     [file version.texi not included]

This is a tutorial that documents GiNaC 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.

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.

GiNaC
*****

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

Introduction
************

The motivation behind GiNaC derives from the observation that most present day computer algebra systems (CAS) are linguistically and semantically impoverished.  Although they are quite powerful tools for learning math and solving particular problems they lack modern 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 the GiNaC home page (http://www.ginac.de/reference/).  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.

License
=======

The GiNaC framework for symbolic computation within the C++ programming language is 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.

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.

How to use it from within C++
=============================

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

     #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;
     }

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

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

(see Package Tools, for tools that help you when creating a software package that uses GiNaC.)

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

     #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;
     }

When run, this will type out

     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

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

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

What it can do for you
======================

After invoking 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 ginsh syntax we refer to its accompanied man page. Suffice to say that assignments and comparisons in ginsh are written as they are in C, i.e. = assigns and == compares.

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

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

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

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

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

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

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:

     > cos(42*Pi);
     1
     > cos(acos(x));
     x
     > acos(cos(x));
     acos(cos(x))

(Note that converting the last input to x would allow one to conclude that 42*Pi is equal to 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 ginsh's notation of double brackets to type them in:

     > 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

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

     > 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

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):

     > diff(tan(x),x);
     tan(x)^2+1
     > series(sin(x),x,0,4);
     x-1/6*x^3+Order(x^4)
     > series(1/tan(x),x,0,4);
     x^(-1)-1/3*x+Order(x^2)
     > series(gamma(x),x,0,3);
     x^(-1)-EulerGamma+(1/12*Pi^2+1/2*EulerGamma^2)*x
     +(-1/3*zeta(3)-1/12*Pi^2*EulerGamma-1/6*EulerGamma^3)*x^2+Order(x^3)
     > evalf(");
     x^(-1.0)-0.5772156649015328606+(0.98905599532797255544)*x
     -(0.90747907608088628905)*x^2+Order(x^(3.0))
     > series(gamma(2*sin(x)-2),x,Pi/2,6);
     -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*EulerGamma^2-1/240)*(x-1/2*Pi)^2
     -EulerGamma-1/12+Order((x-1/2*Pi)^3)

Here we have made use of the ginsh-command " to pop the previously evaluated element from 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 therefore easy:

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

Installation
************

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

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 ISO/IEC 14882:1998(E).  We used GCC for development so if you have a different compiler you are on your own. For the configuration to succeed you need a Posix compliant shell installed in /bin/sh, GNU 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 CLN is extensively used and needs to be installed on your system. Please get it from ftp://ftp.santafe.edu/pub/gnu/ or from Bruno Haible's FTP site (ftp://ftp.ilog.fr/pub/Users/haible/gnu/) (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.

Configuration
=============

To configure GiNaC means to prepare the source distribution for building.  It is done via a shell script called 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 --help option.  The most important ones will be shortly described in what follows:

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

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

  * --libdir=LIBDIR: Use this option in case you want to have the library installed in some other directory than PREFIX/lib/.

  * --includedir=INCLUDEDIR: Use this option in case you want to have the header files installed in some other directory than PREFIX/include/ginac/. For instance, if you specify --includedir=/usr/include you will end up with the header files sitting in the directory /usr/include/ginac/. Note that the subdirectory 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).

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

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

The whole process is illustrated in the following two examples. (Substitute setenv VARIABLE VALUE for export VARIABLE=VALUE if the Berkeley C shell is your login shell.)

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

     $ export CXXFLAGS="-Wall -O2"
     $ ./configure

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

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

Building GiNaC
==============

After proper configuration you should just build the whole library by typing

     $ make

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 CXXFLAGS you entered.  Optimization may be very time-consuming.

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

     $ make check

This will compile some sample programs, run them and compare the output to reference output. Each of the checks should return a message 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, not a sanity check of your system.  Another intent is to allow people to fiddle around with optimization.  If CLN was installed all right this step is unlikely to return any errors.

Generally, the top-level Makefile runs recursively to the subdirectories.  It is therfore safe to go into any subdirectory (doc/, ginsh/, ...) and simply type make TARGET there in case something went wrong.

Installing GiNaC
================

To install GiNaC on your system, simply type

     $ make install

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):

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

  * All the header files will be installed into PREFIX/include/ginac/ (or INCLUDEDIR/ginac/, if specified).

  * All documentation (HTML and Postscript) will be stuffed into PREFIX/share/doc/GiNaC/ (or DATADIR/doc/GiNaC/, if DATADIR was specified).

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

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.

Expressions
===========

The most common class of objects a user deals with is the expression 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:

     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

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

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

The Class Hierarchy
===================

GiNaC's class hierarchy consists of several classes representing mathematical objects, all of which (except for ex and some helpers) are internally derived from one abstract base class called basic.  You do not have to deal with objects of class basic, instead you'll be dealing with symbols, numbers, containers of expressions and so on.  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 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.  The oval classes are atomic ones and the squared classes are containers. The dashed line symbolizes a `points to' or `handles' relationship while the solid lines stand for `inherits from' relationship in the class hierarchy:

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 expairseq, which is a container for a sequence of pairs each consisting of one expression and a number (numeric).  What is visible to the user are the derived classes add and mul, representing sums of terms and products, respectively. See Internal Structures, where these two classes are described in more detail.

Symbols
=======

Symbols are for symbolic manipulation what atoms are for chemistry.  You can declare objects of class symbol as any other object simply by saying 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 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 symbol x("x"),y("x"); but it is likely to lead into trouble.  Here, x and y are different symbols and statements like x-y will not be simplified to zero although the output 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 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 .subs() method.

Numbers
=======

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

The user can construct an object of class 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:

     #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...
         // ...
     }

Note that all those constructors are explicit which means you are not allowed to write numeric two=2;.  This is because the basic objects to be handled by GiNaC are the expressions ex and we want to keep things simple and wish objects like pow(x,2) to be handled the same way as pow(x,a), which means that we need to allow a general 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 numeric r(3/2). This would, however, call C's built-in operator / for integers first and result in a numeric holding a plain integer 1.  Never use the operator / on integers unless you know exactly what you are doing!  Use the constructor from two integers instead, as shown in the example above.  Writing 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, i.e. how `long' they are.  The situation is different for floating point numbers.  Their accuracy is controlled by one global variable, called Digits.  (For those readers who know about Maple: it behaves very much like Maple's Digits).  All objects of class numeric that are constructed from then on will be stored with a precision matching that number of decimal digits:

     #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;
     }

The above example prints the following output to screen:

     in 17 digits:
     0.333333333333333333
     3.14159265358979324
     in 60 digits:
     0.333333333333333333333333333333333333333333333333333333333333333333
     3.14159265358979323846264338327950288419716939937510582097494459231

It should be clear that objects of class 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 ex.

Tests on numbers
----------------

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

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

     #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!
         // ...
     }

Note that the variable answer is constructed here as an integer by numeric's copy constructor but in an intermediate step it holds a rational number represented as integer numerator and integer denominator.  When multiplied by 10, the denominator becomes unity and the result is automatically converted to a pure integer again. Internally, the underlying CLN is responsible for this behaviour and we refer the reader to 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.

Method   Returns true if the object is...
.is_zero()
...equal to zero
.is_positive()
...not complex and greater than 0
.is_integer()
...a (non-complex) integer
.is_pos_integer()
...an integer and greater than 0
.is_nonneg_integer()
...an integer and greater equal 0
.is_even()
...an even integer
.is_odd()
...an odd integer
.is_prime()
...a prime integer (probabilistic primality test)
.is_rational()
...an exact rational number (integers are rational, too)
.is_real()
...a real integer, rational or float (i.e. is not complex)
.is_cinteger()
...a (complex) integer, such as 2-3*I
.is_crational()
...an exact (complex) rational number (such as 2/3+7/2*I)

Constants
=========

Constants behave pretty much like symbols except that they return some specific number when the method .evalf() is called.

The predefined known constants are:

Name   Common Name   Numerical Value (to 35 digits)
Pi
Archimedes' constant
3.14159265358979323846264338327950288
Catalan
Catalan's constant
0.91596559417721901505460351493238411
EulerGamma
Euler's (or Euler-Mascheroni) constant
0.57721566490153286060651209008240243

Fundamental containers: the power, add and mul classes
======================================================

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 +, -, * and / have been overloaded to achieve this goal.  When you run the following program, the constructor for an object of type mul is automatically called to hold the product of a and b and then the constructor for an object of type add is called to hold the sum of that mul object and the number one:

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

     int main()
     {
         symbol a("a"), b("b");
         ex MyTerm = 1+a*b;
         // ...
     }

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

  * Due to C's operator precedence, 2*x^2 would be parsed as (2*x)^2.

  * Due to the binding of the operator ^, x^a^b would result in (x^a)^b. This would be confusing since most (though not all) other CAS interpret this as x^(a^b).

  * Also, expressions involving integer exponents are very frequently used, which makes it even more dangerous to overload ^ since it is then hard to distinguish between the semantics as exponentiation and the one for exclusive or.  (It would be embarassing to return 1 where one has requested 2^3.)

All effects are contrary to mathematical notation and differ from the way most other CAS handle exponentiation, therefore overloading ^ is ruled out for GiNaC's C++ part.  The situation is different in ginsh, there the exponentiation-^ exists.  (Also note that the other frequently used exponentiation operator ** 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 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 pow(pow(x,2),3) to x^6 automatically are only performed when this is mathematically possible.  If we replace the outer exponent three in the example by some symbols a, the simplification is not safe and will not be performed, since a might be 1/2 and x negative.

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

Built-in functions
==================

There are quite a number of useful functions hard-wired into GiNaC.  For instance, all trigonometric and hyperbolic functions are implemented. They are all objects of class 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:

     #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;
         // ...
     }

This program shows how the function returns itself twice and finally an expression that may be really useful:

     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)

For functions that have a branch cut in the complex plane GiNaC follows the conventions for C++ as defined in the ANSI standard.  In particular: the natural logarithm (log) and the square root (sqrt) both have their branch cuts running along the negative real axis where the points on the axis itself belong to the upper part.

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

Relations
=========

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

See Built-in functions, for examples where various applications of the .subs() method show how objects of class relational are used as arguments.  There they provide an intuitive syntax for substitutions.

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:

     #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;
         // ...
     }

The general rule is that wherever methods accept one or more parameters (ARG1, ARG2, ...) the order of arguments the function wrapper accepts is the same but preceded by the object to act on (OBJECT, ARG1, ARG2, ...).  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 A:=x+1; subs(x=2,A); GiNaC would require A=x+1; subs(A,x==2); (after proper declaration of A and x).  On the other hand, since MapleV returns 3 on A:=x^2+3; coeff(A,x,0); (GiNaC: 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, 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 ex and sometimes calling a function cannot be avoided.

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 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 (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 (4*y + z)*x + 20*y^2 + 21*y*z + 4*z^2 and 20*y^2 + (21*z + 4*x)*y + 4*z^2 + x*z.

To bring an expression into expanded form, its method .expand() may be called.  In our example above, this corresponds to 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!

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 collect() accomplishes this task.  Here is its declaration:

     ex ex::collect(symbol const & s);

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

     int ex::degree(symbol const & s);
     int ex::ldegree(symbol const & s);

where degree() returns the highest coefficient and ldegree() the lowest one.  (These two methods work also reliably on non-expanded input polynomials).  An application is illustrated in the next example, where a multivariate polynomial is analyzed:

     #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;
         // ...
     }

When run, it returns an output in the following fashion:

     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

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.

Polynomial Arithmetic
=====================

GCD and LCM
-----------

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

     ex gcd(const ex & a, const ex & b);
     ex lcm(const ex & a, const ex & b);

The functions gcd() and lcm() accept two expressions a and b as arguments and return a new expression, their greatest common divisor or least common multiple, respectively.  If the polynomials a and b are coprime gcd(a,b) returns 1 and lcm(a,b) returns the product of a and b.

     #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
         // ...
     }

The normal method
-----------------

While in common symbolic code gcd() and lcm() are not too heavily used, simplification is called for frequently.  Therefore .normal(), which provides some basic form of simplification, has become a method of class ex, just like .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 t1 and t2 are indeed simplified in this little program:

     #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;
         // ...
     }

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 P_a/P_b = (4*y+z)/(y+3*z).

Symbolic Differentiation
========================

GiNaC's objects know how to differentiate themselves.  Thus, a polynomial (class add) knows that its derivative is the sum of the derivatives of all the monomials:

     #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
         // ...
     }

If a second integer parameter N is given, the diff method returns the Nth derivative.

If 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 1/cosh(x).  Since the derivative of cosh(x) is sinh(x) and the derivative of pow(x,-1) is -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 Nth Euler number is the coefficient of x^n/n! in the expansion of 1/cosh(x).  We may use this identity to code a function that generates Euler numbers in just three lines:

     #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;
     }

When you run it, it produces the sequence 1, -1, 5, -61, 1385, -50521.  We increment the loop variable i by two since all odd Euler numbers vanish anyways.

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:

     #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;
         // ...
     }

As an instructive application, let us calculate the numerical value of Archimedes' constant (for which there already exists the built-in constant Pi) using Méchain's amazing formula Pi==16*atan(1/5)-4*atan(1/239). We may expand the arcus tangent around 0 and insert the fractions 1/5 and 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 series_to_poly() to simply strip the order term off:

     #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;
     }

When you run this program, it will type out:

     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

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.

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 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 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 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 CLN are much better suited.

Symbolic functions
==================

The easiest and most instructive way to start with is probably to implement your own function.  Objects of class 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 ex as arguments and returning one 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 eval it.  It could look something like this:

     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();
     }

The last line returns cos(x) if we don't know what else to do and stops a potential recursive evaluation by saying .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 numeric:

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

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

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

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 ingrediences for 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:

     REGISTER_FUNCTION(cos, cos_eval_method, cos_evalf_method, cos_diff, NULL);

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.  (If set to NULL it defaults to simple Taylor expansion, which is correct if there are no poles involved.  The way GiNaC handles poles in case there are any is best understood by studying one of the examples, like the Gamma 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:

     DECLARE_FUNCTION_1P(cos)

The suffix _1P stands for one parameter.  Of course, this implementation of 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!

A Comparison With Other CAS
***************************

This chapter will give you some information on how GiNaC compares to other, traditional Computer Algebra Systems, like Maple, Mathematica or Reduce, where it has advantages and disadvantages over these systems.

Advantages
==========

GiNaC has several advantages over traditional Computer Algebra Systems, like

  * 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.

  * structured data types: you can build up structured data types using structs or classes together with STL features instead of using unnamed lists of lists of lists.

  * 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.

  * development tools: powerful development tools exist for C++, like fancy editors (e.g. with automatic indentation and syntax highlighting), debuggers, visualization tools, documentation tools...

  * modularization: C++ programs can easily be split into modules by separating interface and implementation.

  * 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.

  * 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.

  * 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. Maple) to providing a method for `embedding' the system (i.e. Yacas).

  * 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.

Disadvantages
=============

Of course it also has some disadvantages:

  * 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 shell that exposes GiNaC's types to a command line can readily be written (the tiny ginsh that is part of the distribution being an example) and second, as a more consistent approach we are working on an integration with the CINT C++ interpreter.

  * advanced features: GiNaC cannot compete with a program like Reduce which exists for more than 30 years now or 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).

  * 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 (GCC). (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, dynamic_casts and STL, so ANSI compliance is meant literally. Recent GCC versions starting at 2.95, although itself not yet ANSI compliant, support all needed features.

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 + operator is rarely meaningful, it is perfectly suited for algebraic expressions.  Writing 3x+5y as 3*x+5*y instead of 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.

Internal Structures
*******************

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 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:

     #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
         // ...
     }

The line e2 = e1; creates a second expression pointing to the object held already by e1.  The time involved for this operation is therefore constant, no matter how large e1 was.  Actual copying, however, must take place in the line e2 += 1; because e1 and e2 are not handles for the same object any more. This concept is called copy-on-write semantics.  It increases performance considerably whenever one object occurs multiple times and represents a simple garbage collection scheme because when an 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:

     #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)
         // ...
     }

Here, e1 will actually be referenced three times while 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 e1 is involved when e3 is constructed.  So, when e3 is constructed it will print as 3*(x+3*y)^2*cos((x+3*y)^3) but the argument of cos() only holds a reference to e2 and the factor in front is just 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 C++-FAQ lite (http://www.cerfnet.com/~mpcline/c++-faq-lite/) by Marshall Cline.  Chapter 16 covers this issue and presents an implementation which is pretty close to the one in GiNaC.

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 unexpanded symbolic expression 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:

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:

The number 3 above the symbol d shows that mul objects are treated similarly where the coefficients are interpreted as 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 2 and the additive term -3 look somewhat out of place in this representation, however, since they are still carrying a trivial exponent and multiplicative factor 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 2*d^3*(4*a+5*b-3):

This also allows for a better handling of numeric radicals, since sqrt(2) can now be carried along calculations.  Now it should be clear, why both classes add and 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 add and mul, but the data structure is inherited from expairseq.

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.

ginac-config
============

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.

ginac-config takes the following flags:

--version
     Prints out the version of GiNaC installed.

--cppflags
     Prints '-I' flags pointing to the installed header files.

--libs
     Prints out the linker flags necessary to link a program against GiNaC.

--prefix[=PREFIX]
     If PREFIX is specified, overrides the configured value of $prefix. (And of exec-prefix, unless --exec-prefix is also specified) Otherwise, prints out the configured value of $prefix.

--exec-prefix[=PREFIX]
     If PREFIX is specified, overrides the configured value of $exec_prefix. Otherwise, prints out the configured value of $exec_prefix.

Typically, 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:

     c++ -o simple `ginac-config --cppflags` simple.cpp `ginac-config --libs`

This command line might expand to (for example):

     cc -o simple -I/usr/local/include simple.cpp -L/usr/local/lib \
       -lginac -lcln -lstdc++

Not only is the form using ginac-config easier to type, it will work on any system, no matter how GiNaC was configured.

AM_PATH_GINAC
=============

For packages configured using GNU automake, GiNaC also provides a macro to automate the process of checking for GiNaC.

     AM_PATH_GINAC([MINIMUM-VERSION, [ACTION-IF-FOUND [, ACTION-IF-NOT-FOUND]]])

This macro:

  * Determines the location of GiNaC using ginac-config, which is either found in the user's path, or from the environment variable GINACLIB_CONFIG.

  * Tests the installed libraries to make sure that their version is later than MINIMUM-VERSION. (A default version will be used if not specified)

  * If the required version was found, sets the GINACLIB_CPPFLAGS variable to the output of ginac-config --cppflags and the GINACLIB_LIBS variable to the output of ginac-config --libs, and calls AC_SUBST() for these variables so they can be used in generated makefiles, and then executes ACTION-IF-FOUND.

  * If the required version was not found, sets GINACLIB_CPPFLAGS and GINACLIB_LIBS to empty strings, and executes ACTION-IF-NOT-FOUND.

This macro is in file ginac.m4 which is installed in $datadir/aclocal. Note that if automake was installed with a different --prefix than GiNaC, you will either have to manually move ginac.m4 to automake's $datadir/aclocal, or give aclocal the -I option when running it.

Configuring a package that uses AM_PATH_GINAC
---------------------------------------------

Simply make sure that ginac-config is in your path, and run the configure script.

Notes:

  * The directory where the GiNaC libraries are installed needs to be found by your system's dynamic linker.

     This is generally done by

          editing /etc/ld.so.conf and running ldconfig

     or by

          setting the environment variable LD_LIBRARY_PATH,

     or, as a last resort,

          giving a -R or -rpath flag (depending on your linker) when
          running configure, for instance:

               LDFLAGS=-R/home/cbauer/lib ./configure

  * You can also specify a ginac-config not in your path by setting the GINACLIB_CONFIG environment variable to the name of the executable

  * If you move the GiNaC package from its installed location, you will need either need to modify ginac-config script manually to point to the new location or rebuild GiNaC.

Advanced note:

  * configure flags

          --with-ginac-prefix=PREFIX
          --with-ginac-exec-prefix=PREFIX

     are provided to override the prefix and exec-prefix that were stored in the ginac-config shell script by GiNaC's configure. You are generally better off configuring GiNaC with the right path to begin with.

Example of a package using AM_PATH_GINAC
----------------------------------------

The following shows how to build a simple package using automake and the AM_PATH_GINAC macro. The program used here is simple.cpp:

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

     int main(void)
     {
         symbol x("x");
         ex a = sin(x);
         cout << "Derivative of " << a << " is " << a.diff(x) << endl;
         return 0;
     }

You should first read the introductory portions of the automake Manual, if you are not already familiar with it.

Two files are needed, configure.in, which is used to build the configure script:

     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(0.4.0, [
       LIBS="$LIBS $GINACLIB_LIBS"
       CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
     ], AC_MSG_ERROR([need to have GiNaC installed]))

     AC_OUTPUT(Makefile)

The only command in this which is not standard for automake is the AM_PATH_GINAC macro.

That command does the following:

     If a GiNaC version greater than 0.4.0 is found, adds $GINACLIB_LIBS to
     $LIBS and $GINACLIB_CPPFLAGS to $CPPFLAGS. Otherwise, dies
     with the error message `need to have GiNaC installed'

And the Makefile.am, which will be used to build the Makefile.

     ## Process this file with automake to produce Makefile.in
     bin_PROGRAMS = simple
     simple_SOURCES = simple.cpp

This Makefile.am, says that we are building a single executable, from a single sourcefile simple.cpp. Since every program we are building uses GiNaC we simply added the GiNaC options to $LIBS and $CPPFLAGS, but in other circumstances, we might want to specify them on a per-program basis: for instance by adding the lines:

     simple_LDADD = $(GINACLIB_LIBS)
     INCLUDES = $(GINACLIB_CPPFLAGS)

to the Makefile.am.

To try this example out, create a new directory and add the three files above to it.

Now execute the following commands:

     $ automake --add-missing
     $ aclocal
     $ autoconf

You now have a package that can be built in the normal fashion

     $ ./configure
     $ make
     $ make install

Bibliography
************

  - ISO/IEC 14882:1998: Programming Languages: C++

  - CLN: A Class Library for Numbers, Bruno Haible

  - The C++ Programming Language, Bjarne Stroustrup, 3rd Edition, ISBN 0-201-88954-4, Addison Wesley

  - C++ FAQs, Marshall Cline, ISBN 0-201-58958-3, 1995, Addison Wesley

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

  - 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