since
-@code{sqrt(2)} can now be carried along calculations. Now it should be
-clear, why both classes @code{add} and @code{mul} are derived from the
-same abstract class: the data representation is the same, only the
-semantics differs. In the class hierarchy, methods for polynomial
-expansion and such are reimplemented for @code{add} and @code{mul}, but
-the data structure is inherited from @code{expairseq}.
+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 @math{sin(2*x)}
+@item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
+@item @code{matrix} @tab @math{n}x@math{m} 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
+@end multitable
+@end cartouche
@node Symbols, Numbers, The Class Hierarchy, Basic Concepts
@c node-name, next, previous, up
@section Symbols
-
-Symbols are for symbolic manipulation what atoms are for
-chemistry. You can declare objects of class @code{symbol}
-as any other object simply by saying @code{symbol x,y;}.
-There is, however, a catch in here having to do with the fact that C++
-is a compiled language. The information about the symbol's name is
-thrown away by the compiler but at a later stage you may want to print
-expressions holding your symbols. In order to avoid confusion GiNaC's
-symbols are able to know their own name. This is accomplished by
-declaring its name for output at construction time in the fashion
-@code{symbol x("x");}. If you declare a symbol using the
-default constructor (i.e. without string argument) the system will
-deal out a unique name. That name may not be suitable for printing
-but for internal routines when no output is desired it is often
-enough. We'll come across examples of such symbols later in this
-tutorial.
-
-This implies that the stings passed to symbols at construction
-time may not be used for comparing two of them. It is perfectly
-legitimate to write @code{symbol x("x"),y("x");} but it is
-likely to lead into trouble. Here, @code{x} and
-@code{y} are different symbols and statements like
-@code{x-y} will not be simplified to zero although the
-output @code{x-x} looks funny. Such output may also occur
-when there are two different symbols in two scopes, for instance when
-you call a function that declares a symbol with a name already
-existent in a symbol in the calling function. Again, comparing them
-(using @code{operator==} for instance) will always reveal
-their difference. Watch out, please.
-
-Although symbols can be assigned expressions for internal
-reasons, you should not do it (and we are not going to tell you how it
-is done). If you want to replace a symbol with something else in an
-expression, you can use the expression's @code{.subs()}
-method.
+@cindex @code{symbol} (class)
+@cindex hierarchy of classes
+
+@cindex atom
+Symbols are for symbolic manipulation what atoms are for chemistry. You
+can declare objects of class @code{symbol} as any other object simply by
+saying @code{symbol x,y;}. There is, however, a catch in here having to
+do with the fact that C++ is a compiled language. The information about
+the symbol's name is thrown away by the compiler but at a later stage
+you may want to print expressions holding your symbols. In order to
+avoid confusion GiNaC's symbols are able to know their own name. This
+is accomplished by declaring its name for output at construction time in
+the fashion @code{symbol x("x");}. If you declare a symbol using the
+default constructor (i.e. without string argument) the system will deal
+out a unique name. That name may not be suitable for printing but for
+internal routines when no output is desired it is often enough. We'll
+come across examples of such symbols later in this tutorial.
+
+This implies that the strings passed to symbols at construction time may
+not be used for comparing two of them. It is perfectly legitimate to
+write @code{symbol x("x"),y("x");} but it is likely to lead into
+trouble. Here, @code{x} and @code{y} are different symbols and
+statements like @code{x-y} will not be simplified to zero although the
+output @code{x-x} looks funny. Such output may also occur when there
+are two different symbols in two scopes, for instance when you call a
+function that declares a symbol with a name already existent in a symbol
+in the calling function. Again, comparing them (using @code{operator==}
+for instance) will always reveal their difference. Watch out, please.
+
+@cindex @code{subs()}
+Although symbols can be assigned expressions for internal reasons, you
+should not do it (and we are not going to tell you how it is done). If
+you want to replace a symbol with something else in an expression, you
+can use the expression's @code{.subs()} method (@xref{Substituting Expressions},
+for more information).
@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
-@acronym{CLN}. The classes therein serve as foundation
-classes for GiNaC. @acronym{CLN} stands for Class Library
-for Numbers or alternatively for Common Lisp Numbers. In order to
-find out more about @acronym{CLN}'s internals the reader is
-refered to the documentation of that library. Suffice to say that it
-is by itself build on top of another library, the GNU Multiple
-Precision library @acronym{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. @acronym{CLN} extends
-@acronym{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: construction
-from C-integer, construction of fractions from two integers, construction
-from C-float and construction from a string.
-
-@subheading Construction of numbers
+@acronym{CLN}. The classes therein serve as foundation classes for
+GiNaC. @acronym{CLN} stands for Class Library for Numbers or
+alternatively for Common Lisp Numbers. In order to find out more about
+@acronym{CLN}'s internals the reader is refered to the documentation of
+that library. @inforef{Introduction, , cln}, for more
+information. Suffice to say that it is by itself build on top of another
+library, the GNU Multiple Precision library @acronym{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. @acronym{CLN} extends
+@acronym{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 <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("3.1415926535897932385"); // floating point number
-
- cout << two*p << endl; // floating point 6.283...
- // ...
+ numeric two(2); // exact integer 2
+ numeric r(2,3); // exact fraction 2/3
+ numeric e(2.71828); // floating point number
+ numeric p("3.1415926535897932385"); // floating point number
+ // Trott's constant in scientific notation:
+ numeric trott("1.0841015122311136151E-2");
+
+ std::cout << two*p << std::endl; // floating point 6.283...
@}
@end example
not allowed to write @code{numeric two=2;}. This is because the basic
objects to be handled by GiNaC are the expressions @code{ex} and we want
to keep things simple and wish objects like @code{pow(x,2)} to be
-handled the same way as @code{pow(x,a)}, which means that we need to allow
-a general @code{ex} as base and exponent. Therefore there is an implicit
-constructor from C-integers directly to expressions handling numerics at
-work in most of our examples. This design really becomes convenient when
-one declares own functions having more than one parameter but it forbids
-using implicit constructors because that would lead to ambiguities.
+handled the same way as @code{pow(x,a)}, which means that we need to
+allow a general @code{ex} as base and exponent. Therefore there is an
+implicit constructor from C-integers directly to expressions handling
+numerics at work in most of our examples. This design really becomes
+convenient when one declares own functions having more than one
+parameter but it forbids using implicit constructors because that would
+lead to compile-time ambiguities.
It may be tempting to construct numbers 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
-@code{/} on integers!} Use the constructor from two integers instead, as
-shown in the example above. Writing @code{numeric(1)/2} may look funny but
-works also.
-
-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. The situation is
-different for floating point numbers. Their accuracy is handled 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:
-
-@subheading Controlling the precision of floating point numbers
+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 <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
void foo()
@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.)
+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 check what comes out:
+As an example, let's construct some rational number, multiply it with
+some multiple of its denominator and test what comes out:
-@subheading Sample test on objects of type numeric
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
// some very important constants:
const numeric twentyone(21);
const numeric ten(10);
-const numeric fife(5);
+const numeric five(5);
int main()
@{
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
-@acronym{CLN} is responsible for this behaviour and we refer
-the reader to @acronym{CLN}'s documentation. Suffice to say
-that the same behaviour 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.
+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 @acronym{CLN} is responsible for this
+behaviour and we refer the reader to @acronym{CLN}'s documentation.
+Suffice to say that the same behaviour 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 .33 .66
-@item Method @tab Returns true if@dots{}
+@multitable @columnfractions .30 .70
+@item @strong{Method} @tab @strong{Returns true if the object is@dots{}}
@item @code{.is_zero()}
-@tab object is equal to zero
+@tab @dots{}equal to zero
@item @code{.is_positive()}
-@tab object is not complex and greater than 0
+@tab @dots{}not complex and greater than 0
@item @code{.is_integer()}
-@tab object is a (non-complex) integer
+@tab @dots{}a (non-complex) integer
@item @code{.is_pos_integer()}
-@tab object is an integer and greater than 0
+@tab @dots{}an integer and greater than 0
@item @code{.is_nonneg_integer()}
-@tab object is an integer and greater equal 0
+@tab @dots{}an integer and greater equal 0
@item @code{.is_even()}
-@tab object is an even integer
+@tab @dots{}an even integer
@item @code{.is_odd()}
-@tab object is an odd integer
+@tab @dots{}an odd integer
@item @code{.is_prime()}
-@tab object is a prime integer (probabilistic primality test)
+@tab @dots{}a prime integer (probabilistic primality test)
@item @code{.is_rational()}
-@tab object is an exact rational number (integers are rational, too, as are complex extensions like @math{2/3+7/2*I})
+@tab @dots{}an exact rational number (integers are rational, too)
@item @code{.is_real()}
-@tab object is a real integer, rational or float (i.e. is not complex)
+@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
-@node Constants, Fundamental operations, Numbers, Basic Concepts
+@node Constants, Fundamental containers, Numbers, Basic Concepts
@c node-name, next, previous, up
@section Constants
+@cindex @code{constant} (class)
-Constants behave pretty much like symbols except that that they return
-some specific number when the method @code{.evalf()} is called.
+@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 .29 .57
-@item Name @tab Common Name @tab Numerical Value (35 digits)
+@multitable @columnfractions .14 .30 .56
+@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{EulerGamma}
+@item @code{Euler}
@tab Euler's (or Euler-Mascheroni) constant
@tab 0.57721566490153286060651209008240243
@end multitable
@end cartouche
-@node Fundamental operations, Built-in functions, Constants, Basic Concepts
+@node Fundamental containers, Lists, Constants, Basic Concepts
@c node-name, next, previous, up
-@section Fundamental operations: the @code{power}, @code{add} and @code{mul} classes
+@section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
+@cindex polynomial
+@cindex @code{add}
+@cindex @code{mul}
+@cindex @code{power}
+
+Simple polynomial 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:
-Simple polynomial expressions are written down in GiNaC pretty
-much like in other CAS. The necessary operators @code{+}, @code{-},
-@code{*} and @code{/} have been overloaded to achieve this goal.
-When you run the following program, 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:
-
-@subheading Construction of @code{add} and @code{mul} objects
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
+ ...
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 effects:
+have several counterintuitive and undesired effects:
@itemize @bullet
@item
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}.
+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}.
The general rule is that when you construct such objects, GiNaC
automatically creates them in canonical form, which might differ from
-the form you typed in your program. This allows for rapid comparison
-of expressions, since after all @code{a-a} is simply zero.
-Note, that the canonical form is not necessarily lexicographical
-ordering or in any way easily guessable. It is only guaranteed that
-constructing the same expression twice, either implicitly or
-explicitly, results in the same canonical form.
+the form you typed in your program. This allows for rapid comparison of
+expressions, since after all @code{a-a} is simply zero. Note, that the
+canonical form is not necessarily lexicographical ordering or in any way
+easily guessable. It is only guaranteed that constructing the same
+expression twice, either implicitly or explicitly, results in the same
+canonical form.
-@node Built-in functions, Important Algorithms, Fundamental operations, Basic Concepts
+@node Lists, Mathematical functions, Fundamental containers, Basic Concepts
@c node-name, next, previous, up
-@section Built-in functions
-
-There are quite a number of useful functions built into GiNaC.
-They 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 functions
-may be halted, as it does in the next example:
+@section Lists of expressions
+@cindex @code{lst} (class)
+@cindex lists
+@cindex @code{nops()}
+@cindex @code{op()}
+@cindex @code{append()}
+@cindex @code{prepend()}
+
+The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
+These are sometimes used to supply a variable number of arguments of the same
+type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
+should have a basic understanding about them.
+
+Lists of up to 15 expressions can be directly constructed from single
+expressions:
-@subheading Evaluation of built-in functions
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
@{
symbol x("x"), y("y");
-
- ex foo = x+y/2;
- cout << "gamma(" << foo << ") -> " << gamma(foo) << endl;
- ex bar = foo.subs(y==1);
- cout << "gamma(" << bar << ") -> " << gamma(bar) << endl;
- ex foobar= bar.subs(x==7);
- cout << "gamma(" << foobar << ") -> " << gamma(foobar) << endl;
+ lst l(x, 2, y, x+y);
+ // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
// ...
-@}
@end example
-This program will type out two times a function and then an
-expression that may be really useful:
+Use the @code{nops()} method to determine the size (number of expressions) of
+a list and the @code{op()} method to access individual elements:
@example
-gamma(x+(1/2)*y) -> gamma(x+(1/2)*y)
-gamma(x+1/2) -> gamma(x+1/2)
-gamma(15/2) -> (135135/128)*Pi^(1/2)
+ // ...
+ cout << l.nops() << endl; // prints '4'
+ cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
+ // ...
@end example
-Most of these functions can be differentiated, series expanded so on.
-Read the next chapter in order to learn more about this..
-
-
-@node Important Algorithms, Polynomial Expansion, Built-in functions, Top
-@c node-name, next, previous, up
-@chapter Important Algorithms
-
-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:
+Finally you can append or prepend an expression to a list with the
+@code{append()} and @code{prepend()} methods:
-@subheading Methods vs. wrapper functions
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- ex x = numeric(1.0);
-
- cout << "As method: " << sin(x).evalf() << endl;
- cout << "As function: " << evalf(sin(x)) << endl;
// ...
+ l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
+ l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
@}
@end example
-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 always 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, a chain of function wrappers is much harder
-to read than a chain 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
-* Polynomial Expansion::
-* Collecting expressions::
-* Polynomial Arithmetic::
-* Symbolic Differentiation::
-* Series Expansion::
-@end menu
+@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 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.
-@node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
+@node Relations, Indexed objects, Mathematical functions, Basic Concepts
@c node-name, next, previous, up
-@section Polynomial Expansion
+@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.
+
+
+@node Indexed objects, Non-commutative objects, Relations, Basic Concepts
+@c node-name, next, previous, up
+@section Indexed objects
-A polynomial in one or more variables has many equivalent
-representations. Some useful ones serve a specific purpose. Consider
-for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 + 21*y*z + 4*z^2}.
-It is equivalent to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}.
-Other representations are the recursive ones where one collects for
-exponents in one of the three variable. Since the factors are
-themselves polynomials in the remaining two variables the procedure
-can be repeated. In our expample, two possibilies would be
-@math{(4*y + z)*x + 20*y^2 + 21*y*z + 4*z^2} and
-@math{20*y^2 + (21*z + 4*x)*y + 4*z^2 + x*z}.
+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.
-To bring an expression into expanded form, its method
-@code{.expand()} may be called. In our example above,
-this corresponds to @math{4*x*y + x*z + 20*y^2 + 21*y*z + 4*z^2}.
-Again, since the canonical form in GiNaC is not easily guessable you
-should be prepared to see different orderings of terms in such sums!
+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
-@node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
-@c node-name, next, previous, up
-@section Collecting expressions
+Indexed expressions in GiNaC are constructed of two special types of objects,
+@dfn{index objects} and @dfn{indexed objects}.
-Another useful representation of multivariate polynomials is as
-a univariate polynomial in one of the variables with the coefficients
-being polynomials in the remaining variables. The method
-@code{collect()} accomplishes this task:
+@itemize @bullet
-@example
-#include <ginac/ginac.h>
-ex ex::collect(symbol const & s);
-@end example
+@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}.
-Note that the original polynomial needs to be in expanded form in
-order to be able to find the coefficients properly. The range of
-occuring coefficients can be checked using the two methods
+@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.
-@example
-#include <ginac/ginac.h>
-int ex::degree(symbol const & s);
-int ex::ldegree(symbol const & s);
-@end example
+@end itemize
+
+@strong{Note:} 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.
-where @code{degree()} returns the highest coefficient and
-@code{ldegree()} the lowest one. These two methods work
-also reliably on non-expanded input polynomials. This is illustrated
-in the following example:
+A simple example shall illustrate the concepts:
-@subheading Collecting expressions in multivariate polynomials
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@{
- symbol x("x"), y("y");
- ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
- - pow(x+y,2) + 2*pow(y+2,2) - 8;
- ex Poly = PolyInp.expand();
-
- for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
- cout << "The x^" << i << "-coefficient is "
- << Poly.coeff(x,i) << endl;
- @}
- cout << "As polynomial in y: "
- << Poly.collect(y) << endl;
- // ...
-@}
+ 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
+ ...
@end example
-When run, it returns an output in the following fashion:
+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}.
+
+Note the difference between the indices @code{i} and @code{j} which are of
+class @code{idx}, and the index values which are the sybols @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
-The x^0-coefficient is y^2+11*y
-The x^1-coefficient is 5*y^2-2*y
-The x^2-coefficient is -1
-The x^3-coefficient is 4*y
-As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
+symbol i("i"), j("j");
+e = indexed(A, i, j); // ERROR: indices must be of type idx
@end example
-As always, the exact output may vary between different versions of
-GiNaC or even from run to run since the internal canonical ordering is
-not within the user's sphere of influence.
+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
-@node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
-@c node-name, next, previous, up
-@section Polynomial Arithmetic
-
-@subsection GCD and LCM
+@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).
-The functions for polynomial greatest common divisor and least common
-multiple have the synopsis:
+In fact, base expressions, index values and index dimensions can be
+arbitrary expressions:
@example
-#include <ginac/normal.h>
-ex gcd(const ex & a, const ex & b);
-ex lcm(const ex & a, const ex & b);
+ ...
+ cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
+ // -> (B+A).(1+2*i)
+ ...
@end example
-The functions @code{gcd()} and @code{lcm()} accept two expressions
-@code{a} and @code{b} as arguments and return
-a new expression, their greatest common divisor or least common
-multiple, respectively. If the polynomials @code{a} and
-@code{b} are coprime @code{gcd(a,b)} returns 1 and @code{lcm(a,b)}
-returns the product of @code{a} and @code{b}.
+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_dimension()}
+The methods
-@subheading Polynomal GCD/LCM
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+ex idx::get_value(void);
+ex idx::get_dimension(void);
+@end example
-int main()
-@{
- symbol x("x"), y("y"), z("z");
- ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
- ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
+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.
- ex P_gcd = gcd(P_a, P_b);
- // x + 5*y + 4*z
- ex P_lcm = lcm(P_a, P_b);
- // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
- // ...
-@}
+There are also the methods
+
+@example
+bool idx::is_numeric(void);
+bool idx::is_symbolic(void);
+bool idx::is_dim_numeric(void);
+bool idx::is_dim_symbolic(void);
@end example
-@subsection The @code{normal} method
+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.
-While in common symbolic code @code{gcd()} and @code{lcm()} are not too
-heavily used, simplification occurs frequently. Therefore @code{.normal()},
-which provides some basic form of simplification, has become a method of
-class @code{ex}, just like @code{.expand()}.
-It converts a rational function into an equivalent rational function
-where numererator and denominator are coprime. This means, it finds
-the GCD of numerator and denominator and cancels it. If it encounters
-some object which does not belong to the domain of rationals (a
-function for instance), that object is replaced by a temporary symbol.
-This means that both expressions @code{t1} and
-@code{t2} are indeed simplified in this little program:
+@cindex @code{varidx} (class)
+If you need co- and contravariant indices, use the @code{varidx} class:
-@subheading Cancellation of polynomial GCD (with obstacles)
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- symbol x("x");
- ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
- ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
- cout << "t1 is " << t1.normal() << endl;
- cout << "t2 is " << t2.normal() << endl;
- // ...
-@}
+ ...
+ 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
-Of course this works for multivariate polynomials too, so the ratio of
-the sample-polynomials from the section about GCD and LCM above would
-be normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
-
+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
-@node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
-@c node-name, next, previous, up
-@section Symbolic Differentiation
+@example
+bool varidx::is_covariant(void);
+bool varidx::is_contravariant(void);
+@end example
-GiNaC's objects know how to differentiate themselves. Thus, a
-polynomial (class @code{add}) knows that its derivative is
-the sum of the derivatives of all the monomials:
+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
-@subheading Simple polynomial differentiation
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+ex varidx::toggle_variance(void);
+@end example
-int main()
-@{
- symbol x("x"), y("y"), z("z");
- ex P = pow(x, 5) + pow(x, 2) + y;
+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.
- cout << P.diff(x,2) << endl; // 20*x^3 + 2
- cout << P.diff(y) << endl; // 1
- cout << P.diff(z) << endl; // 0
- // ...
-@}
+@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
-If a second integer parameter @var{n} is given, the @code{diff} method
-returns the @var{n}th derivative.
+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
-If @emph{every} object and every function is told
-what its derivative is, all derivatives of composed objects can be
-calculated using the chain rule and the product rule. Consider, for
-instance the expression @code{1/cosh(x)}. Since the
-derivative of @code{cosh(x)} is @code{sinh(x)}
-and the derivative of @code{pow(x,-1)} is
-@code{-pow(x,-2)}, GiNaC can readily compute the
-composition. It turns out that the composition is the generating
-function for Euler Numbers, i.e. the so called
-@var{n}th Euler number is the coefficient of @code{x^n/n!} in
-the expansion of @code{1/cosh(x)}. We may use this identity to code a
-function that generates Euler numbers in just three lines:
-
-@subheading Differentiation with nontrivial functions: Euler numbers
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+bool spinidx::is_dotted(void);
+bool spinidx::is_undotted(void);
+@end example
-ex EulerNumber(unsigned n)
-@{
- symbol x;
- ex generator = pow(cosh(x),-1);
- return generator.diff(x,n).subs(x==0);
-@}
+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
-int main()
-@{
- for (unsigned i=0; i<11; i+=2)
- cout << EulerNumber(i) << endl;
- return 0;
-@}
+@example
+ex spinidx::toggle_dot(void);
+ex spinidx::toggle_variance_dot(void);
@end example
-When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
-@code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
-@code{i} by two since all odd Euler numbers vanish anyways.
+create a new index with the same value and dimension but opposite dottedness
+and the same or opposite variance.
+@subsection Substituting indices
-@node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms
-@c node-name, next, previous, up
-@section Series Expansion
+@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}).
-Expressions know how to expand themselves as a Taylor series or
-(more generally) a Laurent series. As in most conventional Computer
-Algebra Systems no distinction is made between those two. There is a
-class of its own for storing such series as well as a class for
-storing the order of the series. A sample program could read:
+You have two possibilities here. You can either substitute the whole index
+by another index or expression:
-@subheading Series expansion
@example
-#include <ginac/ginac.h>
+ ...
+ 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
+
+Indexed objects can be declared as being totally symmetric or antisymmetric
+with respect to their indices. In this case, GiNaC will automatically bring
+the indices into a canonical order which allows for some immediate
+simplifications:
+
+@example
+ ...
+ cout << indexed(A, indexed::symmetric, i, j)
+ + indexed(A, indexed::symmetric, j, i) << endl;
+ // -> 2*A.j.i
+ cout << indexed(B, indexed::antisymmetric, i, j)
+ + indexed(B, indexed::antisymmetric, j, j) << endl;
+ // -> -B.j.i
+ cout << indexed(B, indexed::antisymmetric, i, j)
+ + indexed(B, indexed::antisymmetric, j, 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 dimension 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{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(void);
+ex ex::simplify_indexed(const scalar_products & sp);
+@end example
+
+that performs some more expensive operations:
+
+@itemize
+@item it checks the consistency of free indices in sums in the same way
+ @code{get_free_indices()} does
+@item it tries to give dumy 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 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. After @code{sp.add(A, B, 0)},
+@code{simplify_indexed()} will replace all scalar products of indexed
+objects that have the symbols @code{A} and @code{B} as base expressions
+with the single value 0. The number, type and dimension of the indices
+don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
+
+@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) << endl;
+ 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 behaviour 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).
+
+@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");
+
+ matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(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 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 ususally carry
+indices.
+
+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 commute 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 commute 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. Symbols
+always commute and it's not possible to construct non-commutative products
+using symbols to represent the algebra elements or generators. User-defined
+functions can, however, be specified as being non-commutative.
+
+@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(void) const;
+unsigned ex::return_type_tinfo(void) 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
+@item @code{return_types::commutative}: Commutes 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 commute
+ 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 commute with any other @code{noncommutative} or
+ @code{noncommutative_composite} expressions.
+@end itemize
+
+The value returned by the @code{return_type_tinfo()} method is valid only
+when the return type of the expression is @code{noncommutative}. It is a
+value that is unique to the class of the object and usually one of the
+constants in @file{tinfos.h}, or derived therefrom.
+
+Here are a couple of examples:
+
+@cartouche
+@multitable @columnfractions 0.33 0.33 0.34
+@item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
+@item @code{42} @tab @code{commutative} @tab -
+@item @code{2*x-y} @tab @code{commutative} @tab -
+@item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
+@item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
+@item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
+@item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
+@end multitable
+@end cartouche
+
+Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
+@code{TINFO_clifford} for objects with a representation label of zero.
+Other representation labels yield a different @code{return_type_tinfo()},
+but it's the same for any two objects with the same label. This is also true
+for color objects.
+
+As a last note, powers of non-commutative objects are not allowed in GiNaC.
+You can use the function
+
+@example
+ex ncpow(const ex & basis, unsigned exponent)
+@end example
+
+instead which returns an expanded product (e.g. @code{ncpow(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
+
+@cindex @code{dirac_gamma()}
+Clifford algebra elements (also called Dirac gamma matrices, although 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 commute 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
+
+@cindex @code{dirac_gamma5()}
+and there's a special element @samp{gamma5} that commutes with all other
+gammas 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_gamma6()}
+@cindex @code{dirac_gamma7()}
+The two additional functions
+
+@example
+ex dirac_gamma6(unsigned char rl = 0);
+ex dirac_gamma7(unsigned char rl = 0);
+@end example
+
+return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
+respectively.
+
+@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 of the form @samp{e.mu gamma~mu} with a new and unique index
+whose dimension is given by the @code{dim} argument.
+
+In products of dirac gammas, superfluous unity elements are automatically
+removed, squares are replaced by their values and @samp{gamma5} is
+anticommuted 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*gamma~symbol10*gamma.mu)*a.symbol10
+ e = e.simplify_indexed();
+ cout << e << endl;
+ // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
+ cout << e.subs(D == 4) << endl;
+ // -> -2*gamma~symbol10*a.symbol10
+ // [ == -2 * dirac_slash(a, D) ]
+ ...
+@}
+@end example
+
+@cindex @code{dirac_trace()}
+To calculate the trace of an expression containing strings of Dirac gammas
+you use the function
+
+@example
+ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
+@end example
+
+This function takes the trace of all gammas with the specified representation
+label; 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 usual trace only in @math{D = 4} dimensions.
+In particular, the functional is not cyclic in @math{D != 4} 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
+@cite{The Role of gamma5 in Dimensional Regularization}.
+
+The value of the trace itself is also usually different in 4 and in
+@math{D != 4} 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), true);
+ 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*eta~mu~nu
+@}
+@end example
+
+
+@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 commute 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
+
+@cindex @code{color_d()}
+@cindex @code{color_f()}
+and 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}.
+
+@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 the
+function
+
+@example
+ex color_trace(const ex & e, unsigned char rl = 0);
+@end example
+
+This function takes the trace of all color @samp{T} objects with the
+specified representation label; @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::
+* Substituting Expressions::
+* Pattern Matching and Advanced Substitutions::
+* Polynomial Arithmetic:: Working with polynomials.
+* Rational Expressions:: Working with rational functions.
+* Symbolic Differentiation::
+* Series Expansion:: Taylor and Laurent expansion.
+* Built-in Functions:: List of predefined mathematical functions.
+* Input/Output:: Input and output of expressions.
+@end menu
+
+
+@node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
+@c node-name, next, previous, up
+@section Getting information about expressions
+
+@subsection Checking expression types
+@cindex @code{is_ex_of_type()}
+@cindex @code{ex_to_numeric()}
+@cindex @code{ex_to_@dots{}}
+@cindex @code{Converting 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 (the first one is actually a macro):
+
+@example
+bool is_ex_of_type(const ex & e, TYPENAME t);
+bool ex::info(unsigned flag);
+unsigned ex::return_type(void) const;
+unsigned ex::return_type_tinfo(void) const;
+@end example
+
+When the test made by @code{is_ex_of_type()} returns true, it is safe to
+call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} 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_ex_of_type(e, numeric))
+ numeric n = ex_to_numeric(e);
+ @dots{}
+@}
+@end example
+
+@code{is_ex_of_type()} 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_ex_of_type(e1, numeric); // true
+ is_ex_of_type(e2, numeric); // false
+ is_ex_of_type(e1, add); // false
+ is_ex_of_type(e2, add); // true
+ is_ex_of_type(e1, mul); // false
+ is_ex_of_type(e2, mul); // false
+@}
+@end example
+
+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_ex_of_type(..., numeric)})
+@item @code{real}
+@tab @dots{}a real integer, rational or float (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_ex_of_type(..., 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_ex_of_type(..., symbol)})
+@item @code{list}
+@tab @dots{}a list (same as @code{is_ex_of_type(..., 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)})
+@item @code{algebraic}
+@tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
+@end multitable
+@end cartouche
+
+To determine whether an expression is commutative or non-commutative and if
+so, with which other expressions it would commute, 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 @code{nops()}
+@cindex @code{op()}
+@cindex container
+@cindex @code{relational} (class)
+
+GiNaC provides the two methods
+
+@example
+unsigned ex::nops();
+ex ex::op(unsigned i);
+@end example
+
+for accessing the subexpressions in the container-like GiNaC classes like
+@code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
+determines the number of subexpressions (@samp{operands}) contained, while
+@code{op()} 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.
+
+The left-hand 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 (@xref{Relations}.)
+which is not evaluated until (explicitly or implicitely) 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.
+
+@strong{Warning:} You will also find an @code{ex::compare()} method in the
+GiNaC header files. This method is however only to be used internally by
+GiNaC to establish a canonical sort order for terms, and using it to compare
+expressions will give very surprising results.
+
+
+@node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, 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);
+ex ex::subs(const lst & syms, const lst & repls);
+@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^2-4*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 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
+@code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
+
+@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, Polynomial Arithmetic, Substituting Expressions, Methods and Functions
+@c node-name, next, previous, up
+@section Pattern matching and advanced substitutions
+
+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
+@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
+
+@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, lst & 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 @code{repls}
+object as a list of relations of the form @samp{wildcard == expression}.
+If @code{match()} returns false, the state of @code{repls} is undefined.
+For reproducible results, the list should be empty when passed to
+@code{match()}, but it is also possible to find similarities in multiple
+expressions by passing in the result of a previous match.
+
+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 global wildcard in the pattern, in
+ which case this wildcard matches the remaining terms.
+@end itemize
+
+In general, having more than one single wildcard as a term of a sum or a
+factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
+amgiguous results.
+
+Here are some examples in @command{ginsh} to demonstrate how it works (the
+@code{match()} function in @command{ginsh} returns @samp{FAIL} if the
+match fails, and the list of wildcard replacements otherwise):
+
+@example
+> match((x+y)^a,(x+y)^a);
+[]
+> match((x+y)^a,(x+y)^b);
+FAIL
+> match((x+y)^a,$1^$2);
+[$1==x+y,$2==a]
+> match((x+y)^a,$1^$1);
+FAIL
+> match((x+y)^(x+y),$1^$1);
+[$1==x+y]
+> match((x+y)^(x+y),$1^$2);
+[$1==x+y,$2==x+y]
+> match((a+b)*(a+c),($1+b)*($1+c));
+[$1==a]
+> match((a+b)*(a+c),(a+$1)*(a+$2));
+[$1==c,$2==b]
+ (Unpredictable. The result might also be [$1==c,$2==b].)
+> match((a+b)*(a+c),($1+$2)*($1+$3));
+ (The result is undefined. Due to the sequential nature of the algorithm
+ and the re-ordering of terms in GiNaC, the match for the first factor
+ may be [$1==a,$2==b] in which case the match for the second factor
+ succeeds, or it may be [$1==b,$2==a] which causes the second match to
+ fail.)
+> match(a*(x+y)+a*z+b,a*$1+$2);
+ (This is also ambiguous and may return either [$1==z,$2==a*(x+y)+b] or
+ [$1=x+y,$2=a*z+b].)
+> match(a+b+c+d+e+f,c);
+FAIL
+> match(a+b+c+d+e+f,c+$0);
+[$0==a+e+b+f+d]
+> match(a+b+c+d+e+f,c+e+$0);
+[$0==a+b+f+d]
+> match(a+b,a+b+$0);
+[$0==0]
+> match(a*b^2,a^$1*b^$2);
+FAIL
+ (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
+ even if a==a^1.)
+> match(x*atan2(x,x^2),$0*atan2($0,$0^2));
+[$0==x]
+> match(atan2(y,x^2),atan2(y,$0));
+[$0==x^2]
+@end example
+
+@cindex @code{has()}
+A more general way to look for patterns in expressions is provided by the
+member function
+
+@example
+bool ex::has(const ex & pattern);
+@end example
+
+This function checks whether a pattern is matched by an expression itself or
+by any of its subexpressions.
+
+Again some examples in @command{ginsh} for illustration (in @command{ginsh},
+@code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
+
+@example
+> has(x*sin(x+y+2*a),y);
+1
+> has(x*sin(x+y+2*a+y),x+y);
+0
+ (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
+ has the subexpressions "x", "y" and "2*a".)
+> has(x*sin(x+y+2*a+y),x+y+$1);
+1
+ (But this is possible.)
+> has(x*sin(2*(x+y)+2*a),x+y);
+0
+ (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
+ which "x+y" is not a subexpression.)
+> has(x+1,x^$1);
+0
+ (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
+ "x^something".)
+> has(4*x^2-x+3,$1*x);
+1
+> has(4*x^2+x+3,$1*x);
+0
+ (Another possible pitfall. The first expression matches because the term
+ "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
+ contains a linear term you should use the coeff() function instead.)
+@end example
+
+@cindex @code{subs()}
+Probably the most useful application of patterns is to use them for
+substituting expressions with the @code{subs()} method. Wildcards can be
+used in the search patterns as well as in the replacement expressions, where
+they get replaced by the expressions matched by them. @code{subs()} doesn't
+know anything about algebra; it performs purely syntactic substitutions.
+
+Some examples:
+
+@example
+> subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
+b^3+a^3+(x+y)^3
+> subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
+b^4+a^4+(x+y)^4
+> subs((a+b+c)^2,a+b=x);
+(a+b+c)^2
+> subs((a+b+c)^2,a+b+$1==x+$1);
+(x+c)^2
+> subs(a+2*b,a+b=x);
+a+2*b
+> subs(4*x^3-2*x^2+5*x-1,x==a);
+-1+5*a-2*a^2+4*a^3
+> subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
+-1+5*x-2*a^2+4*a^3
+> subs(sin(1+sin(x)),sin($1)==cos($1));
+cos(1+cos(x))
+> expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
+a+b
+@end example
+
+The last example would be written in C++ in this way:
+
+@example
+@{
+ symbol a("a"), b("b"), x("x"), y("y");
+ e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
+ e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
+ cout << e.expand() << endl;
+ // -> a+b
+@}
+@end example
+
+
+@node Polynomial Arithmetic, Rational Expressions, Pattern Matching and Advanced Substitutions, Methods and Functions
+@c node-name, next, previous, up
+@section Polynomial arithmetic
+
+@subsection Expanding and collecting
+@cindex @code{expand()}
+@cindex @code{collect()}
+
+A polynomial in one or more variables has many equivalent
+representations. Some useful ones serve a specific purpose. Consider
+for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
+21*y*z + 4*z^2} (written down here in output-style). It is equivalent
+to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
+representations are the recursive ones where one collects for exponents
+in one of the three variable. Since the factors are themselves
+polynomials in the remaining two variables the procedure can be
+repeated. In our expample, two possibilities would be @math{(4*y + z)*x
++ 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
+x*z}.
+
+To bring an expression into expanded form, its method
+
+@example
+ex ex::expand();
+@end example
+
+may be called. In our example above, this corresponds to @math{4*x*y +
+x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
+GiNaC is not easily guessable you should be prepared to see different
+orderings of terms in such sums!
+
+Another useful representation of multivariate polynomials is as a
+univariate polynomial in one of the variables with the coefficients
+being polynomials in the remaining variables. The method
+@code{collect()} accomplishes this task:
+
+@example
+ex ex::collect(const ex & s, bool distributed = false);
+@end example
+
+The first argument to @code{collect()} can also be a list of objects in which
+case the result is either a recursively collected polynomial, or a polynomial
+in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
+by the @code{distributed} flag.
+
+Note that the original polynomial needs to be in expanded form in order
+for @code{collect()} to be able to find the coefficients properly.
+
+@subsection Degree and coefficients
+@cindex @code{degree()}
+@cindex @code{ldegree()}
+@cindex @code{coeff()}
+
+The degree and low degree of a polynomial can be obtained using the two
+methods
+
+@example
+int ex::degree(const ex & s);
+int ex::ldegree(const ex & s);
+@end example
+
+which also work reliably on non-expanded input polynomials (they even work
+on rational functions, returning the asymptotic degree). To extract
+a coefficient with a certain power from an expanded polynomial you use
+
+@example
+ex ex::coeff(const ex & s, int n);
+@end example
+
+You can also obtain the leading and trailing coefficients with the methods
+
+@example
+ex ex::lcoeff(const ex & s);
+ex ex::tcoeff(const ex & s);
+@end example
+
+which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
+respectively.
+
+An application is illustrated in the next example, where a multivariate
+polynomial is analyzed:
+
+@example
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x"), y("y");
+ ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
+ - pow(x+y,2) + 2*pow(y+2,2) - 8;
+ ex Poly = PolyInp.expand();
+
+ for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
+ cout << "The x^" << i << "-coefficient is "
+ << Poly.coeff(x,i) << endl;
+ @}
+ cout << "As polynomial in y: "
+ << Poly.collect(y) << endl;
+@}
+@end example
+
+When run, it returns an output in the following fashion:
+
+@example
+The x^0-coefficient is y^2+11*y
+The x^1-coefficient is 5*y^2-2*y
+The x^2-coefficient is -1
+The x^3-coefficient is 4*y
+As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
+@end example
+
+As always, the exact output may vary between different versions of GiNaC
+or even from run to run since the internal canonical ordering is not
+within the user's sphere of influence.
+
+@code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
+@code{tcoeff()} and @code{collect()} can also be used to a certain degree
+with non-polynomial expressions as they not only work with symbols but with
+constants, functions and indexed objects as well:
+
+@example
+@{
+ symbol a("a"), b("b"), c("c");
+ idx i(symbol("i"), 3);
+
+ ex e = pow(sin(x) - cos(x), 4);
+ cout << e.degree(cos(x)) << endl;
+ // -> 4
+ cout << e.expand().coeff(sin(x), 3) << endl;
+ // -> -4*cos(x)
+
+ e = indexed(a+b, i) * indexed(b+c, i);
+ e = e.expand(expand_options::expand_indexed);
+ cout << e.collect(indexed(b, i)) << endl;
+ // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
+@}
+@end example
+
+
+@subsection Polynomial division
+@cindex polynomial division
+@cindex quotient
+@cindex remainder
+@cindex pseudo-remainder
+@cindex @code{quo()}
+@cindex @code{rem()}
+@cindex @code{prem()}
+@cindex @code{divide()}
+
+The two functions
+
+@example
+ex quo(const ex & a, const ex & b, const symbol & x);
+ex rem(const ex & a, const ex & b, const symbol & x);
+@end example
+
+compute the quotient and remainder of univariate polynomials in the variable
+@samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
+
+The additional function
+
+@example
+ex prem(const ex & a, const ex & b, const symbol & x);
+@end example
+
+computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
+@math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
+
+Exact division of multivariate polynomials is performed by the function
+
+@example
+bool divide(const ex & a, const ex & b, ex & q);
+@end example
+
+If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
+and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
+in which case the value of @code{q} is undefined.
+
+
+@subsection Unit, content and primitive part
+@cindex @code{unit()}
+@cindex @code{content()}
+@cindex @code{primpart()}
+
+The methods
+
+@example
+ex ex::unit(const symbol & x);
+ex ex::content(const symbol & x);
+ex ex::primpart(const symbol & x);
+@end example
+
+return the unit part, content part, and primitive polynomial of a multivariate
+polynomial with respect to the variable @samp{x} (the unit part being the sign
+of the leading coefficient, the content part being the GCD of the coefficients,
+and the primitive polynomial being the input polynomial divided by the unit and
+content parts). The product of unit, content, and primitive part is the
+original polynomial.
+
+
+@subsection GCD and LCM
+@cindex GCD
+@cindex LCM
+@cindex @code{gcd()}
+@cindex @code{lcm()}
+
+The functions for polynomial greatest common divisor and least common
+multiple have the synopsis
+
+@example
+ex gcd(const ex & a, const ex & b);
+ex lcm(const ex & a, const ex & b);
+@end example
+
+The functions @code{gcd()} and @code{lcm()} accept two expressions
+@code{a} and @code{b} as arguments and return a new expression, their
+greatest common divisor or least common multiple, respectively. If the
+polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
+and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
+
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x"), y("y"), z("z");
+ ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
+ ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
+
+ ex P_gcd = gcd(P_a, P_b);
+ // x + 5*y + 4*z
+ ex P_lcm = lcm(P_a, P_b);
+ // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
+@}
+@end example
+
+
+@subsection Square-free decomposition
+@cindex square-free decomposition
+@cindex factorization
+@cindex @code{sqrfree()}
+
+GiNaC still lacks proper factorization support. Some form of
+factorization is, however, easily implemented by noting that factors
+appearing in a polynomial with power two or more also appear in the
+derivative and hence can easily be found by computing the GCD of the
+original polynomial and its derivatives. Any system has an interface
+for this so called square-free factorization. So we provide one, too:
+@example
+ex sqrfree(const ex & a, const lst & l = lst());
+@end example
+Here is an example that by the way illustrates how the result may depend
+on the order of differentiation:
+@example
+ ...
+ symbol x("x"), y("y");
+ ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
+
+ cout << sqrfree(BiVarPol, lst(x,y)) << endl;
+ // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
+
+ cout << sqrfree(BiVarPol, lst(y,x)) << endl;
+ // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
+
+ cout << sqrfree(BiVarPol) << endl;
+ // -> depending on luck, any of the above
+ ...
+@end example
+
+
+@node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
+@c node-name, next, previous, up
+@section Rational expressions
+
+@subsection The @code{normal} method
+@cindex @code{normal()}
+@cindex simplification
+@cindex temporary replacement
+
+Some basic form of simplification of expressions is called for frequently.
+GiNaC provides the method @code{.normal()}, which converts a rational function
+into an equivalent rational function of the form @samp{numerator/denominator}
+where numerator and denominator are coprime. If the input expression is already
+a fraction, it just finds the GCD of numerator and denominator and cancels it,
+otherwise it performs fraction addition and multiplication.
+
+@code{.normal()} can also be used on expressions which are not rational functions
+as it will replace all non-rational objects (like functions or non-integer
+powers) by temporary symbols to bring the expression to the domain of rational
+functions before performing the normalization, and re-substituting these
+symbols afterwards. This algorithm is also available as a separate method
+@code{.to_rational()}, described below.
+
+This means that both expressions @code{t1} and @code{t2} are indeed
+simplified in this little program:
+
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x");
+ ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
+ ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
+ std::cout << "t1 is " << t1.normal() << std::endl;
+ std::cout << "t2 is " << t2.normal() << std::endl;
+@}
+@end example
+
+Of course this works for multivariate polynomials too, so the ratio of
+the sample-polynomials from the section about GCD and LCM above would be
+normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
+
+
+@subsection Numerator and denominator
+@cindex numerator
+@cindex denominator
+@cindex @code{numer()}
+@cindex @code{denom()}
+
+The numerator and denominator of an expression can be obtained with
+
+@example
+ex ex::numer();
+ex ex::denom();
+@end example
+
+These functions will first normalize the expression as described above and
+then return the numerator or denominator, respectively.
+
+
+@subsection Converting to a rational expression
+@cindex @code{to_rational()}
+
+Some of the methods described so far only work on polynomials or rational
+functions. GiNaC provides a way to extend the domain of these functions to
+general expressions by using the temporary replacement algorithm described
+above. You do this by calling
+
+@example
+ex ex::to_rational(lst &l);
+@end example
+
+on the expression to be converted. The supplied @code{lst} will be filled
+with the generated temporary symbols and their replacement expressions in
+a format that can be used directly for the @code{subs()} method. It can also
+already contain a list of replacements from an earlier application of
+@code{.to_rational()}, so it's possible to use it on multiple expressions
+and get consistent results.
+
+For example,
+
+@example
+@{
+ symbol x("x");
+ ex a = pow(sin(x), 2) - pow(cos(x), 2);
+ ex b = sin(x) + cos(x);
+ ex q;
+ lst l;
+ divide(a.to_rational(l), b.to_rational(l), q);
+ cout << q.subs(l) << endl;
+@}
+@end example
+
+will print @samp{sin(x)-cos(x)}.
+
+
+@node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
+@c node-name, next, previous, up
+@section Symbolic differentiation
+@cindex differentiation
+@cindex @code{diff()}
+@cindex chain rule
+@cindex product rule
+
+GiNaC's objects know how to differentiate themselves. Thus, a
+polynomial (class @code{add}) knows that its derivative is the sum of
+the derivatives of all the monomials:
+
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x"), y("y"), z("z");
+ ex P = pow(x, 5) + pow(x, 2) + y;
+
+ cout << P.diff(x,2) << endl; // 20*x^3 + 2
+ cout << P.diff(y) << endl; // 1
+ cout << P.diff(z) << endl; // 0
+@}
+@end example
+
+If a second integer parameter @var{n} is given, the @code{diff} method
+returns the @var{n}th derivative.
+
+If @emph{every} object and every function is told what its derivative
+is, all derivatives of composed objects can be calculated using the
+chain rule and the product rule. Consider, for instance the expression
+@code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
+@code{sinh(x)} and the derivative of @code{pow(x,-1)} is
+@code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
+out that the composition is the generating function for Euler Numbers,
+i.e. the so called @var{n}th Euler number is the coefficient of
+@code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
+identity to code a function that generates Euler numbers in just three
+lines:
+
+@cindex Euler numbers
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+ex EulerNumber(unsigned n)
+@{
+ symbol x;
+ const ex generator = pow(cosh(x),-1);
+ return generator.diff(x,n).subs(x==0);
+@}
+
+int main()
+@{
+ for (unsigned i=0; i<11; i+=2)
+ std::cout << EulerNumber(i) << std::endl;
+ return 0;
+@}
+@end example
+
+When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
+@code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
+@code{i} by two since all odd Euler numbers vanish anyways.
+
+
+@node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
+@c node-name, next, previous, up
+@section Series expansion
+@cindex @code{series()}
+@cindex Taylor expansion
+@cindex Laurent expansion
+@cindex @code{pseries} (class)
+
+Expressions know how to expand themselves as a Taylor series or (more
+generally) a Laurent series. As in most conventional Computer Algebra
+Systems, no distinction is made between those two. There is a class of
+its own for storing such series (@code{class pseries}) and a built-in
+function (called @code{Order}) for storing the order term of the series.
+As a consequence, if you want to work with series, i.e. multiply two
+series, you need to call the method @code{ex::series} again to convert
+it to a series object with the usual structure (expansion plus order
+term). A sample application from special relativity could read:
+
+@example
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ symbol v("v"), c("c");
+
+ ex gamma = 1/sqrt(1 - pow(v/c,2));
+ ex mass_nonrel = gamma.series(v==0, 10);
+
+ cout << "the relativistic mass increase with v is " << endl
+ << mass_nonrel << endl;
+
+ cout << "the inverse square of this series is " << endl
+ << pow(mass_nonrel,-2).series(v==0, 10) << endl;
+@}
+@end example
+
+Only calling the series method makes the last output simplify to
+@math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
+series raised to the power @math{-2}.
+
+@cindex M@'echain's formula
+As another instructive application, let us calculate the numerical
+value of Archimedes' constant
+@tex
+$\pi$
+@end tex
+(for which there already exists the built-in constant @code{Pi})
+using M@'echain's amazing formula
+@tex
+$\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
+@end tex
+@ifnottex
+@math{Pi==16*atan(1/5)-4*atan(1/239)}.
+@end ifnottex
+We may expand the arcus tangent around @code{0} and insert the fractions
+@code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
+carries an order term with it and the question arises what the system is
+supposed to do when the fractions are plugged into that order term. The
+solution is to use the function @code{series_to_poly()} to simply strip
+the order term off:
+
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+ex mechain_pi(int degr)
+@{
+ symbol x;
+ ex pi_expansion = series_to_poly(atan(x).series(x,degr));
+ ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
+ -4*pi_expansion.subs(x==numeric(1,239));
+ return pi_approx;
+@}
+
+int main()
+@{
+ using std::cout; // just for fun, another way of...
+ using std::endl; // ...dealing with this namespace std.
+ ex pi_frac;
+ for (int i=2; i<12; i+=2) @{
+ pi_frac = mechain_pi(i);
+ cout << i << ":\t" << pi_frac << endl
+ << "\t" << pi_frac.evalf() << endl;
+ @}
+ return 0;
+@}
+@end example
+
+Note how we just called @code{.series(x,degr)} instead of
+@code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
+method @code{series()}: if the first argument is a symbol the expression
+is expanded in that symbol around point @code{0}. When you run this
+program, it will type out:
+
+@example
+2: 3804/1195
+ 3.1832635983263598326
+4: 5359397032/1706489875
+ 3.1405970293260603143
+6: 38279241713339684/12184551018734375
+ 3.141621029325034425
+8: 76528487109180192540976/24359780855939418203125
+ 3.141591772182177295
+10: 327853873402258685803048818236/104359128170408663038552734375
+ 3.1415926824043995174
+@end example
+
+
+@node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
+@c node-name, next, previous, up
+@section Predefined mathematical functions
+
+GiNaC contains the following predefined mathematical functions:
+
+@cartouche
+@multitable @columnfractions .30 .70
+@item @strong{Name} @tab @strong{Function}
+@item @code{abs(x)}
+@tab absolute value
+@item @code{csgn(x)}
+@tab complex sign
+@item @code{sqrt(x)}
+@tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
+@item @code{sin(x)}
+@tab sine
+@item @code{cos(x)}
+@tab cosine
+@item @code{tan(x)}
+@tab tangent
+@item @code{asin(x)}
+@tab inverse sine
+@item @code{acos(x)}
+@tab inverse cosine
+@item @code{atan(x)}
+@tab inverse tangent
+@item @code{atan2(y, x)}
+@tab inverse tangent with two arguments
+@item @code{sinh(x)}
+@tab hyperbolic sine
+@item @code{cosh(x)}
+@tab hyperbolic cosine
+@item @code{tanh(x)}
+@tab hyperbolic tangent
+@item @code{asinh(x)}
+@tab inverse hyperbolic sine
+@item @code{acosh(x)}
+@tab inverse hyperbolic cosine
+@item @code{atanh(x)}
+@tab inverse hyperbolic tangent
+@item @code{exp(x)}
+@tab exponential function
+@item @code{log(x)}
+@tab natural logarithm
+@item @code{Li2(x)}
+@tab Dilogarithm
+@item @code{zeta(x)}
+@tab Riemann's zeta function
+@item @code{zeta(n, x)}
+@tab derivatives of Riemann's zeta function
+@item @code{tgamma(x)}
+@tab Gamma function
+@item @code{lgamma(x)}
+@tab logarithm of Gamma function
+@item @code{beta(x, y)}
+@tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
+@item @code{psi(x)}
+@tab psi (digamma) function
+@item @code{psi(n, x)}
+@tab derivatives of psi function (polygamma functions)
+@item @code{factorial(n)}
+@tab factorial function
+@item @code{binomial(n, m)}
+@tab binomial coefficients
+@item @code{Order(x)}
+@tab order term function in truncated power series
+@item @code{Derivative(x, l)}
+@tab inert partial differentiation operator (used internally)
+@end multitable
+@end cartouche
+
+@cindex branch cut
+For functions that have a branch cut in the complex plane GiNaC follows
+the conventions for C++ as defined in the ANSI standard as far as
+possible. In particular: the natural logarithm (@code{log}) and the
+square root (@code{sqrt}) both have their branch cuts running along the
+negative real axis where the points on the axis itself belong to the
+upper part (i.e. continuous with quadrant II). The inverse
+trigonometric and hyperbolic functions are not defined for complex
+arguments by the C++ standard, however. In GiNaC we follow the
+conventions used by CLN, which in turn follow the carefully designed
+definitions in the Common Lisp standard. It should be noted that this
+convention is identical to the one used by the C99 standard and by most
+serious CAS. It is to be expected that future revisions of the C++
+standard incorporate these functions in the complex domain in a manner
+compatible with C99.
+
+
+@node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
+@c node-name, next, previous, up
+@section Input and output of expressions
+@cindex I/O
+
+@subsection Expression output
+@cindex printing
+@cindex output of expressions
+
+The easiest way to print an expression is to write it to a stream:
+
+@example
+@{
+ symbol x("x");
+ ex e = 4.5+pow(x,2)*3/2;
+ cout << e << endl; // prints '(4.5)+3/2*x^2'
+ // ...
+@end example
+
+The output format is identical to the @command{ginsh} input syntax and
+to that used by most computer algebra systems, but not directly pastable
+into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
+is printed as @samp{x^2}).
+
+It is possible to print expressions in a number of different formats with
+the method
+
+@example
+void ex::print(const print_context & c, unsigned level = 0);
+@end example
+
+@cindex @code{print_context} (class)
+The type of @code{print_context} object passed in determines the format
+of the output. The possible types are defined in @file{ginac/print.h}.
+All constructors of @code{print_context} and derived classes take an
+@code{ostream &} as their first argument.
+
+To print an expression in a way that can be directly used in a C or C++
+program, you pass a @code{print_csrc} object like this:
+
+@example
+ // ...
+ cout << "float f = ";
+ e.print(print_csrc_float(cout));
+ cout << ";\n";
+
+ cout << "double d = ";
+ e.print(print_csrc_double(cout));
+ cout << ";\n";
+
+ cout << "cl_N n = ";
+ e.print(print_csrc_cl_N(cout));
+ cout << ";\n";
+ // ...
+@end example
+
+The three possible types mostly affect the way in which floating point
+numbers are written.
+
+The above example will produce (note the @code{x^2} being converted to @code{x*x}):
+
+@example
+float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
+double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
+cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
+@end example
+
+The @code{print_context} type @code{print_tree} provides a dump of the
+internal structure of an expression for debugging purposes:
+
+@example
+ // ...
+ e.print(print_tree(cout));
+@}
+@end example
+
+produces
+
+@example
+add, hash=0x0, flags=0x3, nops=2
+ power, hash=0x9, flags=0x3, nops=2
+ x (symbol), serial=3, hash=0x44a113a6, flags=0xf
+ 2 (numeric), hash=0x80000042, flags=0xf
+ 3/2 (numeric), hash=0x80000061, flags=0xf
+ -----
+ overall_coeff
+ 4.5L0 (numeric), hash=0x8000004b, flags=0xf
+ =====
+@end example
+
+This kind of output is also available in @command{ginsh} as the @code{print()}
+function.
+
+Another useful output format is for LaTeX parsing in mathematical mode.
+It is rather similar to the default @code{print_context} but provides
+some braces needed by LaTeX for delimiting boxes and also converts some
+common objects to conventional LaTeX names. It is possible to give symbols
+a special name for LaTeX output by supplying it as a second argument to
+the @code{symbol} constructor.
+
+For example, the code snippet
+
+@example
+ // ...
+ symbol x("x");
+ ex foo = lgamma(x).series(x==0,3);
+ foo.print(print_latex(std::cout));
+@end example
+
+will print out:
+
+@example
+ @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
+@end example
+
+If you need any fancy special output format, e.g. for interfacing GiNaC
+with other algebra systems or for producing code for different
+programming languages, you can always traverse the expression tree yourself:
+
+@example
+static void my_print(const ex & e)
+@{
+ if (is_ex_of_type(e, function))
+ cout << ex_to_function(e).get_name();
+ else
+ cout << e.bp->class_name();
+ cout << "(";
+ unsigned n = e.nops();
+ if (n)
+ for (unsigned i=0; i<n; i++) @{
+ my_print(e.op(i));
+ if (i != n-1)
+ cout << ",";
+ @}
+ else
+ cout << e;
+ cout << ")";
+@}
+
+int main(void)
+@{
+ my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
+ return 0;
+@}
+@end example
+
+This will produce
+
+@example
+add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
+symbol(y))),numeric(-2)))
+@end example
+
+If you need an output format that makes it possible to accurately
+reconstruct an expression by feeding the output to a suitable parser or
+object factory, you should consider storing the expression in an
+@code{archive} object and reading the object properties from there.
+See the section on archiving for more information.
+
+
+@subsection Expression input
+@cindex input of expressions
+
+GiNaC provides no way to directly read an expression from a stream because
+you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
+and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
+@code{y} you defined in your program and there is no way to specify the
+desired symbols to the @code{>>} stream input operator.
+
+Instead, GiNaC lets you construct an expression from a string, specifying the
+list of symbols to be used:
+
+@example
+@{
+ symbol x("x"), y("y");
+ ex e("2*x+sin(y)", lst(x, y));
+@}
+@end example
+
+The input syntax is the same as that used by @command{ginsh} and the stream
+output operator @code{<<}. The symbols in the string are matched by name to
+the symbols in the list and if GiNaC encounters a symbol not specified in
+the list it will throw an exception.
+
+With this constructor, it's also easy to implement interactive GiNaC programs:
+
+@example
+#include <iostream>
+#include <string>
+#include <stdexcept>
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x");
+ string s;
+
+ cout << "Enter an expression containing 'x': ";
+ getline(cin, s);
+
+ try @{
+ ex e(s, lst(x));
+ cout << "The derivative of " << e << " with respect to x is ";
+ cout << e.diff(x) << ".\n";
+ @} catch (exception &p) @{
+ cerr << p.what() << endl;
+ @}
+@}
+@end example
+
+
+@subsection Archiving
+@cindex @code{archive} (class)
+@cindex archiving
+
+GiNaC allows creating @dfn{archives} of expressions which can be stored
+to or retrieved from files. To create an archive, you declare an object
+of class @code{archive} and archive expressions in it, giving each
+expression a unique name:
+
+@example
+#include <fstream>
+using namespace std;
+#include <ginac/ginac.h>
using namespace GiNaC;
int main()
@{
- symbol x("x");
- numeric point(0);
- ex MyExpr1 = sin(x);
- ex MyExpr2 = 1/(x - pow(x, 2) - pow(x, 3));
- ex MyTailor, MySeries;
-
- MyTailor = MyExpr1.series(x, point, 5);
- cout << MyExpr1 << " == " << MyTailor
- << " for small " << x << endl;
- MySeries = MyExpr2.series(x, point, 7);
- cout << MyExpr2 << " == " << MySeries
- << " for small " << x << endl;
+ symbol x("x"), y("y"), z("z");
+
+ ex foo = sin(x + 2*y) + 3*z + 41;
+ ex bar = foo + 1;
+
+ archive a;
+ a.archive_ex(foo, "foo");
+ a.archive_ex(bar, "the second one");
+ // ...
+@end example
+
+The archive can then be written to a file:
+
+@example
+ // ...
+ ofstream out("foobar.gar");
+ out << a;
+ out.close();
+ // ...
+@end example
+
+The file @file{foobar.gar} contains all information that is needed to
+reconstruct the expressions @code{foo} and @code{bar}.
+
+@cindex @command{viewgar}
+The tool @command{viewgar} that comes with GiNaC can be used to view
+the contents of GiNaC archive files:
+
+@example
+$ viewgar foobar.gar
+foo = 41+sin(x+2*y)+3*z
+the second one = 42+sin(x+2*y)+3*z
+@end example
+
+The point of writing archive files is of course that they can later be
+read in again:
+
+@example
+ // ...
+ archive a2;
+ ifstream in("foobar.gar");
+ in >> a2;
+ // ...
+@end example
+
+And the stored expressions can be retrieved by their name:
+
+@example
// ...
+ lst syms(x, y);
+
+ ex ex1 = a2.unarchive_ex(syms, "foo");
+ ex ex2 = a2.unarchive_ex(syms, "the second one");
+
+ cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
+ cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
+ cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
+@}
+@end example
+
+Note that you have to supply a list of the symbols which are to be inserted
+in the expressions. Symbols in archives are stored by their name only and
+if you don't specify which symbols you have, unarchiving the expression will
+create new symbols with that name. E.g. if you hadn't included @code{x} in
+the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
+have had no effect because the @code{x} in @code{ex1} would have been a
+different symbol than the @code{x} which was defined at the beginning of
+the program, altough both would appear as @samp{x} when printed.
+
+You can also use the information stored in an @code{archive} object to
+output expressions in a format suitable for exact reconstruction. The
+@code{archive} and @code{archive_node} classes have a couple of member
+functions that let you access the stored properties:
+
+@example
+static void my_print2(const archive_node & n)
+@{
+ string class_name;
+ n.find_string("class", class_name);
+ cout << class_name << "(";
+
+ archive_node::propinfovector p;
+ n.get_properties(p);
+
+ unsigned num = p.size();
+ for (unsigned i=0; i<num; i++) @{
+ const string &name = p[i].name;
+ if (name == "class")
+ continue;
+ cout << name << "=";
+
+ unsigned count = p[i].count;
+ if (count > 1)
+ cout << "@{";
+
+ for (unsigned j=0; j<count; j++) @{
+ switch (p[i].type) @{
+ case archive_node::PTYPE_BOOL: @{
+ bool x;
+ n.find_bool(name, x);
+ cout << (x ? "true" : "false");
+ break;
+ @}
+ case archive_node::PTYPE_UNSIGNED: @{
+ unsigned x;
+ n.find_unsigned(name, x);
+ cout << x;
+ break;
+ @}
+ case archive_node::PTYPE_STRING: @{
+ string x;
+ n.find_string(name, x);
+ cout << '\"' << x << '\"';
+ break;
+ @}
+ case archive_node::PTYPE_NODE: @{
+ const archive_node &x = n.find_ex_node(name, j);
+ my_print2(x);
+ break;
+ @}
+ @}
+
+ if (j != count-1)
+ cout << ",";
+ @}
+
+ if (count > 1)
+ cout << "@}";
+
+ if (i != num-1)
+ cout << ",";
+ @}
+
+ cout << ")";
+@}
+
+int main(void)
+@{
+ ex e = pow(2, x) - y;
+ archive ar(e, "e");
+ my_print2(ar.get_top_node(0)); cout << endl;
+ return 0;
+@}
+@end example
+
+This will produce:
+
+@example
+add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
+symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
+overall_coeff=numeric(number="0"))
+@end example
+
+Be warned, however, that the set of properties and their meaning for each
+class may change between GiNaC versions.
+
+
+@node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
+@c node-name, next, previous, up
+@chapter Extending GiNaC
+
+By reading so far you should have gotten a fairly good understanding of
+GiNaC's design-patterns. From here on you should start reading the
+sources. All we can do now is issue some recommendations how to tackle
+GiNaC's many loose ends in order to fulfill everybody's dreams. If you
+develop some useful extension please don't hesitate to contact the GiNaC
+authors---they will happily incorporate them into future versions.
+
+@menu
+* What does not belong into GiNaC:: What to avoid.
+* Symbolic functions:: Implementing symbolic functions.
+* Adding classes:: Defining new algebraic classes.
+@end menu
+
+
+@node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
+@c node-name, next, previous, up
+@section What doesn't belong into GiNaC
+
+@cindex @command{ginsh}
+First of all, GiNaC's name must be read literally. It is designed to be
+a library for use within C++. The tiny @command{ginsh} accompanying
+GiNaC makes this even more clear: it doesn't even attempt to provide a
+language. There are no loops or conditional expressions in
+@command{ginsh}, it is merely a window into the library for the
+programmer to test stuff (or to show off). Still, the design of a
+complete CAS with a language of its own, graphical capabilites and all
+this on top of GiNaC is possible and is without doubt a nice project for
+the future.
+
+There are many built-in functions in GiNaC that do not know how to
+evaluate themselves numerically to a precision declared at runtime
+(using @code{Digits}). Some may be evaluated at certain points, but not
+generally. This ought to be fixed. However, doing numerical
+computations with GiNaC's quite abstract classes is doomed to be
+inefficient. For this purpose, the underlying foundation classes
+provided by @acronym{CLN} are much better suited.
+
+
+@node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
+@c node-name, next, previous, up
+@section Symbolic functions
+
+The easiest and most instructive way to start with is probably to
+implement your own function. GiNaC's functions are objects of class
+@code{function}. The preprocessor is then used to convert the function
+names to objects with a corresponding serial number that is used
+internally to identify them. You usually need not worry about this
+number. New functions may be inserted into the system via a kind of
+`registry'. It is your responsibility to care for some functions that
+are called when the user invokes certain methods. These are usual
+C++-functions accepting a number of @code{ex} as arguments and returning
+one @code{ex}. As an example, if we have a look at a simplified
+implementation of the cosine trigonometric function, we first need a
+function that is called when one wishes to @code{eval} it. It could
+look something like this:
+
+@example
+static ex cos_eval_method(const ex & x)
+@{
+ // if (!x%(2*Pi)) return 1
+ // if (!x%Pi) return -1
+ // if (!x%Pi/2) return 0
+ // care for other cases...
+ return cos(x).hold();
+@}
+@end example
+
+@cindex @code{hold()}
+@cindex evaluation
+The last line returns @code{cos(x)} if we don't know what else to do and
+stops a potential recursive evaluation by saying @code{.hold()}, which
+sets a flag to the expression signaling that it has been evaluated. We
+should also implement a method for numerical evaluation and since we are
+lazy we sweep the problem under the rug by calling someone else's
+function that does so, in this case the one in class @code{numeric}:
+
+@example
+static ex cos_evalf(const ex & x)
+@{
+ return cos(ex_to_numeric(x));
+@}
+@end example
+
+Differentiation will surely turn up and so we need to tell @code{cos}
+what the first derivative is (higher derivatives (@code{.diff(x,3)} for
+instance are then handled automatically by @code{basic::diff} and
+@code{ex::diff}):
+
+@example
+static ex cos_deriv(const ex & x, unsigned diff_param)
+@{
+ return -sin(x);
+@}
+@end example
+
+@cindex product rule
+The second parameter is obligatory but uninteresting at this point. It
+specifies which parameter to differentiate in a partial derivative in
+case the function has more than one parameter and its main application
+is for correct handling of the chain rule. For Taylor expansion, it is
+enough to know how to differentiate. But if the function you want to
+implement does have a pole somewhere in the complex plane, you need to
+write another method for Laurent expansion around that point.
+
+Now that all the ingredients for @code{cos} have been set up, we need
+to tell the system about it. This is done by a macro and we are not
+going to descibe how it expands, please consult your preprocessor if you
+are curious:
+
+@example
+REGISTER_FUNCTION(cos, eval_func(cos_eval).
+ evalf_func(cos_evalf).
+ derivative_func(cos_deriv));
+@end example
+
+The first argument is the function's name used for calling it and for
+output. The second binds the corresponding methods as options to this
+object. Options are separated by a dot and can be given in an arbitrary
+order. GiNaC functions understand several more options which are always
+specified as @code{.option(params)}, for example a method for series
+expansion @code{.series_func(cos_series)}. Again, if no series
+expansion method is given, GiNaC defaults to simple Taylor expansion,
+which is correct if there are no poles involved as is the case for the
+@code{cos} function. The way GiNaC handles poles in case there are any
+is best understood by studying one of the examples, like the Gamma
+(@code{tgamma}) function for instance. (In essence the function first
+checks if there is a pole at the evaluation point and falls back to
+Taylor expansion if there isn't. Then, the pole is regularized by some
+suitable transformation.) Also, the new function needs to be declared
+somewhere. This may also be done by a convenient preprocessor macro:
+
+@example
+DECLARE_FUNCTION_1P(cos)
+@end example
+
+The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
+implementation of @code{cos} is very incomplete and lacks several safety
+mechanisms. Please, have a look at the real implementation in GiNaC.
+(By the way: in case you are worrying about all the macros above we can
+assure you that functions are GiNaC's most macro-intense classes. We
+have done our best to avoid macros where we can.)
+
+
+@node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
+@c node-name, next, previous, up
+@section Adding classes
+
+If you are doing some very specialized things with GiNaC you may find that
+you have to implement your own algebraic classes to fit your needs. This
+section will explain how to do this by giving the example of a simple
+'string' class. After reading this section you will know how to properly
+declare a GiNaC class and what the minimum required member functions are
+that you have to implement. We only cover the implementation of a 'leaf'
+class here (i.e. one that doesn't contain subexpressions). Creating a
+container class like, for example, a class representing tensor products is
+more involved but this section should give you enough information so you can
+consult the source to GiNaC's predefined classes if you want to implement
+something more complicated.
+
+@subsection GiNaC's run-time type information system
+
+@cindex hierarchy of classes
+@cindex RTTI
+All algebraic classes (that is, all classes that can appear in expressions)
+in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
+@code{basic *} (which is essentially what an @code{ex} is) represents a
+generic pointer to an algebraic class. Occasionally it is necessary to find
+out what the class of an object pointed to by a @code{basic *} really is.
+Also, for the unarchiving of expressions it must be possible to find the
+@code{unarchive()} function of a class given the class name (as a string). A
+system that provides this kind of information is called a run-time type
+information (RTTI) system. The C++ language provides such a thing (see the
+standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
+implements its own, simpler RTTI.
+
+The RTTI in GiNaC is based on two mechanisms:
+
+@itemize @bullet
+
+@item
+The @code{basic} class declares a member variable @code{tinfo_key} which
+holds an unsigned integer that identifies the object's class. These numbers
+are defined in the @file{tinfos.h} header file for the built-in GiNaC
+classes. They all start with @code{TINFO_}.
+
+@item
+By means of some clever tricks with static members, GiNaC maintains a list
+of information for all classes derived from @code{basic}. The information
+available includes the class names, the @code{tinfo_key}s, and pointers
+to the unarchiving functions. This class registry is defined in the
+@file{registrar.h} header file.
+
+@end itemize
+
+The disadvantage of this proprietary RTTI implementation is that there's
+a little more to do when implementing new classes (C++'s RTTI works more
+or less automatic) but don't worry, most of the work is simplified by
+macros.
+
+@subsection A minimalistic example
+
+Now we will start implementing a new class @code{mystring} that allows
+placing character strings in algebraic expressions (this is not very useful,
+but it's just an example). This class will be a direct subclass of
+@code{basic}. You can use this sample implementation as a starting point
+for your own classes.
+
+The code snippets given here assume that you have included some header files
+as follows:
+
+@example
+#include <iostream>
+#include <string>
+#include <stdexcept>
+using namespace std;
+
+#include <ginac/ginac.h>
+using namespace GiNaC;
+@end example
+
+The first thing we have to do is to define a @code{tinfo_key} for our new
+class. This can be any arbitrary unsigned number that is not already taken
+by one of the existing classes but it's better to come up with something
+that is unlikely to clash with keys that might be added in the future. The
+numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
+which is not a requirement but we are going to stick with this scheme:
+
+@example
+const unsigned TINFO_mystring = 0x42420001U;
+@end example
+
+Now we can write down the class declaration. The class stores a C++
+@code{string} and the user shall be able to construct a @code{mystring}
+object from a C or C++ string:
+
+@example
+class mystring : public basic
+@{
+ GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
+
+public:
+ mystring(const string &s);
+ mystring(const char *s);
+
+private:
+ string str;
+@};
+
+GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
+@end example
+
+The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
+macros are defined in @file{registrar.h}. They take the name of the class
+and its direct superclass as arguments and insert all required declarations
+for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
+the first line after the opening brace of the class definition. The
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
+source (at global scope, of course, not inside a function).
+
+@code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
+declarations of the default and copy constructor, the destructor, the
+assignment operator and a couple of other functions that are required. It
+also defines a type @code{inherited} which refers to the superclass so you
+don't have to modify your code every time you shuffle around the class
+hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
+constructor, the destructor and the assignment operator.
+
+Now there are nine member functions we have to implement to get a working
+class:
+
+@itemize
+
+@item
+@code{mystring()}, the default constructor.
+
+@item
+@code{void destroy(bool call_parent)}, which is used in the destructor and the
+assignment operator to free dynamically allocated members. The @code{call_parent}
+specifies whether the @code{destroy()} function of the superclass is to be
+called also.
+
+@item
+@code{void copy(const mystring &other)}, which is used in the copy constructor
+and assignment operator to copy the member variables over from another
+object of the same class.
+
+@item
+@code{void archive(archive_node &n)}, the archiving function. This stores all
+information needed to reconstruct an object of this class inside an
+@code{archive_node}.
+
+@item
+@code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
+constructor. This constructs an instance of the class from the information
+found in an @code{archive_node}.
+
+@item
+@code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
+unarchiving function. It constructs a new instance by calling the unarchiving
+constructor.
+
+@item
+@code{int compare_same_type(const basic &other)}, which is used internally
+by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
+-1, depending on the relative order of this object and the @code{other}
+object. If it returns 0, the objects are considered equal.
+@strong{Note:} This has nothing to do with the (numeric) ordering
+relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
+for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
+may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
+must provide a @code{compare_same_type()} function, even those representing
+objects for which no reasonable algebraic ordering relationship can be
+defined.
+
+@item
+And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
+which are the two constructors we declared.
+
+@end itemize
+
+Let's proceed step-by-step. The default constructor looks like this:
+
+@example
+mystring::mystring() : inherited(TINFO_mystring)
+@{
+ // dynamically allocate resources here if required
+@}
+@end example
+
+The golden rule is that in all constructors you have to set the
+@code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
+it will be set by the constructor of the superclass and all hell will break
+loose in the RTTI. For your convenience, the @code{basic} class provides
+a constructor that takes a @code{tinfo_key} value, which we are using here
+(remember that in our case @code{inherited = basic}). If the superclass
+didn't have such a constructor, we would have to set the @code{tinfo_key}
+to the right value manually.
+
+In the default constructor you should set all other member variables to
+reasonable default values (we don't need that here since our @code{str}
+member gets set to an empty string automatically). The constructor(s) are of
+course also the right place to allocate any dynamic resources you require.
+
+Next, the @code{destroy()} function:
+
+@example
+void mystring::destroy(bool call_parent)
+@{
+ // free dynamically allocated resources here if required
+ if (call_parent)
+ inherited::destroy(call_parent);
+@}
+@end example
+
+This function is where we free all dynamically allocated resources. We don't
+have any so we're not doing anything here, but if we had, for example, used
+a C-style @code{char *} to store our string, this would be the place to
+@code{delete[]} the string storage. If @code{call_parent} is true, we have
+to call the @code{destroy()} function of the superclass after we're done
+(to mimic C++'s automatic invocation of superclass destructors where
+@code{destroy()} is called from outside a destructor).
+
+The @code{copy()} function just copies over the member variables from
+another object:
+
+@example
+void mystring::copy(const mystring &other)
+@{
+ inherited::copy(other);
+ str = other.str;
+@}
+@end example
+
+We can simply overwrite the member variables here. There's no need to worry
+about dynamically allocated storage. The assignment operator (which is
+automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
+recall) calls @code{destroy()} before it calls @code{copy()}. You have to
+explicitly call the @code{copy()} function of the superclass here so
+all the member variables will get copied.
+
+Next are the three functions for archiving. You have to implement them even
+if you don't plan to use archives, but the minimum required implementation
+is really simple. First, the archiving function:
+
+@example
+void mystring::archive(archive_node &n) const
+@{
+ inherited::archive(n);
+ n.add_string("string", str);
+@}
+@end example
+
+The only thing that is really required is calling the @code{archive()}
+function of the superclass. Optionally, you can store all information you
+deem necessary for representing the object into the passed
+@code{archive_node}. We are just storing our string here. For more
+information on how the archiving works, consult the @file{archive.h} header
+file.
+
+The unarchiving constructor is basically the inverse of the archiving
+function:
+
+@example
+mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+@{
+ n.find_string("string", str);
+@}
+@end example
+
+If you don't need archiving, just leave this function empty (but you must
+invoke the unarchiving constructor of the superclass). Note that we don't
+have to set the @code{tinfo_key} here because it is done automatically
+by the unarchiving constructor of the @code{basic} class.
+
+Finally, the unarchiving function:
+
+@example
+ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
+@{
+ return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
+@}
+@end example
+
+You don't have to understand how exactly this works. Just copy these four
+lines into your code literally (replacing the class name, of course). It
+calls the unarchiving constructor of the class and unless you are doing
+something very special (like matching @code{archive_node}s to global
+objects) you don't need a different implementation. For those who are
+interested: setting the @code{dynallocated} flag puts the object under
+the control of GiNaC's garbage collection. It will get deleted automatically
+once it is no longer referenced.
+
+Our @code{compare_same_type()} function uses a provided function to compare
+the string members:
+
+@example
+int mystring::compare_same_type(const basic &other) const
+@{
+ const mystring &o = static_cast<const mystring &>(other);
+ int cmpval = str.compare(o.str);
+ if (cmpval == 0)
+ return 0;
+ else if (cmpval < 0)
+ return -1;
+ else
+ return 1;
+@}
+@end example
+
+Although this function takes a @code{basic &}, it will always be a reference
+to an object of exactly the same class (objects of different classes are not
+comparable), so the cast is safe. If this function returns 0, the two objects
+are considered equal (in the sense that @math{A-B=0}), so you should compare
+all relevant member variables.
+
+Now the only thing missing is our two new constructors:
+
+@example
+mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
+@{
+ // dynamically allocate resources here if required
+@}
+
+mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
+@{
+ // dynamically allocate resources here if required
+@}
+@end example
+
+No surprises here. We set the @code{str} member from the argument and
+remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
+
+That's it! We now have a minimal working GiNaC class that can store
+strings in algebraic expressions. Let's confirm that the RTTI works:
+
+@example
+ex e = mystring("Hello, world!");
+cout << is_ex_of_type(e, mystring) << endl;
+ // -> 1 (true)
+
+cout << e.bp->class_name() << endl;
+ // -> mystring
+@end example
+
+Obviously it does. Let's see what the expression @code{e} looks like:
+
+@example
+cout << e << endl;
+ // -> [mystring object]
+@end example
+
+Hm, not exactly what we expect, but of course the @code{mystring} class
+doesn't yet know how to print itself. This is done in the @code{print()}
+member function. Let's say that we wanted to print the string surrounded
+by double quotes:
+
+@example
+class mystring : public basic
+@{
+ ...
+public:
+ void print(const print_context &c, unsigned level = 0) const;
+ ...
+@};
+
+void mystring::print(const print_context &c, unsigned level) const
+@{
+ // print_context::s is a reference to an ostream
+ c.s << '\"' << str << '\"';
@}
@end example
-As an instructive application, let us calculate the numerical
-value of Archimedes' constant (for which there already exists the
-built-in constant @code{Pi}) using M@'echain's
-mysterious formula @code{Pi==16*atan(1/5)-4*atan(1/239)}.
-We may expand the arcus tangent around @code{0} and insert
-the fractions @code{1/5} and @code{1/239}.
-But, as we have seen, a series in GiNaC carries an order term with it.
-The function @code{series_to_poly()} may be used to strip
-this off:
+The @code{level} argument is only required for container classes to
+correctly parenthesize the output. Let's try again to print the expression:
+
+@example
+cout << e << endl;
+ // -> "Hello, world!"
+@end example
+
+Much better. The @code{mystring} class can be used in arbitrary expressions:
+
+@example
+e += mystring("GiNaC rulez");
+cout << e << endl;
+ // -> "GiNaC rulez"+"Hello, world!"
+@end example
+
+(note that GiNaC's automatic term reordering is in effect here), or even
+
+@example
+e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
+cout << e << endl;
+ // -> "One string"^(2*sin(-"Another string"+Pi))
+@end example
+
+Whether this makes sense is debatable but remember that this is only an
+example. At least it allows you to implement your own symbolic algorithms
+for your objects.
+
+Note that GiNaC's algebraic rules remain unchanged:
+
+@example
+e = mystring("Wow") * mystring("Wow");
+cout << e << endl;
+ // -> "Wow"^2
+
+e = pow(mystring("First")-mystring("Second"), 2);
+cout << e.expand() << endl;
+ // -> -2*"First"*"Second"+"First"^2+"Second"^2
+@end example
+
+There's no way to, for example, make GiNaC's @code{add} class perform string
+concatenation. You would have to implement this yourself.
+
+@subsection Automatic evaluation
-@subheading Series expansion using M@'echain's formula for @code{Pi}
-@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+@cindex @code{hold()}
+@cindex evaluation
+When dealing with objects that are just a little more complicated than the
+simple string objects we have implemented, chances are that you will want to
+have some automatic simplifications or canonicalizations performed on them.
+This is done in the evaluation member function @code{eval()}. Let's say that
+we wanted all strings automatically converted to lowercase with
+non-alphabetic characters stripped, and empty strings removed:
-ex mechain_pi(int degr)
+@example
+class mystring : public basic
@{
- symbol x;
- ex pi_expansion = series_to_poly(atan(x).series(x,0,degr));
- ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
- -4*pi_expansion.subs(x==numeric(1,239));
- return pi_approx;
-@}
+ ...
+public:
+ ex eval(int level = 0) const;
+ ...
+@};
-int main()
+ex mystring::eval(int level) const
@{
- ex pi_frac;
- for (int i=2; i<12; i+=2) @{
- pi_frac = mechain_pi(i);
- cout << i << ":\t" << pi_frac << endl
- << "\t" << pi_frac.evalf() << endl;
+ string new_str;
+ for (int i=0; i<str.length(); i++) @{
+ char c = str[i];
+ if (c >= 'A' && c <= 'Z')
+ new_str += tolower(c);
+ else if (c >= 'a' && c <= 'z')
+ new_str += c;
@}
- return 0;
+
+ if (new_str.length() == 0)
+ return 0;
+ else
+ return mystring(new_str).hold();
@}
@end example
-When you run this program, it will type out:
+The @code{level} argument is used to limit the recursion depth of the
+evaluation. We don't have any subexpressions in the @code{mystring} class
+so we are not concerned with this. If we had, we would call the @code{eval()}
+functions of the subexpressions with @code{level - 1} as the argument if
+@code{level != 1}. The @code{hold()} member function sets a flag in the
+object that prevents further evaluation. Otherwise we might end up in an
+endless loop. When you want to return the object unmodified, use
+@code{return this->hold();}.
+
+Let's confirm that it works:
@example
-2: 3804/1195
- 3.1832635983263598326
-4: 5359397032/1706489875
- 3.1405970293260603143
-6: 38279241713339684/12184551018734375
- 3.141621029325034425
-8: 76528487109180192540976/24359780855939418203125
- 3.141591772182177295
-10: 327853873402258685803048818236/104359128170408663038552734375
- 3.1415926824043995174
+ex e = mystring("Hello, world!") + mystring("!?#");
+cout << e << endl;
+ // -> "helloworld"
+
+e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
+cout << e << endl;
+ // -> 3*"wow"
+@end example
+
+@subsection Other member functions
+
+We have implemented only a small set of member functions to make the class
+work in the GiNaC framework. For a real algebraic class, there are probably
+some more functions that you will want to re-implement, such as
+@code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
+or the header file of the class you want to make a subclass of to see
+what's there. One member function that you will most likely want to
+implement for terminal classes like the described string class is
+@code{calcchash()} that returns an @code{unsigned} hash value for the object
+which will allow GiNaC to compare and canonicalize expressions much more
+efficiently.
+
+You can, of course, also add your own new member functions. In this case you
+will probably want to define a little helper function like
+
+@example
+inline const mystring &ex_to_mystring(const ex &e)
+@{
+ return static_cast<const mystring &>(*e.bp);
+@}
@end example
+that let's you get at the object inside an expression (after you have
+verified that the type is correct) so you can call member functions that are
+specific to the class.
+
+That's it. May the source be with you!
+
-@node Extending GiNaC, What does not belong into GiNaC, Series Expansion, Top
+@node A Comparison With Other CAS, Advantages, Adding classes, Top
@c node-name, next, previous, up
-@chapter Extending GiNaC
+@chapter A Comparison With Other CAS
+@cindex advocacy
-By reading so far you should have gotten a fairly good
-understanding of GiNaC's design-patterns. From here on you should
-start reading the sources. All we can do now is issue some
-recommendations how to tackle GiNaC's many loose ends in order to
-fulfill everybody's dreams.
+This chapter will give you some information on how GiNaC compares to
+other, traditional Computer Algebra Systems, like @emph{Maple},
+@emph{Mathematica} or @emph{Reduce}, where it has advantages and
+disadvantages over these systems.
@menu
-* What does not belong into GiNaC:: What to avoid.
-* Symbolic functions:: Implementing symbolic functions.
+* Advantages:: Stengths of the GiNaC approach.
+* Disadvantages:: Weaknesses of the GiNaC approach.
+* Why C++?:: Attractiveness of C++.
@end menu
+@node Advantages, Disadvantages, A Comparison With Other CAS, A Comparison With Other CAS
+@c node-name, next, previous, up
+@section Advantages
-@node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
+GiNaC has several advantages over traditional Computer
+Algebra Systems, like
+
+@itemize @bullet
+
+@item
+familiar language: all common CAS implement their own proprietary
+grammar which you have to learn first (and maybe learn again when your
+vendor decides to `enhance' it). With GiNaC you can write your program
+in common C++, which is standardized.
+
+@cindex STL
+@item
+structured data types: you can build up structured data types using
+@code{struct}s or @code{class}es together with STL features instead of
+using unnamed lists of lists of lists.
+
+@item
+strongly typed: in CAS, you usually have only one kind of variables
+which can hold contents of an arbitrary type. This 4GL like feature is
+nice for novice programmers, but dangerous.
+
+@item
+development tools: powerful development tools exist for C++, like fancy
+editors (e.g. with automatic indentation and syntax highlighting),
+debuggers, visualization tools, documentation generators...
+
+@item
+modularization: C++ programs can easily be split into modules by
+separating interface and implementation.
+
+@item
+price: GiNaC is distributed under the GNU Public License which means
+that it is free and available with source code. And there are excellent
+C++-compilers for free, too.
+
+@item
+extendable: you can add your own classes to GiNaC, thus extending it on
+a very low level. Compare this to a traditional CAS that you can
+usually only extend on a high level by writing in the language defined
+by the parser. In particular, it turns out to be almost impossible to
+fix bugs in a traditional system.
+
+@item
+multiple interfaces: Though real GiNaC programs have to be written in
+some editor, then be compiled, linked and executed, there are more ways
+to work with the GiNaC engine. Many people want to play with
+expressions interactively, as in traditional CASs. Currently, two such
+windows into GiNaC have been implemented and many more are possible: the
+tiny @command{ginsh} that is part of the distribution exposes GiNaC's
+types to a command line and second, as a more consistent approach, an
+interactive interface to the @acronym{Cint} C++ interpreter has been put
+together (called @acronym{GiNaC-cint}) that allows an interactive
+scripting interface consistent with the C++ language.
+
+@item
+seemless integration: it is somewhere between difficult and impossible
+to call CAS functions from within a program written in C++ or any other
+programming language and vice versa. With GiNaC, your symbolic routines
+are part of your program. You can easily call third party libraries,
+e.g. for numerical evaluation or graphical interaction. All other
+approaches are much more cumbersome: they range from simply ignoring the
+problem (i.e. @emph{Maple}) to providing a method for `embedding' the
+system (i.e. @emph{Yacas}).
+
+@item
+efficiency: often large parts of a program do not need symbolic
+calculations at all. Why use large integers for loop variables or
+arbitrary precision arithmetics where @code{int} and @code{double} are
+sufficient? For pure symbolic applications, GiNaC is comparable in
+speed with other CAS.
+
+@end itemize
+
+
+@node Disadvantages, Why C++?, Advantages, A Comparison With Other CAS
@c node-name, next, previous, up
-@section What doesn't belong into GiNaC
+@section Disadvantages
+
+Of course it also has some disadvantages:
+
+@itemize @bullet
+
+@item
+advanced features: GiNaC cannot compete with a program like
+@emph{Reduce} which exists for more than 30 years now or @emph{Maple}
+which grows since 1981 by the work of dozens of programmers, with
+respect to mathematical features. Integration, factorization,
+non-trivial simplifications, limits etc. are missing in GiNaC (and are
+not planned for the near future).
+
+@item
+portability: While the GiNaC library itself is designed to avoid any
+platform dependent features (it should compile on any ANSI compliant C++
+compiler), the currently used version of the CLN library (fast large
+integer and arbitrary precision arithmetics) can be compiled only on
+systems with a recently new C++ compiler from the GNU Compiler
+Collection (@acronym{GCC}).@footnote{This is because CLN uses
+PROVIDE/REQUIRE like macros to let the compiler gather all static
+initializations, which works for GNU C++ only.} GiNaC uses recent
+language features like explicit constructors, mutable members, RTTI,
+@code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
+Recent @acronym{GCC} versions starting at 2.95, although itself not yet
+ANSI compliant, support all needed features.
+
+@end itemize
+
-First of all, GiNaC's name must be read literally. It is
-designed to be a library for use within C++. The tiny
-@command{ginsh} accompanying GiNaC makes this even more
-clear: it doesn't even attempt to provide a language. There are no
-loops or conditional expressions in @command{ginsh}, it is
-merely a window into the library for the programmer to test stuff (or
-to show off). Still, the design of a complete CAS with a language of
-its own, graphical capabilites and all this on top of GiNaC is
-possible and is without doubt a nice project for the future.
-
-There are many built-in functions in GiNaC that do not know how
-to evaluate themselves numerically to a precision declared at runtime
-(using @code{Digits}). Some may be evaluated at certain
-points, but not generally. This ought to be fixed. However, doing
-numerical computations with GiNaC's quite abstract classes is doomed
-to be inefficient. For this purpose, the underlying bignum-package
-@acronym{CLN} is much better suited.
-
-
-@node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
+@node Why C++?, Internal Structures, Disadvantages, A Comparison With Other CAS
@c node-name, next, previous, up
-@section Symbolic functions
+@section Why C++?
-The easiest and most instructive way to start with is probably
-to implement your own function. Objects of class
-@code{function} are inserted into the system via a kind of
-"registry". They get a serial number that is used internally to
-identify them but you usually need not worry about this. What you
-have to care for are functions that are called when the user invokes
-certain methods. These are usual C++-functions accepting a number of
-@code{ex} as arguments and returning one
-@code{ex}. As an example, if we have a look at a
-simplified implementation of the cosine trigonometric function, we
-first need a function that is called when one wishes to
-@code{eval} it. It could look something like this:
-
-@example
-static ex cos_eval_method(ex const & x)
-@{
- // if x%2*Pi return 1
- // if x%Pi return -1
- // if x%Pi/2 return 0
- // care for other cases...
- return cos(x).hold();
-@}
-@end example
+Why did we choose to implement GiNaC in C++ instead of Java or any other
+language? C++ is not perfect: type checking is not strict (casting is
+possible), separation between interface and implementation is not
+complete, object oriented design is not enforced. The main reason is
+the often scolded feature of operator overloading in C++. While it may
+be true that operating on classes with a @code{+} operator is rarely
+meaningful, it is perfectly suited for algebraic expressions. Writing
+@math{3x+5y} as @code{3*x+5*y} instead of
+@code{x.times(3).plus(y.times(5))} looks much more natural.
+Furthermore, the main developers are more familiar with C++ than with
+any other programming language.
+
+
+@node Internal Structures, Expressions are reference counted, Why C++? , Top
+@c node-name, next, previous, up
+@appendix Internal Structures
+
+@menu
+* Expressions are reference counted::
+* Internal representation of products and sums::
+@end menu
-The last line returns @code{cos(x)} if we don't know what
-else to do and stops a potential recursive evaluation by saying
-@code{.hold()}. We should also implement a method for
-numerical evaluation and since we are lazy we sweep the problem under
-the rug by calling someone else's function that does so, in this case
-the one in class @code{numeric}:
+@node Expressions are reference counted, Internal representation of products and sums, Internal Structures, Internal Structures
+@c node-name, next, previous, up
+@appendixsection Expressions are reference counted
+
+@cindex reference counting
+@cindex copy-on-write
+@cindex garbage collection
+An expression is extremely light-weight since internally it works like a
+handle to the actual representation and really holds nothing more than a
+pointer to some other object. What this means in practice is that
+whenever you create two @code{ex} and set the second equal to the first
+no copying process is involved. Instead, the copying takes place as soon
+as you try to change the second. Consider the simple sequence of code:
@example
-static ex cos_evalf_method(ex const & x)
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
@{
- return sin(ex_to_numeric(x));
+ symbol x("x"), y("y"), z("z");
+ ex e1, e2;
+
+ e1 = sin(x + 2*y) + 3*z + 41;
+ e2 = e1; // e2 points to same object as e1
+ cout << e2 << endl; // prints sin(x+2*y)+3*z+41
+ e2 += 1; // e2 is copied into a new object
+ cout << e2 << endl; // prints sin(x+2*y)+3*z+42
@}
@end example
-Differentiation will surely turn up and so we need to tell
-@code{sin} how to differentiate itself:
+The line @code{e2 = e1;} creates a second expression pointing to the
+object held already by @code{e1}. The time involved for this operation
+is therefore constant, no matter how large @code{e1} was. Actual
+copying, however, must take place in the line @code{e2 += 1;} because
+@code{e1} and @code{e2} are not handles for the same object any more.
+This concept is called @dfn{copy-on-write semantics}. It increases
+performance considerably whenever one object occurs multiple times and
+represents a simple garbage collection scheme because when an @code{ex}
+runs out of scope its destructor checks whether other expressions handle
+the object it points to too and deletes the object from memory if that
+turns out not to be the case. A slightly less trivial example of
+differentiation using the chain-rule should make clear how powerful this
+can be:
@example
-static ex cos_diff_method(ex const & x, unsigned diff_param)
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
@{
- return cos(x);
+ symbol x("x"), y("y");
+
+ ex e1 = x + 3*y;
+ ex e2 = pow(e1, 3);
+ ex e3 = diff(sin(e2), x); // first derivative of sin(e2) by x
+ cout << e1 << endl // prints x+3*y
+ << e2 << endl // prints (x+3*y)^3
+ << e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
@}
@end example
-The second parameter is obligatory but uninteresting at this point.
-It is used for correct handling of the product rule only. For Taylor
-expansion, it is enough to know how to differentiate. But if the
-function you want to implement does have a pole somewhere in the
-complex plane, you need to write another method for Laurent expansion
-around that point.
+Here, @code{e1} will actually be referenced three times while @code{e2}
+will be referenced two times. When the power of an expression is built,
+that expression needs not be copied. Likewise, since the derivative of
+a power of an expression can be easily expressed in terms of that
+expression, no copying of @code{e1} is involved when @code{e3} is
+constructed. So, when @code{e3} is constructed it will print as
+@code{3*(x+3*y)^2*cos((x+3*y)^3)} but the argument of @code{cos()} only
+holds a reference to @code{e2} and the factor in front is just
+@code{3*e1^2}.
+
+As a user of GiNaC, you cannot see this mechanism of copy-on-write
+semantics. When you insert an expression into a second expression, the
+result behaves exactly as if the contents of the first expression were
+inserted. But it may be useful to remember that this is not what
+happens. Knowing this will enable you to write much more efficient
+code. If you still have an uncertain feeling with copy-on-write
+semantics, we recommend you have a look at the
+@uref{http://www.cerfnet.com/~mpcline/c++-faq-lite/, C++-FAQ lite} by
+Marshall Cline. Chapter 16 covers this issue and presents an
+implementation which is pretty close to the one in GiNaC.
+
+
+@node Internal representation of products and sums, Package Tools, Expressions are reference counted, Internal Structures
+@c node-name, next, previous, up
+@appendixsection Internal representation of products and sums
+
+@cindex representation
+@cindex @code{add}
+@cindex @code{mul}
+@cindex @code{power}
+Although it should be completely transparent for the user of
+GiNaC a short discussion of this topic helps to understand the sources
+and also explain performance to a large degree. Consider the
+unexpanded symbolic expression
+@tex
+$2d^3 \left( 4a + 5b - 3 \right)$
+@end tex
+@ifnottex
+@math{2*d^3*(4*a+5*b-3)}
+@end ifnottex
+which could naively be represented by a tree of linear containers for
+addition and multiplication, one container for exponentiation with base
+and exponent and some atomic leaves of symbols and numbers in this
+fashion:
+
+@image{repnaive}
+
+@cindex pair-wise representation
+However, doing so results in a rather deeply nested tree which will
+quickly become inefficient to manipulate. We can improve on this by
+representing the sum as a sequence of terms, each one being a pair of a
+purely numeric multiplicative coefficient and its rest. In the same
+spirit we can store the multiplication as a sequence of terms, each
+having a numeric exponent and a possibly complicated base, the tree
+becomes much more flat:
+
+@image{reppair}
+
+The number @code{3} above the symbol @code{d} shows that @code{mul}
+objects are treated similarly where the coefficients are interpreted as
+@emph{exponents} now. Addition of sums of terms or multiplication of
+products with numerical exponents can be coded to be very efficient with
+such a pair-wise representation. Internally, this handling is performed
+by most CAS in this way. It typically speeds up manipulations by an
+order of magnitude. The overall multiplicative factor @code{2} and the
+additive term @code{-3} look somewhat out of place in this
+representation, however, since they are still carrying a trivial
+exponent and multiplicative factor @code{1} respectively. Within GiNaC,
+this is avoided by adding a field that carries an overall numeric
+coefficient. This results in the realistic picture of internal
+representation for
+@tex
+$2d^3 \left( 4a + 5b - 3 \right)$:
+@end tex
+@ifnottex
+@math{2*d^3*(4*a+5*b-3)}:
+@end ifnottex
+
+@image{repreal}
+
+@cindex radical
+This also allows for a better handling of numeric radicals, since
+@code{sqrt(2)} can now be carried along calculations. Now it should be
+clear, why both classes @code{add} and @code{mul} are derived from the
+same abstract class: the data representation is the same, only the
+semantics differs. In the class hierarchy, methods for polynomial
+expansion and the like are reimplemented for @code{add} and @code{mul},
+but the data structure is inherited from @code{expairseq}.
+
+
+@node Package Tools, ginac-config, Internal representation of products and sums, Top
+@c node-name, next, previous, up
+@appendix Package Tools
+
+If you are creating a software package that uses the GiNaC library,
+setting the correct command line options for the compiler and linker
+can be difficult. GiNaC includes two tools to make this process easier.
-Now that everything has been written for @code{cos},
-we need to tell the system about it. This is done by a macro and we
-are not going to descibe how it expands, please consult your
-preprocessor if you are curious:
+@menu
+* ginac-config:: A shell script to detect compiler and linker flags.
+* AM_PATH_GINAC:: Macro for GNU automake.
+@end menu
+
+
+@node ginac-config, AM_PATH_GINAC, Package Tools, Package Tools
+@c node-name, next, previous, up
+@section @command{ginac-config}
+@cindex ginac-config
+
+@command{ginac-config} is a shell script that you can use to determine
+the compiler and linker command line options required to compile and
+link a program with the GiNaC library.
+
+@command{ginac-config} takes the following flags:
+
+@table @samp
+@item --version
+Prints out the version of GiNaC installed.
+@item --cppflags
+Prints '-I' flags pointing to the installed header files.
+@item --libs
+Prints out the linker flags necessary to link a program against GiNaC.
+@item --prefix[=@var{PREFIX}]
+If @var{PREFIX} is specified, overrides the configured value of @env{$prefix}.
+(And of exec-prefix, unless @code{--exec-prefix} is also specified)
+Otherwise, prints out the configured value of @env{$prefix}.
+@item --exec-prefix[=@var{PREFIX}]
+If @var{PREFIX} is specified, overrides the configured value of @env{$exec_prefix}.
+Otherwise, prints out the configured value of @env{$exec_prefix}.
+@end table
+
+Typically, @command{ginac-config} will be used within a configure
+script, as described below. It, however, can also be used directly from
+the command line using backquotes to compile a simple program. For
+example:
@example
-REGISTER_FUNCTION(cos, cos_eval_method, cos_evalf_method, cos_diff, NULL);
+c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`
@end example
-The first argument is the function's name, the second, third and
-fourth bind the corresponding methods to this objects and the fifth is
-a slot for inserting a method for series expansion. Also, the new
-function needs to be declared somewhere. This may also be done by a
-convenient preprocessor macro:
+This command line might expand to (for example):
@example
-DECLARE_FUNCTION_1P(cos)
+cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
+ -lginac -lcln -lstdc++
@end example
-The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
-implementation of @code{cos} is very incomplete and lacks several safety
-mechanisms. Please, have a look at the real implementation in GiNaC.
-(By the way: in case you are worrying about all the macros above we
-can assure you that functions are GiNaC's most macro-intense classes.
-We have done our best to avoid them where we can.)
-
-That's it. May the source be with you!
+Not only is the form using @command{ginac-config} easier to type, it will
+work on any system, no matter how GiNaC was configured.
-@node A Comparison With Other CAS, Bibliography, Symbolic functions, Top
+@node AM_PATH_GINAC, Configure script options, ginac-config, Package Tools
@c node-name, next, previous, up
-@chapter A Comparison With Other CAS
+@section @samp{AM_PATH_GINAC}
+@cindex AM_PATH_GINAC
-This chapter will give you some information on how GiNaC
-compares to other, traditional Computer Algebra Systems, like
-@emph{Maple}, @emph{Mathematica} or @emph{Reduce}, where it has
-advantages and disadvantages over these systems.
+For packages configured using GNU automake, GiNaC also provides
+a macro to automate the process of checking for GiNaC.
+@example
+AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
+@end example
-@heading Advantages
-
-GiNaC has several advantages over traditional Computer
-Algebra Systems, like
+This macro:
@itemize @bullet
@item
-familiar language: all common CAS implement their own
-proprietary grammar which you have to learn first (and maybe learn
-again when your vendor chooses to "enhance" it). With GiNaC you
-can write your program in common C++, which is standardized.
+Determines the location of GiNaC using @command{ginac-config}, which is
+either found in the user's path, or from the environment variable
+@env{GINACLIB_CONFIG}.
@item
-structured data types: you can build up structured data
-types using @code{struct}s or @code{class}es
-together with STL features instead of using unnamed lists of lists
-of lists.
+Tests the installed libraries to make sure that their version
+is later than @var{MINIMUM-VERSION}. (A default version will be used
+if not specified)
@item
-strongly typed: in CAS, you usually have only one kind of
-variables which can hold contents of an arbitrary type. This
-4GL like feature is nice for novice programmers, but dangerous.
-
-@item
-development tools: powerful development tools exist for
-C++, like fancy editors (e.g. with automatic
-indentation and syntax highlighting), debuggers, visualization
-tools, documentation tools...
+If the required version was found, sets the @env{GINACLIB_CPPFLAGS} variable
+to the output of @command{ginac-config --cppflags} and the @env{GINACLIB_LIBS}
+variable to the output of @command{ginac-config --libs}, and calls
+@samp{AC_SUBST()} for these variables so they can be used in generated
+makefiles, and then executes @var{ACTION-IF-FOUND}.
@item
-modularization: C++ programs can
-easily be split into modules by separating interface and
-implementation.
+If the required version was not found, sets @env{GINACLIB_CPPFLAGS} and
+@env{GINACLIB_LIBS} to empty strings, and executes @var{ACTION-IF-NOT-FOUND}.
-@item
-price: GiNaC is distributed under the GNU Public License
-which means that it is free and available with source code. And
-there are excellent C++-compilers for free, too.
-
-@item
-extendable: you can add your own classes to GiNaC, thus
-extending it on a very low level. Compare this to a traditional
-CAS that you can usually only extend on a high level by writing in
-the language defined by the parser. In particular, it turns out
-to be almost impossible to fix bugs in a traditional system.
+@end itemize
-@item
-seemless integration: it is somewhere between difficult
-and impossible to call CAS functions from within a program
-written in C++ or any other programming
-language and vice versa. With GiNaC, your symbolic routines
-are part of your program. You can easily call third party
-libraries, e.g. for numerical evaluation or graphical
-interaction. All other approaches are much more cumbersome: they
-range from simply ignoring the problem (i.e. @emph{Maple}) to providing a
-method for "embedding" the system (i.e. @emph{Yacas}).
+This macro is in file @file{ginac.m4} which is installed in
+@file{$datadir/aclocal}. Note that if automake was installed with a
+different @samp{--prefix} than GiNaC, you will either have to manually
+move @file{ginac.m4} to automake's @file{$datadir/aclocal}, or give
+aclocal the @samp{-I} option when running it.
-@item
-efficiency: often large parts of a program do not need
-symbolic calculations at all. Why use large integers for loop
-variables or arbitrary precision arithmetics where double
-accuracy is sufficient? For pure symbolic applications,
-GiNaC is comparable in speed with other CAS.
+@menu
+* Configure script options:: Configuring a package that uses AM_PATH_GINAC.
+* Example package:: Example of a package using AM_PATH_GINAC.
+@end menu
-@end itemize
+@node Configure script options, Example package, AM_PATH_GINAC, AM_PATH_GINAC
+@c node-name, next, previous, up
+@subsection Configuring a package that uses @samp{AM_PATH_GINAC}
-@heading Disadvantages
+Simply make sure that @command{ginac-config} is in your path, and run
+the configure script.
-Of course it also has some disadvantages
+Notes:
@itemize @bullet
@item
-not interactive: GiNaC programs have to be written in
-an editor, compiled and executed. You cannot play with
-expressions interactively. However, such an extension is not
-inherently forbidden by design. In fact, two interactive
-interfaces are possible: First, a simple shell that exposes GiNaC's
-types to a command line can readily be written (and has been
-written) and second, as a more consistent approach we plan
-an integration with the @acronym{CINT} C++ interpreter.
+The directory where the GiNaC libraries are installed needs
+to be found by your system's dynamic linker.
+
+This is generally done by
+
+@display
+editing @file{/etc/ld.so.conf} and running @command{ldconfig}
+@end display
+
+or by
+
+@display
+setting the environment variable @env{LD_LIBRARY_PATH},
+@end display
+
+or, as a last resort,
+
+@display
+giving a @samp{-R} or @samp{-rpath} flag (depending on your linker) when
+running configure, for instance:
+
+@example
+LDFLAGS=-R/home/cbauer/lib ./configure
+@end example
+@end display
@item
-advanced features: GiNaC cannot compete with a program
-like @emph{Reduce} which exists for more than
-30 years now or @emph{Maple} which grows since
-1981 by the work of dozens of programmers, with respect to
-mathematical features. Integration, factorization, non-trivial
-simplifications, limits etc. are missing in GiNaC (and are not
-planned for the near future).
+You can also specify a @command{ginac-config} not in your path by
+setting the @env{GINACLIB_CONFIG} environment variable to the
+name of the executable
@item
-portability: While the GiNaC library itself is designed
-to avoid any platform dependent features (it should compile
-on any ANSI compliant C++ compiler), the
-currently used version of the CLN library (fast large integer and
-arbitrary precision arithmetics) can be compiled only on systems
-with a recently new C++ compiler from the
-GNU Compiler Collection (@acronym{GCC}). GiNaC uses
-recent language features like explicit constructors, mutable
-members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
-literally. Recent @acronym{GCC} versions starting at
-2.95, although itself not yet ANSI compliant, support all needed
-features.
-
+If you move the GiNaC package from its installed location,
+you will either need to modify @command{ginac-config} script
+manually to point to the new location or rebuild GiNaC.
+
@end itemize
+Advanced note:
-@heading Why C++?
+@itemize @bullet
+@item
+configure flags
+
+@example
+--with-ginac-prefix=@var{PREFIX}
+--with-ginac-exec-prefix=@var{PREFIX}
+@end example
-Why did we choose to implement GiNaC in C++ instead of Java or any other
-language? C++ is not perfect: type checking is not strict
-(casting is possible), separation between interface and implementation
-is not complete, object oriented design is not enforced. The main
-reason is the often scolded feature of operator overloading in
-C++. While it may be true that operating on classes
-with a @code{+} operator is rarely meaningful, it is
-perfectly suited for algebraic expressions. Writing @math{3x+5y} as
-@code{3*x+5*y} instead of @code{x.times(3).plus(y.times(5))} looks much more
-natural. Furthermore, the main developers are more familiar with
-C++ than with any other programming language.
-
-
-@node Bibliography, Concept Index, A Comparison With Other CAS, Top
+are provided to override the prefix and exec-prefix that were stored
+in the @command{ginac-config} shell script by GiNaC's configure. You are
+generally better off configuring GiNaC with the right path to begin with.
+@end itemize
+
+
+@node Example package, Bibliography, Configure script options, AM_PATH_GINAC
+@c node-name, next, previous, up
+@subsection Example of a package using @samp{AM_PATH_GINAC}
+
+The following shows how to build a simple package using automake
+and the @samp{AM_PATH_GINAC} macro. The program used here is @file{simple.cpp}:
+
+@example
+#include <ginac/ginac.h>
+
+int main(void)
+@{
+ GiNaC::symbol x("x");
+ GiNaC::ex a = GiNaC::sin(x);
+ std::cout << "Derivative of " << a
+ << " is " << a.diff(x) << std::endl;
+ return 0;
+@}
+@end example
+
+You should first read the introductory portions of the automake
+Manual, if you are not already familiar with it.
+
+Two files are needed, @file{configure.in}, which is used to build the
+configure script:
+
+@example
+dnl Process this file with autoconf to produce a configure script.
+AC_INIT(simple.cpp)
+AM_INIT_AUTOMAKE(simple.cpp, 1.0.0)
+
+AC_PROG_CXX
+AC_PROG_INSTALL
+AC_LANG_CPLUSPLUS
+
+AM_PATH_GINAC(0.7.0, [
+ LIBS="$LIBS $GINACLIB_LIBS"
+ CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
+], AC_MSG_ERROR([need to have GiNaC installed]))
+
+AC_OUTPUT(Makefile)
+@end example
+
+The only command in this which is not standard for automake
+is the @samp{AM_PATH_GINAC} macro.
+
+That command does the following: If a GiNaC version greater or equal
+than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
+and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
+the error message `need to have GiNaC installed'
+
+And the @file{Makefile.am}, which will be used to build the Makefile.
+
+@example
+## Process this file with automake to produce Makefile.in
+bin_PROGRAMS = simple
+simple_SOURCES = simple.cpp
+@end example
+
+This @file{Makefile.am}, says that we are building a single executable,
+from a single sourcefile @file{simple.cpp}. Since every program
+we are building uses GiNaC we simply added the GiNaC options
+to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
+want to specify them on a per-program basis: for instance by
+adding the lines:
+
+@example
+simple_LDADD = $(GINACLIB_LIBS)
+INCLUDES = $(GINACLIB_CPPFLAGS)
+@end example
+
+to the @file{Makefile.am}.
+
+To try this example out, create a new directory and add the three
+files above to it.
+
+Now execute the following commands:
+
+@example
+$ automake --add-missing
+$ aclocal
+$ autoconf
+@end example
+
+You now have a package that can be built in the normal fashion
+
+@example
+$ ./configure
+$ make
+$ make install
+@end example
+
+
+@node Bibliography, Concept Index, Example package, Top
@c node-name, next, previous, up
-@chapter Bibliography
+@appendix Bibliography
@itemize @minus{}
J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
Academic Press, London
+@item
+@cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
+
@end itemize
@printindex cp
@bye
-
git://www.ginac.de / ginac.git / blobdiff