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

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

@include version.texi

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

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

Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany

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

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

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

@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

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

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

@page
@contents

@page


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

This file 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.
* Extending GiNaC::              How to extend the library.
* A Comparison With Other CAS::  Compares GiNaC to traditional CAS.
* Bibliography::
* Concept Index::
@end menu


@node Introduction, A Tour of GiNaC, Top, Top
@c    node-name, next, previous, up
@chapter Introduction

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
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.

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

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

You should have received a copy of the GNU General Public License
along with this program; see the file COPYING.  If not, write to the
Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA.


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

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

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


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

The GiNaC open framework for symbolic computation within the C++ programming
language does not try to define a language of it's 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
pointless) bivariate polynomial with some large coefficients:

@subheading My first GiNaC program (a bivariate polynomial)
@example
#include <ginac/ginac.h>
using namespace GiNaC;

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

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

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

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

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

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 GiNaC;

ex HermitePoly(symbol x, int deg)
@{
    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);
@}

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

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

    return 0;
@}
@end example

When run, this will type out

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

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

In order to show some more examples of what GiNaC can do we
will now use @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

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

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

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

All numbers occuring in GiNaC's expressions can be converted into floating
point numbers with the @code{evalf} method, to arbitrary accuracy:

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

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

@example
> a=Pi^2+x;
x+Pi^2
> evalf(a);
x+9.869604401089358619L0
> x=2;
2
> evalf(a);
11.869604401089358619L0
@end example

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

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

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

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

@example
> lsolve(a+x*y==z,x);
y^(-1)*(z-a);
lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
[x==19/8,y==-1/40]
> M = [[ [[1, 3]], [[-3, 2]] ]];
[[ [[1,3]], [[-3,2]] ]]
> determinant(M);
11
> charpoly(M,lambda);
lambda^2-3*lambda+11
@end example

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

@example
> a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
-3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
> b = x^2 + 4*x*y - y^2;
-y^2+x^2+4*x*y
> 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)
> 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):

@example
> diff(tan(x),x);
tan(x)^2+1
> series(sin(x),x,0,4);
x-1/6*x^3+Order(x^4)
> series(1/tan(x),x,0,4);
x^(-1)-1/3*x+Order(x^2)
@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:

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


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

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

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


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

In order to install GiNaC on your system, some prerequisites need
to be met.  First of all, you need to have a C++-compiler adhering to
the 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
it will refuse to continue.


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

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:

@itemize @bullet

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

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

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

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

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

@end itemize

In addition, you may specify some environment variables.
@env{CXX} holds the path and the name of the C++ compiler
in case you want to override the default in your path.  (The
@command{configure} script searches your path for @command{c++},
@command{g++}, @command{gcc}, @command{CC}, @command{cxx}
and @command{cc++} in that order.)  It may be very useful to
define some compiler flags with the @env{CXXFLAGS} environment
variable, like optimization, debugging information and warning
levels.  If omitted, it defaults to @option{-g -O2}.

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

@subheading Sample sessions of how to call the configure script

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 pursuaded to be picky and full assertions 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 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

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.

Just to make sure GiNaC works properly you may run a simple test
suite 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.


@node Installing GiNaC, Basic Concepts, Building GiNaC, Installation
@c    node-name, next, previous, up
@section Installing GiNaC

To install GiNaC on your system, simply type

@example
$ make install
@end example

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

@itemize @bullet

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

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

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

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

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

@menu
* Expressions::                  The fundamental GiNaC class.
* The Class Hierarchy::          Overview of GiNaC's classes.
* Symbols::                      Symbolic objects.
* Numbers::                      Numerical objects.
* Constants::                    Pre-defined constants.
* Fundamental operations::       The power, add and mul classes
* Built-in functions::           Mathematical functions.
@end menu


@node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
@c    node-name, next, previous, up
@section 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 MyEx3 = (x + 1)/(x - 1);         // rational expression
ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
ex MyEx5 = MyEx4 + 1;               // similar to above
@end example

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.

So much for expressions.  But what exactly are these expressions
handles of?  This will be answered in the following sections.


@node The Class Hierarchy, Symbols, Expressions, Basic Concepts
@c    node-name, next, previous, up
@section The Class Hierarchy

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

@unnumberedsubsec 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}

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}

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 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 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.

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.


@node Numbers, Constants, Symbols, Basic Concepts
@c    node-name, next, previous, up
@section Numbers

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.  @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: construction
from C-integer, construction of fractions from two integers, construction
from C-float and construction from a string.

@subheading Construction of numbers
@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...
    // ...
@}
@end example

Note that all those constructors are @emph{explicit} which means you are
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. 

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

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

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

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

The above example prints the following output to screen:

@example
in 17 digits:
0.333333333333333333
3.14159265358979324
in 60 digits:
0.333333333333333333333333333333333333333333333333333333333333333333
3.14159265358979323846264338327950288419716939937510582097494459231
@end example

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

@subsection Tests on numbers

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

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

@subheading Sample test on objects of type numeric
@example
#include <ginac/ginac.h>
using namespace GiNaC;

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

int main()
@{
    numeric answer = twentyone;

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

Note that the variable @code{answer} is constructed here
as an integer by @code{numeric}'s copy constructor but in
an intermediate step it holds a rational number represented as integer
numerator and integer denominator.  When multiplied by 10, the
denominator becomes unity and the result is automatically converted to
a pure integer again.  Internally, the underlying
@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{}
@item @code{.is_zero()}
@tab object is equal to zero
@item @code{.is_positive()}
@tab object is not complex and greater than 0
@item @code{.is_integer()}
@tab object is a (non-complex) integer
@item @code{.is_pos_integer()}
@tab object is an integer and greater than 0
@item @code{.is_nonneg_integer()}
@tab object is an integer and greater equal 0
@item @code{.is_even()}
@tab object is an even integer
@item @code{.is_odd()}
@tab object is an odd integer
@item @code{.is_prime()}
@tab object is 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})
@item @code{.is_real()}
@tab object is a real integer, rational or float (i.e. is not complex)
@end multitable
@end cartouche


@node Constants, Fundamental operations, Numbers, Basic Concepts
@c    node-name, next, previous, up
@section Constants

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)
@item @code{Pi}
@tab Archimedes' constant
@tab 3.14159265358979323846264338327950288
@item @code{Catalan}
@tab Catalan's constant
@tab 0.91596559417721901505460351493238411
@item @code{EulerGamma}
@tab Euler's (or Euler-Mascheroni) constant
@tab 0.57721566490153286060651209008240243
@end multitable
@end cartouche


@node Fundamental operations, Built-in functions, Constants, Basic Concepts
@c    node-name, next, previous, up
@section Fundamental operations: the @code{power}, @code{add} and @code{mul} classes

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

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:

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

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}.

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.


@node Built-in functions, Important Algorithms, Fundamental operations, 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:

@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;
    // ...
@}
@end example

This program will type out two times a function and then an
expression that may be really useful:

@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)
@end example

Most of these functions can be differentiated, series expanded and 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:

@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;
    // ...
@}
@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 Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
@c    node-name, next, previous, up
@section Polynomial Expansion

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}.

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!


@node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
@c    node-name, next, previous, up
@section Collecting expressions

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
#include <ginac/ginac.h>
ex ex::collect(symbol const & s);
@end example

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

@example
#include <ginac/ginac.h>
int ex::degree(symbol const & s);
int ex::ldegree(symbol const & s);
@end example

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: 

@subheading Collecting expressions in multivariate polynomials
@example
#include <ginac/ginac.h>
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.


@node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
@c    node-name, next, previous, up
@section Polynomial Arithmetic

@subsection GCD and LCM

The functions for polynomial greatest common divisor and least common
multiple have the synopsis:

@example
#include <ginac/normal.h>
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}.

@subheading Polynomal GCD/LCM
@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 The @code{normal} method

While in common symbolic code @code{gcd()} and @code{lcm()} are not too
heavily used, simplification is called for 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
numerator 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:

@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;
    // ...
@}
@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)}.


@node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
@c    node-name, next, previous, up
@section Symbolic Differentiation

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:

@subheading Simple polynomial differentiation
@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:

@subheading Differentiation with nontrivial functions: Euler numbers
@example
#include <ginac/ginac.h>
using namespace GiNaC;

ex EulerNumber(unsigned n)
@{
    symbol x;
    ex generator = pow(cosh(x),-1);
    return generator.diff(x,n).subs(x==0);
@}

int main()
@{
    for (unsigned i=0; i<11; i+=2)
        cout << EulerNumber(i) << 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, Extending GiNaC, Symbolic Differentiation, Important Algorithms
@c    node-name, next, previous, up
@section Series Expansion

Expressions know how to expand themselves as a Taylor series or (more
generally) a Laurent series.  Similar to 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:

@subheading Series expansion
@example
#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;
    // ...
@}
@end example

As an instructive application, let us calculate the numerical
value of Archimedes' constant (for which there already exists the
exbuilt-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:

@subheading Series expansion using M@'echain's formula for @code{Pi}
@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,0,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()
@{
    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

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 Extending GiNaC, What does not belong into GiNaC, Series Expansion, 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.

@menu
* What does not belong into GiNaC::  What to avoid.
* Symbolic functions::               Implementing symbolic functions.
@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

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
@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.  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

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}:

@example
static ex cos_evalf_method(ex const & x)
@{
    return sin(ex_to_numeric(x));
@}
@end example

Differentiation will surely turn up and so we need to tell
@code{sin} how to differentiate itself:

@example
static ex cos_diff_method(ex const & x, unsigned diff_param)
@{
    return cos(x);
@}
@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.

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:

@example
REGISTER_FUNCTION(cos, cos_eval_method, cos_evalf_method, cos_diff, NULL);
@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:

@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 them where we can.)

That's it. May the source be with you!


@node A Comparison With Other CAS, Bibliography, Symbolic functions, Top
@c    node-name, next, previous, up
@chapter A Comparison With Other CAS

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.


@heading Advantages

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 chooses to "enhance" it).  With GiNaC you
can write your program in common C++, which is standardized.

@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 tools...

@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
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 double
accuracy is sufficient?  For pure symbolic applications,
GiNaC is comparable in speed with other CAS.

@end itemize


@heading Disadvantages

Of course it also has some disadvantages:

@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.

@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}).  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


@heading Why C++?

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
@c    node-name, next, previous, up
@chapter Bibliography

@itemize @minus{}

@item
@cite{ISO/IEC 14882:1998: Programming Languages: C++}

@item
@cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}

@item
@cite{The C++ Programming Language}, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley

@item
@cite{C++ FAQs}, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley

@item
@cite{Algorithms for Computer Algebra}, Keith O. Geddes, Stephen R. Czapor,
and George Labahn, ISBN 0-7923-9259-0, 1992, Kluwer Academic Publishers, Norwell, Massachusetts

@item
@cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988, 
Academic Press, London

@end itemize


@node Concept Index, , Bibliography, Top
@c    node-name, next, previous, up
@unnumbered Concept Index

@printindex cp

@bye