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

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

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 1.8.2, an open framework for
symbolic computation within the C++ programming language.

1 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
linguistic structures that allow for the creation of large-scale
projects.  GiNaC is an attempt to overcome this situation by extending a
well established and standardized computer language (C++) by some
fundamental symbolic capabilities, thus allowing for integrated systems
that embed symbolic manipulations together with more established areas
of computer science (like computation-intense numeric applications,
graphical interfaces, etc.)  under one roof.

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

This tutorial is intended for the novice user who is new to GiNaC but
already has some background in C++ programming.  However, since a
hand-made documentation like this one is difficult to keep in sync with
the development, the actual documentation is inside the sources in the
form of comments.  That documentation may be parsed by one of the many
Javadoc-like documentation systems.  If you fail at generating it you
may access it from the GiNaC home page
(https://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.

1.1 License
===========

The GiNaC framework for symbolic computation within the C++ programming
language is Copyright (C) 1999-2021 Johannes Gutenberg University Mainz,
Germany.

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

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

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

2 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.

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

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

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

         cout << poly << endl;
         return 0;
     }

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

     $ c++ hello.cc -o hello -lginac -lcln
     $ ./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 <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;

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

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

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

         return 0;
     }

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.

2.2 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

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

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

The function 'evalf' that was used above converts any number in GiNaC's
expressions into floating point numbers.  This can be done to arbitrary
predefined 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);
     9.869604401089358619+x
     > x=2;
     2
     > evalf(a);
     11.869604401089358619

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
bracket notation 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
     > A = [ [1, 1], [2, -1] ];
     [[1,1],[2,-1]]
     > A+2*M;
     [[1,1],[2,-1]]+2*[[1,3],[-3,2]]
     > evalm(%);
     [[3,7],[-4,3]]
     > B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
     > evalm(B^(2^12345));
     [[1,0,0],[0,1,0],[0,0,1]]

Multivariate polynomials and rational functions may be expanded,
collected, factorized, 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;
     12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
     > b = x^2 + 4*x*y - y^2;
     4*x*y-y^2+x^2
     > expand(a*b);
     8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
     > factor(%);
     (4*x*y+x^2-y^2)^2*(x^2+3*y^2)
     > collect(a+b,x);
     4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
     > collect(a+b,y);
     12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
     > normal(a/b);
     3*y^2+x^2

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

You can differentiate functions and expand them as Taylor or Laurent
series in a very natural syntax (the second argument of 'series' is a
relation defining the evaluation point, the third specifies 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(tgamma(x),x==0,3);
     x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
     (-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
     > evalf(%);
     x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
     -(0.90747907608088628905)*x^2+Order(x^3)
     > series(tgamma(2*sin(x)-2),x==Pi/2,6);
     -(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
     -Euler-1/12+Order((x-1/2*Pi)^3)

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

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

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

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

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

3 Installation
**************

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

3.1 Prerequisites
=================

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

3.2 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'.(1)

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 persuaded 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 -pedantic"
     $ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
     $ ./configure --disable-shared --prefix=$(HOME)

   ---------- Footnotes ----------

   (1) The 'configure' script is itself generated from the file
'configure.ac'.  It is only distributed in packaged releases of GiNaC.
If you got the naked sources, e.g.  from git, you must generate
'configure' along with the various 'Makefile.in' by using the
'autoreconf' utility.  This will require a fair amount of support from
your local toolchain, though.

3.3 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 collection of
regression tests by typing

     $ make check

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

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

     $ make html
     $ make dvi
     $ make ps
     $ make pdf

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

3.4 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 (info) 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(1).

   ---------- Footnotes ----------

   (1) 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.

4 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.

4.1 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 often
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'.

4.1.1 Note: Expressions and STL containers
------------------------------------------

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

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

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

See Information about expressions, for more about comparing and
ordering expressions.

4.2 Automatic evaluation and canonicalization of expressions
============================================================

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

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

This behavior is usually referred to as "automatic" or "anonymous
evaluation".  GiNaC only performs transformations that are

   * at most of complexity O(n log n)
   * algebraically correct, possibly except for a set of measure zero
     (e.g.  x/x is transformed to 1 although this is incorrect for x=0)

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

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

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

Internally, the anonymous evaluator in GiNaC is implemented by the
methods

     ex ex::eval() const;
     ex basic::eval() const;

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

4.3 Error handling
==================

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

The 'pole_error' class has a member function

     int pole_error::degree() const;

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

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

Here is an example for a 'main()' function that catches and prints
exceptions generated by GiNaC:

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

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

4.4 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.

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

<PICTURE MISSING>

The abstract classes shown here (the ones without drop-shadow) are of no
interest for the user.  They are used internally in order to avoid code
duplication if two or more classes derived from them share certain
features.  An example is 'expairseq', 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 and products.  See Internal structures, where
these two classes are described in more detail.  The following table
shortly summarizes what kinds of mathematical objects are stored in the
different classes:

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

4.5 Symbols
===========

Symbolic indeterminates, or "symbols" for short, are for symbolic
manipulation what atoms are for chemistry.

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

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

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

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

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

Now, here is one important property of GiNaC that differentiates it from
other computer algebra programs you may have used: GiNaC does _not_ use
the names of symbols to tell them apart, but a (hidden) serial number
that is unique for each newly created 'symbol' object.  If you want to
use one and the same symbol in different places in your program, you
must only create one 'symbol' object and pass that around.  If you
create another symbol, even if it has the same name, GiNaC will treat it
as a different indeterminate.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

         std::cout << two*p << std::endl;  // floating point 6.283...
         ...

The imaginary unit in GiNaC is a predefined 'numeric' object with the
name 'I':

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

It may be tempting to construct fractions by 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 <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     using namespace GiNaC;

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

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

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

The above example prints the following output to screen:

     in 17 digits:
     0.33333333333333333334
     3.1415926535897932385
     in 60 digits:
     0.33333333333333333333333333333333333333333333333333333333333333333334
     3.1415926535897932384626433832795028841971693993751058209749445923078

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

It should be clear that objects of class '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'.

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

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

     int main()
     {
         numeric answer = twentyone;

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

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 behavior and we
refer the reader to CLN's documentation.  Suffice to say that the same
behavior applies to complex numbers as well as return values of certain
functions.  Complex numbers are automatically converted to real numbers
if the imaginary part becomes zero.  The full set of tests that can be
applied is listed in the following table.

*Method*               *Returns true if the object is...*
'.is_zero()'           ...equal to zero
'.is_positive()'       ...not complex and greater than 0
'.is_negative()'       ...not complex and smaller 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)

4.6.2 Numeric functions
-----------------------

The following functions can be applied to 'numeric' objects and will be
evaluated immediately:

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

Most of these functions are also available as symbolic functions that
can be used in expressions (see Mathematical functions) or, like
'gcd()', as polynomial algorithms.

4.6.3 Converting numbers
------------------------

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

     int numeric::to_int() const;
     long numeric::to_long() const;
     double numeric::to_double() const;
     cln::cl_N numeric::to_cl_N() const;

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

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

4.7 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
'Euler'    Euler's (or             0.57721566490153286060651209008240243
           Euler-Mascheroni)
           constant

4.8 Sums, products and powers
=============================

Simple rational expressions are written down in GiNaC pretty much like
in other CAS or like expressions involving numerical variables in C. The
necessary operators '+', '-', '*' and '/' have been overloaded to
achieve this goal.  When you run the following code snippet, 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:

         ...
         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 and undesired 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 embarrassing 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'.

4.9 Lists of expressions
========================

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

Lists can be constructed from an initializer list of expressions:

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

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

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

As with the standard 'list<T>' container, accessing random elements of a
'lst' is generally an operation of order O(N). Faster read-only
sequential access to the elements of a list is possible with the
iterator types provided by the 'lst' class:

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

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

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

which is one order faster than

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

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

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

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

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

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

You can append or prepend an expression to a list with the 'append()'
and 'prepend()' methods:

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

You can remove the first or last element of a list with 'remove_first()'
and 'remove_last()':

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

You can remove all the elements of a list with 'remove_all()':

         ...
         l.remove_all();     // l is now empty
         ...

You can bring the elements of a list into a canonical order with
'sort()':

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

Finally, you can remove all but the first element of consecutive groups
of elements with 'unique()':

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

4.10 Mathematical functions
===========================

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

These functions (better called _pseudofunctions_) 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, showing how a function returns itself twice and finally an
expression that may be really useful:

         ...
         symbol x("x"), y("y");
         ex foo = x+y/2;
         cout << tgamma(foo) << endl;
          // -> tgamma(x+(1/2)*y)
         ex bar = foo.subs(y==1);
         cout << tgamma(bar) << endl;
          // -> tgamma(x+1/2)
         ex foobar = bar.subs(x==7);
         cout << tgamma(foobar) << endl;
          // -> (135135/128)*Pi^(1/2)
         ...

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

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

4.11 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 Mathematical 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.
They are also used as arguments to the 'ex::series' method, where the
left hand side of the relation specifies the variable to expand in and
the right hand side the expansion point.  They can also be used for
creating systems of equations that are to be solved for unknown
variables.

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

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

4.12 Integrals
==============

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

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

4.13 Matrices
=============

A "matrix" is a two-dimensional array of expressions.  The elements of a
matrix with m rows and n columns are accessed with two 'unsigned'
indices, the first one in the range 0...m-1, the second one in the range
0...n-1.

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

     matrix::matrix(unsigned r, unsigned c);

creates a matrix with 'r' rows and 'c' columns with all elements set to
zero.

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

     {
         matrix m = {{1, -a},
                     {a,  1}};
     }

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

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

The function

     ex lst_to_matrix(const lst & l);

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

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

     ex diag_matrix(const lst & l);
     ex diag_matrix(initializer_list<ex> l);
     ex unit_matrix(unsigned x);
     ex unit_matrix(unsigned r, unsigned c);
     ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
     ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
                        const string & tex_base_name);

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

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

     ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
     ex reduced_matrix(const matrix& m, unsigned r, unsigned c);

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

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

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

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

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

Here are a couple of examples for constructing matrices:

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

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

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

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

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

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

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

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

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

There are three ways to do arithmetic with matrices.  The first (and
most direct one) is to use the methods provided by the 'matrix' class:

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

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

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

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

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

     ex ex::evalm() const;

to obtain the result:

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

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

Finally, you can work with indexed matrices and call
'simplify_indexed()' to perform the arithmetic:

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

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

The 'matrix' class provides a couple of additional methods for computing
determinants, traces, characteristic polynomials and ranks:

     ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
     ex matrix::trace() const;
     ex matrix::charpoly(const ex & lambda) const;
     unsigned matrix::rank(unsigned algo=solve_algo::automatic) const;

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

Linear systems can be solved with:

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

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

To invert a matrix, use the method:

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

The 'algo' argument is optional.  If given, it must be one of
'solve_algo' defined in 'flags.h'.

4.14 Indexed objects
====================

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

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

4.14.1 Indexed quantities and their indices
-------------------------------------------

Indexed expressions in GiNaC are constructed of two special types of
objects, "index objects" and "indexed objects".

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

   * Indexed objects are of class 'indexed' or a subclass.  They contain
     a "base expression" (which is the expression being indexed), and
     one or more indices.

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

A simple example shall illustrate the concepts:

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

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

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

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

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

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

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

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

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

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

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

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

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

The methods

     ex idx::get_value();
     ex idx::get_dim();

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

There are also the methods

     bool idx::is_numeric();
     bool idx::is_symbolic();
     bool idx::is_dim_numeric();
     bool idx::is_dim_symbolic();

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

If you need co- and contravariant indices, use the 'varidx' class:

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

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

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

     bool varidx::is_covariant();
     bool varidx::is_contravariant();

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

     ex varidx::toggle_variance();

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

The 'spinidx' class provides dotted and undotted variant indices, as
used in the Weyl-van-der-Waerden spinor formalism:

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

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

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

     bool spinidx::is_dotted();
     bool spinidx::is_undotted();

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

     ex spinidx::toggle_dot();
     ex spinidx::toggle_variance_dot();

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

4.14.2 Substituting indices
---------------------------

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

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

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

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

Alternatively, you can substitute the _symbol_ of a symbolic index by
another expression:

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

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

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

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

4.14.3 Symmetries
-----------------

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

     symmetry sy_none(...);
     symmetry sy_symm(...);
     symmetry sy_anti(...);
     symmetry sy_cycl(...);

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

Here are some examples of symmetry definitions:

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

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

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

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

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

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

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

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

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

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

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

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

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

4.14.4 Dummy indices
--------------------

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

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

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

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

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

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

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

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

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

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

A dummy index summation like a.i b~i can be expanded for indices with
numeric dimensions (e.g.  3) into the explicit sum like a.1 b~1 + a.2
b~2 + a.3 b~3.  This is performed by the function

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

which takes an expression 'e' and returns the expanded sum for all dummy
indices with numeric dimensions.  If the parameter 'subs_idx' is set to
'true' then all substitutions are made by 'idx' class indices, i.e.
without variance.  In this case the above sum a.i b~i will be expanded
to a.1 b.1 + a.2 b.2 + a.3 b.3.

4.14.5 Simplifying indexed expressions
--------------------------------------

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

     ex ex::simplify_indexed();
     ex ex::simplify_indexed(const scalar_products & sp);

that performs some more expensive operations:

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

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

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

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

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

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

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

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

4.14.6 Predefined tensors
-------------------------

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

4.14.6.1 Delta tensor
.....................

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

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

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

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

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

4.14.6.2 General metric tensor
..............................

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

     {
         symbol A("A");

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

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

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

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

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

4.14.6.3 Minkowski metric tensor
................................

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

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

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

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

4.14.6.4 Spinor metric tensor
.............................

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

     {
         symbol psi("psi");

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

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

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

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

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

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

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

The matrix representation of the spinor metric is '[[0, 1], [-1, 0]]'.

4.14.6.5 Epsilon tensor
.......................

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

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

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

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

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

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

4.14.7 Linear algebra
---------------------

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

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

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

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

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

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

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

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

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

4.15 Non-commutative objects
============================

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

   * Clifford (Dirac) algebra (class 'clifford')
   * su(3) Lie algebra (class 'color')
   * Matrices (unindexed) (class 'matrix')

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

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

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

As can be seen, GiNaC pulls out the overall commutative factor '-16' and
groups the non-commutative factors (the gammas and the su(3) generators)
together while preserving the order of factors within each class
(because Clifford objects commutate with color objects).  The resulting
expression is a _commutative_ product with two factors that are
themselves non-commutative products ('gamma~mu*gamma~nu' and 'T.a*T.b').
For clarification, parentheses are placed around the non-commutative
products in the output.

Non-commutative products are internally represented by objects of the
class 'ncmul', as opposed to commutative products which are handled by
the 'mul' class.  You will normally not have to worry about this
distinction, though.

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

Information about the commutativity of an object or expression can be
obtained with the two member functions

     unsigned      ex::return_type() const;
     return_type_t ex::return_type_tinfo() const;

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

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

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

Here are a couple of examples:

*Expression*                                *'return_type()'*
'42'                                        'commutative'
'2*x-y'                                     'commutative'
'dirac_ONE()'                               'noncommutative'
'dirac_gamma(mu)*dirac_gamma(nu)'           'noncommutative'
'2*color_T(a)'                              'noncommutative'
'dirac_ONE()*color_T(a)'                    'noncommutative_composite'

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

4.15.1 Clifford algebra
-----------------------

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

4.15.1.1 Dirac gamma matrices
.............................

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

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

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

The unity element of a Clifford algebra is constructed by

     ex dirac_ONE(unsigned char rl = 0);

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

There is a special element 'gamma5' that commutates with all other
gammas, has a unit square, and in 4 dimensions equals 'gamma~0 gamma~1
gamma~2 gamma~3', provided by

     ex dirac_gamma5(unsigned char rl = 0);

The chiral projectors '(1+/-gamma5)/2' are also available as proper
objects, constructed by

     ex dirac_gammaL(unsigned char rl = 0);
     ex dirac_gammaR(unsigned char rl = 0);

They observe the relations 'gammaL^2 = gammaL', 'gammaR^2 = gammaR', and
'gammaL gammaR = gammaR gammaL = 0'.

Finally, the function

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

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

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

The 'simplify_indexed()' function performs contractions in gamma
strings, for example

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

To calculate the trace of an expression containing strings of Dirac
gammas you use one of the functions

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

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

The 'dirac_trace()' function is a linear functional that is equal to the
ordinary matrix trace only in D = 4 dimensions.  In particular, the
functional is not cyclic in D != 4 dimensions when acting on expressions
containing 'gamma5', so it's not a proper trace.  This 'gamma5' scheme
is described in greater detail in the article 'The Role of gamma5 in
Dimensional Regularization' (see Bibliography).

The value of the trace itself is also usually different in 4 and in D !=
4 dimensions:

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

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

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

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

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

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

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

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

4.15.1.2 A generic Clifford algebra
...................................

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

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

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

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

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

The method 'clifford::get_metric()' returns a metric defining this
Clifford number.

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

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

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

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

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

will produce four anti-commuting generators of a Clifford algebra with
properties 'pow(e0, 2) = 1', 'pow(e1, 2) = -1', 'pow(e2, 2) = 0' and
'pow(e3, 2) = s'.

A similar effect can be achieved from the function

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

which converts a list or vector 'v = (v~0, v~1, ..., v~n)' into the
Clifford number 'v~0 e.0 + v~1 e.1 + ... + v~n e.n' with 'e.k' directly
supplied in the second form of the procedure.  In the first form the
Clifford unit 'e.k' is generated by the call of 'clifford_unit(mu, metr,
rl)'.  If the number of components supplied by 'v' exceeds the
dimensionality of the Clifford unit 'e' by 1 then function
'lst_to_clifford()' uses the following pseudo-vector representation:
'v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n'

The previous code may be rewritten with the help of 'lst_to_clifford()'
as follows

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

There is the inverse function

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

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

There are several functions for (anti-)automorphisms of Clifford
algebras:

         ex clifford_prime(const ex & e)
         inline ex clifford_star(const ex & e)
         inline ex clifford_bar(const ex & e)

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

The function

         ex clifford_norm(const ex & e);

calculates the norm of a Clifford number from the expression '||e||^2 =
e \bar{e}' The inverse of a Clifford expression is returned by the
function

         ex clifford_inverse(const ex & e);

which calculates it as e^{-1} = \bar{e}/||e||^2 If ||e||=0 then an
exception is raised.

If a Clifford number happens to be a factor of 'dirac_ONE()' then we can
convert it to a "real" (non-Clifford) expression by the function

         ex remove_dirac_ONE(const ex & e);

The function 'canonicalize_clifford()' works for a generic Clifford
algebra in a similar way as for Dirac gammas.

The next provided function is

         ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
                                 const ex & d, const ex & v, const ex & G,
                                 unsigned char rl = 0);
         ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
                                 unsigned char rl = 0);

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

Finally the function

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

can detect a presence of Clifford objects in the expression 'e': if such
objects are found it returns the maximal 'representation_label' of them,
otherwise '-1'.  The optional parameter 'ignore_ONE' indicates if
'dirac_ONE' objects should be ignored during the search.

LaTeX output for Clifford units looks like '\clifford[1]{e}^{{\nu}}',
where '1' is the 'representation_label' and '\nu' is the index of the
corresponding unit.  This provides a flexible typesetting with a
suitable definition of the '\clifford' command.  For example, the
definition
         \newcommand{\clifford}[1][]{}
typesets all Clifford units identically, while the alternative
definition
         \newcommand{\clifford}[2][]{\ifcase #1 #2\or \tilde{#2} \or \breve{#2} \fi}
prints units with 'representation_label=0' as 'e', with
'representation_label=1' as '\tilde{e}' and with
'representation_label=2' as '\breve{e}'.

4.15.2 Color algebra
--------------------

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

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

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

The unity element of a color algebra is constructed by

     ex color_ONE(unsigned char rl = 0);

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

The functions

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

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

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

There's an additional function

     ex color_h(const ex & a, const ex & b, const ex & c);

which returns the linear combination 'color_d(a, b, c)+I*color_f(a, b,
c)'.

The function 'simplify_indexed()' performs some simplifications on
expressions containing color objects:

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

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

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

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

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

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

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

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

To calculate the trace of an expression containing color objects you use
one of the functions

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

These functions take the trace over all color 'T' objects in the
specified set 'rls' or list 'rll' of representation labels, or the
single label 'rl'; 'T's with other labels are left standing.  For
example:

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

5 Methods and functions
***********************

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:

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

5.1 Getting information about expressions
=========================================

5.1.1 Checking expression types
-------------------------------

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

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

When the test made by 'is_a<T>()' returns true, it is safe to call one
of the functions 'ex_to<T>()', where 'T' is one of the class names
(see The class hierarchy, for a list of all classes).  For example,
assuming 'e' is an 'ex':

     {
         ...
         if (is_a<numeric>(e))
             numeric n = ex_to<numeric>(e);
         ...
     }

'is_a<T>(e)' allows you to check whether the top-level object of an
expression 'e' is an instance of the GiNaC class 'T' (see The class hierarchy,
for a list of all classes).  This is most useful, e.g., for
checking whether an expression is a number, a sum, or a product:

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

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

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

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

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

5.1.2 Accessing subexpressions
------------------------------

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

GiNaC provides several ways of accessing subexpressions.  The first way
is to use the two methods

     size_t ex::nops();
     ex ex::op(size_t i);

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

The second way to access subexpressions is via the STL-style
random-access iterator class 'const_iterator' and the methods

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

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

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

     {
         ex e = ...

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

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

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

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

     const_preorder_iterator ex::preorder_begin();
     const_preorder_iterator ex::preorder_end();
     const_postorder_iterator ex::postorder_begin();
     const_postorder_iterator ex::postorder_end();

The following example illustrates the differences between
'const_iterator', 'const_preorder_iterator', and
'const_postorder_iterator':

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

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

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

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

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

     ex ex::lhs();
     ex ex::rhs();

5.1.3 Comparing expressions
---------------------------

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

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

There are also two methods

     bool ex::is_equal(const ex & other);
     bool ex::is_zero();

for checking whether one expression is equal to another, or equal to
zero, respectively.  See also the method 'ex::is_zero_matrix()', see
Matrices.

5.1.4 Ordering expressions
--------------------------

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

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

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

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

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

For example, to define a 'map' that maps expressions to strings you have
to use

     std::map<ex, std::string, ex_is_less> myMap;

Omitting the 'ex_is_less' template parameter will introduce spurious
bugs because the map operates improperly.

Other examples for the use of the functors:

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

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

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

The implementation of 'ex_is_less' uses the member function

     int ex::compare(const ex & other) const;

which returns 0 if '*this' and 'other' are equal, -1 if '*this' sorts
before 'other', and 1 if '*this' sorts after 'other'.

5.2 Numerical evaluation
========================

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

     ex ex::evalf() const;

The accuracy of the evaluation is controlled by the global object
'Digits' which can be assigned an integer value.  The default value of
'Digits' is 17.  See Numbers, for more information and examples.

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

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

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

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

5.3 Substituting expressions
============================

Algebraic objects inside expressions can be replaced with arbitrary
expressions via the '.subs()' method:

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

In the first form, 'subs()' accepts a relational of the form 'object ==
expression' or a 'lst' of such relationals:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

5.4 Pattern matching and advanced substitutions
===============================================

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

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

     ex wild(unsigned label = 0);

which is simply a wrapper for the 'wildcard()' constructor with a
shorter name.

Some examples for patterns:

*Constructed as*                     *Output as*
'wild()'                             '$0'
'pow(x,wild())'                      'x^$0'
'atan2(wild(1),wild(2))'             'atan2($1,$2)'
'indexed(A,idx(wild(),3))'           'A.$0'

Notes:

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

5.4.1 Matching expressions
--------------------------

The most basic application of patterns is to check whether an expression
matches a given pattern.  This is done by the function

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

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

The matching algorithm works as follows:

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

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

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

In general, having more than one single wildcard as a term of a sum or a
factor of a product (such as 'a+$0+$1') will lead to unpredictable or
ambiguous results.

Here are some examples in 'ginsh' to demonstrate how it works (the
'match()' function in 'ginsh' returns 'FAIL' if the match fails, and the
list of wildcard replacements otherwise):

     > match((x+y)^a,(x+y)^a);
     {}
     > match((x+y)^a,(x+y)^b);
     FAIL
     > match((x+y)^a,$1^$2);
     {$1==x+y,$2==a}
     > match((x+y)^a,$1^$1);
     FAIL
     > match((x+y)^(x+y),$1^$1);
     {$1==x+y}
     > match((x+y)^(x+y),$1^$2);
     {$1==x+y,$2==x+y}
     > match((a+b)*(a+c),($1+b)*($1+c));
     {$1==a}
     > match((a+b)*(a+c),(a+$1)*(a+$2));
     {$1==b,$2==c}
       (Unpredictable. The result might also be [$1==c,$2==b].)
     > match((a+b)*(a+c),($1+$2)*($1+$3));
       (The result is undefined. Due to the sequential nature of the algorithm
        and the re-ordering of terms in GiNaC, the match for the first factor
        may be {$1==a,$2==b} in which case the match for the second factor
        succeeds, or it may be {$1==b,$2==a} which causes the second match to
        fail.)
     > match(a*(x+y)+a*z+b,a*$1+$2);
       (This is also ambiguous and may return either {$1==z,$2==a*(x+y)+b} or
        {$1=x+y,$2=a*z+b}.)
     > match(a+b+c+d+e+f,c);
     FAIL
     > match(a+b+c+d+e+f,c+$0);
     {$0==a+e+b+f+d}
     > match(a+b+c+d+e+f,c+e+$0);
     {$0==a+b+f+d}
     > match(a+b,a+b+$0);
     {$0==0}
     > match(a*b^2,a^$1*b^$2);
     FAIL
       (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
        even though a==a^1.)
     > match(x*atan2(x,x^2),$0*atan2($0,$0^2));
     {$0==x}
     > match(atan2(y,x^2),atan2(y,$0));
     {$0==x^2}

5.4.2 Matching parts of expressions
-----------------------------------

A more general way to look for patterns in expressions is provided by
the member function

     bool ex::has(const ex & pattern);

This function checks whether a pattern is matched by an expression
itself or by any of its subexpressions.

Again some examples in 'ginsh' for illustration (in 'ginsh', 'has()'
returns '1' for 'true' and '0' for 'false'):

     > has(x*sin(x+y+2*a),y);
     1
     > has(x*sin(x+y+2*a),x+y);
     0
       (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
        has the subexpressions "x", "y" and "2*a".)
     > has(x*sin(x+y+2*a),x+y+$1);
     1
       (But this is possible.)
     > has(x*sin(2*(x+y)+2*a),x+y);
     0
       (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
        which "x+y" is not a subexpression.)
     > has(x+1,x^$1);
     0
       (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
        "x^something".)
     > has(4*x^2-x+3,$1*x);
     1
     > has(4*x^2+x+3,$1*x);
     0
       (Another possible pitfall. The first expression matches because the term
        "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
        contains a linear term you should use the coeff() function instead.)

The method

     bool ex::find(const ex & pattern, exset& found);

works a bit like 'has()' but it doesn't stop upon finding the first
match.  Instead, it appends all found matches to the specified list.  If
there are multiple occurrences of the same expression, it is entered
only once to the list.  'find()' returns false if no matches were found
(in 'ginsh', it returns an empty list):

     > find(1+x+x^2+x^3,x);
     {x}
     > find(1+x+x^2+x^3,y);
     {}
     > find(1+x+x^2+x^3,x^$1);
     {x^3,x^2}
       (Note the absence of "x".)
     > expand((sin(x)+sin(y))*(a+b));
     sin(y)*a+sin(x)*b+sin(x)*a+sin(y)*b
     > find(%,sin($1));
     {sin(y),sin(x)}

5.4.3 Substituting expressions
------------------------------

Probably the most useful application of patterns is to use them for
substituting expressions with the 'subs()' method.  Wildcards can be
used in the search patterns as well as in the replacement expressions,
where they get replaced by the expressions matched by them.  'subs()'
doesn't know anything about algebra; it performs purely syntactic
substitutions.

Some examples:

     > subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
     b^3+a^3+(x+y)^3
     > subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
     b^4+a^4+(x+y)^4
     > subs((a+b+c)^2,a+b==x);
     (a+b+c)^2
     > subs((a+b+c)^2,a+b+$1==x+$1);
     (x+c)^2
     > subs(a+2*b,a+b==x);
     a+2*b
     > subs(4*x^3-2*x^2+5*x-1,x==a);
     -1+5*a-2*a^2+4*a^3
     > subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
     -1+5*x-2*a^2+4*a^3
     > subs(sin(1+sin(x)),sin($1)==cos($1));
     cos(1+cos(x))
     > expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
     a+b

The last example would be written in C++ in this way:

     {
         symbol a("a"), b("b"), x("x"), y("y");
         e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
         e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
         cout << e.expand() << endl;
          // -> a+b
     }

5.4.4 The option algebraic
--------------------------

Both 'has()' and 'subs()' take an optional argument to pass them extra
options.  This section describes what happens if you give the former the
option 'has_options::algebraic' or the latter 'subs_options::algebraic'.
In that case the matching condition for powers and multiplications is
changed in such a way that they become more intuitive.  Intuition says
that 'x*y' is a part of 'x*y*z'.  If you use these options you will find
that '(x*y*z).has(x*y, has_options::algebraic)' indeed returns true.
Besides matching some of the factors of a product also powers match as
often as is possible without getting negative exponents.  For example
'(x^5*y^2*z).subs(x^2*y^2==c, subs_options::algebraic)' will return
'x*c^2*z'.  This also works with negative powers:
'(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)' will
return 'x^(-1)*c^2*z'.

*Please notice:* this only works for multiplications and not for
locating 'x+y' within 'x+y+z'.

5.5 Applying a function on subexpressions
=========================================

Sometimes you may want to perform an operation on specific parts of an
expression while leaving the general structure of it intact.  An example
of this would be a matrix trace operation: the trace of a sum is the sum
of the traces of the individual terms.  That is, the trace should "map"
on the sum, by applying itself to each of the sum's operands.  It is
possible to do this manually which usually results in code like this:

     ex calc_trace(ex e)
     {
         if (is_a<matrix>(e))
             return ex_to<matrix>(e).trace();
         else if (is_a<add>(e)) {
             ex sum = 0;
             for (size_t i=0; i<e.nops(); i++)
                 sum += calc_trace(e.op(i));
             return sum;
         } else if (is_a<mul>)(e)) {
             ...
         } else {
             ...
         }
     }

This is, however, slightly inefficient (if the sum is very large it can
take a long time to add the terms one-by-one), and its applicability is
limited to a rather small class of expressions.  If 'calc_trace()' is
called with a relation or a list as its argument, you will probably want
the trace to be taken on both sides of the relation or of all elements
of the list.

GiNaC offers the 'map()' method to aid in the implementation of such
operations:

     ex ex::map(map_function & f) const;
     ex ex::map(ex (*f)(const ex & e)) const;

In the first (preferred) form, 'map()' takes a function object that is
subclassed from the 'map_function' class.  In the second form, it takes
a pointer to a function that accepts and returns an expression.  'map()'
constructs a new expression of the same type, applying the specified
function on all subexpressions (in the sense of 'op()'),
non-recursively.

The use of a function object makes it possible to supply more arguments
to the function that is being mapped, or to keep local state
information.  The 'map_function' class declares a virtual function call
operator that you can overload.  Here is a sample implementation of
'calc_trace()' that uses 'map()' in a recursive fashion:

     struct calc_trace : public map_function {
         ex operator()(const ex &e)
         {
             if (is_a<matrix>(e))
                 return ex_to<matrix>(e).trace();
             else if (is_a<mul>(e)) {
                 ...
             } else
                 return e.map(*this);
         }
     };

This function object could then be used like this:

     {
         ex M = ... // expression with matrices
         calc_trace do_trace;
         ex tr = do_trace(M);
     }

Here is another example for you to meditate over.  It removes quadratic
terms in a variable from an expanded polynomial:

     struct map_rem_quad : public map_function {
         ex var;
         map_rem_quad(const ex & var_) : var(var_) {}

         ex operator()(const ex & e)
         {
             if (is_a<add>(e) || is_a<mul>(e))
          	    return e.map(*this);
             else if (is_a<power>(e) &&
                      e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
                 return 0;
             else
                 return e;
         }
     };

     ...

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

         ex e;
         for (int i=0; i<8; i++)
             e += pow(x, i) * pow(y, 8-i) * (i+1);
         cout << e << endl;
          // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8

         map_rem_quad rem_quad(x);
         cout << rem_quad(e) << endl;
          // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
     }

'ginsh' offers a slightly different implementation of 'map()' that
allows applying algebraic functions to operands.  The second argument to
'map()' is an expression containing the wildcard '$0' which acts as the
placeholder for the operands:

     > map(a*b,sin($0));
     sin(a)*sin(b)
     > map(a+2*b,sin($0));
     sin(a)+sin(2*b)
     > map({a,b,c},$0^2+$0);
     {a^2+a,b^2+b,c^2+c}

Note that it is only possible to use algebraic functions in the second
argument.  You can not use functions like 'diff()', 'op()', 'subs()'
etc.  because these are evaluated immediately:

     > map({a,b,c},diff($0,a));
     {0,0,0}
       This is because "diff($0,a)" evaluates to "0", so the command is equivalent
       to "map({a,b,c},0)".

5.6 Visitors and tree traversal
===============================

Suppose that you need a function that returns a list of all indices
appearing in an arbitrary expression.  The indices can have any
dimension, and for indices with variance you always want the covariant
version returned.

You can't use 'get_free_indices()' because you also want to include
dummy indices in the list, and you can't use 'find()' as it needs
specific index dimensions (and it would require two passes: one for
indices with variance, one for plain ones).

The obvious solution to this problem is a tree traversal with a type
switch, such as the following:

     void gather_indices_helper(const ex & e, lst & l)
     {
         if (is_a<varidx>(e)) {
             const varidx & vi = ex_to<varidx>(e);
             l.append(vi.is_covariant() ? vi : vi.toggle_variance());
         } else if (is_a<idx>(e)) {
             l.append(e);
         } else {
             size_t n = e.nops();
             for (size_t i = 0; i < n; ++i)
                 gather_indices_helper(e.op(i), l);
         }
     }

     lst gather_indices(const ex & e)
     {
         lst l;
         gather_indices_helper(e, l);
         l.sort();
         l.unique();
         return l;
     }

This works fine but fans of object-oriented programming will feel
uncomfortable with the type switch.  One reason is that there is a
possibility for subtle bugs regarding derived classes.  If we had, for
example, written

         if (is_a<idx>(e)) {
           ...
         } else if (is_a<varidx>(e)) {
           ...

in 'gather_indices_helper', the code wouldn't have worked because the
first line "absorbs" all classes derived from 'idx', including 'varidx',
so the special case for 'varidx' would never have been executed.

Also, for a large number of classes, a type switch like the above can
get unwieldy and inefficient (it's a linear search, after all).
'gather_indices_helper' only checks for two classes, but if you had to
write a function that required a different implementation for nearly
every GiNaC class, the result would be very hard to maintain and extend.

The cleanest approach to the problem would be to add a new virtual
function to GiNaC's class hierarchy.  In our example, there would be
specializations for 'idx' and 'varidx' while the default implementation
in 'basic' performed the tree traversal.  Unfortunately, in C++ it's
impossible to add virtual member functions to existing classes without
changing their source and recompiling everything.  GiNaC comes with
source, so you could actually do this, but for a small algorithm like
the one presented this would be impractical.

One solution to this dilemma is the "Visitor" design pattern, which is
implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
variation, described in detail in
<https://condor.depaul.edu/dmumaugh/OOT/Design-Principles/acv.pdf>).
Instead of adding virtual functions to the class hierarchy to implement
operations, GiNaC provides a single "bouncing" method 'accept()' that
takes an instance of a special 'visitor' class and redirects execution
to the one 'visit()' virtual function of the visitor that matches the
type of object that 'accept()' was being invoked on.

Visitors in GiNaC must derive from the global 'visitor' class as well as
from the class 'T::visitor' of each class 'T' they want to visit, and
implement the member functions 'void visit(const T &)' for each class.

A call of

     void ex::accept(visitor & v) const;

will then dispatch to the correct 'visit()' member function of the
specified visitor 'v' for the type of GiNaC object at the root of the
expression tree (e.g.  a 'symbol', an 'idx' or a 'mul').

Here is an example of a visitor:

     class my_visitor
      : public visitor,          // this is required
        public add::visitor,     // visit add objects
        public numeric::visitor, // visit numeric objects
        public basic::visitor    // visit basic objects
     {
         void visit(const add & x)
         { cout << "called with an add object" << endl; }

         void visit(const numeric & x)
         { cout << "called with a numeric object" << endl; }

         void visit(const basic & x)
         { cout << "called with a basic object" << endl; }
     };

which can be used as follows:

     ...
         symbol x("x");
         ex e1 = 42;
         ex e2 = 4*x-3;
         ex e3 = 8*x;

         my_visitor v;
         e1.accept(v);
          // prints "called with a numeric object"
         e2.accept(v);
          // prints "called with an add object"
         e3.accept(v);
          // prints "called with a basic object"
     ...

The 'visit(const basic &)' method gets called for all objects that are
not 'numeric' or 'add' and acts as an (optional) default.

From a conceptual point of view, the 'visit()' methods of the visitor
behave like a newly added virtual function of the visited hierarchy.  In
addition, visitors can store state in member variables, and they can be
extended by deriving a new visitor from an existing one, thus building
hierarchies of visitors.

We can now rewrite our index example from above with a visitor:

     class gather_indices_visitor
      : public visitor, public idx::visitor, public varidx::visitor
     {
         lst l;

         void visit(const idx & i)
         {
             l.append(i);
         }

         void visit(const varidx & vi)
         {
             l.append(vi.is_covariant() ? vi : vi.toggle_variance());
         }

     public:
         const lst & get_result() // utility function
         {
             l.sort();
             l.unique();
             return l;
         }
     };

What's missing is the tree traversal.  We could implement it in
'visit(const basic &)', but GiNaC has predefined methods for this:

     void ex::traverse_preorder(visitor & v) const;
     void ex::traverse_postorder(visitor & v) const;
     void ex::traverse(visitor & v) const;

'traverse_preorder()' visits a node _before_ visiting its
subexpressions, while 'traverse_postorder()' visits a node _after_
visiting its subexpressions.  'traverse()' is a synonym for
'traverse_preorder()'.

Here is a new implementation of 'gather_indices()' that uses the visitor
and 'traverse()':

     lst gather_indices(const ex & e)
     {
         gather_indices_visitor v;
         e.traverse(v);
         return v.get_result();
     }

Alternatively, you could use pre- or postorder iterators for the tree
traversal:

     lst gather_indices(const ex & e)
     {
         gather_indices_visitor v;
         for (const_preorder_iterator i = e.preorder_begin();
              i != e.preorder_end(); ++i) {
             i->accept(v);
         }
         return v.get_result();
     }

5.7 Polynomial arithmetic
=========================

5.7.1 Testing whether an expression is a polynomial
---------------------------------------------------

Testing whether an expression is a polynomial in one or more variables
can be done with the method
     bool ex::is_polynomial(const ex & vars) const;
In the case of more than one variable, the variables are given as a
list.

     (x*y*sin(y)).is_polynomial(x)         // Returns true.
     (x*y*sin(y)).is_polynomial(lst{x,y})  // Returns false.

5.7.2 Expanding and collecting
------------------------------

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 example,
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

     ex ex::expand(unsigned options = 0);

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 easy to guess you should be prepared to see different orderings of
terms in such sums!

Another useful representation of multivariate polynomials is as a
univariate polynomial in one of the variables with the coefficients
being polynomials in the remaining variables.  The method 'collect()'
accomplishes this task:

     ex ex::collect(const ex & s, bool distributed = false);

The first argument to 'collect()' can also be a list of objects in which
case the result is either a recursively collected polynomial, or a
polynomial in a distributed form with terms like c*x1^e1*...*xn^en, as
specified by the 'distributed' flag.

Note that the original polynomial needs to be in expanded form (for the
variables concerned) in order for 'collect()' to be able to find the
coefficients properly.

The following 'ginsh' transcript shows an application of 'collect()'
together with 'find()':

     > a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
     d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)
     +q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
     > collect(a,{p,q});
     d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
     +(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
     > collect(a,find(a,sin($1)));
     (1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
     > collect(a,{find(a,sin($1)),p,q});
     (1+(1+d)*p+d+q*(1+d))*sin(x)+(1+(1+d)*p+d+q*(1+d))*sin(y)
     > collect(a,{find(a,sin($1)),d});
     (1+q+d*(1+q+p)+p)*sin(y)+(1+q+d*(1+q+p)+p)*sin(x)

Polynomials can often be brought into a more compact form by collecting
common factors from the terms of sums.  This is accomplished by the
function

     ex collect_common_factors(const ex & e);

This function doesn't perform a full factorization but only looks for
factors which are already explicitly present:

     > collect_common_factors(a*x+a*y);
     (x+y)*a
     > collect_common_factors(a*x^2+2*a*x*y+a*y^2);
     a*(2*x*y+y^2+x^2)
     > collect_common_factors(a*(b*(a+c)*x+b*((a+c)*x+(a+c)*y)*y));
     (c+a)*a*(x*y+y^2+x)*b

5.7.3 Degree and coefficients
-----------------------------

The degree and low degree of a polynomial in expanded form can be
obtained using the two methods

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

These functions even work on rational functions, returning the
asymptotic degree.  By definition, the degree of zero is zero.  To
extract a coefficient with a certain power from an expanded polynomial
you use

     ex ex::coeff(const ex & s, int n);

You can also obtain the leading and trailing coefficients with the
methods

     ex ex::lcoeff(const ex & s);
     ex ex::tcoeff(const ex & s);

which are equivalent to 'coeff(s, degree(s))' and 'coeff(s,
ldegree(s))', respectively.

An application is illustrated in the next example, where a multivariate
polynomial is analyzed:

     {
         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.

'degree()', 'ldegree()', 'coeff()', 'lcoeff()', 'tcoeff()' and
'collect()' can also be used to a certain degree with non-polynomial
expressions as they not only work with symbols but with constants,
functions and indexed objects as well:

     {
         symbol a("a"), b("b"), c("c"), x("x");
         idx i(symbol("i"), 3);

         ex e = pow(sin(x) - cos(x), 4);
         cout << e.degree(cos(x)) << endl;
          // -> 4
         cout << e.expand().coeff(sin(x), 3) << endl;
          // -> -4*cos(x)

         e = indexed(a+b, i) * indexed(b+c, i);
         e = e.expand(expand_options::expand_indexed);
         cout << e.collect(indexed(b, i)) << endl;
          // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
     }

5.7.4 Polynomial division
-------------------------

The two functions

     ex quo(const ex & a, const ex & b, const ex & x);
     ex rem(const ex & a, const ex & b, const ex & x);

compute the quotient and remainder of univariate polynomials in the
variable 'x'.  The results satisfy a = b*quo(a, b, x) + rem(a, b, x).

The additional function

     ex prem(const ex & a, const ex & b, const ex & x);

computes the pseudo-remainder of 'a' and 'b' which satisfies c*a = b*q +
prem(a, b, x), where c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1).

Exact division of multivariate polynomials is performed by the function

     bool divide(const ex & a, const ex & b, ex & q);

If 'b' divides 'a' over the rationals, this function returns 'true' and
returns the quotient in the variable 'q'.  Otherwise it returns 'false'
in which case the value of 'q' is undefined.

5.7.5 Unit, content and primitive part
--------------------------------------

The methods

     ex ex::unit(const ex & x);
     ex ex::content(const ex & x);
     ex ex::primpart(const ex & x);
     ex ex::primpart(const ex & x, const ex & c);

return the unit part, content part, and primitive polynomial of a
multivariate polynomial with respect to the variable 'x' (the unit part
being the sign of the leading coefficient, the content part being the
GCD of the coefficients, and the primitive polynomial being the input
polynomial divided by the unit and content parts).  The second variant
of 'primpart()' expects the previously calculated content part of the
polynomial in 'c', which enables it to work faster in the case where the
content part has already been computed.  The product of unit, content,
and primitive part is the original polynomial.

Additionally, the method

     void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);

computes the unit, content, and primitive parts in one go, returning
them in 'u', 'c', and 'p', respectively.

5.7.6 GCD, LCM and resultant
----------------------------

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'.  Note that all the coefficients must be rationals.

     #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 resultant of two expressions only makes sense with polynomials.  It
is always computed with respect to a specific symbol within the
expressions.  The function has the interface

     ex resultant(const ex & a, const ex & b, const ex & s);

Resultants are symmetric in 'a' and 'b'.  The following example computes
the resultant of two expressions with respect to 'x' and 'y',
respectively:

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

     int main()
     {
         symbol x("x"), y("y");

         ex e1 = x+pow(y,2), e2 = 2*pow(x,3)-1; // x+y^2, 2*x^3-1
         ex r;

         r = resultant(e1, e2, x);
         // -> 1+2*y^6
         r = resultant(e1, e2, y);
         // -> 1-4*x^3+4*x^6
     }

5.7.7 Square-free decomposition
-------------------------------

Square-free decomposition is available in GiNaC:
     ex sqrfree(const ex & a, const lst & l = lst{});
Here is an example that by the way illustrates how the exact form of the
result may slightly depend on the order of differentiation, calling for
some care with subsequent processing of the result:
         ...
         symbol x("x"), y("y");
         ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));

         cout << sqrfree(BiVarPol, lst{x,y}) << endl;
          // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)

         cout << sqrfree(BiVarPol, lst{y,x}) << endl;
          // -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)

         cout << sqrfree(BiVarPol) << endl;
          // -> depending on luck, any of the above
         ...
Note also, how factors with the same exponents are not fully factorized
with this method.

5.7.8 Polynomial factorization
------------------------------

Polynomials can also be fully factored with a call to the function
     ex factor(const ex & a, unsigned int options = 0);
The factorization works for univariate and multivariate polynomials with
rational coefficients.  The following code snippet shows its
capabilities:
         ...
         cout << factor(pow(x,2)-1) << endl;
          // -> (1+x)*(-1+x)
         cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
          // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
         cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
          // -> -1+sin(-1+x^2)+x^2
         ...
The results are as expected except for the last one where no
factorization seems to have been done.  This is due to the default
option 'factor_options::polynomial' (equals zero) to 'factor()', which
tells GiNaC to try a factorization only if the expression is a valid
polynomial.  In the shown example this is not the case, because one term
is a function.

There exists a second option 'factor_options::all', which tells GiNaC to
ignore non-polynomial parts of an expression and also to look inside
function arguments.  With this option the example gives:
         ...
         cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
              << endl;
          // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
         ...
GiNaC's factorization functions cannot handle algebraic extensions.
Therefore the following example does not factor:
         ...
         cout << factor(pow(x,2)-2) << endl;
          // -> -2+x^2  and not  (x-sqrt(2))*(x+sqrt(2))
         ...
Factorization is useful in many applications.  A lot of algorithms in
computer algebra depend on the ability to factor a polynomial.  Of
course, factorization can also be used to simplify expressions, but it
is costly and applying it to complicated expressions (high degrees or
many terms) may consume far too much time.  So usually, looking for a
GCD at strategic points in a calculation is the cheaper and more
appropriate alternative.

5.8 Rational expressions
========================

5.8.1 The 'normal' method
-------------------------

Some basic form of simplification of expressions is called for
frequently.  GiNaC provides the method '.normal()', which converts a
rational function into an equivalent rational function of the form
'numerator/denominator' where numerator and denominator are coprime.  If
the input expression is already a fraction, it just finds the GCD of
numerator and denominator and cancels it, otherwise it performs fraction
addition and multiplication.

'.normal()' can also be used on expressions which are not rational
functions as it will replace all non-rational objects (like functions or
non-integer powers) by temporary symbols to bring the expression to the
domain of rational functions before performing the normalization, and
re-substituting these symbols afterwards.  This algorithm is also
available as a separate method '.to_rational()', described below.

This means that both expressions 't1' and 't2' are indeed simplified in
this little code snippet:

     {
         symbol x("x");
         ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
         ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
         std::cout << "t1 is " << t1.normal() << std::endl;
         std::cout << "t2 is " << t2.normal() << std::endl;
     }

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)'.

5.8.2 Numerator and denominator
-------------------------------

The numerator and denominator of an expression can be obtained with

     ex ex::numer();
     ex ex::denom();
     ex ex::numer_denom();

These functions will first normalize the expression as described above
and then return the numerator, denominator, or both as a list,
respectively.  If you need both numerator and denominator, call
'numer_denom()': it is faster than using 'numer()' and 'denom()'
separately.  And even more important: a separate evaluation of 'numer()'
and 'denom()' may result in a spurious sign, e.g.  for $x/(x^2-1)$
'numer()' may return $x$ and 'denom()' $1-x^2$.

5.8.3 Converting to a polynomial or rational expression
-------------------------------------------------------

Some of the methods described so far only work on polynomials or
rational functions.  GiNaC provides a way to extend the domain of these
functions to general expressions by using the temporary replacement
algorithm described above.  You do this by calling

     ex ex::to_polynomial(exmap & m);
or
     ex ex::to_rational(exmap & m);

on the expression to be converted.  The supplied 'exmap' will be filled
with the generated temporary symbols and their replacement expressions
in a format that can be used directly for the 'subs()' method.  It can
also already contain a list of replacements from an earlier application
of '.to_polynomial()' or '.to_rational()', so it's possible to use it on
multiple expressions and get consistent results.

The difference between '.to_polynomial()' and '.to_rational()' is
probably best illustrated with an example:

     {
         symbol x("x"), y("y");
         ex a = 2*x/sin(x) - y/(3*sin(x));
         cout << a << endl;

         exmap mp;
         ex p = a.to_polynomial(mp);
         cout << " = " << p << "\n   with " << mp << endl;
          // = symbol3*symbol2*y+2*symbol2*x
          //   with {symbol2==sin(x)^(-1),symbol3==-1/3}

         exmap mr;
         ex r = a.to_rational(mr);
         cout << " = " << r << "\n   with " << mr << endl;
          // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
          //   with {symbol4==sin(x)}
     }

The following more useful example will print 'sin(x)-cos(x)':

     {
         symbol x("x");
         ex a = pow(sin(x), 2) - pow(cos(x), 2);
         ex b = sin(x) + cos(x);
         ex q;
         exmap m;
         divide(a.to_polynomial(m), b.to_polynomial(m), q);
         cout << q.subs(m) << endl;
     }

5.9 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:

     {
         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
          // -> 1
         cout << P.diff(z) << endl;    // 0
          // -> 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;
         const ex generator = pow(cosh(x),-1);
         return generator.diff(x,n).subs(x==0);
     }

     int main()
     {
         for (unsigned i=0; i<11; i+=2)
             std::cout << EulerNumber(i) << std::endl;
         return 0;
     }

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.

5.10 Series expansion
=====================

Expressions know how to expand themselves as a Taylor series or (more
generally) a Laurent series.  As in most conventional Computer Algebra
Systems, no distinction is made between those two.  There is a class of
its own for storing such series ('class pseries') and a built-in
function (called 'Order') for storing the order term of the series.  As
a consequence, if you want to work with series, i.e.  multiply two
series, you need to call the method 'ex::series' again to convert it to
a series object with the usual structure (expansion plus order term).  A
sample application from special relativity could read:

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

     int main()
     {
         symbol v("v"), c("c");

         ex gamma = 1/sqrt(1 - pow(v/c,2));
         ex mass_nonrel = gamma.series(v==0, 10);

         cout << "the relativistic mass increase with v is " << endl
              << mass_nonrel << endl;

         cout << "the inverse square of this series is " << endl
              << pow(mass_nonrel,-2).series(v==0, 10) << endl;
     }

Only calling the series method makes the last output simplify to
1-v^2/c^2+O(v^10), without that call we would just have a long series
raised to the power -2.

As another instructive application, let us calculate the numerical value
of Archimedes' constant Pi (for which there already exists the built-in
constant 'Pi') using John Machin's amazing formula
Pi==16*atan(1/5)-4*atan(1/239).  This equation (and similar ones) were
used for over 200 years for computing digits of pi (see 'Pi Unleashed').
We may expand the arcus tangent around '0' and insert the fractions
'1/5' and '1/239'.  However, 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 machin_pi(int degr)
     {
         symbol x;
         ex pi_expansion = series_to_poly(atan(x).series(x,degr));
         ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
                        -4*pi_expansion.subs(x==numeric(1,239));
         return pi_approx;
     }

     int main()
     {
         using std::cout;  // just for fun, another way of...
         using std::endl;  // ...dealing with this namespace std.
         ex pi_frac;
         for (int i=2; i<12; i+=2) {
             pi_frac = machin_pi(i);
             cout << i << ":\t" << pi_frac << endl
                  << "\t" << pi_frac.evalf() << endl;
         }
         return 0;
     }

Note how we just called '.series(x,degr)' instead of
'.series(x==0,degr)'.  This is a simple shortcut for 'ex''s method
'series()': if the first argument is a symbol the expression is expanded
in that symbol around point '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

5.11 Symmetrization
===================

The three methods

     ex ex::symmetrize(const lst & l);
     ex ex::antisymmetrize(const lst & l);
     ex ex::symmetrize_cyclic(const lst & l);

symmetrize an expression by returning the sum over all symmetric,
antisymmetric or cyclic permutations of the specified list of objects,
weighted by the number of permutations.

The three additional methods

     ex ex::symmetrize();
     ex ex::antisymmetrize();
     ex ex::symmetrize_cyclic();

symmetrize or antisymmetrize an expression over its free indices.

Symmetrization is most useful with indexed expressions but can be used
with almost any kind of object (anything that is 'subs()'able):

     {
         idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
         symbol A("A"), B("B"), a("a"), b("b"), c("c");

         cout << ex(indexed(A, i, j)).symmetrize() << endl;
          // -> 1/2*A.j.i+1/2*A.i.j
         cout << ex(indexed(A, i, j, k)).antisymmetrize(lst{i, j}) << endl;
          // -> -1/2*A.j.i.k+1/2*A.i.j.k
         cout << ex(lst{a, b, c}).symmetrize_cyclic(lst{a, b, c}) << endl;
          // -> 1/3*{a,b,c}+1/3*{b,c,a}+1/3*{c,a,b}
     }

5.12 Predefined mathematical functions
======================================

5.12.1 Overview
---------------

GiNaC contains the following predefined mathematical functions:

*Name*                 *Function*
'abs(x)'               absolute value
'step(x)'              step function
'csgn(x)'              complex sign
'conjugate(x)'         complex conjugation
'real_part(x)'         real part
'imag_part(x)'         imaginary part
'sqrt(x)'              square root (not a GiNaC function, rather an
                       alias for 'pow(x, numeric(1, 2))')
'sin(x)'               sine
'cos(x)'               cosine
'tan(x)'               tangent
'asin(x)'              inverse sine
'acos(x)'              inverse cosine
'atan(x)'              inverse tangent
'atan2(y, x)'          inverse tangent with two arguments
'sinh(x)'              hyperbolic sine
'cosh(x)'              hyperbolic cosine
'tanh(x)'              hyperbolic tangent
'asinh(x)'             inverse hyperbolic sine
'acosh(x)'             inverse hyperbolic cosine
'atanh(x)'             inverse hyperbolic tangent
'exp(x)'               exponential function
'log(x)'               natural logarithm
'eta(x,y)'             Eta function: 'eta(x,y) = log(x*y) - log(x) -
                       log(y)'
'Li2(x)'               dilogarithm
'Li(m, x)'             classical polylogarithm as well as multiple
                       polylogarithm
'G(a, y)'              multiple polylogarithm
'G(a, s, y)'           multiple polylogarithm with explicit signs for
                       the imaginary parts
'S(n, p, x)'           Nielsen's generalized polylogarithm
'H(m, x)'              harmonic polylogarithm
'zeta(m)'              Riemann's zeta function as well as multiple zeta
                       value
'zeta(m, s)'           alternating Euler sum
'zetaderiv(n, x)'      derivatives of Riemann's zeta function
'iterated_integral(a,  iterated integral
y)'
'iterated_integral(a,  iterated integral with explicit truncation
y, N)'                 parameter
'tgamma(x)'            gamma function
'lgamma(x)'            logarithm of gamma function
'beta(x, y)'           beta function
                       ('tgamma(x)*tgamma(y)/tgamma(x+y)')
'psi(x)'               psi (digamma) function
'psi(n, x)'            derivatives of psi function (polygamma
                       functions)
'EllipticK(x)'         complete elliptic integral of the first kind
'EllipticE(x)'         complete elliptic integral of the second kind
'factorial(n)'         factorial function n!
'binomial(n, k)'       binomial coefficients
'Order(x)'             order term function in truncated power series

For functions that have a branch cut in the complex plane, GiNaC follows
the conventions of C/C++ for systems that do not support a signed zero.
In particular: the natural logarithm ('log') and the square root
('sqrt') both have their branch cuts running along the negative real
axis.  The 'asin', 'acos', and 'atanh' functions all have two branch
cuts starting at +/-1 and running away towards infinity along the real
axis.  The 'atan' and 'asinh' functions have two branch cuts starting at
+/-i and running away towards infinity along the imaginary axis.  The
'acosh' function has one branch cut starting at +1 and running towards
-infinity.  These functions are continuous as the branch cut is
approached coming around the finite endpoint of the cut in a counter
clockwise direction.

5.12.2 Expanding functions
--------------------------

GiNaC knows several expansion laws for trancedent functions, e.g.
'exp(a+b)=exp(a) exp(b), |zw|=|z| |w|' or 'log(cd)=log(c)+log(d)' (for
positive 'c, d' ).  In order to use these rules you need to call
'expand()' method with the option
'expand_options::expand_transcendental'.  Another relevant option is
'expand_options::expand_function_args'.  Their usage and interaction can
be seen from the following example:
     {
     	symbol x("x"),  y("y");
     	ex e=exp(pow(x+y,2));
     	cout << e.expand() << endl;
     	// -> exp((x+y)^2)
     	cout << e.expand(expand_options::expand_transcendental) << endl;
     	// -> exp((x+y)^2)
     	cout << e.expand(expand_options::expand_function_args) << endl;
     	// -> exp(2*x*y+x^2+y^2)
     	cout << e.expand(expand_options::expand_function_args
     			| expand_options::expand_transcendental) << endl;
     	// -> exp(y^2)*exp(2*x*y)*exp(x^2)
     }
If both flags are set (as in the last call), then GiNaC tries to get the
maximal expansion.  For example, for the exponent GiNaC firstly expands
the argument and then the function.  For the logarithm and absolute
value, GiNaC uses the opposite order: firstly expands the function and
then its argument.  Of course, a user can fine-tune this behavior by
sequential calls of several 'expand()' methods with desired flags.

5.12.3 Multiple polylogarithms
------------------------------

The multiple polylogarithm is the most generic member of a family of
functions, to which others like the harmonic polylogarithm, Nielsen's
generalized polylogarithm and the multiple zeta value belong.  Each of
these functions can also be written as a multiple polylogarithm with
specific parameters.  This whole family of functions is therefore often
referred to simply as multiple polylogarithms, containing 'Li', 'G',
'H', 'S' and 'zeta'.  The multiple polylogarithm itself comes in two
variants: 'Li' and 'G'.  While 'Li' and 'G' in principle represent the
same function, the different notations are more natural to the series
representation or the integral representation, respectively.

To facilitate the discussion of these functions we distinguish between
indices and arguments as parameters.  In the table above indices are
printed as 'm', 's', 'n' or 'p', whereas arguments are printed as 'x',
'a' and 'y'.

To define a 'Li', 'H' or 'zeta' with a depth greater than one, you have
to pass a GiNaC 'lst' for the indices 'm' and 's', and in the case of
'Li' for the argument 'x' as well.  The parameter 'a' of 'G' must always
be a 'lst' containing the arguments in expanded form.  If 'G' is used
with a third parameter 's', 's' must have the same length as 'a'.  It
contains then the signs of the imaginary parts of the arguments.  If 's'
is not given, the signs default to +1.  Note that 'Li' and 'zeta' are
polymorphic in this respect.  They can stand in for the classical
polylogarithm and Riemann's zeta function (if depth is one), as well as
for the multiple polylogarithm and the multiple zeta value,
respectively.  Note also, that GiNaC doesn't check whether the 'lst's
for two parameters do have the same length.  It is up to the user to
ensure this, otherwise evaluating will result in undefined behavior.

The functions print in LaTeX format as
'\mbox{Li}_{m_1,m_2,...,m_k}(x_1,x_2,...,x_k)', '\mbox{S}_{n,p}(x)',
'\mbox{H}_{m_1,m_2,...,m_k}(x)' and '\zeta(m_1,m_2,...,m_k)' (with the
dots replaced by actual parameters).  If 'zeta' is an alternating zeta
sum, i.e.  'zeta(m,s)', the indices with negative sign are printed with
a line above, e.g.  '\zeta(5,\overline{2})'.  The order of indices and
arguments in the GiNaC 'lst's and in the output is the same.

Definitions and analytical as well as numerical properties of multiple
polylogarithms are too numerous to be covered here.  Instead, the user
is referred to the publications listed at the end of this section.  The
implementation in GiNaC adheres to the definitions and conventions
therein, except for a few differences which will be explicitly stated in
the following.

One difference is about the order of the indices and arguments.  For
GiNaC we adopt the convention that the indices and arguments are
understood to be in the same order as in which they appear in the series
representation.  This means 'Li_{m_1,m_2,m_3}(x,1,1) =
H_{m_1,m_2,m_3}(x)' and 'Li_{2,1}(1,1) = zeta(2,1) = zeta(3)', but
'zeta(1,2)' evaluates to infinity.  So in comparison to the older ones
of the referenced publications the order of indices and arguments for
'Li' is reversed.

The functions only evaluate if the indices are integers greater than
zero, except for the indices 's' in 'zeta' and 'G' as well as 'm' in
'H'.  Since 's' will be interpreted as the sequence of signs for the
corresponding indices 'm' or the sign of the imaginary part for the
corresponding arguments 'a', it must contain 1 or -1, e.g.
'zeta(lst{3,4}, lst{-1,1})' means 'zeta(\overline{3},4)' and
'G(lst{a,b}, lst{-1,1}, c)' means 'G(a-0\epsilon,b+0\epsilon;c)'.  The
definition of 'H' allows indices to be 0, 1 or -1 (in expanded notation)
or equally to be any integer (in compact notation).  With GiNaC expanded
and compact notation can be mixed, e.g.  'lst{0,0,-1,0,1,0,0}',
'lst{0,0,-1,2,0,0}' and 'lst{-3,2,0,0}' are equivalent as indices.  The
anonymous evaluator 'eval()' tries to reduce the functions, if possible,
to the least-generic multiple polylogarithm.  If all arguments are unit,
it returns 'zeta'.  Arguments equal to zero get considered, too.
Riemann's zeta function 'zeta' (with depth one) evaluates also for
negative integers and positive even integers.  For example:

     > Li({3,1},{x,1});
     S(2,2,x)
     > H({-3,2},1);
     -zeta({3,2},{-1,-1})
     > S(3,1,1);
     1/90*Pi^4

It is easy to tell for a given function into which other function it can
be rewritten, may it be a less-generic or a more-generic one, except for
harmonic polylogarithms 'H' with negative indices or trailing zeros (the
example above gives a hint).  Signs can quickly be messed up, for
example.  Therefore GiNaC offers a C++ function 'convert_H_to_Li()' to
deal with the upgrade of a 'H' to a multiple polylogarithm 'Li'
('eval()' already cares for the possible downgrade):

     > convert_H_to_Li({0,-2,-1,3},x);
     Li({3,1,3},{-x,1,-1})
     > convert_H_to_Li({2,-1,0},x);
     -Li({2,1},{x,-1})*log(x)+2*Li({3,1},{x,-1})+Li({2,2},{x,-1})

Every function can be numerically evaluated for arbitrary real or
complex arguments.  The precision is arbitrary and can be set through
the global variable 'Digits':

     > Digits=100;
     100
     > evalf(zeta({3,1,3,1}));
     0.005229569563530960100930652283899231589890420784634635522547448972148869544...

Note that the convention for arguments on the branch cut in GiNaC as
stated above is different from the one Remiddi and Vermaseren have
chosen for the harmonic polylogarithm.

If a function evaluates to infinity, no exceptions are raised, but the
function is returned unevaluated, e.g.  'zeta(1)'.  In long expressions
this helps a lot with debugging, because you can easily spot the
divergencies.  But on the other hand, you have to make sure for
yourself, that no illegal cancellations of divergencies happen.

Useful publications:

'Nested Sums, Expansion of Transcendental Functions and Multi-Scale
Multi-Loop Integrals', S.Moch, P.Uwer, S.Weinzierl, hep-ph/0110083

'Harmonic Polylogarithms', E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys.
A15 (2000), pp.  725-754

'Special Values of Multiple Polylogarithms', J.Borwein, D.Bradley,
D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc.  353/3 (2001), pp.
907-941

'Numerical Evaluation of Multiple Polylogarithms', J.Vollinga,
S.Weinzierl, hep-ph/0410259

5.12.4 Iterated integrals
-------------------------

Multiple polylogarithms are a particular example of iterated integrals.
An iterated integral is defined by the function
'iterated_integral(a,y)'.  The variable 'y' gives the upper integration
limit for the outermost integration, by convention the lower integration
limit is always set to zero.  The variable 'a' must be a GiNaC 'lst'
containing sub-classes of 'integration_kernel' as elements.  The depth
of the iterated integral corresponds to the number of elements of 'a'.
The available integrands for iterated integrals are (for a more detailed
description the user is referred to the publications listed at the end
of this section)
*Class*                       *Description*
'integration_kernel()'        Base class, represents the one-form dy
'basic_log_kernel()'          Logarithmic one-form dy/y
'multiple_polylog_kernel(z_j)'The one-form dy/(y-z_j)
'ELi_kernel(n, m, x, y)'      The one form ELi_{n;m}(x;y;q) dq/q
'Ebar_kernel(n, m, x, y)'     The one form \overline{E}_{n;m}(x;y;q)
                              dq/q
'Kronecker_dtau_kernel(k,     The one form C_k K (k-1)/(2 \pi i)^k
z_j, K, C_k)'                 g^{(k)}(z_j,K \tau) dq/q
'Kronecker_dz_kernel(k,       The one form C_k (2 \pi i)^{2-k}
z_j, tau, K, C_k)'            g^{(k-1)}(z-z_j,K \tau) dz
'Eisenstein_kernel(k, N, a,   The one form C_k E_{k,N,a,b,K}(\tau) dq/q
b, K, C_k)'
'Eisenstein_h_kernel(k, N,    The one form C_k h_{k,N,r,s}(\tau) dq/q
r, s, C_k)'
'modular_form_kernel(k, P,    The one form C_k P dq/q
C_k)'
'user_defined_kernel(f, y)'   The one form f(y) dy
All parameters are assumed to be such that all integration kernels have
a convergent Laurent expansion around zero with at most a simple pole at
zero.  The iterated integral may also be called with an optional third
parameter 'iterated_integral(a,y,N_trunc)', in which case the numerical
evaluation will truncate the series expansion at order 'N_trunc'.

The classes 'Eisenstein_kernel()', 'Eisenstein_h_kernel()' and
'modular_form_kernel()' provide a method 'q_expansion_modular_form(q,
order)', which can used to obtain the q-expansion of
E_{k,N,a,b,K}(\tau), h_{k,N,r,s}(\tau) or P to the specified order.

Useful publications:

'Numerical evaluation of iterated integrals related to elliptic Feynman
integrals', M.Walden, S.Weinzierl, arXiv:2010.05271

5.13 Complex expressions
========================

For dealing with complex expressions there are the methods

     ex ex::conjugate();
     ex ex::real_part();
     ex ex::imag_part();

that return respectively the complex conjugate, the real part and the
imaginary part of an expression.  Complex conjugation works as expected
for all built-in functions and objects.  Taking real and imaginary parts
has not yet been implemented for all built-in functions.  In cases where
it is not known how to conjugate or take a real/imaginary part one of
the functions 'conjugate', 'real_part' or 'imag_part' is returned.  For
instance, in case of a complex symbol 'x' (symbols are complex by
default), one could not simplify 'conjugate(x)'.  In the case of strings
of gamma matrices, the 'conjugate' method takes the Dirac conjugate.

For example,
     {
         varidx a(symbol("a"), 4), b(symbol("b"), 4);
         symbol x("x");
         realsymbol y("y");

         cout << (3*I*x*y + sin(2*Pi*I*y)).conjugate() << endl;
          // -> -3*I*conjugate(x)*y+sin(-2*I*Pi*y)
         cout << (dirac_gamma(a)*dirac_gamma(b)*dirac_gamma5()).conjugate() << endl;
          // -> -gamma5*gamma~b*gamma~a
     }

If you declare your own GiNaC functions and you want to conjugate them,
you will have to supply a specialized conjugation method for them (see
Symbolic functions and the GiNaC source-code for 'abs' as an
example).  GiNaC does not automatically conjugate user-supplied
functions by conjugating their arguments because this would be incorrect
on branch cuts.  Also, specialized methods can be provided to take real
and imaginary parts of user-defined functions.

5.14 Solving linear systems of equations
========================================

The function 'lsolve()' provides a convenient wrapper around some matrix
operations that comes in handy when a system of linear equations needs
to be solved:

     ex lsolve(const ex & eqns, const ex & symbols,
               unsigned options = solve_algo::automatic);

Here, 'eqns' is a 'lst' of equalities (i.e.  class 'relational') while
'symbols' is a 'lst' of indeterminates.  (see The class hierarchy,
for an exposition of class 'lst').

It returns the 'lst' of solutions as an expression.  As an example, let
us solve the two equations 'a*x+b*y==3' and 'x-y==b':

     {
         symbol a("a"), b("b"), x("x"), y("y");
         lst eqns = {a*x+b*y==3, x-y==b};
         lst vars = {x, y};
         cout << lsolve(eqns, vars) << endl;
          // -> {x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)}

When the linear equations 'eqns' are underdetermined, the solution will
contain one or more tautological entries like 'x==x', depending on the
rank of the system.  When they are overdetermined, the solution will be
an empty 'lst'.  Note the third optional parameter to 'lsolve()': it
accepts the same parameters as 'matrix::solve()'.  This is because
'lsolve' is just a wrapper around that method.

5.15 Input and output of expressions
====================================

5.15.1 Expression output
------------------------

Expressions can simply be written to any stream:

     {
         symbol x("x");
         ex e = 4.5*I+pow(x,2)*3/2;
         cout << e << endl;    // prints '4.5*I+3/2*x^2'
         // ...

The default output format is identical to the 'ginsh' input syntax and
to that used by most computer algebra systems, but not directly pastable
into a GiNaC C++ program (note that in the above example, 'pow(x,2)' is
printed as 'x^2').

It is possible to print expressions in a number of different formats
with a set of stream manipulators;

     std::ostream & dflt(std::ostream & os);
     std::ostream & latex(std::ostream & os);
     std::ostream & tree(std::ostream & os);
     std::ostream & csrc(std::ostream & os);
     std::ostream & csrc_float(std::ostream & os);
     std::ostream & csrc_double(std::ostream & os);
     std::ostream & csrc_cl_N(std::ostream & os);
     std::ostream & index_dimensions(std::ostream & os);
     std::ostream & no_index_dimensions(std::ostream & os);

The 'tree', 'latex' and 'csrc' formats are also available in 'ginsh' via
the 'print()', 'print_latex()' and 'print_csrc()' functions,
respectively.

All manipulators affect the stream state permanently.  To reset the
output format to the default, use the 'dflt' manipulator:

         // ...
         cout << latex;            // all output to cout will be in LaTeX format from
                                   // now on
         cout << e << endl;        // prints '4.5 i+\frac{3}{2} x^{2}'
         cout << sin(x/2) << endl; // prints '\sin(\frac{1}{2} x)'
         cout << dflt;             // revert to default output format
         cout << e << endl;        // prints '4.5*I+3/2*x^2'
         // ...

If you don't want to affect the format of the stream you're working
with, you can output to a temporary 'ostringstream' like this:

         // ...
         ostringstream s;
         s << latex << e;         // format of cout remains unchanged
         cout << s.str() << endl; // prints '4.5 i+\frac{3}{2} x^{2}'
         // ...

The 'csrc' (an alias for 'csrc_double'), 'csrc_float', 'csrc_double' and
'csrc_cl_N' manipulators set the output to a format that can be directly
used in a C or C++ program.  The three possible formats select the data
types used for numbers ('csrc_cl_N' uses the classes provided by the CLN
library):

         // ...
         cout << "f = " << csrc_float << e << ";\n";
         cout << "d = " << csrc_double << e << ";\n";
         cout << "n = " << csrc_cl_N << e << ";\n";
         // ...

The above example will produce (note the 'x^2' being converted to
'x*x'):

     f = (3.0/2.0)*(x*x)+std::complex<float>(0.0,4.5000000e+00);
     d = (3.0/2.0)*(x*x)+std::complex<double>(0.0,4.5000000000000000e+00);
     n = cln::cl_RA("3/2")*(x*x)+cln::complex(cln::cl_I("0"),cln::cl_F("4.5_17"));

The 'tree' manipulator allows dumping the internal structure of an
expression for debugging purposes:

         // ...
         cout << tree << e;
     }

produces

     add, hash=0x0, flags=0x3, nops=2
         power, hash=0x0, flags=0x3, nops=2
             x (symbol), serial=0, hash=0xc8d5bcdd, flags=0xf
             2 (numeric), hash=0x6526b0fa, flags=0xf
         3/2 (numeric), hash=0xf9828fbd, flags=0xf
         -----
         overall_coeff
         4.5L0i (numeric), hash=0xa40a97e0, flags=0xf
         =====

The 'latex' output format is for LaTeX parsing in mathematical mode.  It
is rather similar to the default format but provides some braces needed
by LaTeX for delimiting boxes and also converts some common objects to
conventional LaTeX names.  It is possible to give symbols a special name
for LaTeX output by supplying it as a second argument to the 'symbol'
constructor.

For example, the code snippet

     {
         symbol x("x", "\\circ");
         ex e = lgamma(x).series(x==0,3);
         cout << latex << e << endl;
     }

will print

         {(-\ln(\circ))}+{(-\gamma_E)} \circ+{(\frac{1}{12} \pi^{2})} \circ^{2}
         +\mathcal{O}(\circ^{3})

Index dimensions are normally hidden in the output.  To make them
visible, use the 'index_dimensions' manipulator.  The dimensions will be
written in square brackets behind each index value in the default and
LaTeX output formats:

     {
         symbol x("x"), y("y");
         varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
         ex e = indexed(x, mu) * indexed(y, nu);

         cout << e << endl;
          // prints 'x~mu*y~nu'
         cout << index_dimensions << e << endl;
          // prints 'x~mu[4]*y~nu[4]'
         cout << no_index_dimensions << e << endl;
          // prints 'x~mu*y~nu'
     }

If you need any fancy special output format, e.g.  for interfacing GiNaC
with other algebra systems or for producing code for different
programming languages, you can always traverse the expression tree
yourself:

     static void my_print(const ex & e)
     {
         if (is_a<function>(e))
             cout << ex_to<function>(e).get_name();
         else
             cout << ex_to<basic>(e).class_name();
         cout << "(";
         size_t n = e.nops();
         if (n)
             for (size_t i=0; i<n; i++) {
                 my_print(e.op(i));
                 if (i != n-1)
                     cout << ",";
             }
         else
             cout << e;
         cout << ")";
     }

     int main()
     {
         my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
         return 0;
     }

This will produce

     add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
     symbol(y))),numeric(-2)))

If you need an output format that makes it possible to accurately
reconstruct an expression by feeding the output to a suitable parser or
object factory, you should consider storing the expression in an
'archive' object and reading the object properties from there.  See the
section on archiving for more information.

5.15.2 Expression input
-----------------------

GiNaC provides no way to directly read an expression from a stream
because you will usually want the user to be able to enter something
like '2*x+sin(y)' and have the 'x' and 'y' correspond to the symbols 'x'
and 'y' you defined in your program and there is no way to specify the
desired symbols to the '>>' stream input operator.

Instead, GiNaC lets you read an expression from a stream or a string,
specifying the mapping between the input strings and symbols to be used:

     {
         symbol x, y;
         symtab table;
         table["x"] = x;
         table["y"] = y;
         parser reader(table);
         ex e = reader("2*x+sin(y)");
     }

The input syntax is the same as that used by 'ginsh' and the stream
output operator '<<'.  Matching between the input strings and
expressions is given by 'table'.  The 'table' in this example instructs
GiNaC to substitute any input substring "x" with symbol 'x'.  Likewise,
the substring "y" will be replaced with symbol 'y'.  It's also possible
to map input (sub)strings to arbitrary expressions:

     {
         symbol x, y;
         symtab table;
         table["x"] = x+log(y)+1;
         parser reader(table);
         ex e = reader("5*x^3 - x^2");
         // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
     }

If no mapping is specified for a particular string GiNaC will create a
symbol with corresponding name.  Later on you can obtain all parser
generated symbols with 'get_syms()' method:

     {
         parser reader;
         ex e = reader("2*x+sin(y)");
         symtab table = reader.get_syms();
         symbol x = ex_to<symbol>(table["x"]);
         symbol y = ex_to<symbol>(table["y"]);
     }

Sometimes you might want to prevent GiNaC from inserting these extra
symbols (for example, you want treat an unexpected string in the input
as an error).

     {
     	symtab table;
     	table["x"] = symbol();
     	parser reader(table);
     	parser.strict = true;
     	ex e;
     	try {
     		e = reader("2*x+sin(y)");
     	} catch (parse_error& err) {
     		cerr << err.what() << endl;
     		// prints "unknown symbol "y" in the input"
     	}
     }

With this parser, it's also easy to implement interactive GiNaC
programs.  When running the following program interactively, remember to
send an EOF marker after the input, e.g.  by pressing Ctrl-D on an empty
line:

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

     int main()
     {
     	cout << "Enter an expression containing 'x': " << flush;
     	parser reader;

     	try {
     		ex e = reader(cin);
     		symtab table = reader.get_syms();
     		symbol x = table.find("x") != table.end() ?
     			   ex_to<symbol>(table["x"]) : symbol("x");
     		cout << "The derivative of " << e << " with respect to x is ";
     		cout << e.diff(x) << "." << endl;
     	} catch (exception &p) {
     		cerr << p.what() << endl;
     	}
     }

5.15.3 Compiling expressions to C function pointers
---------------------------------------------------

Numerical evaluation of algebraic expressions is seamlessly integrated
into GiNaC by help of the CLN library.  While CLN allows for very fast
arbitrary precision numerics, which is more than sufficient for most
users, sometimes only the speed of built-in floating point numbers is
fast enough, e.g.  for Monte Carlo integration.  The only viable option
then is the following: print the expression in C syntax format, manually
add necessary C code, compile that program and run is as a separate
application.  This is not only cumbersome and involves a lot of manual
intervention, but it also separates the algebraic and the numerical
evaluation into different execution stages.

GiNaC offers a couple of functions that help to avoid these
inconveniences and problems.  The functions automatically perform the
printing of a GiNaC expression and the subsequent compiling of its
associated C code.  The created object code is then dynamically linked
to the currently running program.  A function pointer to the C function
that performs the numerical evaluation is returned and can be used
instantly.  This all happens automatically, no user intervention is
needed.

The following example demonstrates the use of 'compile_ex':

         // ...
         symbol x("x");
         ex myexpr = sin(x) / x;

         FUNCP_1P fp;
         compile_ex(myexpr, x, fp);

         cout << fp(3.2) << endl;
         // ...

The function 'compile_ex' is called with the expression to be compiled
and its only free variable 'x'.  Upon successful completion the third
parameter contains a valid function pointer to the corresponding C code
module.  If called like in the last line only built-in double precision
numerics is involved.

The function pointer has to be defined in advance.  GiNaC offers three
function pointer types at the moment:

         typedef double (*FUNCP_1P) (double);
         typedef double (*FUNCP_2P) (double, double);
         typedef void (*FUNCP_CUBA) (const int*, const double[], const int*, double[]);

'FUNCP_2P' allows for two variables in the expression.  'FUNCP_CUBA' is
the correct type to be used with the CUBA library
(<http://www.feynarts.de/cuba>) for numerical integrations.  The details
for the parameters of 'FUNCP_CUBA' are explained in the CUBA manual.

For every function pointer type there is a matching 'compile_ex'
available:

         void compile_ex(const ex& expr, const symbol& sym, FUNCP_1P& fp,
                         const std::string filename = "");
         void compile_ex(const ex& expr, const symbol& sym1, const symbol& sym2,
                         FUNCP_2P& fp, const std::string filename = "");
         void compile_ex(const lst& exprs, const lst& syms, FUNCP_CUBA& fp,
                         const std::string filename = "");

When the last parameter 'filename' is not supplied, 'compile_ex' will
choose a unique random name for the intermediate source and object files
it produces.  On program termination these files will be deleted.  If
one wishes to keep the C code and the object files, one can supply the
'filename' parameter.  The intermediate files will use that filename and
will not be deleted.

'link_ex' is a function that allows to dynamically link an existing
object file and to make it available via a function pointer.  This is
useful if you have already used 'compile_ex' on an expression and want
to avoid the compilation step to be performed over and over again when
you restart your program.  The precondition for this is of course, that
you have chosen a filename when you did call 'compile_ex'.  For every
above mentioned function pointer type there exists a corresponding
'link_ex' function:

         void link_ex(const std::string filename, FUNCP_1P& fp);
         void link_ex(const std::string filename, FUNCP_2P& fp);
         void link_ex(const std::string filename, FUNCP_CUBA& fp);

The complete filename (including the suffix '.so') of the object file
has to be supplied.

The function

         void unlink_ex(const std::string filename);

is supplied for the rare cases when one wishes to close the dynamically
linked object files directly and have the intermediate files (only if
filename has not been given) deleted.  Normally one doesn't need this
function, because all the clean-up will be done automatically upon
(regular) program termination.

All the described functions will throw an exception in case they cannot
perform correctly, like for example when writing the file or starting
the compiler fails.  Since internally the same printing methods as
described in section csrc printing are used, only functions and
objects that are available in standard C will compile successfully (that
excludes polylogarithms for example at the moment).  Another
precondition for success is, of course, that it must be possible to
evaluate the expression numerically.  No free variables despite the ones
supplied to 'compile_ex' should appear in the expression.

'compile_ex' uses the shell script 'ginac-excompiler' to start the C
compiler and produce the object files.  This shell script comes with
GiNaC and will be installed together with GiNaC in the configured
'$LIBEXECDIR' (typically '$PREFIX/libexec' or '$PREFIX/lib/ginac').  You
can also export additional compiler flags via the '$CXXFLAGS' variable:

     setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
     compile_ex(...);

5.15.4 Archiving
----------------

GiNaC allows creating "archives" of expressions which can be stored to
or retrieved from files.  To create an archive, you declare an object of
class 'archive' and archive expressions in it, giving each expression a
unique name:

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

     int main()
     {
         symbol x("x"), y("y"), z("z");

         ex foo = sin(x + 2*y) + 3*z + 41;
         ex bar = foo + 1;

         archive a;
         a.archive_ex(foo, "foo");
         a.archive_ex(bar, "the second one");
         // ...

The archive can then be written to a file:

         // ...
         ofstream out("foobar.gar", ios::binary);
         out << a;
         out.close();
         // ...

The file 'foobar.gar' contains all information that is needed to
reconstruct the expressions 'foo' and 'bar'.  The flag 'ios::binary'
prevents locales setting of your OS tampers the archive file structure.

The tool 'viewgar' that comes with GiNaC can be used to view the
contents of GiNaC archive files:

     $ viewgar foobar.gar
     foo = 41+sin(x+2*y)+3*z
     the second one = 42+sin(x+2*y)+3*z

The point of writing archive files is of course that they can later be
read in again:

         // ...
         archive a2;
         ifstream in("foobar.gar", ios::binary);
         in >> a2;
         // ...

And the stored expressions can be retrieved by their name:

         // ...
         lst syms = {x, y};

         ex ex1 = a2.unarchive_ex(syms, "foo");
         ex ex2 = a2.unarchive_ex(syms, "the second one");

         cout << ex1 << endl;              // prints "41+sin(x+2*y)+3*z"
         cout << ex2 << endl;              // prints "42+sin(x+2*y)+3*z"
         cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
     }

Note that you have to supply a list of the symbols which are to be
inserted in the expressions.  Symbols in archives are stored by their
name only and if you don't specify which symbols you have, unarchiving
the expression will create new symbols with that name.  E.g.  if you
hadn't included 'x' in the 'syms' list above, the 'ex1.subs(x == 2)'
statement would have had no effect because the 'x' in 'ex1' would have
been a different symbol than the 'x' which was defined at the beginning
of the program, although both would appear as 'x' when printed.

You can also use the information stored in an 'archive' object to output
expressions in a format suitable for exact reconstruction.  The
'archive' and 'archive_node' classes have a couple of member functions
that let you access the stored properties:

     static void my_print2(const archive_node & n)
     {
         string class_name;
         n.find_string("class", class_name);
         cout << class_name << "(";

         archive_node::propinfovector p;
         n.get_properties(p);

         size_t num = p.size();
         for (size_t i=0; i<num; i++) {
             const string &name = p[i].name;
             if (name == "class")
                 continue;
             cout << name << "=";

             unsigned count = p[i].count;
             if (count > 1)
                 cout << "{";

             for (unsigned j=0; j<count; j++) {
                 switch (p[i].type) {
                     case archive_node::PTYPE_BOOL: {
                         bool x;
                         n.find_bool(name, x, j);
                         cout << (x ? "true" : "false");
                         break;
                     }
                     case archive_node::PTYPE_UNSIGNED: {
                         unsigned x;
                         n.find_unsigned(name, x, j);
                         cout << x;
                         break;
                     }
                     case archive_node::PTYPE_STRING: {
                         string x;
                         n.find_string(name, x, j);
                         cout << '\"' << x << '\"';
                         break;
                     }
                     case archive_node::PTYPE_NODE: {
                         const archive_node &x = n.find_ex_node(name, j);
                         my_print2(x);
                         break;
                     }
                 }

                 if (j != count-1)
                     cout << ",";
             }

             if (count > 1)
                 cout << "}";

             if (i != num-1)
                 cout << ",";
         }

         cout << ")";
     }

     int main()
     {
         ex e = pow(2, x) - y;
         archive ar(e, "e");
         my_print2(ar.get_top_node(0)); cout << endl;
         return 0;
     }

This will produce:

     add(rest={power(basis=numeric(number="2"),exponent=symbol(name="x")),
     symbol(name="y")},coeff={numeric(number="1"),numeric(number="-1")},
     overall_coeff=numeric(number="0"))

Be warned, however, that the set of properties and their meaning for
each class may change between GiNaC versions.

6 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.

6.1 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 capabilities 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.

6.2 Symbolic functions
======================

The easiest and most instructive way to start extending GiNaC is
probably to create your own symbolic functions.  These are implemented
with the help of two preprocessor macros:

     DECLARE_FUNCTION_<n>P(<name>)
     REGISTER_FUNCTION(<name>, <options>)

The 'DECLARE_FUNCTION' macro will usually appear in a header file.  It
declares a C++ function with the given 'name' that takes exactly 'n'
parameters of type 'ex' and returns a newly constructed GiNaC 'function'
object that represents your function.

The 'REGISTER_FUNCTION' macro implements the function.  It must be
passed the same 'name' as the respective 'DECLARE_FUNCTION' macro, and a
set of options that associate the symbolic function with C++ functions
you provide to implement the various methods such as evaluation,
derivative, series expansion etc.  They also describe additional
attributes the function might have, such as symmetry and commutation
properties, and a name for LaTeX output.  Multiple options are separated
by the member access operator '.' and can be given in an arbitrary
order.

(By the way: in case you are worrying about all the macros above we can
assure you that functions are GiNaC's most macro-intense classes.  We
have done our best to avoid macros where we can.)

6.2.1 A minimal example
-----------------------

Here is an example for the implementation of a function with two
arguments that is not further evaluated:

     DECLARE_FUNCTION_2P(myfcn)

     REGISTER_FUNCTION(myfcn, dummy())

Any code that has seen the 'DECLARE_FUNCTION' line can use 'myfcn()' in
algebraic expressions:

     {
         ...
         symbol x("x");
         ex e = 2*myfcn(42, 1+3*x) - x;
         cout << e << endl;
          // prints '2*myfcn(42,1+3*x)-x'
         ...
     }

The 'dummy()' option in the 'REGISTER_FUNCTION' line signifies "no
options".  A function with no options specified merely acts as a kind of
container for its arguments.  It is a pure "dummy" function with no
associated logic (which is, however, sometimes perfectly sufficient).

Let's now have a look at the implementation of GiNaC's cosine function
for an example of how to make an "intelligent" function.

6.2.2 The cosine function
-------------------------

The GiNaC header file 'inifcns.h' contains the line

     DECLARE_FUNCTION_1P(cos)

which declares to all programs using GiNaC that there is a function
'cos' that takes one 'ex' as an argument.  This is all they need to know
to use this function in expressions.

The implementation of the cosine function is in 'inifcns_trans.cpp'.
Here is its 'REGISTER_FUNCTION' line:

     REGISTER_FUNCTION(cos, eval_func(cos_eval).
                            evalf_func(cos_evalf).
                            derivative_func(cos_deriv).
                            latex_name("\\cos"));

There are four options defined for the cosine function.  One of them
('latex_name') gives the function a proper name for LaTeX output; the
other three indicate the C++ functions in which the "brains" of the
cosine function are defined.

The 'eval_func()' option specifies the C++ function that implements the
'eval()' method, GiNaC's anonymous evaluator.  This function takes the
same number of arguments as the associated symbolic function (one in
this case) and returns the (possibly transformed or in some way
simplified) symbolically evaluated function (see Automatic evaluation,
for a description of the automatic evaluation process).
If no (further) evaluation is to take place, the 'eval_func()' function
must return the original function with '.hold()', to avoid a potential
infinite recursion.  If your symbolic functions produce a segmentation
fault or stack overflow when using them in expressions, you are probably
missing a '.hold()' somewhere.

The 'eval_func()' function for the cosine looks something like this
(actually, it doesn't look like this at all, but it should give you an
idea what is going on):

     static ex cos_eval(const ex & x)
     {
         if ("x is a multiple of 2*Pi")
             return 1;
         else if ("x is a multiple of Pi")
             return -1;
         else if ("x is a multiple of Pi/2")
             return 0;
         // more rules...

         else if ("x has the form 'acos(y)'")
             return y;
         else if ("x has the form 'asin(y)'")
             return sqrt(1-y^2);
         // more rules...

         else
             return cos(x).hold();
     }

This function is called every time the cosine is used in a symbolic
expression:

     {
         ...
         e = cos(Pi);
          // this calls cos_eval(Pi), and inserts its return value into
          // the actual expression
         cout << e << endl;
          // prints '-1'
         ...
     }

In this way, 'cos(4*Pi)' automatically becomes 1, 'cos(asin(a+b))'
becomes 'sqrt(1-(a+b)^2)', etc.  If no reasonable symbolic
transformation can be done, the unmodified function is returned with
'.hold()'.

GiNaC doesn't automatically transform 'cos(2)' to '-0.416146...'.  The
user has to call 'evalf()' for that.  This is implemented in a different
function:

     static ex cos_evalf(const ex & x)
     {
         if (is_a<numeric>(x))
             return cos(ex_to<numeric>(x));
         else
             return cos(x).hold();
     }

Since we are lazy we defer the problem of numeric evaluation to somebody
else, in this case the 'cos()' function for 'numeric' objects, which in
turn hands it over to the 'cos()' function in CLN. The '.hold()' isn't
really needed here, but reminds us that the corresponding 'eval()'
function would require it in this place.

Differentiation will surely turn up and so we need to tell 'cos' what
its first derivative is (higher derivatives, '.diff(x,3)' for instance,
are then handled automatically by 'basic::diff' and 'ex::diff'):

     static ex cos_deriv(const ex & x, unsigned diff_param)
     {
         return -sin(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.

Derivatives of some functions, for example 'abs()' and 'Order()', could
not be evaluated through the chain rule.  In such cases the full
derivative may be specified as shown for 'Order()':

     static ex Order_expl_derivative(const ex & arg, const symbol & s)
     {
     	return Order(arg.diff(s));
     }

That is, we need to supply a procedure, which returns the expression of
derivative with respect to the variable 's' for the argument 'arg'.
This procedure need to be registered with the function through the
option 'expl_derivative_func' (see the next Subsection).  In contrast, a
partial derivative, e.g.  as was defined for 'cos()' above, needs to be
registered through the option 'derivative_func'.

An implementation of the series expansion is not needed for 'cos()' as
it doesn't have any poles and GiNaC can do Taylor expansion by itself
(as long as it knows what the derivative of 'cos()' is).  'tan()', on
the other hand, does have poles and may need to do Laurent expansion:

     static ex tan_series(const ex & x, const relational & rel,
                          int order, unsigned options)
     {
         // Find the actual expansion point
         const ex x_pt = x.subs(rel);

         if ("x_pt is not an odd multiple of Pi/2")
             throw do_taylor();  // tell function::series() to do Taylor expansion

         // On a pole, expand sin()/cos()
         return (sin(x)/cos(x)).series(rel, order+2, options);
     }

The 'series()' implementation of a function _must_ return a 'pseries'
object, otherwise your code will crash.

6.2.3 Function options
----------------------

GiNaC functions understand several more options which are always
specified as '.option(params)'.  None of them are required, but you need
to specify at least one option to 'REGISTER_FUNCTION()'.  There is a
do-nothing option called 'dummy()' which you can use to define functions
without any special options.

     eval_func(<C++ function>)
     evalf_func(<C++ function>)
     derivative_func(<C++ function>)
     expl_derivative_func(<C++ function>)
     series_func(<C++ function>)
     conjugate_func(<C++ function>)

These specify the C++ functions that implement symbolic evaluation,
numeric evaluation, partial derivatives, explicit derivative, and series
expansion, respectively.  They correspond to the GiNaC methods 'eval()',
'evalf()', 'diff()' and 'series()'.

The 'eval_func()' function needs to use '.hold()' if no further
automatic evaluation is desired or possible.

If no 'series_func()' is given, GiNaC defaults to simple Taylor
expansion, which is correct if there are no poles involved.  If the
function has poles in the complex plane, the 'series_func()' needs to
check whether the expansion point is on a pole and fall back to Taylor
expansion if it isn't.  Otherwise, the pole usually needs to be
regularized by some suitable transformation.

     latex_name(const string & n)

specifies the LaTeX code that represents the name of the function in
LaTeX output.  The default is to put the function name in an '\mbox{}'.

     do_not_evalf_params()

This tells 'evalf()' to not recursively evaluate the parameters of the
function before calling the 'evalf_func()'.

     set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)

This allows you to explicitly specify the commutation properties of the
function (see Non-commutative objects, for an explanation of
(non)commutativity in GiNaC). For example, with an object of type
'return_type_t' created like

     return_type_t my_type = make_return_type_t<matrix>();

you can use 'set_return_type(return_types::noncommutative, &my_type)' to
make GiNaC treat your function like a matrix.  By default, functions
inherit the commutation properties of their first argument.  The
utilized template function 'make_return_type_t<>()'

     template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)

can also be called with an argument specifying the representation label
of the non-commutative function (see section on dirac gamma matrices for
more details).

     set_symmetry(const symmetry & s)

specifies the symmetry properties of the function with respect to its
arguments.  See Indexed objects, for an explanation of symmetry
specifications.  GiNaC will automatically rearrange the arguments of
symmetric functions into a canonical order.

Sometimes you may want to have finer control over how functions are
displayed in the output.  For example, the 'abs()' function prints
itself as 'abs(x)' in the default output format, but as '|x|' in LaTeX
mode, and 'fabs(x)' in C source output.  This is achieved with the

     print_func<C>(<C++ function>)

option which is explained in the next section.

6.2.4 Functions with a variable number of arguments
---------------------------------------------------

The 'DECLARE_FUNCTION' and 'REGISTER_FUNCTION' macros define functions
with a fixed number of arguments.  Sometimes, though, you may need to
have a function that accepts a variable number of expressions.  One way
to accomplish this is to pass variable-length lists as arguments.  The
'Li()' function uses this method for multiple polylogarithms.

It is also possible to define functions that accept a different number
of parameters under the same function name, such as the 'psi()' function
which can be called either as 'psi(z)' (the digamma function) or as
'psi(n, z)' (polygamma functions).  These are actually two different
functions in GiNaC that, however, have the same name.  Defining such
functions is not possible with the macros but requires manually fiddling
with GiNaC internals.  If you are interested, please consult the GiNaC
source code for the 'psi()' function ('inifcns.h' and
'inifcns_gamma.cpp').

6.3 GiNaC's expression output system
====================================

GiNaC allows the output of expressions in a variety of different formats
(see Input/output).  This section will explain how expression output
is implemented internally, and how to define your own output formats or
change the output format of built-in algebraic objects.  You will also
want to read this section if you plan to write your own algebraic
classes or functions.

All the different output formats are represented by a hierarchy of
classes rooted in the 'print_context' class, defined in the 'print.h'
header file:

'print_dflt'
     the default output format
'print_latex'
     output in LaTeX mathematical mode
'print_tree'
     a dump of the internal expression structure (for debugging)
'print_csrc'
     the base class for C source output
'print_csrc_float'
     C source output using the 'float' type
'print_csrc_double'
     C source output using the 'double' type
'print_csrc_cl_N'
     C source output using CLN types

The 'print_context' base class provides two public data members:

     class print_context
     {
         ...
     public:
         std::ostream & s;
         unsigned options;
     };

's' is a reference to the stream to output to, while 'options' holds
flags and modifiers.  Currently, there is only one flag defined:
'print_options::print_index_dimensions' instructs the 'idx' class to
print the index dimension which is normally hidden.

When you write something like 'std::cout << e', where 'e' is an object
of class 'ex', GiNaC will construct an appropriate 'print_context'
object (of a class depending on the selected output format), fill in the
's' and 'options' members, and call

     void ex::print(const print_context & c, unsigned level = 0) const;

which in turn forwards the call to the 'print()' method of the top-level
algebraic object contained in the expression.

Unlike other methods, GiNaC classes don't usually override their
'print()' method to implement expression output.  Instead, the default
implementation 'basic::print(c, level)' performs a run-time double
dispatch to a function selected by the dynamic type of the object and
the passed 'print_context'.  To this end, GiNaC maintains a separate
method table for each class, similar to the virtual function table used
for ordinary (single) virtual function dispatch.

The method table contains one slot for each possible 'print_context'
type, indexed by the (internally assigned) serial number of the type.
Slots may be empty, in which case GiNaC will retry the method lookup
with the 'print_context' object's parent class, possibly repeating the
process until it reaches the 'print_context' base class.  If there's
still no method defined, the method table of the algebraic object's
parent class is consulted, and so on, until a matching method is found
(eventually it will reach the combination 'basic/print_context', which
prints the object's class name enclosed in square brackets).

You can think of the print methods of all the different classes and
output formats as being arranged in a two-dimensional matrix with one
axis listing the algebraic classes and the other axis listing the
'print_context' classes.

Subclasses of 'basic' can, of course, also overload 'basic::print()' to
implement printing, but then they won't get any of the benefits of the
double dispatch mechanism (such as the ability for derived classes to
inherit only certain print methods from its parent, or the replacement
of methods at run-time).

6.3.1 Print methods for classes
-------------------------------

The method table for a class is set up either in the definition of the
class, by passing the appropriate 'print_func<C>()' option to
'GINAC_IMPLEMENT_REGISTERED_CLASS_OPT()' (see Adding classes, for an
example), or at run-time using 'set_print_func<T, C>()'.  The latter can
also be used to override existing methods dynamically.

The argument to 'print_func<C>()' and 'set_print_func<T, C>()' can be a
member function of the class (or one of its parent classes), a static
member function, or an ordinary (global) C++ function.  The 'C' template
parameter specifies the appropriate 'print_context' type for which the
method should be invoked, while, in the case of 'set_print_func<>()',
the 'T' parameter specifies the algebraic class (for 'print_func<>()',
the class is the one being implemented by
'GINAC_IMPLEMENT_REGISTERED_CLASS_OPT').

For print methods that are member functions, their first argument must
be of a type convertible to a 'const C &', and the second argument must
be an 'unsigned'.

For static members and global functions, the first argument must be of a
type convertible to a 'const T &', the second argument must be of a type
convertible to a 'const C &', and the third argument must be an
'unsigned'.  A global function will, of course, not have access to
private and protected members of 'T'.

The 'unsigned' argument of the print methods (and of 'ex::print()' and
'basic::print()') is used for proper parenthesizing of the output (and
by 'print_tree' for proper indentation).  It can be used for similar
purposes if you write your own output formats.

The explanations given above may seem complicated, but in practice it's
really simple, as shown in the following example.  Suppose that we want
to display exponents in LaTeX output not as superscripts but with little
upwards-pointing arrows.  This can be achieved in the following way:

     void my_print_power_as_latex(const power & p,
                                  const print_latex & c,
                                  unsigned level)
     {
         // get the precedence of the 'power' class
         unsigned power_prec = p.precedence();

         // if the parent operator has the same or a higher precedence
         // we need parentheses around the power
         if (level >= power_prec)
             c.s << '(';

         // print the basis and exponent, each enclosed in braces, and
         // separated by an uparrow
         c.s << '{';
         p.op(0).print(c, power_prec);
         c.s << "}\\uparrow{";
         p.op(1).print(c, power_prec);
         c.s << '}';

         // don't forget the closing parenthesis
         if (level >= power_prec)
             c.s << ')';
     }

     int main()
     {
         // a sample expression
         symbol x("x"), y("y");
         ex e = -3*pow(x, 3)*pow(y, -2) + pow(x+y, 2) - 1;

         // switch to LaTeX mode
         cout << latex;

         // this prints "-1+{(y+x)}^{2}-3 \frac{x^{3}}{y^{2}}"
         cout << e << endl;

         // now we replace the method for the LaTeX output of powers with
         // our own one
         set_print_func<power, print_latex>(my_print_power_as_latex);

         // this prints "-1+{{(y+x)}}\uparrow{2}-3 \frac{{x}\uparrow{3}}{{y}
         //              \uparrow{2}}"
         cout << e << endl;
     }

Some notes:

   * The first argument of 'my_print_power_as_latex' could also have
     been a 'const basic &', the second one a 'const print_context &'.

   * The above code depends on 'mul' objects converting their operands
     to 'power' objects for the purpose of printing.

   * The output of products including negative powers as fractions is
     also controlled by the 'mul' class.

   * The 'power/print_latex' method provided by GiNaC prints square
     roots using '\sqrt', but the above code doesn't.

It's not possible to restore a method table entry to its previous or
default value.  Once you have called 'set_print_func()', you can only
override it with another call to 'set_print_func()', but you can't
easily go back to the default behavior again (you can, of course, dig
around in the GiNaC sources, find the method that is installed at
startup ('power::do_print_latex' in this case), and 'set_print_func'
that one; that is, after you circumvent the C++ member access
control...).

6.3.2 Print methods for functions
---------------------------------

Symbolic functions employ a print method dispatch mechanism similar to
the one used for classes.  The methods are specified with
'print_func<C>()' function options.  If you don't specify any special
print methods, the function will be printed with its name (or LaTeX
name, if supplied), followed by a comma-separated list of arguments
enclosed in parentheses.

For example, this is what GiNaC's 'abs()' function is defined like:

     static ex abs_eval(const ex & arg) { ... }
     static ex abs_evalf(const ex & arg) { ... }

     static void abs_print_latex(const ex & arg, const print_context & c)
     {
         c.s << "{|"; arg.print(c); c.s << "|}";
     }

     static void abs_print_csrc_float(const ex & arg, const print_context & c)
     {
         c.s << "fabs("; arg.print(c); c.s << ")";
     }

     REGISTER_FUNCTION(abs, eval_func(abs_eval).
                            evalf_func(abs_evalf).
                            print_func<print_latex>(abs_print_latex).
                            print_func<print_csrc_float>(abs_print_csrc_float).
                            print_func<print_csrc_double>(abs_print_csrc_float));

This will display 'abs(x)' as '|x|' in LaTeX mode and 'fabs(x)' in
non-CLN C source output, but as 'abs(x)' in all other formats.

There is currently no equivalent of 'set_print_func()' for functions.

6.3.3 Adding new output formats
-------------------------------

Creating a new output format involves subclassing 'print_context', which
is somewhat similar to adding a new algebraic class (see Adding classes).
There is a macro 'GINAC_DECLARE_PRINT_CONTEXT' that needs
to go into the class definition, and a corresponding macro
'GINAC_IMPLEMENT_PRINT_CONTEXT' that has to appear at global scope.
Every 'print_context' class needs to provide a default constructor and a
constructor from an 'std::ostream' and an 'unsigned' options value.

Here is an example for a user-defined 'print_context' class:

     class print_myformat : public print_dflt
     {
         GINAC_DECLARE_PRINT_CONTEXT(print_myformat, print_dflt)
     public:
         print_myformat(std::ostream & os, unsigned opt = 0)
          : print_dflt(os, opt) {}
     };

     print_myformat::print_myformat() : print_dflt(std::cout) {}

     GINAC_IMPLEMENT_PRINT_CONTEXT(print_myformat, print_dflt)

That's all there is to it.  None of the actual expression output logic
is implemented in this class.  It merely serves as a selector for
choosing a particular format.  The algorithms for printing expressions
in the new format are implemented as print methods, as described above.

'print_myformat' is a subclass of 'print_dflt', so it behaves exactly
like GiNaC's default output format:

     {
         symbol x("x");
         ex e = pow(x, 2) + 1;

         // this prints "1+x^2"
         cout << e << endl;

         // this also prints "1+x^2"
         e.print(print_myformat()); cout << endl;

         ...
     }

To fill 'print_myformat' with life, we need to supply appropriate print
methods with 'set_print_func()', like this:

     // This prints powers with '**' instead of '^'. See the LaTeX output
     // example above for explanations.
     void print_power_as_myformat(const power & p,
                                  const print_myformat & c,
                                  unsigned level)
     {
         unsigned power_prec = p.precedence();
         if (level >= power_prec)
             c.s << '(';
         p.op(0).print(c, power_prec);
         c.s << "**";
         p.op(1).print(c, power_prec);
         if (level >= power_prec)
             c.s << ')';
     }

     {
         ...
         // install a new print method for power objects
         set_print_func<power, print_myformat>(print_power_as_myformat);

         // now this prints "1+x**2"
         e.print(print_myformat()); cout << endl;

         // but the default format is still "1+x^2"
         cout << e << endl;
     }

6.4 Structures
==============

If you are doing some very specialized things with GiNaC, or if you just
need some more organized way to store data in your expressions instead
of anonymous lists, you may want to implement your own algebraic
classes.  ('algebraic class' means any class directly or indirectly
derived from 'basic' that can be used in GiNaC expressions).

GiNaC offers two ways of accomplishing this: either by using the
'structure<T>' template class, or by rolling your own class from
scratch.  This section will discuss the 'structure<T>' template which is
easier to use but more limited, while the implementation of custom GiNaC
classes is the topic of the next section.  However, you may want to read
both sections because many common concepts and member functions are
shared by both concepts, and it will also allow you to decide which
approach is most suited to your needs.

The 'structure<T>' template, defined in the GiNaC header file
'structure.h', wraps a type that you supply (usually a C++ 'struct' or
'class') into a GiNaC object that can be used in expressions.

6.4.1 Example: scalar products
------------------------------

Let's suppose that we need a way to handle some kind of abstract scalar
product of the form '<x|y>' in expressions.  Objects of the scalar
product class have to store their left and right operands, which can in
turn be arbitrary expressions.  Here is a possible way to represent such
a product in a C++ 'struct':

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

     struct sprod_s {
         ex left, right;

         sprod_s() {}
         sprod_s(ex l, ex r) : left(l), right(r) {}
     };

The default constructor is required.  Now, to make a GiNaC class out of
this data structure, we need only one line:

     typedef structure<sprod_s> sprod;

That's it.  This line constructs an algebraic class 'sprod' which
contains objects of type 'sprod_s'.  We can now use 'sprod' in
expressions like any other GiNaC class:

     ...
         symbol a("a"), b("b");
         ex e = sprod(sprod_s(a, b));
     ...

Note the difference between 'sprod' which is the algebraic class, and
'sprod_s' which is the unadorned C++ structure containing the 'left' and
'right' data members.  As shown above, an 'sprod' can be constructed
from an 'sprod_s' object.

If you find the nested 'sprod(sprod_s())' constructor too unwieldy, you
could define a little wrapper function like this:

     inline ex make_sprod(ex left, ex right)
     {
         return sprod(sprod_s(left, right));
     }

The 'sprod_s' object contained in 'sprod' can be accessed with the GiNaC
'ex_to<>()' function followed by the '->' operator or 'get_struct()':

     ...
         cout << ex_to<sprod>(e)->left << endl;
          // -> a
         cout << ex_to<sprod>(e).get_struct().right << endl;
          // -> b
     ...

You only have read access to the members of 'sprod_s'.

The type definition of 'sprod' is enough to write your own algorithms
that deal with scalar products, for example:

     ex swap_sprod(ex p)
     {
         if (is_a<sprod>(p)) {
             const sprod_s & sp = ex_to<sprod>(p).get_struct();
             return make_sprod(sp.right, sp.left);
         } else
             return p;
     }

     ...
         f = swap_sprod(e);
          // f is now <b|a>
     ...

6.4.2 Structure output
----------------------

While the 'sprod' type is useable it still leaves something to be
desired, most notably proper output:

     ...
         cout << e << endl;
          // -> [structure object]
     ...

By default, any structure types you define will be printed as
'[structure object]'.  To override this you can either specialize the
template's 'print()' member function, or specify print methods with
'set_print_func<>()', as described in Printing.  Unfortunately,
it's not possible to supply class options like 'print_func<>()' to
structures, so for a self-contained structure type you need to resort to
overriding the 'print()' function, which is also what we will do here.

The member functions of GiNaC classes are described in more detail in
the next section, but it shouldn't be hard to figure out what's going on
here:

     void sprod::print(const print_context & c, unsigned level) const
     {
         // tree debug output handled by superclass
         if (is_a<print_tree>(c))
             inherited::print(c, level);

         // get the contained sprod_s object
         const sprod_s & sp = get_struct();

         // print_context::s is a reference to an ostream
         c.s << "<" << sp.left << "|" << sp.right << ">";
     }

Now we can print expressions containing scalar products:

     ...
         cout << e << endl;
          // -> <a|b>
         cout << swap_sprod(e) << endl;
          // -> <b|a>
     ...

6.4.3 Comparing structures
--------------------------

The 'sprod' class defined so far still has one important drawback: all
scalar products are treated as being equal because GiNaC doesn't know
how to compare objects of type 'sprod_s'.  This can lead to some
confusing and undesired behavior:

     ...
         cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
          // -> 0
         cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
          // -> 2*<a|b> or 2*<a^2|b^2> (which one is undefined)
     ...

To remedy this, we first need to define the operators '==' and '<' for
objects of type 'sprod_s':

     inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
     {
         return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
     }

     inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
     {
         return lhs.left.compare(rhs.left) < 0
                ? true : lhs.right.compare(rhs.right) < 0;
     }

The ordering established by the '<' operator doesn't have to make any
algebraic sense, but it needs to be well defined.  Note that we can't
use expressions like 'lhs.left == rhs.left' or 'lhs.left < rhs.left' in
the implementation of these operators because they would construct GiNaC
'relational' objects which in the case of '<' do not establish a well
defined ordering (for arbitrary expressions, GiNaC can't decide which
one is algebraically 'less').

Next, we need to change our definition of the 'sprod' type to let GiNaC
know that an ordering relation exists for the embedded objects:

     typedef structure<sprod_s, compare_std_less> sprod;

'sprod' objects then behave as expected:

     ...
         cout << make_sprod(a, b) - make_sprod(a*a, b*b) << endl;
          // -> <a|b>-<a^2|b^2>
         cout << make_sprod(a, b) + make_sprod(a*a, b*b) << endl;
          // -> <a|b>+<a^2|b^2>
         cout << make_sprod(a, b) - make_sprod(a, b) << endl;
          // -> 0
         cout << make_sprod(a, b) + make_sprod(a, b) << endl;
          // -> 2*<a|b>
     ...

The 'compare_std_less' policy parameter tells GiNaC to use the
'std::less' and 'std::equal_to' functors to compare objects of type
'sprod_s'.  By default, these functors forward their work to the
standard '<' and '==' operators, which we have overloaded.
Alternatively, we could have specialized 'std::less' and 'std::equal_to'
for class 'sprod_s'.

GiNaC provides two other comparison policies for 'structure<T>' objects:
the default 'compare_all_equal', and 'compare_bitwise' which does a
bit-wise comparison of the contained 'T' objects.  This should be used
with extreme care because it only works reliably with built-in integral
types, and it also compares any padding (filler bytes of undefined
value) that the 'T' class might have.

6.4.4 Subexpressions
--------------------

Our scalar product class has two subexpressions: the left and right
operands.  It might be a good idea to make them accessible via the
standard 'nops()' and 'op()' methods:

     size_t sprod::nops() const
     {
         return 2;
     }

     ex sprod::op(size_t i) const
     {
         switch (i) {
         case 0:
             return get_struct().left;
         case 1:
             return get_struct().right;
         default:
             throw std::range_error("sprod::op(): no such operand");
         }
     }

Implementing 'nops()' and 'op()' for container types such as 'sprod' has
two other nice side effects:

   * 'has()' works as expected
   * GiNaC generates better hash keys for the objects (the default
     implementation of 'calchash()' takes subexpressions into account)

There is a non-const variant of 'op()' called 'let_op()' that allows
replacing subexpressions:

     ex & sprod::let_op(size_t i)
     {
         // every non-const member function must call this
         ensure_if_modifiable();

         switch (i) {
         case 0:
             return get_struct().left;
         case 1:
             return get_struct().right;
         default:
             throw std::range_error("sprod::let_op(): no such operand");
         }
     }

Once we have provided 'let_op()' we also get 'subs()' and 'map()' for
free.  In fact, every container class that returns a non-null 'nops()'
value must either implement 'let_op()' or provide custom implementations
of 'subs()' and 'map()'.

In turn, the availability of 'map()' enables the recursive behavior of a
couple of other default method implementations, in particular 'evalf()',
'evalm()', 'normal()', 'diff()' and 'expand()'.  Although we probably
want to provide our own version of 'expand()' for scalar products that
turns expressions like '<a+b|c>' into '<a|c>+<b|c>'.  This is left as an
exercise for the reader.

The 'structure<T>' template defines many more member functions that you
can override by specialization to customize the behavior of your
structures.  You are referred to the next section for a description of
some of these (especially 'eval()').  There is, however, one topic that
shall be addressed here, as it demonstrates one peculiarity of the
'structure<T>' template: archiving.

6.4.5 Archiving structures
--------------------------

If you don't know how the archiving of GiNaC objects is implemented, you
should first read the next section and then come back here.  You're
back?  Good.

To implement archiving for structures it is not enough to provide
specializations for the 'archive()' member function and the unarchiving
constructor (the 'unarchive()' function has a default implementation).
You also need to provide a unique name (as a string literal) for each
structure type you define.  This is because in GiNaC archives, the class
of an object is stored as a string, the class name.

By default, this class name (as returned by the 'class_name()' member
function) is 'structure' for all structure classes.  This works as long
as you have only defined one structure type, but if you use two or more
you need to provide a different name for each by specializing the
'get_class_name()' member function.  Here is a sample implementation for
enabling archiving of the scalar product type defined above:

     const char *sprod::get_class_name() { return "sprod"; }

     void sprod::archive(archive_node & n) const
     {
         inherited::archive(n);
         n.add_ex("left", get_struct().left);
         n.add_ex("right", get_struct().right);
     }

     sprod::structure(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
     {
         n.find_ex("left", get_struct().left, sym_lst);
         n.find_ex("right", get_struct().right, sym_lst);
     }

Note that the unarchiving constructor is 'sprod::structure' and not
'sprod::sprod', and that we don't need to supply an 'sprod::unarchive()'
function.

6.5 Adding classes
==================

The 'structure<T>' template provides an way to extend GiNaC with custom
algebraic classes that is easy to use but has its limitations, the most
severe of which being that you can't add any new member functions to
structures.  To be able to do this, you need to write a new class
definition from scratch.

This section will explain how to implement new algebraic classes in
GiNaC by giving the example of a simple 'string' class.  After reading
this section you will know how to properly declare a GiNaC class and
what the minimum required member functions are that you have to
implement.  We only cover the implementation of a 'leaf' class here
(i.e.  one that doesn't contain subexpressions).  Creating a container
class like, for example, a class representing tensor products is more
involved but this section should give you enough information so you can
consult the source to GiNaC's predefined classes if you want to
implement something more complicated.

6.5.1 Hierarchy of algebraic classes.
-------------------------------------

All algebraic classes (that is, all classes that can appear in
expressions) in GiNaC are direct or indirect subclasses of the class
'basic'.  So a 'basic *' represents a generic pointer to an algebraic
class.  Working with such pointers directly is cumbersome (think of
memory management), hence GiNaC wraps them into 'ex' (see Expressions are reference counted).
To make such wrapping possible every
algebraic class has to implement several methods.  Visitors (see
Visitors and tree traversal), printing, and (un)archiving (see
Input/output) require helper methods too.  But don't worry, most of
the work is simplified by the following macros (defined in
'registrar.h'):
   * 'GINAC_DECLARE_REGISTERED_CLASS'
   * 'GINAC_IMPLEMENT_REGISTERED_CLASS'
   * 'GINAC_IMPLEMENT_REGISTERED_CLASS_OPT'

The 'GINAC_DECLARE_REGISTERED_CLASS' macro inserts declarations required
for memory management, visitors, printing, and (un)archiving.  It takes
the name of the class and its direct superclass as arguments.  The
'GINAC_DECLARE_REGISTERED_CLASS' should be the first line after the
opening brace of the class definition.

'GINAC_IMPLEMENT_REGISTERED_CLASS' takes the same arguments as
'GINAC_DECLARE_REGISTERED_CLASS'.  It initializes certain static members
of a class so that printing and (un)archiving works.  The
'GINAC_IMPLEMENT_REGISTERED_CLASS' may appear anywhere else in the
source (at global scope, of course, not inside a function).

'GINAC_IMPLEMENT_REGISTERED_CLASS_OPT' is a variant of
'GINAC_IMPLEMENT_REGISTERED_CLASS'.  It allows specifying additional
options, such as custom printing functions.

6.5.2 A minimalistic example
----------------------------

Now we will start implementing a new class 'mystring' that allows
placing character strings in algebraic expressions (this is not very
useful, but it's just an example).  This class will be a direct subclass
of 'basic'.  You can use this sample implementation as a starting point
for your own classes (1).

The code snippets given here assume that you have included some header
files as follows:

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

Now we can write down the class declaration.  The class stores a C++
'string' and the user shall be able to construct a 'mystring' object
from a string:

     class mystring : public basic
     {
         GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)

     public:
         mystring(const string & s);

     private:
         string str;
     };

     GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)

The 'GINAC_DECLARE_REGISTERED_CLASS' macro insert declarations required
for memory management, visitors, printing, and (un)archiving.
'GINAC_IMPLEMENT_REGISTERED_CLASS' initializes certain static members of
a class so that printing and (un)archiving works.

Now there are three member functions we have to implement to get a
working class:

   * 'mystring()', the default constructor.

   * 'int compare_same_type(const basic & other)', which is used
     internally by GiNaC to establish a canonical sort order for terms.
     It returns 0, +1 or -1, depending on the relative order of this
     object and the 'other' object.  If it returns 0, the objects are
     considered equal.  *Please notice:* This has nothing to do with the
     (numeric) ordering relationship expressed by '<', '>=' etc (which
     cannot be defined for non-numeric classes).  For example,
     'numeric(1).compare_same_type(numeric(2))' may return +1 even
     though 1 is clearly smaller than 2.  Every GiNaC class must provide
     a 'compare_same_type()' function, even those representing objects
     for which no reasonable algebraic ordering relationship can be
     defined.

   * And, of course, 'mystring(const string& s)' which is the
     constructor we declared.

Let's proceed step-by-step.  The default constructor looks like this:

     mystring::mystring() { }

In the default constructor you should set all other member variables to
reasonable default values (we don't need that here since our 'str'
member gets set to an empty string automatically).

Our 'compare_same_type()' function uses a provided function to compare
the string members:

     int mystring::compare_same_type(const basic & other) const
     {
         const mystring &o = static_cast<const mystring &>(other);
         int cmpval = str.compare(o.str);
         if (cmpval == 0)
             return 0;
         else if (cmpval < 0)
             return -1;
         else
             return 1;
     }

Although this function takes a 'basic &', it will always be a reference
to an object of exactly the same class (objects of different classes are
not comparable), so the cast is safe.  If this function returns 0, the
two objects are considered equal (in the sense that A-B=0), so you
should compare all relevant member variables.

Now the only thing missing is our constructor:

     mystring::mystring(const string& s) : str(s) { }

No surprises here.  We set the 'str' member from the argument.

That's it!  We now have a minimal working GiNaC class that can store
strings in algebraic expressions.  Let's confirm that the RTTI works:

     ex e = mystring("Hello, world!");
     cout << is_a<mystring>(e) << endl;
      // -> 1 (true)

     cout << ex_to<basic>(e).class_name() << endl;
      // -> mystring

Obviously it does.  Let's see what the expression 'e' looks like:

     cout << e << endl;
      // -> [mystring object]

Hm, not exactly what we expect, but of course the 'mystring' class
doesn't yet know how to print itself.  This can be done either by
implementing the 'print()' member function, or, preferably, by
specifying a 'print_func<>()' class option.  Let's say that we want to
print the string surrounded by double quotes:

     class mystring : public basic
     {
         ...
     protected:
         void do_print(const print_context & c, unsigned level = 0) const;
         ...
     };

     void mystring::do_print(const print_context & c, unsigned level) const
     {
         // print_context::s is a reference to an ostream
         c.s << '\"' << str << '\"';
     }

The 'level' argument is only required for container classes to correctly
parenthesize the output.

Now we need to tell GiNaC that 'mystring' objects should use the
'do_print()' member function for printing themselves.  For this, we
replace the line

     GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)

with

     GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(mystring, basic,
       print_func<print_context>(&mystring::do_print))

Let's try again to print the expression:

     cout << e << endl;
      // -> "Hello, world!"

Much better.  If we wanted to have 'mystring' objects displayed in a
different way depending on the output format (default, LaTeX, etc.), we
would have supplied multiple 'print_func<>()' options with different
template parameters ('print_dflt', 'print_latex', etc.), separated by
dots.  This is similar to the way options are specified for symbolic
functions.  See Printing, for a more in-depth description of the way
expression output is implemented in GiNaC.

The 'mystring' class can be used in arbitrary expressions:

     e += mystring("GiNaC rulez");
     cout << e << endl;
      // -> "GiNaC rulez"+"Hello, world!"

(GiNaC's automatic term reordering is in effect here), or even

     e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
     cout << e << endl;
      // -> "One string"^(2*sin(-"Another string"+Pi))

Whether this makes sense is debatable but remember that this is only an
example.  At least it allows you to implement your own symbolic
algorithms for your objects.

Note that GiNaC's algebraic rules remain unchanged:

     e = mystring("Wow") * mystring("Wow");
     cout << e << endl;
      // -> "Wow"^2

     e = pow(mystring("First")-mystring("Second"), 2);
     cout << e.expand() << endl;
      // -> -2*"First"*"Second"+"First"^2+"Second"^2

There's no way to, for example, make GiNaC's 'add' class perform string
concatenation.  You would have to implement this yourself.

6.5.3 Automatic evaluation
--------------------------

When dealing with objects that are just a little more complicated than
the simple string objects we have implemented, chances are that you will
want to have some automatic simplifications or canonicalizations
performed on them.  This is done in the evaluation member function
'eval()'.  Let's say that we wanted all strings automatically converted
to lowercase with non-alphabetic characters stripped, and empty strings
removed:

     class mystring : public basic
     {
         ...
     public:
         ex eval() const override;
         ...
     };

     ex mystring::eval() const
     {
         string new_str;
         for (size_t i=0; i<str.length(); i++) {
             char c = str[i];
             if (c >= 'A' && c <= 'Z')
                 new_str += tolower(c);
             else if (c >= 'a' && c <= 'z')
                 new_str += c;
         }

         if (new_str.length() == 0)
             return 0;

         return mystring(new_str).hold();
     }

The 'hold()' member function sets a flag in the object that prevents
further evaluation.  Otherwise we might end up in an endless loop.  When
you want to return the object unmodified, use 'return this->hold();'.

If our class had subobjects, we would have to evaluate them first
(unless they are all of type 'ex', which are automatically evaluated).
We don't have any subexpressions in the 'mystring' class, so we are not
concerned with this.

Let's confirm that it works:

     ex e = mystring("Hello, world!") + mystring("!?#");
     cout << e << endl;
      // -> "helloworld"

     e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
     cout << e << endl;
      // -> 3*"wow"

6.5.4 Optional member functions
-------------------------------

We have implemented only a small set of member functions to make the
class work in the GiNaC framework.  There are two functions that are not
strictly required but will make operations with objects of the class
more efficient:

     unsigned calchash() const override;
     bool is_equal_same_type(const basic & other) const override;

The 'calchash()' method returns an 'unsigned' hash value for the object
which will allow GiNaC to compare and canonicalize expressions much more
efficiently.  You should consult the implementation of some of the
built-in GiNaC classes for examples of hash functions.  The default
implementation of 'calchash()' calculates a hash value out of the
'tinfo_key' of the class and all subexpressions that are accessible via
'op()'.

'is_equal_same_type()' works like 'compare_same_type()' but only tests
for equality without establishing an ordering relation, which is often
faster.  The default implementation of 'is_equal_same_type()' just calls
'compare_same_type()' and tests its result for zero.

6.5.5 Other member functions
----------------------------

For a real algebraic class, there are probably some more functions that
you might want to provide:

     bool info(unsigned inf) const override;
     ex evalf() const override;
     ex series(const relational & r, int order, unsigned options = 0) const override;
     ex derivative(const symbol & s) const override;

If your class stores sub-expressions (see the scalar product example in
the previous section) you will probably want to override

     size_t nops() const override;
     ex op(size_t i) const override;
     ex & let_op(size_t i) override;
     ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
     ex map(map_function & f) const override;

'let_op()' is a variant of 'op()' that allows write access.  The default
implementations of 'subs()' and 'map()' use it, so you have to implement
either 'let_op()', or 'subs()' and 'map()'.

You can, of course, also add your own new member functions.  Remember
that the RTTI may be used to get information about what kinds of objects
you are dealing with (the position in the class hierarchy) and that you
can always extract the bare object from an 'ex' by stripping the 'ex'
off using the 'ex_to<mystring>(e)' function when that should become a
need.

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

6.5.6 Upgrading extension classes from older version of GiNaC
-------------------------------------------------------------

GiNaC used to use a custom run time type information system (RTTI). It
was removed from GiNaC. Thus, one needs to rewrite constructors which
set 'tinfo_key' (which does not exist any more).  For example,

     myclass::myclass() : inherited(&myclass::tinfo_static) {}

needs to be rewritten as

     myclass::myclass() {}

   ---------- Footnotes ----------

   (1) The self-contained source for this example is included in GiNaC,
see the 'doc/examples/mystring.cpp' file.

7 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.

7.1 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
     'struct's or 'class'es 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
     generators...

   * 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.

   * multiple interfaces: Though real GiNaC programs have to be written
     in some editor, then be compiled, linked and executed, there are
     more ways to work with the GiNaC engine.  Many people want to play
     with expressions interactively, as in traditional CASs: The tiny
     'ginsh' that comes with the distribution exposes many, but not all,
     of GiNaC's types to a command line.

   * seamless 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 'int' and 'double' are
     sufficient?  For pure symbolic applications, GiNaC is comparable in
     speed with other CAS.

7.2 Disadvantages
=================

Of course it also has some disadvantages:

   * 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, 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 only by compiled without hassle on systems with the C++
     compiler from the GNU Compiler Collection (GCC).(1) GiNaC uses
     recent language features like explicit constructors, mutable
     members, RTTI, 'dynamic_cast's and STL, so ANSI compliance is meant
     literally.

   ---------- Footnotes ----------

   (1) This is because CLN uses PROVIDE/REQUIRE like macros to let the
compiler gather all static initializations, which works for GNU C++
only.  Feel free to contact the authors in case you really believe that
you need to use a different compiler.  We have occasionally used other
compilers and may be able to give you advice.

7.3 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.

Appendix A Internal structures
******************************

A.1 Expressions are reference counted
=====================================

In GiNaC, there is an _intrusive reference-counting_ mechanism at work
where the counter belongs to the algebraic objects derived from class
'basic' but is maintained by the smart pointer class 'ptr', of which
'ex' contains an instance.  If you understood that, you can safely skip
the rest of this passage.

Expressions are extremely light-weight since internally they work like
handles to the actual representation.  They really hold 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 <iostream>
     #include <ginac/ginac.h>
     using namespace std;
     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:

     {
         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's
(https://isocpp.org/faq) chapter on memory management.  It covers this
issue and presents an implementation which is pretty close to the one in
GiNaC.

A.2 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:

<PICTURE MISSING>

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:

<PICTURE MISSING>

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):

<PICTURE MISSING>

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'.

Appendix B 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.  The 'pkg-config' utility makes this process easier.
GiNaC supplies all necessary data in 'ginac.pc' (installed into
'/usr/local/lib/pkgconfig' by default).  To compile a simple program use
(1)
     g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp

This command line might expand to (for example):
     g++ -o simple -lginac -lcln simple.cpp

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

For packages configured using GNU automake, 'pkg-config' also provides
the 'PKG_CHECK_MODULES' macro to automate the process of checking for
libraries

     PKG_CHECK_MODULES(MYAPP, ginac >= MINIMUM_VERSION,
                       [ACTION-IF-FOUND],
                       [ACTION-IF-NOT-FOUND])

This macro:

   * Determines the location of GiNaC using data from 'ginac.pc', which
     is either found in the default 'pkg-config' search path, or from
     the environment variable 'PKG_CONFIG_PATH'.

   * Tests the installed libraries to make sure that their version is
     later than MINIMUM-VERSION.

   * If the required version was found, sets the 'MYAPP_CFLAGS' variable
     to the output of 'pkg-config --cflags ginac' and the 'MYAPP_LIBS'
     variable to the output of 'pkg-config --libs ginac', 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, executes
     ACTION-IF-NOT-FOUND.

   ---------- Footnotes ----------

   (1) If GiNaC is installed into some non-standard directory PREFIX one
should set the PKG_CONFIG_PATH environment variable to
PREFIX/lib/pkgconfig for this to work.

B.1 Configuring a package that uses GiNaC
=========================================

The directory where the GiNaC libraries are installed needs to be found
by your system's dynamic linkers (both compile- and run-time ones).  See
the documentation of your system linker for details.  Also make sure
that 'ginac.pc' is in 'pkg-config''s search path, see pkg-config (*manpages*).

The short summary below describes how to do this on a GNU/Linux system.

Suppose GiNaC is installed into the directory 'PREFIX'.  To tell the
linkers where to find the library one should

   * edit '/etc/ld.so.conf' and run 'ldconfig'.  For example,
          # echo PREFIX/lib >> /etc/ld.so.conf
          # ldconfig

   * or set the environment variables 'LD_LIBRARY_PATH' and
     'LD_RUN_PATH'
          $ export LD_LIBRARY_PATH=PREFIX/lib
          $ export LD_RUN_PATH=PREFIX/lib

   * or give a '-L' and '--rpath' flags when running configure, for
     instance:

          $ LDFLAGS='-Wl,-LPREFIX/lib -Wl,--rpath=PREFIX/lib' ./configure

To tell 'pkg-config' where the 'ginac.pc' file is, set the
'PKG_CONFIG_PATH' environment variable:
     $ export PKG_CONFIG_PATH=PREFIX/lib/pkgconfig

Finally, run the 'configure' script
     $ ./configure

B.2 Example of a package using GiNaC
====================================

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

     #include <iostream>
     #include <ginac/ginac.h>

     int main()
     {
         GiNaC::symbol x("x");
         GiNaC::ex a = GiNaC::sin(x);
         std::cout << "Derivative of " << a
                   << " is " << a.diff(x) << std::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.ac', which is used to build the
configure script:

     dnl Process this file with autoreconf to produce a configure script.
     AC_INIT([simple], 1.0.0, bogus@example.net)
     AC_CONFIG_SRCDIR(simple.cpp)
     AM_INIT_AUTOMAKE([foreign 1.8])

     AC_PROG_CXX
     AC_PROG_INSTALL
     AC_LANG([C++])

     PKG_CHECK_MODULES(SIMPLE, ginac >= 1.3.7)

     AC_OUTPUT(Makefile)

The 'PKG_CHECK_MODULES' macro does the following: If a GiNaC version
greater or equal than 1.3.7 is found, then it defines SIMPLE_CFLAGS and
SIMPLE_LIBS.  Otherwise, it dies with the error message like
     configure: error: Package requirements (ginac >= 1.3.7) were not met:

     Requested 'ginac >= 1.3.7' but version of GiNaC is 1.3.5

     Consider adjusting the PKG_CONFIG_PATH environment variable if you
     installed software in a non-standard prefix.

     Alternatively, you may set the environment variables SIMPLE_CFLAGS
     and SIMPLE_LIBS to avoid the need to call pkg-config.
     See the pkg-config man page for more details.

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
     simple_CPPFLAGS = $(SIMPLE_CFLAGS)
     simple_LDADD = $(SIMPLE_LIBS)

This 'Makefile.am', says that we are building a single executable, from
a single source file 'simple.cpp'.  Since every program we are building
uses GiNaC we could have simply added SIMPLE_CFLAGS to CPPFLAGS and
SIMPLE_LIBS to LIBS.  However, it is more flexible to specify libraries
and complier options on a per-program basis.

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

Now execute the following command:

     $ autoreconf -i

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

     $ ./configure
     $ make
     $ make install

Appendix C Bibliography
***********************

   - 'ISO/IEC 14882:2011: Programming Languages: C++'

   - 'CLN: A Class Library for Numbers', Bruno Haible <haible@ilog.fr>

   - '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', James H. Davenport, Yvon Siret and Evelyne Tournier,
     ISBN 0-12-204230-1, 1988, Academic Press, London

   - 'Computer Algebra Systems - A Practical Guide', Michael J. Wester
     (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester

   - 'The Art of Computer Programming, Vol 2: Seminumerical Algorithms',
     Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley

   - 'Pi Unleashed', Jörg Arndt and Christoph Haenel, ISBN
     3-540-66572-2, 2001, Springer, Heidelberg

   - 'The Role of gamma5 in Dimensional Regularization', Dirk Kreimer,
     hep-ph/9401354