This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2003 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
@page
@vskip 0pt plus 1filll
-Copyright @copyright{} 1999-2001 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2003 Johannes Gutenberg University Mainz, Germany
@sp 2
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
present day computer algebra systems (CAS) are linguistically and
semantically impoverished. Although they are quite powerful tools for
learning math and solving particular problems they lack modern
-linguistical structures that allow for the creation of large-scale
+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
@section License
The GiNaC framework for symbolic computation within the C++ programming
-language is Copyright @copyright{} 1999-2001 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2003 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
pointless) bivariate polynomial with some large coefficients:
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
generates Hermite polynomials in a specified free variable.
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
Linear equation systems can be solved along with basic linear
algebra manipulations over symbolic expressions. In C++ GiNaC offers
a matrix class for this purpose but we can see what it can do using
-@command{ginsh}'s notation of double brackets to type them in:
+@command{ginsh}'s bracket notation to type them in:
@example
> lsolve(a+x*y==z,x);
y^(-1)*(z-a);
-> lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
-[x==19/8,y==-1/40]
-> M = [[ [[1, 3]], [[-3, 2]] ]];
-[[ [[1,3]], [[-3,2]] ]]
+> lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
+@{x==19/8,y==-1/40@}
+> M = [ [1, 3], [-3, 2] ];
+[[1,3],[-3,2]]
> determinant(M);
11
> charpoly(M,lambda);
lambda^2-3*lambda+11
+> A = [ [1, 1], [2, -1] ];
+[[1,1],[2,-1]]
+> A+2*M;
+[[1,1],[2,-1]]+2*[[1,3],[-3,2]]
+> evalm(%);
+[[3,7],[-4,3]]
+> B = [ [0, 0, a], [b, 1, -b], [-1/a, 0, 0] ];
+> evalm(B^(2^12345));
+[[1,0,0],[0,1,0],[0,0,1]]
@end example
Multivariate polynomials and rational functions may be expanded,
> 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(");
+> 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);
-Euler-1/12+Order((x-1/2*Pi)^3)
@end example
-Here we have made use of the @command{ginsh}-command @code{"} to pop the
+Here we have made use of the @command{ginsh}-command @code{%} to pop the
previously evaluated element from @command{ginsh}'s internal stack.
If you ever wanted to convert units in C or C++ and found this is
In order to install GiNaC on your system, some prerequisites need to be
met. First of all, you need to have a C++-compiler adhering to the
-ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
-development so if you have a different compiler you are on your own.
-For the configuration to succeed you need a Posix compliant shell
-installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
-by the built process as well, since some of the source files are
-automatically generated by Perl scripts. Last but not least, Bruno
-Haible's library @acronym{CLN} is extensively used and needs to be
-installed on your system. Please get it either from
-@uref{ftp://ftp.santafe.edu/pub/gnu/}, from
+ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used GCC for development
+so if you have a different compiler you are on your own. For the
+configuration to succeed you need a Posix compliant shell installed in
+@file{/bin/sh}, GNU @command{bash} is fine. Perl is needed by the built
+process as well, since some of the source files are automatically
+generated by Perl scripts. Last but not least, Bruno Haible's library
+CLN is extensively used and needs to be installed on your system.
+Please get it either from @uref{ftp://ftp.santafe.edu/pub/gnu/}, from
@uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
site} (it is covered by GPL) and install it prior to trying to install
@end itemize
-In addition, you may specify some environment variables.
-@env{CXX} holds the path and the name of the C++ compiler
-in case you want to override the default in your path. (The
-@command{configure} script searches your path for @command{c++},
-@command{g++}, @command{gcc}, @command{CC}, @command{cxx}
-and @command{cc++} in that order.) It may be very useful to
-define some compiler flags with the @env{CXXFLAGS} environment
-variable, like optimization, debugging information and warning
-levels. If omitted, it defaults to @option{-g -O2}.
+In addition, you may specify some environment variables. @env{CXX}
+holds the path and the name of the C++ compiler in case you want to
+override the default in your path. (The @command{configure} script
+searches your path for @command{c++}, @command{g++}, @command{gcc},
+@command{CC}, @command{cxx} and @command{cc++} in that order.) It may
+be very useful to define some compiler flags with the @env{CXXFLAGS}
+environment variable, like optimization, debugging information and
+warning levels. If omitted, it defaults to @option{-g
+-O2}.@footnote{The @command{configure} script is itself generated from
+the file @file{configure.ac}. It is only distributed in packaged
+releases of GiNaC. If you got the naked sources, e.g. from CVS, you
+must generate @command{configure} along with the various
+@file{Makefile.in} by using the @command{autogen.sh} script. This will
+require a fair amount of support from your local toolchain, though.}
The whole process is illustrated in the following two
examples. (Substitute @command{setenv @var{VARIABLE} @var{value}} for
@end example
And here is a configuration for a private static GiNaC library with
-several components sitting in custom places (site-wide @acronym{GCC} and
-private @acronym{CLN}). The compiler is pursuaded to be picky and full
-assertions and debugging information are switched on:
+several components sitting in custom places (site-wide GCC and private
+CLN). The compiler is persuaded to be picky and full assertions and
+debugging information are switched on:
@example
$ export CXX=/usr/local/gnu/bin/c++
$ export CPPFLAGS="$(CPPFLAGS) -I$(HOME)/include"
-$ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -ansi -pedantic"
+$ export CXXFLAGS="$(CXXFLAGS) -DDO_GINAC_ASSERT -ggdb -Wall -pedantic"
$ export LDFLAGS="$(LDFLAGS) -L$(HOME)/lib"
$ ./configure --disable-shared --prefix=$(HOME)
@end example
to fiddle around with optimization.
Generally, the top-level Makefile runs recursively to the
-subdirectories. It is therfore safe to go into any subdirectory
-(@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
+subdirectories. It is therefore safe to go into any subdirectory
+(@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
@var{target} there in case something went wrong.
@menu
* Expressions:: The fundamental GiNaC class.
+* Automatic evaluation:: Evaluation and canonicalization.
+* Error handling:: How the library reports errors.
* The Class Hierarchy:: Overview of GiNaC's classes.
* Symbols:: Symbolic objects.
* Numbers:: Numerical objects.
* Constants:: Pre-defined constants.
-* Fundamental containers:: The power, add and mul classes.
+* Fundamental containers:: Sums, products and powers.
* Lists:: Lists of expressions.
* Mathematical functions:: Mathematical functions.
* Relations:: Equality, Inequality and all that.
+* Matrices:: Matrices.
* Indexed objects:: Handling indexed quantities.
* Non-commutative objects:: Algebras with non-commutative products.
@end menu
-@node Expressions, The Class Hierarchy, Basic Concepts, Basic Concepts
+@node Expressions, Automatic evaluation, Basic Concepts, Basic Concepts
@c node-name, next, previous, up
@section Expressions
@cindex expression (class @code{ex})
The most common class of objects a user deals with is the expression
@code{ex}, representing a mathematical object like a variable, number,
-function, sum, product, etc... Expressions may be put together to form
+function, sum, product, etc@dots{} Expressions may be put together to form
new expressions, passed as arguments to functions, and so on. Here is a
little collection of valid expressions:
@code{ex}.
-@node The Class Hierarchy, Symbols, Expressions, Basic Concepts
+@node Automatic evaluation, Error handling, Expressions, Basic Concepts
+@c node-name, next, previous, up
+@section Automatic evaluation and canonicalization of expressions
+@cindex evaluation
+
+GiNaC performs some automatic transformations on expressions, to simplify
+them and put them into a canonical form. Some examples:
+
+@example
+ex MyEx1 = 2*x - 1 + x; // 3*x-1
+ex MyEx2 = x - x; // 0
+ex MyEx3 = cos(2*Pi); // 1
+ex MyEx4 = x*y/x; // y
+@end example
+
+This behavior is usually referred to as @dfn{automatic} or @dfn{anonymous
+evaluation}. GiNaC only performs transformations that are
+
+@itemize @bullet
+@item
+at most of complexity @math{O(n log n)}
+@item
+algebraically correct, possibly except for a set of measure zero (e.g.
+@math{x/x} is transformed to @math{1} although this is incorrect for @math{x=0})
+@end itemize
+
+There are two types of automatic transformations in GiNaC that may not
+behave in an entirely obvious way at first glance:
+
+@itemize
+@item
+The terms of sums and products (and some other things like the arguments of
+symmetric functions, the indices of symmetric tensors etc.) are re-ordered
+into a canonical form that is deterministic, but not lexicographical or in
+any other way easily guessable (it almost always depends on the number and
+order of the symbols you define). However, constructing the same expression
+twice, either implicitly or explicitly, will always result in the same
+canonical form.
+@item
+Expressions of the form 'number times sum' are automatically expanded (this
+has to do with GiNaC's internal representation of sums and products). For
+example
+@example
+ex MyEx5 = 2*(x + y); // 2*x+2*y
+ex MyEx6 = z*(x + y); // z*(x+y)
+@end example
+@end itemize
+
+The general rule is that when you construct expressions, GiNaC automatically
+creates them in canonical form, which might differ from the form you typed in
+your program. This may create some awkward looking output (@samp{-y+x} instead
+of @samp{x-y}) but allows for more efficient operation and usually yields
+some immediate simplifications.
+
+@cindex @code{eval()}
+Internally, the anonymous evaluator in GiNaC is implemented by the methods
+
+@example
+ex ex::eval(int level = 0) const;
+ex basic::eval(int level = 0) const;
+@end example
+
+but unless you are extending GiNaC with your own classes or functions, there
+should never be any reason to call them explicitly. All GiNaC methods that
+transform expressions, like @code{subs()} or @code{normal()}, automatically
+re-evaluate their results.
+
+
+@node Error handling, The Class Hierarchy, Automatic evaluation, Basic Concepts
+@c node-name, next, previous, up
+@section Error handling
+@cindex exceptions
+@cindex @code{pole_error} (class)
+
+GiNaC reports run-time errors by throwing C++ exceptions. All exceptions
+generated by GiNaC are subclassed from the standard @code{exception} class
+defined in the @file{<stdexcept>} header. In addition to the predefined
+@code{logic_error}, @code{domain_error}, @code{out_of_range},
+@code{invalid_argument}, @code{runtime_error}, @code{range_error} and
+@code{overflow_error} types, GiNaC also defines a @code{pole_error}
+exception that gets thrown when trying to evaluate a mathematical function
+at a singularity.
+
+The @code{pole_error} class has a member function
+
+@example
+int pole_error::degree() const;
+@end example
+
+that returns the order of the singularity (or 0 when the pole is
+logarithmic or the order is undefined).
+
+When using GiNaC it is useful to arrange for exceptions to be catched in
+the main program even if you don't want to do any special error handling.
+Otherwise whenever an error occurs in GiNaC, it will be delegated to the
+default exception handler of your C++ compiler's run-time system which
+usually only aborts the program without giving any information what went
+wrong.
+
+Here is an example for a @code{main()} function that catches and prints
+exceptions generated by GiNaC:
+
+@example
+#include <iostream>
+#include <stdexcept>
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ try @{
+ ...
+ // code using GiNaC
+ ...
+ @} catch (exception &p) @{
+ cerr << p.what() << endl;
+ return 1;
+ @}
+ return 0;
+@}
+@end example
+
+
+@node The Class Hierarchy, Symbols, Error handling, Basic Concepts
@c node-name, next, previous, up
@section The Class Hierarchy
@cindex container
@cindex atom
-To get an idea about what kinds of symbolic composits may be built we
+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:
@dots{}
@item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
@item @code{function} @tab A symbolic function like @math{sin(2*x)}
-@item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
-@item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
+@item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
+@item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
@item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
@item @code{indexed} @tab Indexed object like @math{A_ij}
@item @code{tensor} @tab Special tensor like the delta and metric tensors
@item @code{idx} @tab Index of an indexed object
@item @code{varidx} @tab Index with variance
@item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
+@item @code{wildcard} @tab Wildcard for pattern matching
+@item @code{structure} @tab Template for user-defined classes
@end multitable
@end cartouche
+
@node Symbols, Numbers, The Class Hierarchy, Basic Concepts
@c node-name, next, previous, up
@section Symbols
Although symbols can be assigned expressions for internal reasons, you
should not do it (and we are not going to tell you how it is done). If
you want to replace a symbol with something else in an expression, you
-can use the expression's @code{.subs()} method (@xref{Substituting Expressions},
-for more information).
+can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
@node Numbers, Constants, Symbols, Basic Concepts
@cindex CLN
@cindex rational
@cindex fraction
-For storing numerical things, GiNaC uses Bruno Haible's library
-@acronym{CLN}. The classes therein serve as foundation classes for
-GiNaC. @acronym{CLN} stands for Class Library for Numbers or
-alternatively for Common Lisp Numbers. In order to find out more about
-@acronym{CLN}'s internals the reader is refered to the documentation of
-that library. @inforef{Introduction, , cln}, for more
-information. Suffice to say that it is by itself build on top of another
-library, the GNU Multiple Precision library @acronym{GMP}, which is an
+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. @inforef{Introduction, , cln}, for
+more information. Suffice to say that it is by itself build on top of
+another library, the GNU Multiple Precision library GMP, which is an
extremely fast library for arbitrary long integers and rationals as well
as arbitrary precision floating point numbers. It is very commonly used
-by several popular cryptographic applications. @acronym{CLN} extends
-@acronym{GMP} by several useful things: First, it introduces the complex
-number field over either reals (i.e. floating point numbers with
-arbitrary precision) or rationals. Second, it automatically converts
-rationals to integers if the denominator is unity and complex numbers to
-real numbers if the imaginary part vanishes and also correctly treats
-algebraic functions. Third it provides good implementations of
-state-of-the-art algorithms for all trigonometric and hyperbolic
-functions as well as for calculation of some useful constants.
+by several popular cryptographic applications. CLN extends GMP by
+several useful things: First, it introduces the complex number field
+over either reals (i.e. floating point numbers with arbitrary precision)
+or rationals. Second, it automatically converts rationals to integers
+if the denominator is unity and complex numbers to real numbers if the
+imaginary part vanishes and also correctly treats algebraic functions.
+Third it provides good implementations of state-of-the-art algorithms
+for all trigonometric and hyperbolic functions as well as for
+calculation of some useful constants.
The user can construct an object of class @code{numeric} in several
ways. The following example shows the four most important constructors.
integers, construction from C-float and construction from a string:
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace GiNaC;
int main()
@{
- numeric two(2); // exact integer 2
+ numeric two = 2; // exact integer 2
numeric r(2,3); // exact fraction 2/3
numeric e(2.71828); // floating point number
- numeric p("3.1415926535897932385"); // floating point number
+ numeric p = "3.14159265358979323846"; // constructor from string
// Trott's constant in scientific notation:
numeric trott("1.0841015122311136151E-2");
std::cout << two*p << std::endl; // floating point 6.283...
-@}
+ ...
@end example
-Note that all those constructors are @emph{explicit} which means you are
-not allowed to write @code{numeric two=2;}. This is because the basic
-objects to be handled by GiNaC are the expressions @code{ex} and we want
-to keep things simple and wish objects like @code{pow(x,2)} to be
-handled the same way as @code{pow(x,a)}, which means that we need to
-allow a general @code{ex} as base and exponent. Therefore there is an
-implicit constructor from C-integers directly to expressions handling
-numerics at work in most of our examples. This design really becomes
-convenient when one declares own functions having more than one
-parameter but it forbids using implicit constructors because that would
-lead to compile-time ambiguities.
+@cindex @code{I}
+@cindex complex numbers
+The imaginary unit in GiNaC is a predefined @code{numeric} object with the
+name @code{I}:
+
+@example
+ ...
+ numeric z1 = 2-3*I; // exact complex number 2-3i
+ numeric z2 = 5.9+1.6*I; // complex floating point number
+@}
+@end example
-It may be tempting to construct numbers writing @code{numeric r(3/2)}.
+It may be tempting to construct fractions by writing @code{numeric r(3/2)}.
This would, however, call C's built-in operator @code{/} for integers
first and result in a numeric holding a plain integer 1. @strong{Never
use the operator @code{/} on integers} unless you know exactly what you
digits:
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
@example
in 17 digits:
-0.333333333333333333
-3.14159265358979324
+0.33333333333333333334
+3.1415926535897932385
in 60 digits:
-0.333333333333333333333333333333333333333333333333333333333333333333
-3.14159265358979323846264338327950288419716939937510582097494459231
+0.33333333333333333333333333333333333333333333333333333333333333333334
+3.1415926535897932384626433832795028841971693993751058209749445923078
@end example
+@cindex rounding
+Note that the last number is not necessarily rounded as you would
+naively expect it to be rounded in the decimal system. But note also,
+that in both cases you got a couple of extra digits. This is because
+numbers are internally stored by CLN as chunks of binary digits in order
+to match your machine's word size and to not waste precision. Thus, on
+architectures with different word size, the above output might even
+differ with regard to actually computed digits.
+
It should be clear that objects of class @code{numeric} should be used
for constructing numbers or for doing arithmetic with them. The objects
one deals with most of the time are the polymorphic expressions @code{ex}.
some multiple of its denominator and test what comes out:
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
holds a rational number represented as integer numerator and integer
denominator. When multiplied by 10, the denominator becomes unity and
the result is automatically converted to a pure integer again.
-Internally, the underlying @acronym{CLN} is responsible for this
-behaviour and we refer the reader to @acronym{CLN}'s documentation.
-Suffice to say that the same behaviour applies to complex numbers as
-well as return values of certain functions. Complex numbers are
-automatically converted to real numbers if the imaginary part becomes
-zero. The full set of tests that can be applied is listed in the
-following table.
+Internally, the underlying CLN is responsible for this behavior and we
+refer the reader to CLN's documentation. Suffice to say that
+the same behavior applies to complex numbers as well as return values of
+certain functions. Complex numbers are automatically converted to real
+numbers if the imaginary part becomes zero. The full set of tests that
+can be applied is listed in the following table.
@cartouche
@multitable @columnfractions .30 .70
@end multitable
@end cartouche
+@subsection Converting numbers
+
+Sometimes it is desirable to convert a @code{numeric} object back to a
+built-in arithmetic type (@code{int}, @code{double}, etc.). The @code{numeric}
+class provides a couple of methods for this purpose:
+
+@cindex @code{to_int()}
+@cindex @code{to_long()}
+@cindex @code{to_double()}
+@cindex @code{to_cl_N()}
+@example
+int numeric::to_int() const;
+long numeric::to_long() const;
+double numeric::to_double() const;
+cln::cl_N numeric::to_cl_N() const;
+@end example
+
+@code{to_int()} and @code{to_long()} only work when the number they are
+applied on is an exact integer. Otherwise the program will halt with a
+message like @samp{Not a 32-bit integer}. @code{to_double()} applied on a
+rational number will return a floating-point approximation. Both
+@code{to_int()/to_long()} and @code{to_double()} discard the imaginary
+part of complex numbers.
+
@node Constants, Fundamental containers, Numbers, Basic Concepts
@c node-name, next, previous, up
@node Fundamental containers, Lists, Constants, Basic Concepts
@c node-name, next, previous, up
-@section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
+@section Sums, products and powers
@cindex polynomial
@cindex @code{add}
@cindex @code{mul}
@cindex @code{power}
-Simple polynomial expressions are written down in GiNaC pretty much like
+Simple rational expressions are written down in GiNaC pretty much like
in other CAS or like expressions involving numerical variables in C.
The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
been overloaded to achieve this goal. When you run the following
Also, expressions involving integer exponents are very frequently used,
which makes it even more dangerous to overload @code{^} since it is then
hard to distinguish between the semantics as exponentiation and the one
-for exclusive or. (It would be embarassing to return @code{1} where one
+for exclusive or. (It would be embarrassing to return @code{1} where one
has requested @code{2^3}.)
@end itemize
and safe simplifications are carried out like transforming
@code{3*x+4-x} to @code{2*x+4}.
-The general rule is that when you construct such objects, GiNaC
-automatically creates them in canonical form, which might differ from
-the form you typed in your program. This allows for rapid comparison of
-expressions, since after all @code{a-a} is simply zero. Note, that the
-canonical form is not necessarily lexicographical ordering or in any way
-easily guessable. It is only guaranteed that constructing the same
-expression twice, either implicitly or explicitly, results in the same
-canonical form.
-
@node Lists, Mathematical functions, Fundamental containers, Basic Concepts
@c node-name, next, previous, up
@cindex @code{op()}
@cindex @code{append()}
@cindex @code{prepend()}
+@cindex @code{remove_first()}
+@cindex @code{remove_last()}
+@cindex @code{remove_all()}
-The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
-These are sometimes used to supply a variable number of arguments of the same
-type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
-should have a basic understanding about them.
+The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
+expressions. They are not as ubiquitous as in many other computer algebra
+packages, but are sometimes used to supply a variable number of arguments of
+the same type to GiNaC methods such as @code{subs()} and @code{to_rational()},
+so you should have a basic understanding of them.
-Lists of up to 15 expressions can be directly constructed from single
+Lists of up to 16 expressions can be directly constructed from single
expressions:
@example
symbol x("x"), y("y");
lst l(x, 2, y, x+y);
// now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
- // ...
+ ...
@end example
Use the @code{nops()} method to determine the size (number of expressions) of
-a list and the @code{op()} method to access individual elements:
+a list and the @code{op()} method or the @code{[]} operator to access
+individual elements:
@example
- // ...
- cout << l.nops() << endl; // prints '4'
- cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
- // ...
+ ...
+ cout << l.nops() << endl; // prints '4'
+ cout << l.op(2) << " " << l[0] << endl; // prints 'y x'
+ ...
@end example
-Finally you can append or prepend an expression to a list with the
-@code{append()} and @code{prepend()} methods:
+As with the standard @code{list<T>} container, accessing random elements of a
+@code{lst} is generally an operation of order @math{O(N)}. Faster read-only
+sequential access to the elements of a list is possible with the
+iterator types provided by the @code{lst} class:
@example
- // ...
- l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
- l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
+typedef ... lst::const_iterator;
+typedef ... lst::const_reverse_iterator;
+lst::const_iterator lst::begin() const;
+lst::const_iterator lst::end() const;
+lst::const_reverse_iterator lst::rbegin() const;
+lst::const_reverse_iterator lst::rend() const;
+@end example
+
+For example, to print the elements of a list individually you can use:
+
+@example
+ ...
+ // O(N)
+ for (lst::const_iterator i = l.begin(); i != l.end(); ++i)
+ cout << *i << endl;
+ ...
+@end example
+
+which is one order faster than
+
+@example
+ ...
+ // O(N^2)
+ for (size_t i = 0; i < l.nops(); ++i)
+ cout << l.op(i) << endl;
+ ...
+@end example
+
+These iterators also allow you to use some of the algorithms provided by
+the C++ standard library:
+
+@example
+ ...
+ // print the elements of the list (requires #include <iterator>)
+ copy(l.begin(), l.end(), ostream_iterator<ex>(cout, "\n"));
+
+ // sum up the elements of the list (requires #include <numeric>)
+ ex sum = accumulate(l.begin(), l.end(), ex(0));
+ cout << sum << endl; // prints '2+2*x+2*y'
+ ...
+@end example
+
+@code{lst} is one of the few GiNaC classes that allow in-place modifications
+(the only other one is @code{matrix}). You can modify single elements:
+
+@example
+ ...
+ l[1] = 42; // l is now @{x, 42, y, x+y@}
+ l.let_op(1) = 7; // l is now @{x, 7, y, x+y@}
+ ...
+@end example
+
+You can append or prepend an expression to a list with the @code{append()}
+and @code{prepend()} methods:
+
+@example
+ ...
+ l.append(4*x); // l is now @{x, 7, y, x+y, 4*x@}
+ l.prepend(0); // l is now @{0, x, 7, y, x+y, 4*x@}
+ ...
+@end example
+
+You can remove the first or last element of a list with @code{remove_first()}
+and @code{remove_last()}:
+
+@example
+ ...
+ l.remove_first(); // l is now @{x, 7, y, x+y, 4*x@}
+ l.remove_last(); // l is now @{x, 7, y, x+y@}
+ ...
+@end example
+
+You can remove all the elements of a list with @code{remove_all()}:
+
+@example
+ ...
+ l.remove_all(); // l is now empty
+ ...
+@end example
+
+You can bring the elements of a list into a canonical order with @code{sort()}:
+
+@example
+ ...
+ lst l1(x, 2, y, x+y);
+ lst l2(2, x+y, x, y);
+ l1.sort();
+ l2.sort();
+ // l1 and l2 are now equal
+ ...
+@end example
+
+Finally, you can remove all but the first element of consecutive groups of
+elements with @code{unique()}:
+
+@example
+ ...
+ lst l3(x, 2, 2, 2, y, x+y, y+x);
+ l3.unique(); // l3 is now @{x, 2, y, x+y@}
@}
@end example
instance, all trigonometric and hyperbolic functions are implemented
(@xref{Built-in Functions}, for a complete list).
-These functions are all objects of class @code{function}. They accept
-one or more expressions as arguments and return one expression. If the
-arguments are not numerical, the evaluation of the function may be
-halted, as it does in the next example, showing how a function returns
-itself twice and finally an expression that may be really useful:
+These functions (better called @emph{pseudofunctions}) are all objects
+of class @code{function}. They accept one or more expressions as
+arguments and return one expression. If the arguments are not
+numerical, the evaluation of the function may be halted, as it does in
+the next example, showing how a function returns itself twice and
+finally an expression that may be really useful:
@cindex Gamma function
@cindex @code{subs()}
expansion and so on. Read the next chapter in order to learn more about
this.
+It must be noted that these pseudofunctions are created by inline
+functions, where the argument list is templated. This means that
+whenever you call @code{GiNaC::sin(1)} it is equivalent to
+@code{sin(ex(1))} and will therefore not result in a floating point
+number. Unless of course the function prototype is explicitly
+overridden -- which is the case for arguments of type @code{numeric}
+(not wrapped inside an @code{ex}). Hence, in order to obtain a floating
+point number of class @code{numeric} you should call
+@code{sin(numeric(1))}. This is almost the same as calling
+@code{sin(1).evalf()} except that the latter will return a numeric
+wrapped inside an @code{ex}.
-@node Relations, Indexed objects, Mathematical functions, Basic Concepts
+
+@node Relations, Matrices, Mathematical functions, Basic Concepts
@c node-name, next, previous, up
@section Relations
@cindex @code{relational} (class)
@code{expand()} must be called explicitly.
-@node Indexed objects, Non-commutative objects, Relations, Basic Concepts
+@node Matrices, Indexed objects, Relations, Basic Concepts
+@c node-name, next, previous, up
+@section Matrices
+@cindex @code{matrix} (class)
+
+A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
+matrix with @math{m} rows and @math{n} columns are accessed with two
+@code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
+second one in the range 0@dots{}@math{n-1}.
+
+There are a couple of ways to construct matrices, with or without preset
+elements:
+
+@cindex @code{lst_to_matrix()}
+@cindex @code{diag_matrix()}
+@cindex @code{unit_matrix()}
+@cindex @code{symbolic_matrix()}
+@example
+matrix::matrix(unsigned r, unsigned c);
+matrix::matrix(unsigned r, unsigned c, const lst & l);
+ex lst_to_matrix(const lst & l);
+ex diag_matrix(const lst & l);
+ex unit_matrix(unsigned x);
+ex unit_matrix(unsigned r, unsigned c);
+ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
+ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
+@end example
+
+The first two functions are @code{matrix} constructors which create a matrix
+with @samp{r} rows and @samp{c} columns. The matrix elements can be
+initialized from a (flat) list of expressions @samp{l}. Otherwise they are
+all set to zero. The @code{lst_to_matrix()} function constructs a matrix
+from a list of lists, each list representing a matrix row. @code{diag_matrix()}
+constructs a diagonal matrix given the list of diagonal elements.
+@code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r} by @samp{c})
+unit matrix. And finally, @code{symbolic_matrix} constructs a matrix filled
+with newly generated symbols made of the specified base name and the
+position of each element in the matrix.
+
+Matrix elements can be accessed and set using the parenthesis (function call)
+operator:
+
+@example
+const ex & matrix::operator()(unsigned r, unsigned c) const;
+ex & matrix::operator()(unsigned r, unsigned c);
+@end example
+
+It is also possible to access the matrix elements in a linear fashion with
+the @code{op()} method. But C++-style subscripting with square brackets
+@samp{[]} is not available.
+
+Here are a couple of examples of constructing matrices:
+
+@example
+@{
+ symbol a("a"), b("b");
+
+ matrix M(2, 2);
+ M(0, 0) = a;
+ M(1, 1) = b;
+ cout << M << endl;
+ // -> [[a,0],[0,b]]
+
+ cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
+ // -> [[a,0],[0,b]]
+
+ cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
+ // -> [[a,0],[0,b]]
+
+ cout << diag_matrix(lst(a, b)) << endl;
+ // -> [[a,0],[0,b]]
+
+ cout << unit_matrix(3) << endl;
+ // -> [[1,0,0],[0,1,0],[0,0,1]]
+
+ cout << symbolic_matrix(2, 3, "x") << endl;
+ // -> [[x00,x01,x02],[x10,x11,x12]]
+@}
+@end example
+
+@cindex @code{transpose()}
+There are three ways to do arithmetic with matrices. The first (and most
+direct one) is to use the methods provided by the @code{matrix} class:
+
+@example
+matrix matrix::add(const matrix & other) const;
+matrix matrix::sub(const matrix & other) const;
+matrix matrix::mul(const matrix & other) const;
+matrix matrix::mul_scalar(const ex & other) const;
+matrix matrix::pow(const ex & expn) const;
+matrix matrix::transpose() const;
+@end example
+
+All of these methods return the result as a new matrix object. Here is an
+example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
+and @math{C}:
+
+@example
+@{
+ matrix A(2, 2, lst(1, 2, 3, 4));
+ matrix B(2, 2, lst(-1, 0, 2, 1));
+ matrix C(2, 2, lst(8, 4, 2, 1));
+
+ matrix result = A.mul(B).sub(C.mul_scalar(2));
+ cout << result << endl;
+ // -> [[-13,-6],[1,2]]
+ ...
+@}
+@end example
+
+@cindex @code{evalm()}
+The second (and probably the most natural) way is to construct an expression
+containing matrices with the usual arithmetic operators and @code{pow()}.
+For efficiency reasons, expressions with sums, products and powers of
+matrices are not automatically evaluated in GiNaC. You have to call the
+method
+
+@example
+ex ex::evalm() const;
+@end example
+
+to obtain the result:
+
+@example
+@{
+ ...
+ ex e = A*B - 2*C;
+ cout << e << endl;
+ // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
+ cout << e.evalm() << endl;
+ // -> [[-13,-6],[1,2]]
+ ...
+@}
+@end example
+
+The non-commutativity of the product @code{A*B} in this example is
+automatically recognized by GiNaC. There is no need to use a special
+operator here. @xref{Non-commutative objects}, for more information about
+dealing with non-commutative expressions.
+
+Finally, you can work with indexed matrices and call @code{simplify_indexed()}
+to perform the arithmetic:
+
+@example
+@{
+ ...
+ idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
+ e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
+ cout << e << endl;
+ // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
+ cout << e.simplify_indexed() << endl;
+ // -> [[-13,-6],[1,2]].i.j
+@}
+@end example
+
+Using indices is most useful when working with rectangular matrices and
+one-dimensional vectors because you don't have to worry about having to
+transpose matrices before multiplying them. @xref{Indexed objects}, for
+more information about using matrices with indices, and about indices in
+general.
+
+The @code{matrix} class provides a couple of additional methods for
+computing determinants, traces, and characteristic polynomials:
+
+@cindex @code{determinant()}
+@cindex @code{trace()}
+@cindex @code{charpoly()}
+@example
+ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
+ex matrix::trace() const;
+ex matrix::charpoly(const ex & lambda) const;
+@end example
+
+The @samp{algo} argument of @code{determinant()} allows to select
+between different algorithms for calculating the determinant. The
+asymptotic speed (as parametrized by the matrix size) can greatly differ
+between those algorithms, depending on the nature of the matrix'
+entries. The possible values are defined in the @file{flags.h} header
+file. By default, GiNaC uses a heuristic to automatically select an
+algorithm that is likely (but not guaranteed) to give the result most
+quickly.
+
+@cindex @code{inverse()}
+@cindex @code{solve()}
+Matrices may also be inverted using the @code{ex matrix::inverse()}
+method and linear systems may be solved with:
+
+@example
+matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
+@end example
+
+Assuming the matrix object this method is applied on is an @code{m}
+times @code{n} matrix, then @code{vars} must be a @code{n} times
+@code{p} matrix of symbolic indeterminates and @code{rhs} a @code{m}
+times @code{p} matrix. The returned matrix then has dimension @code{n}
+times @code{p} and in the case of an underdetermined system will still
+contain some of the indeterminates from @code{vars}. If the system is
+overdetermined, an exception is thrown.
+
+
+@node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
@c node-name, next, previous, up
@section Indexed objects
A simple example shall illustrate the concepts:
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
symbol A("A");
cout << indexed(A, i, j) << endl;
// -> A.i.j
+ cout << index_dimensions << indexed(A, i, j) << endl;
+ // -> A.i[3].j[3]
+ cout << dflt; // reset cout to default output format (dimensions hidden)
...
@end example
the symbol @code{A} as its base expression and the two indices @code{i} and
@code{j}.
+The dimensions of indices are normally not visible in the output, but one
+can request them to be printed with the @code{index_dimensions} manipulator,
+as shown above.
+
Note the difference between the indices @code{i} and @code{j} which are of
-class @code{idx}, and the index values which are the sybols @code{i_sym}
+class @code{idx}, and the index values which are the symbols @code{i_sym}
and @code{j_sym}. The indices of indexed objects cannot directly be symbols
or numbers but must be index objects. For example, the following is not
correct and will raise an exception:
The methods
@example
-ex idx::get_value(void);
-ex idx::get_dimension(void);
+ex idx::get_value();
+ex idx::get_dimension();
@end example
return the value and dimension of an @code{idx} object. If you have an index
in an expression, such as returned by calling @code{.op()} on an indexed
object, you can get a reference to the @code{idx} object with the function
-@code{ex_to_idx()} on the expression.
+@code{ex_to<idx>()} on the expression.
There are also the methods
@example
-bool idx::is_numeric(void);
-bool idx::is_symbolic(void);
-bool idx::is_dim_numeric(void);
-bool idx::is_dim_symbolic(void);
+bool idx::is_numeric();
+bool idx::is_symbolic();
+bool idx::is_dim_numeric();
+bool idx::is_dim_symbolic();
@end example
for checking whether the value and dimension are numeric or symbolic
constructor. The two methods
@example
-bool varidx::is_covariant(void);
-bool varidx::is_contravariant(void);
+bool varidx::is_covariant();
+bool varidx::is_contravariant();
@end example
-allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
+allow you to check the variance of a @code{varidx} object (use @code{ex_to<varidx>()}
to get the object reference from an expression). There's also the very useful
method
@example
-ex varidx::toggle_variance(void);
+ex varidx::toggle_variance();
@end example
which makes a new index with the same value and dimension but the opposite
methods
@example
-bool spinidx::is_dotted(void);
-bool spinidx::is_undotted(void);
+bool spinidx::is_dotted();
+bool spinidx::is_undotted();
@end example
allow you to check whether or not a @code{spinidx} object is dotted (use
-@code{ex_to_spinidx()} to get the object reference from an expression).
+@code{ex_to<spinidx>()} to get the object reference from an expression).
Finally, the two methods
@example
-ex spinidx::toggle_dot(void);
-ex spinidx::toggle_variance_dot(void);
+ex spinidx::toggle_dot();
+ex spinidx::toggle_variance_dot();
@end example
create a new index with the same value and dimension but opposite dottedness
@end example
@subsection Symmetries
+@cindex @code{symmetry} (class)
+@cindex @code{sy_none()}
+@cindex @code{sy_symm()}
+@cindex @code{sy_anti()}
+@cindex @code{sy_cycl()}
+
+Indexed objects can have certain symmetry properties with respect to their
+indices. Symmetries are specified as a tree of objects of class @code{symmetry}
+that is constructed with the helper functions
+
+@example
+symmetry sy_none(...);
+symmetry sy_symm(...);
+symmetry sy_anti(...);
+symmetry sy_cycl(...);
+@end example
+
+@code{sy_none()} stands for no symmetry, @code{sy_symm()} and @code{sy_anti()}
+specify fully symmetric or antisymmetric, respectively, and @code{sy_cycl()}
+represents a cyclic symmetry. Each of these functions accepts up to four
+arguments which can be either symmetry objects themselves or unsigned integer
+numbers that represent an index position (counting from 0). A symmetry
+specification that consists of only a single @code{sy_symm()}, @code{sy_anti()}
+or @code{sy_cycl()} with no arguments specifies the respective symmetry for
+all indices.
+
+Here are some examples of symmetry definitions:
+
+@example
+ ...
+ // No symmetry:
+ e = indexed(A, i, j);
+ e = indexed(A, sy_none(), i, j); // equivalent
+ e = indexed(A, sy_none(0, 1), i, j); // equivalent
+
+ // Symmetric in all three indices:
+ e = indexed(A, sy_symm(), i, j, k);
+ e = indexed(A, sy_symm(0, 1, 2), i, j, k); // equivalent
+ e = indexed(A, sy_symm(2, 0, 1), i, j, k); // same symmetry, but yields a
+ // different canonical order
+
+ // Symmetric in the first two indices only:
+ e = indexed(A, sy_symm(0, 1), i, j, k);
+ e = indexed(A, sy_none(sy_symm(0, 1), 2), i, j, k); // equivalent
+
+ // Antisymmetric in the first and last index only (index ranges need not
+ // be contiguous):
+ e = indexed(A, sy_anti(0, 2), i, j, k);
+ e = indexed(A, sy_none(sy_anti(0, 2), 1), i, j, k); // equivalent
+
+ // An example of a mixed symmetry: antisymmetric in the first two and
+ // last two indices, symmetric when swapping the first and last index
+ // pairs (like the Riemann curvature tensor):
+ e = indexed(A, sy_symm(sy_anti(0, 1), sy_anti(2, 3)), i, j, k, l);
+
+ // Cyclic symmetry in all three indices:
+ e = indexed(A, sy_cycl(), i, j, k);
+ e = indexed(A, sy_cycl(0, 1, 2), i, j, k); // equivalent
+
+ // The following examples are invalid constructions that will throw
+ // an exception at run time.
+
+ // An index may not appear multiple times:
+ e = indexed(A, sy_symm(0, 0, 1), i, j, k); // ERROR
+ e = indexed(A, sy_none(sy_symm(0, 1), sy_anti(0, 2)), i, j, k); // ERROR
+
+ // Every child of sy_symm(), sy_anti() and sy_cycl() must refer to the
+ // same number of indices:
+ e = indexed(A, sy_symm(sy_anti(0, 1), 2), i, j, k); // ERROR
+
+ // And of course, you cannot specify indices which are not there:
+ e = indexed(A, sy_symm(0, 1, 2, 3), i, j, k); // ERROR
+ ...
+@end example
-Indexed objects can be declared as being totally symmetric or antisymmetric
-with respect to their indices. In this case, GiNaC will automatically bring
-the indices into a canonical order which allows for some immediate
-simplifications:
+If you need to specify more than four indices, you have to use the
+@code{.add()} method of the @code{symmetry} class. For example, to specify
+full symmetry in the first six indices you would write
+@code{sy_symm(0, 1, 2, 3).add(4).add(5)}.
+
+If an indexed object has a symmetry, GiNaC will automatically bring the
+indices into a canonical order which allows for some immediate simplifications:
@example
...
- cout << indexed(A, indexed::symmetric, i, j)
- + indexed(A, indexed::symmetric, j, i) << endl;
+ cout << indexed(A, sy_symm(), i, j)
+ + indexed(A, sy_symm(), j, i) << endl;
// -> 2*A.j.i
- cout << indexed(B, indexed::antisymmetric, i, j)
- + indexed(B, indexed::antisymmetric, j, j) << endl;
- // -> -B.j.i
- cout << indexed(B, indexed::antisymmetric, i, j)
- + indexed(B, indexed::antisymmetric, j, i) << endl;
+ cout << indexed(B, sy_anti(), i, j)
+ + indexed(B, sy_anti(), j, i) << endl;
+ // -> 0
+ cout << indexed(B, sy_anti(), i, j, k)
+ - indexed(B, sy_anti(), j, k, i) << endl;
// -> 0
...
@end example
@cindex @code{get_free_indices()}
-@cindex Dummy index
+@cindex dummy index
@subsection Dummy indices
GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
dummy nor free indices.
To be recognized as a dummy index pair, the two indices must be of the same
-class and dimension and their value must be the same single symbol (an index
-like @samp{2*n+1} is never a dummy index). If the indices are of class
+class and their value must be the same single symbol (an index like
+@samp{2*n+1} is never a dummy index). If the indices are of class
@code{varidx} they must also be of opposite variance; if they are of class
@code{spinidx} they must be both dotted or both undotted.
there is the method
@example
-ex ex::simplify_indexed(void);
+ex ex::simplify_indexed();
ex ex::simplify_indexed(const scalar_products & sp);
@end example
@itemize
@item it checks the consistency of free indices in sums in the same way
@code{get_free_indices()} does
+@item it tries to give dummy indices that appear in different terms of a sum
+ the same name to allow simplifications like @math{a_i*b_i-a_j*b_j=0}
@item it (symbolically) calculates all possible dummy index summations/contractions
with the predefined tensors (this will be explained in more detail in the
next section)
+@item it detects contractions that vanish for symmetry reasons, for example
+ the contraction of a symmetric and a totally antisymmetric tensor
@item as a special case of dummy index summation, it can replace scalar products
of two tensors with a user-defined value
@end itemize
@code{simplify_indexed()} will replace all scalar products of indexed
objects that have the symbols @code{A} and @code{B} as base expressions
with the single value 0. The number, type and dimension of the indices
-doesn't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
+don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
@cindex @code{expand()}
The example above also illustrates a feature of the @code{expand()} method:
@subsubsection Delta tensor
The delta tensor takes two indices, is symmetric and has the matrix
-representation @code{diag(1,1,1,...)}. It is constructed by the function
+representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
@code{delta_tensor()}:
@example
@}
@end example
-The matrix representation of the spinor metric is @code{[[ [[ 0, 1 ]], [[ -1, 0 ]]}.
+The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
@cindex @code{epsilon_tensor()}
@cindex @code{lorentz_eps()}
The epsilon tensor is totally antisymmetric, its number of indices is equal
to the dimension of the index space (the indices must all be of the same
numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
-defined to be 1. Its behaviour with indices that have a variance also
+defined to be 1. Its behavior with indices that have a variance also
depends on the signature of the metric. Epsilon tensors are output as
@samp{eps}.
dimensions, the last function creates an epsilon tensor in a 4-dimensional
Minkowski space (the last @code{bool} argument specifies whether the metric
has negative or positive signature, as in the case of the Minkowski metric
-tensor).
+tensor):
+
+@example
+@{
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4),
+ sig(symbol("sig"), 4), lam(symbol("lam"), 4), bet(symbol("bet"), 4);
+ e = lorentz_eps(mu, nu, rho, sig) *
+ lorentz_eps(mu.toggle_variance(), nu.toggle_variance(), lam, bet);
+ cout << simplify_indexed(e) << endl;
+ // -> 2*eta~bet~rho*eta~sig~lam-2*eta~sig~bet*eta~rho~lam
+
+ idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3);
+ symbol A("A"), B("B");
+ e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(B, k);
+ cout << simplify_indexed(e) << endl;
+ // -> -B.k*A.j*eps.i.k.j
+ e = epsilon_tensor(i, j, k) * indexed(A, j) * indexed(A, k);
+ cout << simplify_indexed(e) << endl;
+ // -> 0
+@}
+@end example
@subsection Linear algebra
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(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
cout << indexed(A, i, i) << endl;
ex e = indexed(A, i, j) * indexed(X, j);
cout << e.simplify_indexed() << endl;
- // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
+ // -> [[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
+ // -> [[3*y+3*x,6*y+2*x]].j
@}
@end example
You can of course obtain the same results with the @code{matrix::add()},
-@code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
-don't have to worry about transposing matrices.
+@code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
+but with indices you don't have to worry about transposing matrices.
Matrix indices always start at 0 and their dimension must match the number
of rows/columns of the matrix. Matrices with one row or one column are
@end itemize
The @code{clifford} and @code{color} classes are subclasses of
-@code{indexed} because the elements of these algebras ususally carry
-indices.
+@code{indexed} because the elements of these algebras usually carry
+indices. The @code{matrix} class is described in more detail in
+@ref{Matrices}.
Unlike most computer algebra systems, GiNaC does not primarily provide an
operator (often denoted @samp{&*}) for representing inert products of
obtained with the two member functions
@example
-unsigned ex::return_type(void) const;
-unsigned ex::return_type_tinfo(void) const;
+unsigned ex::return_type() const;
+unsigned ex::return_type_tinfo() const;
@end example
The @code{return_type()} function returns one of three values (defined in
but it's the same for any two objects with the same label. This is also true
for color objects.
+A last note: With the exception of matrices, positive integer powers of
+non-commutative objects are automatically expanded in GiNaC. For example,
+@code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
+non-commutative expressions).
+
@cindex @code{clifford} (class)
@subsection Clifford algebra
ex dirac_ONE(unsigned char rl = 0);
@end example
+@strong{Note:} You must always use @code{dirac_ONE()} when referring to
+multiples of the unity element, even though it's customary to omit it.
+E.g. instead of @code{dirac_gamma(mu)*(dirac_slash(q,4)+m)} you have to
+write @code{dirac_gamma(mu)*(dirac_slash(q,4)+m*dirac_ONE())}. Otherwise,
+GiNaC will complain and/or produce incorrect results.
+
@cindex @code{dirac_gamma5()}
-and there's a special element @samp{gamma5} that commutes with all other
-gammas and in 4 dimensions equals @samp{gamma~0 gamma~1 gamma~2 gamma~3},
-provided by
+There is a special element @samp{gamma5} that commutes with all other
+gammas, has a unit square, and in 4 dimensions equals
+@samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
@example
ex dirac_gamma5(unsigned char rl = 0);
@end example
-@cindex @code{dirac_gamma6()}
-@cindex @code{dirac_gamma7()}
-The two additional functions
+@cindex @code{dirac_gammaL()}
+@cindex @code{dirac_gammaR()}
+The chiral projectors @samp{(1+/-gamma5)/2} are also available as proper
+objects, constructed by
@example
-ex dirac_gamma6(unsigned char rl = 0);
-ex dirac_gamma7(unsigned char rl = 0);
+ex dirac_gammaL(unsigned char rl = 0);
+ex dirac_gammaR(unsigned char rl = 0);
@end example
-return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
-respectively.
+They observe the relations @samp{gammaL^2 = gammaL}, @samp{gammaR^2 = gammaR},
+and @samp{gammaL gammaR = gammaR gammaL = 0}.
@cindex @code{dirac_slash()}
Finally, the function
ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
@end example
-creates a term of the form @samp{e.mu gamma~mu} with a new and unique index
-whose dimension is given by the @code{dim} argument.
+creates a term that represents a contraction of @samp{e} with the Dirac
+Lorentz vector (it behaves like a term of the form @samp{e.mu gamma~mu}
+with a unique index whose dimension is given by the @code{dim} argument).
+Such slashed expressions are printed with a trailing backslash, e.g. @samp{e\}.
-The @code{simplify_indexed()} function performs contractions in gamma strings
-if possible, for example
+In products of dirac gammas, superfluous unity elements are automatically
+removed, squares are replaced by their values, and @samp{gamma5}, @samp{gammaL}
+and @samp{gammaR} are moved to the front.
+
+The @code{simplify_indexed()} function performs contractions in gamma strings,
+for example
@example
@{
ex e = dirac_gamma(mu) * dirac_slash(a, D)
* dirac_gamma(mu.toggle_variance());
cout << e << endl;
- // -> (gamma~mu*gamma~symbol10*gamma.mu)*a.symbol10
+ // -> gamma~mu*a\*gamma.mu
e = e.simplify_indexed();
cout << e << endl;
- // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
+ // -> -D*a\+2*a\
cout << e.subs(D == 4) << endl;
- // -> -2*gamma~symbol10*a.symbol10
- // [ == -2 * dirac_slash(a, D) ]
+ // -> -2*a\
...
@}
@end example
you use the function
@example
-ex dirac_trace(const ex & e, unsigned char rl = 0);
+ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
@end example
This function takes the trace of all gammas with the specified representation
-label; gammas with other labels are left standing. The @code{dirac_trace()}
-function is a linear functional that is equal to the usual trace only in
-@math{D = 4} dimensions. In particular, the functional is not cyclic in
-@math{D != 4} dimensions when acting on expressions containing @samp{gamma5},
-so it's not a proper trace. This @samp{gamma5} scheme is described in greater
-detail in @cite{The Role of gamma5 in Dimensional Regularization}.
+label; gammas with other labels are left standing. The last argument to
+@code{dirac_trace()} is the value to be returned for the trace of the unity
+element, which defaults to 4. The @code{dirac_trace()} function is a linear
+functional that is equal to the usual trace only in @math{D = 4} dimensions.
+In particular, the functional is not cyclic in @math{D != 4} dimensions when
+acting on expressions containing @samp{gamma5}, so it's not a proper trace.
+This @samp{gamma5} scheme is described in greater detail in
+@cite{The Role of gamma5 in Dimensional Regularization}.
The value of the trace itself is also usually different in 4 and in
@math{D != 4} dimensions:
dirac_gamma(mu.toggle_variance()) *
(dirac_slash(l, D) + m * dirac_ONE());
e = dirac_trace(e).simplify_indexed(sp);
- e = e.collect(lst(l, ldotq, m), true);
+ e = e.collect(lst(l, ldotq, m));
cout << e << endl;
// -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
@}
@end example
+The @code{canonicalize_clifford()} function reorders all gamma products that
+appear in an expression to a canonical (but not necessarily simple) form.
+You can use this to compare two expressions or for further simplifications:
+
+@example
+@{
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
+ ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
+ cout << e << endl;
+ // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
+
+ e = canonicalize_clifford(e);
+ cout << e << endl;
+ // -> 2*eta~mu~nu
+@}
+@end example
+
@cindex @code{color} (class)
@subsection Color algebra
ex color_ONE(unsigned char rl = 0);
@end example
+@strong{Note:} You must always use @code{color_ONE()} when referring to
+multiples of the unity element, even though it's customary to omit it.
+E.g. instead of @code{color_T(a)*(color_T(b)*indexed(X,b)+1)} you have to
+write @code{color_T(a)*(color_T(b)*indexed(X,b)+color_ONE())}. Otherwise,
+GiNaC may produce incorrect results.
+
@cindex @code{color_d()}
@cindex @code{color_f()}
-and the functions
+The functions
@example
ex color_d(const ex & a, const ex & b, const ex & 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
@menu
* Information About Expressions::
+* Numerical Evaluation::
* Substituting Expressions::
+* Pattern Matching and Advanced Substitutions::
+* Applying a Function on Subexpressions::
+* Visitors and Tree Traversal::
* Polynomial Arithmetic:: Working with polynomials.
* Rational Expressions:: Working with rational functions.
* Symbolic Differentiation::
* Series Expansion:: Taylor and Laurent expansion.
+* Symmetrization::
* Built-in Functions:: List of predefined mathematical functions.
+* Solving Linear Systems of Equations::
* Input/Output:: Input and output of expressions.
@end menu
-@node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
+@node Information About Expressions, Numerical Evaluation, Methods and Functions, Methods and Functions
@c node-name, next, previous, up
@section Getting information about expressions
@subsection Checking expression types
-@cindex @code{is_ex_of_type()}
-@cindex @code{ex_to_numeric()}
-@cindex @code{ex_to_@dots{}}
-@cindex @code{Converting ex to other classes}
+@cindex @code{is_a<@dots{}>()}
+@cindex @code{is_exactly_a<@dots{}>()}
+@cindex @code{ex_to<@dots{}>()}
+@cindex Converting @code{ex} to other classes
@cindex @code{info()}
@cindex @code{return_type()}
@cindex @code{return_type_tinfo()}
Sometimes it's useful to check whether a given expression is a plain number,
a sum, a polynomial with integer coefficients, or of some other specific type.
-GiNaC provides a couple of functions for this (the first one is actually a macro):
+GiNaC provides a couple of functions for this:
@example
-bool is_ex_of_type(const ex & e, TYPENAME t);
+bool is_a<T>(const ex & e);
+bool is_exactly_a<T>(const ex & e);
bool ex::info(unsigned flag);
-unsigned ex::return_type(void) const;
-unsigned ex::return_type_tinfo(void) const;
+unsigned ex::return_type() const;
+unsigned ex::return_type_tinfo() const;
@end example
-When the test made by @code{is_ex_of_type()} returns true, it is safe to
-call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
-one of the class names (@xref{The Class Hierarchy}, for a list of all
-classes). For example, assuming @code{e} is an @code{ex}:
+When the test made by @code{is_a<T>()} returns true, it is safe to call
+one of the functions @code{ex_to<T>()}, where @code{T} is one of the
+class names (@xref{The Class Hierarchy}, for a list of all classes). For
+example, assuming @code{e} is an @code{ex}:
@example
@{
@dots{}
- if (is_ex_of_type(e, numeric))
- numeric n = ex_to_numeric(e);
+ if (is_a<numeric>(e))
+ numeric n = ex_to<numeric>(e);
@dots{}
@}
@end example
-@code{is_ex_of_type()} allows you to check whether the top-level object of
-an expression @samp{e} is an instance of the GiNaC class @samp{t}
+@code{is_a<T>(e)} allows you to check whether the top-level object of
+an expression @samp{e} is an instance of the GiNaC class @samp{T}
(@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
e.g., for checking whether an expression is a number, a sum, or a product:
symbol x("x");
ex e1 = 42;
ex e2 = 4*x - 3;
- is_ex_of_type(e1, numeric); // true
- is_ex_of_type(e2, numeric); // false
- is_ex_of_type(e1, add); // false
- is_ex_of_type(e2, add); // true
- is_ex_of_type(e1, mul); // false
- is_ex_of_type(e2, mul); // false
+ is_a<numeric>(e1); // true
+ is_a<numeric>(e2); // false
+ is_a<add>(e1); // false
+ is_a<add>(e2); // true
+ is_a<mul>(e1); // false
+ is_a<mul>(e2); // false
@}
@end example
+In contrast, @code{is_exactly_a<T>(e)} allows you to check whether the
+top-level object of an expression @samp{e} is an instance of the GiNaC
+class @samp{T}, not including parent classes.
+
The @code{info()} method is used for checking certain attributes of
expressions. The possible values for the @code{flag} argument are defined
in @file{ginac/flags.h}, the most important being explained in the following
@multitable @columnfractions .30 .70
@item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
@item @code{numeric}
-@tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
+@tab @dots{}a number (same as @code{is_<numeric>(...)})
@item @code{real}
@tab @dots{}a real integer, rational or float (i.e. is not complex)
@item @code{rational}
@item @code{prime}
@tab @dots{}a prime integer (probabilistic primality test)
@item @code{relation}
-@tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
+@tab @dots{}a relation (same as @code{is_a<relational>(...)})
@item @code{relation_equal}
@tab @dots{}a @code{==} relation
@item @code{relation_not_equal}
@item @code{relation_greater_or_equal}
@tab @dots{}a @code{>=} relation
@item @code{symbol}
-@tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
+@tab @dots{}a symbol (same as @code{is_a<symbol>(...)})
@item @code{list}
-@tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
+@tab @dots{}a list (same as @code{is_a<lst>(...)})
@item @code{polynomial}
@tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
@item @code{integer_polynomial}
@subsection Accessing subexpressions
@cindex @code{nops()}
@cindex @code{op()}
-@cindex @code{has()}
@cindex container
@cindex @code{relational} (class)
GiNaC provides the two methods
@example
-unsigned ex::nops();
-ex ex::op(unsigned i);
+size_t ex::nops();
+ex ex::op(size_t i);
@end example
for accessing the subexpressions in the container-like GiNaC classes like
ex ex::rhs();
@end example
-Finally, the method
-
-@example
-bool ex::has(const ex & other);
-@end example
-
-checks whether an expression contains the given subexpression @code{other}.
-This only works reliably if @code{other} is of an atomic class such as a
-@code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
-@code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
-
@subsection Comparing expressions
@cindex @code{is_equal()}
@code{false}.
Actually, if you construct an expression like @code{a == b}, this will be
-represented by an object of the @code{relational} class (@xref{Relations}.)
-which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
+represented by an object of the @code{relational} class (@pxref{Relations})
+which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
There are also two methods
for checking whether one expression is equal to another, or equal to zero,
respectively.
-@strong{Warning:} You will also find an @code{ex::compare()} method in the
-GiNaC header files. This method is however only to be used internally by
-GiNaC to establish a canonical sort order for terms, and using it to compare
-expressions will give very surprising results.
+
+@subsection Ordering expressions
+@cindex @code{ex_is_less} (class)
+@cindex @code{ex_is_equal} (class)
+@cindex @code{compare()}
+
+Sometimes it is necessary to establish a mathematically well-defined ordering
+on a set of arbitrary expressions, for example to use expressions as keys
+in a @code{std::map<>} container, or to bring a vector of expressions into
+a canonical order (which is done internally by GiNaC for sums and products).
+
+The operators @code{<}, @code{>} etc. described in the last section cannot
+be used for this, as they don't implement an ordering relation in the
+mathematical sense. In particular, they are not guaranteed to be
+antisymmetric: if @samp{a} and @samp{b} are different expressions, and
+@code{a < b} yields @code{false}, then @code{b < a} doesn't necessarily
+yield @code{true}.
+
+By default, STL classes and algorithms use the @code{<} and @code{==}
+operators to compare objects, which are unsuitable for expressions, but GiNaC
+provides two functors that can be supplied as proper binary comparison
+predicates to the STL:
+
+@example
+class ex_is_less : public std::binary_function<ex, ex, bool> @{
+public:
+ bool operator()(const ex &lh, const ex &rh) const;
+@};
+
+class ex_is_equal : public std::binary_function<ex, ex, bool> @{
+public:
+ bool operator()(const ex &lh, const ex &rh) const;
+@};
+@end example
+
+For example, to define a @code{map} that maps expressions to strings you
+have to use
+
+@example
+std::map<ex, std::string, ex_is_less> myMap;
+@end example
+
+Omitting the @code{ex_is_less} template parameter will introduce spurious
+bugs because the map operates improperly.
+
+Other examples for the use of the functors:
+
+@example
+std::vector<ex> v;
+// fill vector
+...
+
+// sort vector
+std::sort(v.begin(), v.end(), ex_is_less());
+
+// count the number of expressions equal to '1'
+unsigned num_ones = std::count_if(v.begin(), v.end(),
+ std::bind2nd(ex_is_equal(), 1));
+@end example
+
+The implementation of @code{ex_is_less} uses the member function
+
+@example
+int ex::compare(const ex & other) const;
+@end example
+
+which returns @math{0} if @code{*this} and @code{other} are equal, @math{-1}
+if @code{*this} sorts before @code{other}, and @math{1} if @code{*this} sorts
+after @code{other}.
+
+
+@node Numerical Evaluation, Substituting Expressions, Information About Expressions, Methods and Functions
+@c node-name, next, previous, up
+@section Numercial Evaluation
+@cindex @code{evalf()}
+
+GiNaC keeps algebraic expressions, numbers and constants in their exact form.
+To evaluate them using floating-point arithmetic you need to call
+
+@example
+ex ex::evalf(int level = 0) const;
+@end example
+
+@cindex @code{Digits}
+The accuracy of the evaluation is controlled by the global object @code{Digits}
+which can be assigned an integer value. The default value of @code{Digits}
+is 17. @xref{Numbers}, for more information and examples.
+
+To evaluate an expression to a @code{double} floating-point number you can
+call @code{evalf()} followed by @code{numeric::to_double()}, like this:
+
+@example
+@{
+ // Approximate sin(x/Pi)
+ symbol x("x");
+ ex e = series(sin(x/Pi), x == 0, 6);
+
+ // Evaluate numerically at x=0.1
+ ex f = evalf(e.subs(x == 0.1));
+
+ // ex_to<numeric> is an unsafe cast, so check the type first
+ if (is_a<numeric>(f)) @{
+ double d = ex_to<numeric>(f).to_double();
+ cout << d << endl;
+ // -> 0.0318256
+ @} else
+ // error
+@}
+@end example
-@node Substituting Expressions, Polynomial Arithmetic, Information About Expressions, Methods and Functions
+@node Substituting Expressions, Pattern Matching and Advanced Substitutions, Numerical Evaluation, Methods and Functions
@c node-name, next, previous, up
@section Substituting expressions
@cindex @code{subs()}
expressions via the @code{.subs()} method:
@example
-ex ex::subs(const ex & e);
-ex ex::subs(const lst & syms, const lst & repls);
+ex ex::subs(const ex & e, unsigned options = 0);
+ex ex::subs(const lst & syms, const lst & repls, unsigned options = 0);
@end example
In the first form, @code{subs()} accepts a relational of the form
@}
@end example
+If you specify multiple substitutions, they are performed in parallel, so e.g.
+@code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
+
+The second form of @code{subs()} takes two lists, one for the objects to be
+replaced and one for the expressions to be substituted (both lists must
+contain the same number of elements). Using this form, you would write
+@code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
+
+The optional last argument to @code{subs()} is a combination of
+@code{subs_options} flags. There are two options available:
+@code{subs_options::no_pattern} disables pattern matching, which makes
+large @code{subs()} operations significantly faster if you are not using
+patterns. The second option, @code{subs_options::algebraic} enables
+algebraic substitutions in products and powers.
+@ref{Pattern Matching and Advanced Substitutions}, for more information
+about patterns and algebraic substitutions.
+
@code{subs()} performs syntactic substitution of any complete algebraic
object; it does not try to match sub-expressions as is demonstrated by the
following example:
cout << e1.subs(x+y == 4) << endl;
// -> 16
- ex e2 = sin(x)*cos(x);
+ ex e2 = sin(x)*sin(y)*cos(x);
cout << e2.subs(sin(x) == cos(x)) << endl;
- // -> cos(x)^2
+ // -> cos(x)^2*sin(y)
ex e3 = x+y+z;
cout << e3.subs(x+y == 4) << endl;
@}
@end example
-If you specify multiple substitutions, they are performed in parallel, so e.g.
-@code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
-
-The second form of @code{subs()} takes two lists, one for the objects to be
-replaced and one for the expressions to be substituted (both lists must
-contain the same number of elements). Using this form, you would write
-@code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
+A more powerful form of substitution using wildcards is described in the
+next section.
-@node Polynomial Arithmetic, Rational Expressions, Substituting Expressions, Methods and Functions
+@node Pattern Matching and Advanced Substitutions, Applying a Function on Subexpressions, Substituting Expressions, Methods and Functions
@c node-name, next, previous, up
-@section Polynomial arithmetic
-
-@subsection Expanding and collecting
-@cindex @code{expand()}
-@cindex @code{collect()}
+@section Pattern matching and advanced substitutions
+@cindex @code{wildcard} (class)
+@cindex Pattern matching
-A polynomial in one or more variables has many equivalent
-representations. Some useful ones serve a specific purpose. Consider
-for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
-21*y*z + 4*z^2} (written down here in output-style). It is equivalent
-to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
-representations are the recursive ones where one collects for exponents
-in one of the three variable. Since the factors are themselves
-polynomials in the remaining two variables the procedure can be
-repeated. In our expample, two possibilities would be @math{(4*y + z)*x
-+ 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
-x*z}.
+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.
-To bring an expression into expanded form, its method
+A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
+A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
+represents an arbitrary expression. Every wildcard has a @dfn{label} which is
+an unsigned integer number to allow having multiple different wildcards in a
+pattern. Wildcards are printed as @samp{$label} (this is also the way they
+are specified in @command{ginsh}). In C++ code, wildcard objects are created
+with the call
@example
-ex ex::expand();
+ex wild(unsigned label = 0);
@end example
-may be called. In our example above, this corresponds to @math{4*x*y +
-x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
-GiNaC is not easily guessable you should be prepared to see different
-orderings of terms in such sums!
+which is simply a wrapper for the @code{wildcard()} constructor with a shorter
+name.
-Another useful representation of multivariate polynomials is as a
-univariate polynomial in one of the variables with the coefficients
-being polynomials in the remaining variables. The method
-@code{collect()} accomplishes this task:
+Some examples for patterns:
+
+@multitable @columnfractions .5 .5
+@item @strong{Constructed as} @tab @strong{Output as}
+@item @code{wild()} @tab @samp{$0}
+@item @code{pow(x,wild())} @tab @samp{x^$0}
+@item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
+@item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
+@end multitable
+
+Notes:
+
+@itemize
+@item Wildcards behave like symbols and are subject to the same algebraic
+ rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
+@item As shown in the last example, to use wildcards for indices you have to
+ use them as the value of an @code{idx} object. This is because indices must
+ always be of class @code{idx} (or a subclass).
+@item Wildcards only represent expressions or subexpressions. It is not
+ possible to use them as placeholders for other properties like index
+ dimension or variance, representation labels, symmetry of indexed objects
+ etc.
+@item Because wildcards are commutative, it is not possible to use wildcards
+ as part of noncommutative products.
+@item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
+ are also valid patterns.
+@end itemize
+
+@subsection Matching expressions
+@cindex @code{match()}
+The most basic application of patterns is to check whether an expression
+matches a given pattern. This is done by the function
@example
-ex ex::collect(const ex & s, bool distributed = false);
+bool ex::match(const ex & pattern);
+bool ex::match(const ex & pattern, lst & repls);
@end example
-The first argument to @code{collect()} can also be a list of objects in which
-case the result is either a recursively collected polynomial, or a polynomial
-in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
-by the @code{distributed} flag.
+This function returns @code{true} when the expression matches the pattern
+and @code{false} if it doesn't. If used in the second form, the actual
+subexpressions matched by the wildcards get returned in the @code{repls}
+object as a list of relations of the form @samp{wildcard == expression}.
+If @code{match()} returns false, the state of @code{repls} is undefined.
+For reproducible results, the list should be empty when passed to
+@code{match()}, but it is also possible to find similarities in multiple
+expressions by passing in the result of a previous match.
-Note that the original polynomial needs to be in expanded form in order
-for @code{collect()} to be able to find the coefficients properly.
+The matching algorithm works as follows:
-@subsection Degree and coefficients
-@cindex @code{degree()}
-@cindex @code{ldegree()}
-@cindex @code{coeff()}
+@itemize
+@item A single wildcard matches any expression. If one wildcard appears
+ multiple times in a pattern, it must match the same expression in all
+ places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
+ @samp{x*(x+1)} but not @samp{x*(y+1)}).
+@item If the expression is not of the same class as the pattern, the match
+ fails (i.e. a sum only matches a sum, a function only matches a function,
+ etc.).
+@item If the pattern is a function, it only matches the same function
+ (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
+@item Except for sums and products, the match fails if the number of
+ subexpressions (@code{nops()}) is not equal to the number of subexpressions
+ of the pattern.
+@item If there are no subexpressions, the expressions and the pattern must
+ be equal (in the sense of @code{is_equal()}).
+@item Except for sums and products, each subexpression (@code{op()}) must
+ match the corresponding subexpression of the pattern.
+@end itemize
-The degree and low degree of a polynomial can be obtained using the two
-methods
+Sums (@code{add}) and products (@code{mul}) are treated in a special way to
+account for their commutativity and associativity:
+
+@itemize
+@item If the pattern contains a term or factor that is a single wildcard,
+ this one is used as the @dfn{global wildcard}. If there is more than one
+ such wildcard, one of them is chosen as the global wildcard in a random
+ way.
+@item Every term/factor of the pattern, except the global wildcard, is
+ matched against every term of the expression in sequence. If no match is
+ found, the whole match fails. Terms that did match are not considered in
+ further matches.
+@item If there are no unmatched terms left, the match succeeds. Otherwise
+ the match fails unless there is a global wildcard in the pattern, in
+ which case this wildcard matches the remaining terms.
+@end itemize
+
+In general, having more than one single wildcard as a term of a sum or a
+factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
+ambiguous results.
+
+Here are some examples in @command{ginsh} to demonstrate how it works (the
+@code{match()} function in @command{ginsh} returns @samp{FAIL} if the
+match fails, and the list of wildcard replacements otherwise):
+
+@example
+> match((x+y)^a,(x+y)^a);
+@{@}
+> match((x+y)^a,(x+y)^b);
+FAIL
+> match((x+y)^a,$1^$2);
+@{$1==x+y,$2==a@}
+> match((x+y)^a,$1^$1);
+FAIL
+> match((x+y)^(x+y),$1^$1);
+@{$1==x+y@}
+> match((x+y)^(x+y),$1^$2);
+@{$1==x+y,$2==x+y@}
+> match((a+b)*(a+c),($1+b)*($1+c));
+@{$1==a@}
+> match((a+b)*(a+c),(a+$1)*(a+$2));
+@{$1==c,$2==b@}
+ (Unpredictable. The result might also be [$1==c,$2==b].)
+> match((a+b)*(a+c),($1+$2)*($1+$3));
+ (The result is undefined. Due to the sequential nature of the algorithm
+ and the re-ordering of terms in GiNaC, the match for the first factor
+ may be @{$1==a,$2==b@} in which case the match for the second factor
+ succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
+ fail.)
+> match(a*(x+y)+a*z+b,a*$1+$2);
+ (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
+ @{$1=x+y,$2=a*z+b@}.)
+> match(a+b+c+d+e+f,c);
+FAIL
+> match(a+b+c+d+e+f,c+$0);
+@{$0==a+e+b+f+d@}
+> match(a+b+c+d+e+f,c+e+$0);
+@{$0==a+b+f+d@}
+> match(a+b,a+b+$0);
+@{$0==0@}
+> match(a*b^2,a^$1*b^$2);
+FAIL
+ (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
+ even 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@}
+@end example
+
+@subsection Matching parts of expressions
+@cindex @code{has()}
+A more general way to look for patterns in expressions is provided by the
+member function
@example
-int ex::degree(const ex & s);
-int ex::ldegree(const ex & s);
+bool ex::has(const ex & pattern);
@end example
-which also work reliably on non-expanded input polynomials (they even work
-on rational functions, returning the asymptotic degree). To extract
-a coefficient with a certain power from an expanded polynomial you use
+This function checks whether a pattern is matched by an expression itself or
+by any of its subexpressions.
+
+Again some examples in @command{ginsh} for illustration (in @command{ginsh},
+@code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
@example
-ex ex::coeff(const ex & s, int n);
+> 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.)
@end example
-You can also obtain the leading and trailing coefficients with the methods
+@cindex @code{find()}
+The method
@example
-ex ex::lcoeff(const ex & s);
-ex ex::tcoeff(const ex & s);
+bool ex::find(const ex & pattern, lst & found);
@end example
-which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
-respectively.
-
-An application is illustrated in the next example, where a multivariate
-polynomial is analyzed:
+works a bit like @code{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. @code{find()} returns false if no matches were found (in
+@command{ginsh}, it returns an empty list):
@example
-#include <ginac/ginac.h>
-using namespace std;
-using namespace GiNaC;
+> 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)@}
+@end example
-int main()
+@subsection Substituting expressions
+@cindex @code{subs()}
+Probably the most useful application of patterns is to use them for
+substituting expressions with the @code{subs()} method. Wildcards can be
+used in the search patterns as well as in the replacement expressions, where
+they get replaced by the expressions matched by them. @code{subs()} doesn't
+know anything about algebra; it performs purely syntactic substitutions.
+
+Some examples:
+
+@example
+> subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
+b^3+a^3+(x+y)^3
+> subs(a^4+b^4+(x+y)^4,$1^2==$1^3);
+b^4+a^4+(x+y)^4
+> subs((a+b+c)^2,a+b==x);
+(a+b+c)^2
+> subs((a+b+c)^2,a+b+$1==x+$1);
+(x+c)^2
+> subs(a+2*b,a+b==x);
+a+2*b
+> subs(4*x^3-2*x^2+5*x-1,x==a);
+-1+5*a-2*a^2+4*a^3
+> subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
+-1+5*x-2*a^2+4*a^3
+> subs(sin(1+sin(x)),sin($1)==cos($1));
+cos(1+cos(x))
+> expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
+a+b
+@end example
+
+The last example would be written in C++ in this way:
+
+@example
@{
- symbol x("x"), y("y");
- ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
- - pow(x+y,2) + 2*pow(y+2,2) - 8;
- ex Poly = PolyInp.expand();
-
- for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
- cout << "The x^" << i << "-coefficient is "
- << Poly.coeff(x,i) << endl;
- @}
- cout << "As polynomial in y: "
- << Poly.collect(y) << endl;
+ symbol a("a"), b("b"), x("x"), y("y");
+ e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
+ e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
+ cout << e.expand() << endl;
+ // -> a+b
@}
@end example
-When run, it returns an output in the following fashion:
+@subsection Algebraic substitutions
+Supplying the @code{subs_options::algebraic} option to @code{subs()}
+enables smarter, algebraic substitutions in products and powers. If you want
+to substitute some factors of a product, you only need to list these factors
+in your pattern. Furthermore, if an (integer) power of some expression occurs
+in your pattern and in the expression that you want the substitution to occur
+in, it can be substituted as many times as possible, without getting negative
+powers.
+
+An example clarifies it all (hopefully):
@example
-The x^0-coefficient is y^2+11*y
-The x^1-coefficient is 5*y^2-2*y
-The x^2-coefficient is -1
-The x^3-coefficient is 4*y
-As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
-@end example
+cout << (a*a*a*a+b*b*b*b+pow(x+y,4)).subs(wild()*wild()==pow(wild(),3),
+ subs_options::algebraic) << endl;
+// --> (y+x)^6+b^6+a^6
-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.
+cout << ((a+b+c)*(a+b+c)).subs(a+b==x,subs_options::algebraic) << endl;
+// --> (c+b+a)^2
+// Powers and products are smart, but addition is just the same.
-@code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
-@code{tcoeff()} and @code{collect()} can also be used to a certain degree
-with non-polynomial expressions as they not only work with symbols but with
-constants, functions and indexed objects as well:
+cout << ((a+b+c)*(a+b+c)).subs(a+b+wild()==x+wild(), subs_options::algebraic)
+ << endl;
+// --> (x+c)^2
+// As I said: addition is just the same.
-@example
-@{
- symbol a("a"), b("b"), c("c");
- idx i(symbol("i"), 3);
+cout << (pow(a,5)*pow(b,7)+2*b).subs(b*b*a==x,subs_options::algebraic) << endl;
+// --> x^3*b*a^2+2*b
- 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)
+cout << (pow(a,-5)*pow(b,-7)+2*b).subs(1/(b*b*a)==x,subs_options::algebraic)
+ << endl;
+// --> 2*b+x^3*b^(-1)*a^(-2)
- 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
-@}
+cout << (4*x*x*x-2*x*x+5*x-1).subs(x==a,subs_options::algebraic) << endl;
+// --> -1-2*a^2+4*a^3+5*a
+
+cout << (4*x*x*x-2*x*x+5*x-1).subs(pow(x,wild())==pow(a,wild()),
+ subs_options::algebraic) << endl;
+// --> -1+5*x+4*x^3-2*x^2
+// You should not really need this kind of patterns very often now.
+// But perhaps this it's-not-a-bug-it's-a-feature (c/sh)ould still change.
+
+cout << ex(sin(1+sin(x))).subs(sin(wild())==cos(wild()),
+ subs_options::algebraic) << endl;
+// --> cos(1+cos(x))
+
+cout << expand((a*sin(x+y)*sin(x+y)+a*cos(x+y)*cos(x+y)+b)
+ .subs((pow(cos(wild()),2)==1-pow(sin(wild()),2)),
+ subs_options::algebraic)) << endl;
+// --> b+a
@end example
-@subsection Polynomial division
-@cindex polynomial division
-@cindex quotient
-@cindex remainder
-@cindex pseudo-remainder
-@cindex @code{quo()}
-@cindex @code{rem()}
-@cindex @code{prem()}
-@cindex @code{divide()}
+@node Applying a Function on Subexpressions, Visitors and Tree Traversal, Pattern Matching and Advanced Substitutions, Methods and Functions
+@c node-name, next, previous, up
+@section Applying a Function on Subexpressions
+@cindex tree traversal
+@cindex @code{map()}
-The two functions
+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 @dfn{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:
@example
-ex quo(const ex & a, const ex & b, const symbol & x);
-ex rem(const ex & a, const ex & b, const symbol & x);
+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 @{
+ ...
+ @}
+@}
@end example
-compute the quotient and remainder of univariate polynomials in the variable
-@samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
+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 @code{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.
-The additional function
+GiNaC offers the @code{map()} method to aid in the implementation of such
+operations:
@example
-ex prem(const ex & a, const ex & b, const symbol & x);
+ex ex::map(map_function & f) const;
+ex ex::map(ex (*f)(const ex & e)) const;
@end example
-computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
-@math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
+In the first (preferred) form, @code{map()} takes a function object that
+is subclassed from the @code{map_function} class. In the second form, it
+takes a pointer to a function that accepts and returns an expression.
+@code{map()} constructs a new expression of the same type, applying the
+specified function on all subexpressions (in the sense of @code{op()}),
+non-recursively.
-Exact division of multivariate polynomials is performed by the function
+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 @code{map_function} class declares a virtual function call operator
+that you can overload. Here is a sample implementation of @code{calc_trace()}
+that uses @code{map()} in a recursive fashion:
@example
-bool divide(const ex & a, const ex & b, ex & q);
+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);
+ @}
+@};
@end example
-If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
-and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
-in which case the value of @code{q} is undefined.
-
+This function object could then be used like this:
-@subsection Unit, content and primitive part
-@cindex @code{unit()}
-@cindex @code{content()}
-@cindex @code{primpart()}
+@example
+@{
+ ex M = ... // expression with matrices
+ calc_trace do_trace;
+ ex tr = do_trace(M);
+@}
+@end example
-The methods
+Here is another example for you to meditate over. It removes quadratic
+terms in a variable from an expanded polynomial:
@example
-ex ex::unit(const symbol & x);
-ex ex::content(const symbol & x);
-ex ex::primpart(const symbol & x);
-@end example
+struct map_rem_quad : public map_function @{
+ ex var;
+ map_rem_quad(const ex & var_) : var(var_) @{@}
-return the unit part, content part, and primitive polynomial of a multivariate
-polynomial with respect to the variable @samp{x} (the unit part being the sign
-of the leading coefficient, the content part being the GCD of the coefficients,
-and the primitive polynomial being the input polynomial divided by the unit and
-content parts). The product of unit, content, and primitive part is the
-original polynomial.
+ 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;
+ @}
+@};
+...
-@subsection GCD and LCM
-@cindex GCD
-@cindex LCM
-@cindex @code{gcd()}
-@cindex @code{lcm()}
+@{
+ symbol x("x"), y("y");
-The functions for polynomial greatest common divisor and least common
-multiple have the synopsis
+ 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
+@}
+@end example
+
+@command{ginsh} offers a slightly different implementation of @code{map()}
+that allows applying algebraic functions to operands. The second argument
+to @code{map()} is an expression containing the wildcard @samp{$0} which
+acts as the placeholder for the operands:
@example
-ex gcd(const ex & a, const ex & b);
-ex lcm(const ex & a, const ex & b);
+> 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@}
@end example
-The functions @code{gcd()} and @code{lcm()} accept two expressions
-@code{a} and @code{b} as arguments and return a new expression, their
-greatest common divisor or least common multiple, respectively. If the
-polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
-and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
+Note that it is only possible to use algebraic functions in the second
+argument. You can not use functions like @samp{diff()}, @samp{op()},
+@samp{subs()} etc. because these are evaluated immediately:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+> 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)".
+@end example
-int main()
+
+@node Visitors and Tree Traversal, Polynomial Arithmetic, Applying a Function on Subexpressions, Methods and Functions
+@c node-name, next, previous, up
+@section Visitors and Tree Traversal
+@cindex tree traversal
+@cindex @code{visitor} (class)
+@cindex @code{accept()}
+@cindex @code{visit()}
+@cindex @code{traverse()}
+@cindex @code{traverse_preorder()}
+@cindex @code{traverse_postorder()}
+
+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 @code{get_free_indices()} because you also want to include
+dummy indices in the list, and you can't use @code{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:
+
+@example
+void gather_indices_helper(const ex & e, lst & l)
@{
- 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);
+ 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);
+ @}
+@}
- 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
+lst gather_indices(const ex & e)
+@{
+ lst l;
+ gather_indices_helper(e, l);
+ l.sort();
+ l.unique();
+ return l;
@}
@end example
+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
-@subsection Square-free decomposition
-@cindex square-free decomposition
-@cindex factorization
-@cindex @code{sqrfree()}
-
-GiNaC still lacks proper factorization support. Some form of
-factorization is, however, easily implemented by noting that factors
-appearing in a polynomial with power two or more also appear in the
-derivative and hence can easily be found by computing the GCD of the
-original polynomial and its derivatives. Any system has an interface
-for this so called square-free factorization. So we provide one, too:
@example
-ex sqrfree(const ex & a, const lst & l = lst());
+ if (is_a<idx>(e)) @{
+ ...
+ @} else if (is_a<varidx>(e)) @{
+ ...
@end example
-Here is an example that by the way illustrates how the result may depend
-on the order of differentiation:
-@example
- ...
- symbol x("x"), y("y");
- ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+y,2) * (x-y));
- cout << sqrfree(BiVarPol, lst(x,y)) << endl;
- // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
+in @code{gather_indices_helper}, the code wouldn't have worked because the
+first line "absorbs" all classes derived from @code{idx}, including
+@code{varidx}, so the special case for @code{varidx} would never have been
+executed.
- cout << sqrfree(BiVarPol, lst(y,x)) << endl;
- // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
+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).
+@code{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.
- cout << sqrfree(BiVarPol) << endl;
- // -> depending on luck, any of the above
- ...
-@end example
+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 @code{idx} and @code{varidx} while the default implementation in
+@code{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 @dfn{Visitor} design pattern,
+which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
+variation, described in detail in
+@uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
+virtual functions to the class hierarchy to implement operations, GiNaC
+provides a single "bouncing" method @code{accept()} that takes an instance
+of a special @code{visitor} class and redirects execution to the one
+@code{visit()} virtual function of the visitor that matches the type of
+object that @code{accept()} was being invoked on.
-@node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
-@c node-name, next, previous, up
-@section Rational expressions
+Visitors in GiNaC must derive from the global @code{visitor} class as well
+as from the class @code{T::visitor} of each class @code{T} they want to
+visit, and implement the member functions @code{void visit(const T &)} for
+each class.
-@subsection The @code{normal} method
-@cindex @code{normal()}
-@cindex simplification
-@cindex temporary replacement
+A call of
-Some basic form of simplification of expressions is called for frequently.
-GiNaC provides the method @code{.normal()}, which converts a rational function
-into an equivalent rational function of the form @samp{numerator/denominator}
-where numerator and denominator are coprime. If the input expression is already
-a fraction, it just finds the GCD of numerator and denominator and cancels it,
-otherwise it performs fraction addition and multiplication.
+@example
+void ex::accept(visitor & v) const;
+@end example
-@code{.normal()} can also be used on expressions which are not rational functions
-as it will replace all non-rational objects (like functions or non-integer
-powers) by temporary symbols to bring the expression to the domain of rational
-functions before performing the normalization, and re-substituting these
-symbols afterwards. This algorithm is also available as a separate method
-@code{.to_rational()}, described below.
+will then dispatch to the correct @code{visit()} member function of the
+specified visitor @code{v} for the type of GiNaC object at the root of the
+expression tree (e.g. a @code{symbol}, an @code{idx} or a @code{mul}).
-This means that both expressions @code{t1} and @code{t2} are indeed
-simplified in this little program:
+Here is an example of a visitor:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
+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
@{
- 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;
-@}
+ 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; @}
+@};
@end example
-Of course this works for multivariate polynomials too, so the ratio of
-the sample-polynomials from the section about GCD and LCM above would be
-normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
+which can be used as follows:
+
+@example
+...
+ 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"
+...
+@end example
+The @code{visit(const basic &)} method gets called for all objects that are
+not @code{numeric} or @code{add} and acts as an (optional) default.
-@subsection Numerator and denominator
-@cindex numerator
-@cindex denominator
-@cindex @code{numer()}
-@cindex @code{denom()}
+From a conceptual point of view, the @code{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.
-The numerator and denominator of an expression can be obtained with
+We can now rewrite our index example from above with a visitor:
@example
-ex ex::numer();
-ex ex::denom();
-@end example
+class gather_indices_visitor
+ : public visitor, public idx::visitor, public varidx::visitor
+@{
+ lst l;
-These functions will first normalize the expression as described above and
-then return the numerator or denominator, respectively.
+ void visit(const idx & i)
+ @{
+ l.append(i);
+ @}
+ void visit(const varidx & vi)
+ @{
+ l.append(vi.is_covariant() ? vi : vi.toggle_variance());
+ @}
-@subsection Converting to a rational expression
-@cindex @code{to_rational()}
+public:
+ const lst & get_result() // utility function
+ @{
+ l.sort();
+ l.unique();
+ return l;
+ @}
+@};
+@end example
-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
+What's missing is the tree traversal. We could implement it in
+@code{visit(const basic &)}, but GiNaC has predefined methods for this:
@example
-ex ex::to_rational(lst &l);
+void ex::traverse_preorder(visitor & v) const;
+void ex::traverse_postorder(visitor & v) const;
+void ex::traverse(visitor & v) const;
@end example
-on the expression to be converted. The supplied @code{lst} will be filled
-with the generated temporary symbols and their replacement expressions in
-a format that can be used directly for the @code{subs()} method. It can also
-already contain a list of replacements from an earlier application of
-@code{.to_rational()}, so it's possible to use it on multiple expressions
-and get consistent results.
+@code{traverse_preorder()} visits a node @emph{before} visiting its
+subexpressions, while @code{traverse_postorder()} visits a node @emph{after}
+visiting its subexpressions. @code{traverse()} is a synonym for
+@code{traverse_preorder()}.
-For example,
+Here is a new implementation of @code{gather_indices()} that uses the visitor
+and @code{traverse()}:
@example
+lst gather_indices(const ex & e)
@{
- symbol x("x");
- ex a = pow(sin(x), 2) - pow(cos(x), 2);
- ex b = sin(x) + cos(x);
- ex q;
- lst l;
- divide(a.to_rational(l), b.to_rational(l), q);
- cout << q.subs(l) << endl;
+ gather_indices_visitor v;
+ e.traverse(v);
+ return v.get_result();
@}
@end example
-will print @samp{sin(x)-cos(x)}.
-
-@node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
+@node Polynomial Arithmetic, Rational Expressions, Visitors and Tree Traversal, Methods and Functions
@c node-name, next, previous, up
-@section Symbolic differentiation
-@cindex differentiation
-@cindex @code{diff()}
-@cindex chain rule
-@cindex product rule
+@section Polynomial arithmetic
-GiNaC's objects know how to differentiate themselves. Thus, a
-polynomial (class @code{add}) knows that its derivative is the sum of
-the derivatives of all the monomials:
+@subsection Expanding and collecting
+@cindex @code{expand()}
+@cindex @code{collect()}
+@cindex @code{collect_common_factors()}
+
+A polynomial in one or more variables has many equivalent
+representations. Some useful ones serve a specific purpose. Consider
+for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
+21*y*z + 4*z^2} (written down here in output-style). It is equivalent
+to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
+representations are the recursive ones where one collects for exponents
+in one of the three variable. Since the factors are themselves
+polynomials in the remaining two variables the procedure can be
+repeated. In our example, two possibilities would be @math{(4*y + z)*x
++ 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
+x*z}.
+
+To bring an expression into expanded form, its method
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+ex ex::expand(unsigned options = 0);
+@end example
-int main()
-@{
- symbol x("x"), y("y"), z("z");
- ex P = pow(x, 5) + pow(x, 2) + y;
+may be called. In our example above, this corresponds to @math{4*x*y +
+x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
+GiNaC is not easily guessable you should be prepared to see different
+orderings of terms in such sums!
- cout << P.diff(x,2) << endl; // 20*x^3 + 2
- cout << P.diff(y) << endl; // 1
- cout << P.diff(z) << endl; // 0
-@}
+Another useful representation of multivariate polynomials is as a
+univariate polynomial in one of the variables with the coefficients
+being polynomials in the remaining variables. The method
+@code{collect()} accomplishes this task:
+
+@example
+ex ex::collect(const ex & s, bool distributed = false);
@end example
-If a second integer parameter @var{n} is given, the @code{diff} method
-returns the @var{n}th derivative.
+The first argument to @code{collect()} can also be a list of objects in which
+case the result is either a recursively collected polynomial, or a polynomial
+in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
+by the @code{distributed} flag.
-If @emph{every} object and every function is told what its derivative
-is, all derivatives of composed objects can be calculated using the
-chain rule and the product rule. Consider, for instance the expression
-@code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
-@code{sinh(x)} and the derivative of @code{pow(x,-1)} is
-@code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
-out that the composition is the generating function for Euler Numbers,
-i.e. the so called @var{n}th Euler number is the coefficient of
-@code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
-identity to code a function that generates Euler numbers in just three
-lines:
+Note that the original polynomial needs to be in expanded form (for the
+variables concerned) in order for @code{collect()} to be able to find the
+coefficients properly.
+
+The following @command{ginsh} transcript shows an application of @code{collect()}
+together with @code{find()}:
-@cindex Euler numbers
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+> 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)
+@end example
-ex EulerNumber(unsigned n)
-@{
- symbol x;
- const ex generator = pow(cosh(x),-1);
- return generator.diff(x,n).subs(x==0);
-@}
+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
-int main()
-@{
- for (unsigned i=0; i<11; i+=2)
- std::cout << EulerNumber(i) << std::endl;
- return 0;
-@}
+@example
+ex collect_common_factors(const ex & e);
@end example
-When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
-@code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
-@code{i} by two since all odd Euler numbers vanish anyways.
+This function doesn't perform a full factorization but only looks for
+factors which are already explicitly present:
+@example
+> 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
+@end example
-@node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
-@c node-name, next, previous, up
-@section Series expansion
-@cindex @code{series()}
-@cindex Taylor expansion
-@cindex Laurent expansion
-@cindex @code{pseries} (class)
+@subsection Degree and coefficients
+@cindex @code{degree()}
+@cindex @code{ldegree()}
+@cindex @code{coeff()}
-Expressions know how to expand themselves as a Taylor series or (more
-generally) a Laurent series. As in most conventional Computer Algebra
-Systems, no distinction is made between those two. There is a class of
-its own for storing such series (@code{class pseries}) and a built-in
-function (called @code{Order}) for storing the order term of the series.
-As a consequence, if you want to work with series, i.e. multiply two
-series, you need to call the method @code{ex::series} again to convert
-it to a series object with the usual structure (expansion plus order
-term). A sample application from special relativity could read:
+The degree and low degree of a polynomial can be obtained using the two
+methods
@example
-#include <ginac/ginac.h>
-using namespace std;
-using namespace GiNaC;
+int ex::degree(const ex & s);
+int ex::ldegree(const ex & s);
+@end example
-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;
-@}
+which also work reliably on non-expanded input polynomials (they even work
+on rational functions, returning the asymptotic degree). To extract
+a coefficient with a certain power from an expanded polynomial you use
+
+@example
+ex ex::coeff(const ex & s, int n);
@end example
-Only calling the series method makes the last output simplify to
-@math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
-series raised to the power @math{-2}.
+You can also obtain the leading and trailing coefficients with the methods
-@cindex M@'echain's formula
-As another instructive application, let us calculate the numerical
+@example
+ex ex::lcoeff(const ex & s);
+ex ex::tcoeff(const ex & s);
+@end example
+
+which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
+respectively.
+
+An application is illustrated in the next example, where a multivariate
+polynomial is analyzed:
+
+@example
+@{
+ symbol x("x"), y("y");
+ ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
+ - pow(x+y,2) + 2*pow(y+2,2) - 8;
+ ex Poly = PolyInp.expand();
+
+ for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
+ cout << "The x^" << i << "-coefficient is "
+ << Poly.coeff(x,i) << endl;
+ @}
+ cout << "As polynomial in y: "
+ << Poly.collect(y) << endl;
+@}
+@end example
+
+When run, it returns an output in the following fashion:
+
+@example
+The x^0-coefficient is y^2+11*y
+The x^1-coefficient is 5*y^2-2*y
+The x^2-coefficient is -1
+The x^3-coefficient is 4*y
+As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
+@end example
+
+As always, the exact output may vary between different versions of GiNaC
+or even from run to run since the internal canonical ordering is not
+within the user's sphere of influence.
+
+@code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
+@code{tcoeff()} and @code{collect()} can also be used to a certain degree
+with non-polynomial expressions as they not only work with symbols but with
+constants, functions and indexed objects as well:
+
+@example
+@{
+ symbol a("a"), b("b"), c("c");
+ idx i(symbol("i"), 3);
+
+ ex e = pow(sin(x) - cos(x), 4);
+ cout << e.degree(cos(x)) << endl;
+ // -> 4
+ cout << e.expand().coeff(sin(x), 3) << endl;
+ // -> -4*cos(x)
+
+ e = indexed(a+b, i) * indexed(b+c, i);
+ e = e.expand(expand_options::expand_indexed);
+ cout << e.collect(indexed(b, i)) << endl;
+ // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
+@}
+@end example
+
+
+@subsection Polynomial division
+@cindex polynomial division
+@cindex quotient
+@cindex remainder
+@cindex pseudo-remainder
+@cindex @code{quo()}
+@cindex @code{rem()}
+@cindex @code{prem()}
+@cindex @code{divide()}
+
+The two functions
+
+@example
+ex quo(const ex & a, const ex & b, const ex & x);
+ex rem(const ex & a, const ex & b, const ex & x);
+@end example
+
+compute the quotient and remainder of univariate polynomials in the variable
+@samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
+
+The additional function
+
+@example
+ex prem(const ex & a, const ex & b, const ex & x);
+@end example
+
+computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
+@math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
+
+Exact division of multivariate polynomials is performed by the function
+
+@example
+bool divide(const ex & a, const ex & b, ex & q);
+@end example
+
+If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
+and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
+in which case the value of @code{q} is undefined.
+
+
+@subsection Unit, content and primitive part
+@cindex @code{unit()}
+@cindex @code{content()}
+@cindex @code{primpart()}
+
+The methods
+
+@example
+ex ex::unit(const ex & x);
+ex ex::content(const ex & x);
+ex ex::primpart(const ex & x);
+@end example
+
+return the unit part, content part, and primitive polynomial of a multivariate
+polynomial with respect to the variable @samp{x} (the unit part being the sign
+of the leading coefficient, the content part being the GCD of the coefficients,
+and the primitive polynomial being the input polynomial divided by the unit and
+content parts). The product of unit, content, and primitive part is the
+original polynomial.
+
+
+@subsection GCD and LCM
+@cindex GCD
+@cindex LCM
+@cindex @code{gcd()}
+@cindex @code{lcm()}
+
+The functions for polynomial greatest common divisor and least common
+multiple have the synopsis
+
+@example
+ex gcd(const ex & a, const ex & b);
+ex lcm(const ex & a, const ex & b);
+@end example
+
+The functions @code{gcd()} and @code{lcm()} accept two expressions
+@code{a} and @code{b} as arguments and return a new expression, their
+greatest common divisor or least common multiple, respectively. If the
+polynomials @code{a} and @code{b} are coprime @code{gcd(a,b)} returns 1
+and @code{lcm(a,b)} returns the product of @code{a} and @code{b}.
+
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x"), y("y"), z("z");
+ ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
+ ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
+
+ ex P_gcd = gcd(P_a, P_b);
+ // x + 5*y + 4*z
+ ex P_lcm = lcm(P_a, P_b);
+ // 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
+@}
+@end example
+
+
+@subsection Square-free decomposition
+@cindex square-free decomposition
+@cindex factorization
+@cindex @code{sqrfree()}
+
+GiNaC still lacks proper factorization support. Some form of
+factorization is, however, easily implemented by noting that factors
+appearing in a polynomial with power two or more also appear in the
+derivative and hence can easily be found by computing the GCD of the
+original polynomial and its derivatives. Any decent system has an
+interface for this so called square-free factorization. So we provide
+one, too:
+@example
+ex sqrfree(const ex & a, const lst & l = lst());
+@end example
+Here is an example that by the way illustrates how the exact form of the
+result may slightly depend on the order of differentiation, calling for
+some care with subsequent processing of the result:
+@example
+ ...
+ 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
+ ...
+@end example
+Note also, how factors with the same exponents are not fully factorized
+with this method.
+
+
+@node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
+@c node-name, next, previous, up
+@section Rational expressions
+
+@subsection The @code{normal} method
+@cindex @code{normal()}
+@cindex simplification
+@cindex temporary replacement
+
+Some basic form of simplification of expressions is called for frequently.
+GiNaC provides the method @code{.normal()}, which converts a rational function
+into an equivalent rational function of the form @samp{numerator/denominator}
+where numerator and denominator are coprime. If the input expression is already
+a fraction, it just finds the GCD of numerator and denominator and cancels it,
+otherwise it performs fraction addition and multiplication.
+
+@code{.normal()} can also be used on expressions which are not rational functions
+as it will replace all non-rational objects (like functions or non-integer
+powers) by temporary symbols to bring the expression to the domain of rational
+functions before performing the normalization, and re-substituting these
+symbols afterwards. This algorithm is also available as a separate method
+@code{.to_rational()}, described below.
+
+This means that both expressions @code{t1} and @code{t2} are indeed
+simplified in this little code snippet:
+
+@example
+@{
+ symbol x("x");
+ ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
+ ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
+ std::cout << "t1 is " << t1.normal() << std::endl;
+ std::cout << "t2 is " << t2.normal() << std::endl;
+@}
+@end example
+
+Of course this works for multivariate polynomials too, so the ratio of
+the sample-polynomials from the section about GCD and LCM above would be
+normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
+
+
+@subsection Numerator and denominator
+@cindex numerator
+@cindex denominator
+@cindex @code{numer()}
+@cindex @code{denom()}
+@cindex @code{numer_denom()}
+
+The numerator and denominator of an expression can be obtained with
+
+@example
+ex ex::numer();
+ex ex::denom();
+ex ex::numer_denom();
+@end example
+
+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, calling @code{numer_denom()} is
+faster than using @code{numer()} and @code{denom()} separately.
+
+
+@subsection Converting to a polynomial or rational expression
+@cindex @code{to_polynomial()}
+@cindex @code{to_rational()}
+
+Some of the methods described so far only work on polynomials or rational
+functions. GiNaC provides a way to extend the domain of these functions to
+general expressions by using the temporary replacement algorithm described
+above. You do this by calling
+
+@example
+ex ex::to_polynomial(lst &l);
+@end example
+or
+@example
+ex ex::to_rational(lst &l);
+@end example
+
+on the expression to be converted. The supplied @code{lst} will be filled
+with the generated temporary symbols and their replacement expressions in
+a format that can be used directly for the @code{subs()} method. It can also
+already contain a list of replacements from an earlier application of
+@code{.to_polynomial()} or @code{.to_rational()}, so it's possible to use
+it on multiple expressions and get consistent results.
+
+The difference betwerrn @code{.to_polynomial()} and @code{.to_rational()}
+is probably best illustrated with an example:
+
+@example
+@{
+ symbol x("x"), y("y");
+ ex a = 2*x/sin(x) - y/(3*sin(x));
+ cout << a << endl;
+
+ lst lp;
+ ex p = a.to_polynomial(lp);
+ cout << " = " << p << "\n with " << lp << endl;
+ // = symbol3*symbol2*y+2*symbol2*x
+ // with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
+
+ lst lr;
+ ex r = a.to_rational(lr);
+ cout << " = " << r << "\n with " << lr << endl;
+ // = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
+ // with @{symbol4==sin(x)@}
+@}
+@end example
+
+The following more useful example will print @samp{sin(x)-cos(x)}:
+
+@example
+@{
+ symbol x("x");
+ ex a = pow(sin(x), 2) - pow(cos(x), 2);
+ ex b = sin(x) + cos(x);
+ ex q;
+ lst l;
+ divide(a.to_polynomial(l), b.to_polynomial(l), q);
+ cout << q.subs(l) << endl;
+@}
+@end example
+
+
+@node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
+@c node-name, next, previous, up
+@section Symbolic differentiation
+@cindex differentiation
+@cindex @code{diff()}
+@cindex chain rule
+@cindex product rule
+
+GiNaC's objects know how to differentiate themselves. Thus, a
+polynomial (class @code{add}) knows that its derivative is the sum of
+the derivatives of all the monomials:
+
+@example
+@{
+ 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
+@}
+@end example
+
+If a second integer parameter @var{n} is given, the @code{diff} method
+returns the @var{n}th derivative.
+
+If @emph{every} object and every function is told what its derivative
+is, all derivatives of composed objects can be calculated using the
+chain rule and the product rule. Consider, for instance the expression
+@code{1/cosh(x)}. Since the derivative of @code{cosh(x)} is
+@code{sinh(x)} and the derivative of @code{pow(x,-1)} is
+@code{-pow(x,-2)}, GiNaC can readily compute the composition. It turns
+out that the composition is the generating function for Euler Numbers,
+i.e. the so called @var{n}th Euler number is the coefficient of
+@code{x^n/n!} in the expansion of @code{1/cosh(x)}. We may use this
+identity to code a function that generates Euler numbers in just three
+lines:
+
+@cindex Euler numbers
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+ex EulerNumber(unsigned n)
+@{
+ symbol x;
+ const ex generator = pow(cosh(x),-1);
+ return generator.diff(x,n).subs(x==0);
+@}
+
+int main()
+@{
+ for (unsigned i=0; i<11; i+=2)
+ std::cout << EulerNumber(i) << std::endl;
+ return 0;
+@}
+@end example
+
+When you run it, it produces the sequence @code{1}, @code{-1}, @code{5},
+@code{-61}, @code{1385}, @code{-50521}. We increment the loop variable
+@code{i} by two since all odd Euler numbers vanish anyways.
+
+
+@node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
+@c node-name, next, previous, up
+@section Series expansion
+@cindex @code{series()}
+@cindex Taylor expansion
+@cindex Laurent expansion
+@cindex @code{pseries} (class)
+@cindex @code{Order()}
+
+Expressions know how to expand themselves as a Taylor series or (more
+generally) a Laurent series. As in most conventional Computer Algebra
+Systems, no distinction is made between those two. There is a class of
+its own for storing such series (@code{class pseries}) and a built-in
+function (called @code{Order}) for storing the order term of the series.
+As a consequence, if you want to work with series, i.e. multiply two
+series, you need to call the method @code{ex::series} again to convert
+it to a series object with the usual structure (expansion plus order
+term). A sample application from special relativity could read:
+
+@example
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ symbol v("v"), c("c");
+
+ ex gamma = 1/sqrt(1 - pow(v/c,2));
+ ex mass_nonrel = gamma.series(v==0, 10);
+
+ cout << "the relativistic mass increase with v is " << endl
+ << mass_nonrel << endl;
+
+ cout << "the inverse square of this series is " << endl
+ << pow(mass_nonrel,-2).series(v==0, 10) << endl;
+@}
+@end example
+
+Only calling the series method makes the last output simplify to
+@math{1-v^2/c^2+O(v^10)}, without that call we would just have a long
+series raised to the power @math{-2}.
+
+@cindex Machin's formula
+As another instructive application, let us calculate the numerical
value of Archimedes' constant
@tex
$\pi$
@end tex
(for which there already exists the built-in constant @code{Pi})
-using M@'echain's amazing formula
+using John Machin's amazing formula
@tex
$\pi=16$~atan~$\!\left(1 \over 5 \right)-4$~atan~$\!\left(1 \over 239 \right)$.
@end tex
@ifnottex
@math{Pi==16*atan(1/5)-4*atan(1/239)}.
@end ifnottex
-We may expand the arcus tangent around @code{0} and insert the fractions
-@code{1/5} and @code{1/239}. But, as we have seen, a series in GiNaC
-carries an order term with it and the question arises what the system is
-supposed to do when the fractions are plugged into that order term. The
-solution is to use the function @code{series_to_poly()} to simply strip
-the order term off:
+This equation (and similar ones) were used for over 200 years for
+computing digits of pi (see @cite{Pi Unleashed}). We may expand the
+arcus tangent around @code{0} and insert the fractions @code{1/5} and
+@code{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 @code{series_to_poly()} to simply strip the order
+term off:
+
+@example
+#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;
+@}
+@end example
+
+Note how we just called @code{.series(x,degr)} instead of
+@code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
+method @code{series()}: if the first argument is a symbol the expression
+is expanded in that symbol around point @code{0}. When you run this
+program, it will type out:
+
+@example
+2: 3804/1195
+ 3.1832635983263598326
+4: 5359397032/1706489875
+ 3.1405970293260603143
+6: 38279241713339684/12184551018734375
+ 3.141621029325034425
+8: 76528487109180192540976/24359780855939418203125
+ 3.141591772182177295
+10: 327853873402258685803048818236/104359128170408663038552734375
+ 3.1415926824043995174
+@end example
+
+
+@node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
+@c node-name, next, previous, up
+@section Symmetrization
+@cindex @code{symmetrize()}
+@cindex @code{antisymmetrize()}
+@cindex @code{symmetrize_cyclic()}
+
+The three methods
+
+@example
+ex ex::symmetrize(const lst & l);
+ex ex::antisymmetrize(const lst & l);
+ex ex::symmetrize_cyclic(const lst & l);
+@end example
+
+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
+
+@example
+ex ex::symmetrize();
+ex ex::antisymmetrize();
+ex ex::symmetrize_cyclic();
+@end example
+
+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 @code{subs()}able):
+
+@example
+@{
+ 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 << indexed(A, i, j).symmetrize() << endl;
+ // -> 1/2*A.j.i+1/2*A.i.j
+ cout << indexed(A, i, j, k).antisymmetrize(lst(i, j)) << endl;
+ // -> -1/2*A.j.i.k+1/2*A.i.j.k
+ cout << 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@}
+@}
+@end example
+
+
+@node Built-in Functions, Solving Linear Systems of Equations, Symmetrization, Methods and Functions
+@c node-name, next, previous, up
+@section Predefined mathematical functions
+
+GiNaC contains the following predefined mathematical functions:
+
+@cartouche
+@multitable @columnfractions .30 .70
+@item @strong{Name} @tab @strong{Function}
+@item @code{abs(x)}
+@tab absolute value
+@cindex @code{abs()}
+@item @code{csgn(x)}
+@tab complex sign
+@cindex @code{csgn()}
+@item @code{sqrt(x)}
+@tab square root (not a GiNaC function, rather an alias for @code{pow(x, numeric(1, 2))})
+@cindex @code{sqrt()}
+@item @code{sin(x)}
+@tab sine
+@cindex @code{sin()}
+@item @code{cos(x)}
+@tab cosine
+@cindex @code{cos()}
+@item @code{tan(x)}
+@tab tangent
+@cindex @code{tan()}
+@item @code{asin(x)}
+@tab inverse sine
+@cindex @code{asin()}
+@item @code{acos(x)}
+@tab inverse cosine
+@cindex @code{acos()}
+@item @code{atan(x)}
+@tab inverse tangent
+@cindex @code{atan()}
+@item @code{atan2(y, x)}
+@tab inverse tangent with two arguments
+@item @code{sinh(x)}
+@tab hyperbolic sine
+@cindex @code{sinh()}
+@item @code{cosh(x)}
+@tab hyperbolic cosine
+@cindex @code{cosh()}
+@item @code{tanh(x)}
+@tab hyperbolic tangent
+@cindex @code{tanh()}
+@item @code{asinh(x)}
+@tab inverse hyperbolic sine
+@cindex @code{asinh()}
+@item @code{acosh(x)}
+@tab inverse hyperbolic cosine
+@cindex @code{acosh()}
+@item @code{atanh(x)}
+@tab inverse hyperbolic tangent
+@cindex @code{atanh()}
+@item @code{exp(x)}
+@tab exponential function
+@cindex @code{exp()}
+@item @code{log(x)}
+@tab natural logarithm
+@cindex @code{log()}
+@item @code{Li2(x)}
+@tab Dilogarithm
+@cindex @code{Li2()}
+@item @code{zeta(x)}
+@tab Riemann's zeta function
+@cindex @code{zeta()}
+@item @code{zeta(n, x)}
+@tab derivatives of Riemann's zeta function
+@item @code{tgamma(x)}
+@tab Gamma function
+@cindex @code{tgamma()}
+@cindex Gamma function
+@item @code{lgamma(x)}
+@tab logarithm of Gamma function
+@cindex @code{lgamma()}
+@item @code{beta(x, y)}
+@tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
+@cindex @code{beta()}
+@item @code{psi(x)}
+@tab psi (digamma) function
+@cindex @code{psi()}
+@item @code{psi(n, x)}
+@tab derivatives of psi function (polygamma functions)
+@item @code{factorial(n)}
+@tab factorial function
+@cindex @code{factorial()}
+@item @code{binomial(n, m)}
+@tab binomial coefficients
+@cindex @code{binomial()}
+@item @code{Order(x)}
+@tab order term function in truncated power series
+@cindex @code{Order()}
+@item @code{Li(n,x)}
+@tab polylogarithm
+@cindex @code{Li()}
+@item @code{S(n,p,x)}
+@tab Nielsen's generalized polylogarithm
+@cindex @code{S()}
+@item @code{H(m_lst,x)}
+@tab harmonic polylogarithm
+@cindex @code{H()}
+@item @code{Li(m_lst,x_lst)}
+@tab multiple polylogarithm
+@cindex @code{Li()}
+@item @code{mZeta(m_lst)}
+@tab multiple zeta value
+@cindex @code{mZeta()}
+@end multitable
+@end cartouche
+
+@cindex branch cut
+For functions that have a branch cut in the complex plane GiNaC follows
+the conventions for C++ as defined in the ANSI standard as far as
+possible. In particular: the natural logarithm (@code{log}) and the
+square root (@code{sqrt}) both have their branch cuts running along the
+negative real axis where the points on the axis itself belong to the
+upper part (i.e. continuous with quadrant II). The inverse
+trigonometric and hyperbolic functions are not defined for complex
+arguments by the C++ standard, however. In GiNaC we follow the
+conventions used by CLN, which in turn follow the carefully designed
+definitions in the Common Lisp standard. It should be noted that this
+convention is identical to the one used by the C99 standard and by most
+serious CAS. It is to be expected that future revisions of the C++
+standard incorporate these functions in the complex domain in a manner
+compatible with C99.
+
+
+@node Solving Linear Systems of Equations, Input/Output, Built-in Functions, Methods and Functions
+@c node-name, next, previous, up
+@section Solving Linear Systems of Equations
+@cindex @code{lsolve()}
+
+The function @code{lsolve()} provides a convenient wrapper around some
+matrix operations that comes in handy when a system of linear equations
+needs to be solved:
+
+@example
+ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
+@end example
+
+Here, @code{eqns} is a @code{lst} of equalities (i.e. class
+@code{relational}) while @code{symbols} is a @code{lst} of
+indeterminates. (@xref{The Class Hierarchy}, for an exposition of class
+@code{lst}).
+
+It returns the @code{lst} of solutions as an expression. As an example,
+let us solve the two equations @code{a*x+b*y==3} and @code{x-y==b}:
+
+@example
+@{
+ symbol a("a"), b("b"), x("x"), y("y");
+ lst eqns;
+ eqns.append(a*x+b*y==3).append(x-y==b);
+ lst vars;
+ vars.append(x).append(y);
+ cout << lsolve(eqns, vars) << endl;
+ // -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
+@end example
+
+When the linear equations @code{eqns} are underdetermined, the solution
+will contain one or more tautological entries like @code{x==x},
+depending on the rank of the system. When they are overdetermined, the
+solution will be an empty @code{lst}. Note the third optional parameter
+to @code{lsolve()}: it accepts the same parameters as
+@code{matrix::solve()}. This is because @code{lsolve} is just a wrapper
+around that method.
+
+
+@node Input/Output, Extending GiNaC, Solving Linear Systems of Equations, Methods and Functions
+@c node-name, next, previous, up
+@section Input and output of expressions
+@cindex I/O
+
+@subsection Expression output
+@cindex printing
+@cindex output of expressions
+
+Expressions can simply be written to any stream:
+
+@example
+@{
+ symbol x("x");
+ ex e = 4.5*I+pow(x,2)*3/2;
+ cout << e << endl; // prints '4.5*I+3/2*x^2'
+ // ...
+@end example
+
+The default output format is identical to the @command{ginsh} input syntax and
+to that used by most computer algebra systems, but not directly pastable
+into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
+is printed as @samp{x^2}).
+
+It is possible to print expressions in a number of different formats with
+a set of stream manipulators;
+
+@example
+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);
+@end example
+
+The @code{tree}, @code{latex} and @code{csrc} formats are also available in
+@command{ginsh} via the @code{print()}, @code{print_latex()} and
+@code{print_csrc()} functions, respectively.
+
+@cindex @code{dflt}
+All manipulators affect the stream state permanently. To reset the output
+format to the default, use the @code{dflt} manipulator:
+
+@example
+ // ...
+ 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'
+ // ...
+@end example
+
+If you don't want to affect the format of the stream you're working with,
+you can output to a temporary @code{ostringstream} like this:
+
+@example
+ // ...
+ ostringstream s;
+ s << latex << e; // format of cout remains unchanged
+ cout << s.str() << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
+ // ...
+@end example
+
+@cindex @code{csrc}
+@cindex @code{csrc_float}
+@cindex @code{csrc_double}
+@cindex @code{csrc_cl_N}
+The @code{csrc} (an alias for @code{csrc_double}), @code{csrc_float},
+@code{csrc_double} and @code{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 (@code{csrc_cl_N} uses the
+classes provided by the CLN library):
+
+@example
+ // ...
+ cout << "f = " << csrc_float << e << ";\n";
+ cout << "d = " << csrc_double << e << ";\n";
+ cout << "n = " << csrc_cl_N << e << ";\n";
+ // ...
+@end example
+
+The above example will produce (note the @code{x^2} being converted to
+@code{x*x}):
+
+@example
+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"));
+@end example
+
+@cindex @code{tree}
+The @code{tree} manipulator allows dumping the internal structure of an
+expression for debugging purposes:
+
+@example
+ // ...
+ cout << tree << e;
+@}
+@end example
+
+produces
+
+@example
+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
+ =====
+@end example
+
+@cindex @code{latex}
+The @code{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 @code{symbol}
+constructor.
+
+For example, the code snippet
+
+@example
+@{
+ symbol x("x", "\\circ");
+ ex e = lgamma(x).series(x==0,3);
+ cout << latex << e << endl;
+@}
+@end example
+
+will print
+
+@example
+ @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
+@end example
+
+@cindex @code{index_dimensions}
+@cindex @code{no_index_dimensions}
+Index dimensions are normally hidden in the output. To make them visible, use
+the @code{index_dimensions} manipulator. The dimensions will be written in
+square brackets behind each index value in the default and LaTeX output
+formats:
+
+@example
+@{
+ 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'
+@}
+@end example
+
+
+@cindex Tree traversal
+If you need any fancy special output format, e.g. for interfacing GiNaC
+with other algebra systems or for producing code for different
+programming languages, you can always traverse the expression tree yourself:
+
+@example
+static void my_print(const ex & e)
+@{
+ if (is_a<function>(e))
+ cout << ex_to<function>(e).get_name();
+ else
+ cout << e.bp->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;
+@}
+@end example
+
+This will produce
+
+@example
+add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
+symbol(y))),numeric(-2)))
+@end example
+
+If you need an output format that makes it possible to accurately
+reconstruct an expression by feeding the output to a suitable parser or
+object factory, you should consider storing the expression in an
+@code{archive} object and reading the object properties from there.
+See the section on archiving for more information.
+
+
+@subsection Expression input
+@cindex input of expressions
+
+GiNaC provides no way to directly read an expression from a stream because
+you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
+and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
+@code{y} you defined in your program and there is no way to specify the
+desired symbols to the @code{>>} stream input operator.
+
+Instead, GiNaC lets you construct an expression from a string, specifying the
+list of symbols to be used:
+
+@example
+@{
+ symbol x("x"), y("y");
+ ex e("2*x+sin(y)", lst(x, y));
+@}
+@end example
+
+The input syntax is the same as that used by @command{ginsh} and the stream
+output operator @code{<<}. The symbols in the string are matched by name to
+the symbols in the list and if GiNaC encounters a symbol not specified in
+the list it will throw an exception.
+
+With this constructor, it's also easy to implement interactive GiNaC programs:
@example
+#include <iostream>
+#include <string>
+#include <stdexcept>
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
-ex mechain_pi(int degr)
+int main()
@{
- 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;
+ symbol x("x");
+ string s;
+
+ cout << "Enter an expression containing 'x': ";
+ getline(cin, s);
+
+ try @{
+ ex e(s, lst(x));
+ cout << "The derivative of " << e << " with respect to x is ";
+ cout << e.diff(x) << ".\n";
+ @} catch (exception &p) @{
+ cerr << p.what() << endl;
+ @}
@}
+@end example
+
+
+@subsection Archiving
+@cindex @code{archive} (class)
+@cindex archiving
+
+GiNaC allows creating @dfn{archives} of expressions which can be stored
+to or retrieved from files. To create an archive, you declare an object
+of class @code{archive} and archive expressions in it, giving each
+expression a unique name:
+
+@example
+#include <fstream>
+using namespace std;
+#include <ginac/ginac.h>
+using namespace GiNaC;
int main()
@{
- using std::cout; // just for fun, another way of...
- using std::endl; // ...dealing with this namespace std.
- ex pi_frac;
- for (int i=2; i<12; i+=2) @{
- pi_frac = mechain_pi(i);
- cout << i << ":\t" << pi_frac << endl
- << "\t" << pi_frac.evalf() << endl;
- @}
- return 0;
-@}
+ symbol x("x"), y("y"), z("z");
+
+ ex foo = sin(x + 2*y) + 3*z + 41;
+ ex bar = foo + 1;
+
+ archive a;
+ a.archive_ex(foo, "foo");
+ a.archive_ex(bar, "the second one");
+ // ...
@end example
-Note how we just called @code{.series(x,degr)} instead of
-@code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
-method @code{series()}: if the first argument is a symbol the expression
-is expanded in that symbol around point @code{0}. When you run this
-program, it will type out:
+The archive can then be written to a file:
@example
-2: 3804/1195
- 3.1832635983263598326
-4: 5359397032/1706489875
- 3.1405970293260603143
-6: 38279241713339684/12184551018734375
- 3.141621029325034425
-8: 76528487109180192540976/24359780855939418203125
- 3.141591772182177295
-10: 327853873402258685803048818236/104359128170408663038552734375
- 3.1415926824043995174
+ // ...
+ ofstream out("foobar.gar");
+ out << a;
+ out.close();
+ // ...
@end example
+The file @file{foobar.gar} contains all information that is needed to
+reconstruct the expressions @code{foo} and @code{bar}.
-@node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
-@c node-name, next, previous, up
-@section Predefined mathematical functions
+@cindex @command{viewgar}
+The tool @command{viewgar} that comes with GiNaC can be used to view
+the contents of GiNaC archive files:
-GiNaC contains the following predefined mathematical functions:
+@example
+$ viewgar foobar.gar
+foo = 41+sin(x+2*y)+3*z
+the second one = 42+sin(x+2*y)+3*z
+@end example
-@cartouche
-@multitable @columnfractions .30 .70
-@item @strong{Name} @tab @strong{Function}
-@item @code{abs(x)}
-@tab absolute value
-@item @code{csgn(x)}
-@tab complex sign
-@item @code{sqrt(x)}
-@tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
-@item @code{sin(x)}
-@tab sine
-@item @code{cos(x)}
-@tab cosine
-@item @code{tan(x)}
-@tab tangent
-@item @code{asin(x)}
-@tab inverse sine
-@item @code{acos(x)}
-@tab inverse cosine
-@item @code{atan(x)}
-@tab inverse tangent
-@item @code{atan2(y, x)}
-@tab inverse tangent with two arguments
-@item @code{sinh(x)}
-@tab hyperbolic sine
-@item @code{cosh(x)}
-@tab hyperbolic cosine
-@item @code{tanh(x)}
-@tab hyperbolic tangent
-@item @code{asinh(x)}
-@tab inverse hyperbolic sine
-@item @code{acosh(x)}
-@tab inverse hyperbolic cosine
-@item @code{atanh(x)}
-@tab inverse hyperbolic tangent
-@item @code{exp(x)}
-@tab exponential function
-@item @code{log(x)}
-@tab natural logarithm
-@item @code{Li2(x)}
-@tab Dilogarithm
-@item @code{zeta(x)}
-@tab Riemann's zeta function
-@item @code{zeta(n, x)}
-@tab derivatives of Riemann's zeta function
-@item @code{tgamma(x)}
-@tab Gamma function
-@item @code{lgamma(x)}
-@tab logarithm of Gamma function
-@item @code{beta(x, y)}
-@tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
-@item @code{psi(x)}
-@tab psi (digamma) function
-@item @code{psi(n, x)}
-@tab derivatives of psi function (polygamma functions)
-@item @code{factorial(n)}
-@tab factorial function
-@item @code{binomial(n, m)}
-@tab binomial coefficients
-@item @code{Order(x)}
-@tab order term function in truncated power series
-@item @code{Derivative(x, l)}
-@tab inert partial differentiation operator (used internally)
-@end multitable
-@end cartouche
+The point of writing archive files is of course that they can later be
+read in again:
-@cindex branch cut
-For functions that have a branch cut in the complex plane GiNaC follows
-the conventions for C++ as defined in the ANSI standard as far as
-possible. In particular: the natural logarithm (@code{log}) and the
-square root (@code{sqrt}) both have their branch cuts running along the
-negative real axis where the points on the axis itself belong to the
-upper part (i.e. continuous with quadrant II). The inverse
-trigonometric and hyperbolic functions are not defined for complex
-arguments by the C++ standard, however. In GiNaC we follow the
-conventions used by CLN, which in turn follow the carefully designed
-definitions in the Common Lisp standard. It should be noted that this
-convention is identical to the one used by the C99 standard and by most
-serious CAS. It is to be expected that future revisions of the C++
-standard incorporate these functions in the complex domain in a manner
-compatible with C99.
+@example
+ // ...
+ archive a2;
+ ifstream in("foobar.gar");
+ in >> a2;
+ // ...
+@end example
+
+And the stored expressions can be retrieved by their name:
+
+@example
+ // ...
+ lst syms(x, y);
+
+ ex ex1 = a2.unarchive_ex(syms, "foo");
+ ex ex2 = a2.unarchive_ex(syms, "the second one");
+
+ cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
+ cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
+ cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
+@}
+@end example
+
+Note that you have to supply a list of the symbols which are to be inserted
+in the expressions. Symbols in archives are stored by their name only and
+if you don't specify which symbols you have, unarchiving the expression will
+create new symbols with that name. E.g. if you hadn't included @code{x} in
+the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
+have had no effect because the @code{x} in @code{ex1} would have been a
+different symbol than the @code{x} which was defined at the beginning of
+the program, although both would appear as @samp{x} when printed.
+
+You can also use the information stored in an @code{archive} object to
+output expressions in a format suitable for exact reconstruction. The
+@code{archive} and @code{archive_node} classes have a couple of member
+functions that let you access the stored properties:
+
+@example
+static void my_print2(const archive_node & n)
+@{
+ string class_name;
+ n.find_string("class", class_name);
+ cout << class_name << "(";
+
+ archive_node::propinfovector p;
+ n.get_properties(p);
+
+ 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 << ",";
+ @}
-@node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
-@c node-name, next, previous, up
-@section Input and output of expressions
-@cindex I/O
+ if (count > 1)
+ cout << "@}";
-@subsection Expression output
-@cindex printing
-@cindex output of expressions
+ if (i != num-1)
+ cout << ",";
+ @}
-The easiest way to print an expression is to write it to a stream:
+ cout << ")";
+@}
-@example
+int main()
@{
- symbol x("x");
- ex e = 4.5+pow(x,2)*3/2;
- cout << e << endl; // prints '(4.5)+3/2*x^2'
- // ...
+ ex e = pow(2, x) - y;
+ archive ar(e, "e");
+ my_print2(ar.get_top_node(0)); cout << endl;
+ return 0;
+@}
@end example
-The output format is identical to the @command{ginsh} input syntax and
-to that used by most computer algebra systems, but not directly pastable
-into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
-is printed as @samp{x^2}).
-
-It is possible to print expressions in a number of different formats with
-the method
+This will produce:
@example
-void ex::print(const print_context & c, unsigned level = 0);
+add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
+symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
+overall_coeff=numeric(number="0"))
@end example
-@cindex @code{print_context} (class)
-The type of @code{print_context} object passed in determines the format
-of the output. The possible types are defined in @file{ginac/print.h}.
-All constructors of @code{print_context} and derived classes take an
-@code{ostream &} as their first argument.
+Be warned, however, that the set of properties and their meaning for each
+class may change between GiNaC versions.
-To print an expression in a way that can be directly used in a C or C++
-program, you pass a @code{print_csrc} object like this:
-@example
- // ...
- cout << "float f = ";
- e.print(print_csrc_float(cout));
- cout << ";\n";
+@node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
+@c node-name, next, previous, up
+@chapter Extending GiNaC
- cout << "double d = ";
- e.print(print_csrc_double(cout));
- cout << ";\n";
+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.
- cout << "cl_N n = ";
- e.print(print_csrc_cl_N(cout));
- cout << ";\n";
- // ...
-@end example
+@menu
+* What does not belong into GiNaC:: What to avoid.
+* Symbolic functions:: Implementing symbolic functions.
+* Structures:: Defining new algebraic classes (the easy way).
+* Adding classes:: Defining new algebraic classes (the hard way).
+@end menu
-The three possible types mostly affect the way in which floating point
-numbers are written.
-The above example will produce (note the @code{x^2} being converted to @code{x*x}):
+@node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
+@c node-name, next, previous, up
+@section What doesn't belong into GiNaC
+
+@cindex @command{ginsh}
+First of all, GiNaC's name must be read literally. It is designed to be
+a library for use within C++. The tiny @command{ginsh} accompanying
+GiNaC makes this even more clear: it doesn't even attempt to provide a
+language. There are no loops or conditional expressions in
+@command{ginsh}, it is merely a window into the library for the
+programmer to test stuff (or to show off). Still, the design of a
+complete CAS with a language of its own, graphical 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 @code{Digits}). Some may be evaluated at certain points, but not
+generally. This ought to be fixed. However, doing numerical
+computations with GiNaC's quite abstract classes is doomed to be
+inefficient. For this purpose, the underlying foundation classes
+provided by CLN are much better suited.
+
+
+@node Symbolic functions, Structures, What does not belong into GiNaC, Extending GiNaC
+@c node-name, next, previous, up
+@section Symbolic functions
+The easiest and most instructive way to start extending GiNaC is probably to
+create your own symbolic functions. These are implemented with the help of
+two preprocessor macros:
+
+@cindex @code{DECLARE_FUNCTION}
+@cindex @code{REGISTER_FUNCTION}
@example
-float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
-double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
-cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
+DECLARE_FUNCTION_<n>P(<name>)
+REGISTER_FUNCTION(<name>, <options>)
@end example
-The @code{print_context} type @code{print_tree} provides a dump of the
-internal structure of an expression for debugging purposes:
+The @code{DECLARE_FUNCTION} macro will usually appear in a header file. It
+declares a C++ function with the given @samp{name} that takes exactly @samp{n}
+parameters of type @code{ex} and returns a newly constructed GiNaC
+@code{function} object that represents your function.
+
+The @code{REGISTER_FUNCTION} macro implements the function. It must be passed
+the same @samp{name} as the respective @code{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
+@samp{.} 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.)
+
+@subsection A minimal example
+
+Here is an example for the implementation of a function with two arguments
+that is not further evaluated:
@example
- // ...
- e.print(print_tree(cout));
+DECLARE_FUNCTION_2P(myfcn)
+
+static ex myfcn_eval(const ex & x, const ex & y)
+@{
+ return myfcn(x, y).hold();
@}
+
+REGISTER_FUNCTION(myfcn, eval_func(myfcn_eval))
@end example
-produces
+Any code that has seen the @code{DECLARE_FUNCTION} line can use @code{myfcn()}
+in algebraic expressions:
@example
-add, hash=0x0, flags=0x3, nops=2
- power, hash=0x9, flags=0x3, nops=2
- x (symbol), serial=3, hash=0x44a113a6, flags=0xf
- 2 (numeric), hash=0x80000042, flags=0xf
- 3/2 (numeric), hash=0x80000061, flags=0xf
- -----
- overall_coeff
- 4.5L0 (numeric), hash=0x8000004b, flags=0xf
- =====
+@{
+ ...
+ symbol x("x");
+ ex e = 2*myfcn(42, 3*x+1) - x;
+ // this calls myfcn_eval(42, 3*x+1), and inserts its return value into
+ // the actual expression
+ cout << e << endl;
+ // prints '2*myfcn(42,1+3*x)-x'
+ ...
+@}
@end example
-This kind of output is also available in @command{ginsh} as the @code{print()}
-function.
+@cindex @code{hold()}
+@cindex evaluation
+The @code{eval_func()} option specifies the C++ function that implements
+the @code{eval()} method, GiNaC's anonymous evaluator. This function takes
+the same number of arguments as the associated symbolic function (two in this
+case) and returns the (possibly transformed or in some way simplified)
+symbolically evaluated function (@xref{Automatic evaluation}, for a description
+of the automatic evaluation process). If no (further) evaluation is to take
+place, the @code{eval_func()} function must return the original function
+with @code{.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 @code{.hold()}
+somewhere.
-Another useful output format is for LaTeX parsing in mathematical mode.
-It is rather similar to the default @code{print_context} but provides
-some braces needed by LaTeX for delimiting boxes and also converts some
-common objects to conventional LaTeX names. It is possible to give symbols
-a special name for LaTeX output by supplying it as a second argument to
-the @code{symbol} constructor.
+There is not much you can do with the @code{myfcn} function. It merely acts
+as a kind of container for its arguments (which is, however, sometimes
+perfectly sufficient). Let's have a look at the implementation of GiNaC's
+cosine function.
-For example, the code snippet
+@subsection The cosine function
+
+The GiNaC header file @file{inifcns.h} contains the line
@example
- // ...
- symbol x("x");
- ex foo = lgamma(x).series(x==0,3);
- foo.print(print_latex(std::cout));
+DECLARE_FUNCTION_1P(cos)
@end example
-will print out:
+which declares to all programs using GiNaC that there is a function @samp{cos}
+that takes one @code{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 @file{inifcns_trans.cpp}. The
+@code{eval_func()} function looks something like this (actually, it doesn't
+look like this at all, but it should give you an idea what is going on):
@example
- @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
+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();
+@}
@end example
-If you need any fancy special output format, e.g. for interfacing GiNaC
-with other algebra systems or for producing code for different
-programming languages, you can always traverse the expression tree yourself:
+In this way, @code{cos(4*Pi)} automatically becomes @math{1},
+@code{cos(asin(a+b))} becomes @code{sqrt(1-(a+b)^2)}, etc. If no reasonable
+symbolic transformation can be done, the unmodified function is returned
+with @code{.hold()}.
+
+GiNaC doesn't automatically transform @code{cos(2)} to @samp{-0.416146...}.
+The user has to call @code{evalf()} for that. This is implemented in a
+different function:
@example
-static void my_print(const ex & e)
+static ex cos_evalf(const ex & x)
@{
- if (is_ex_of_type(e, function))
- cout << ex_to_function(e).get_name();
- else
- cout << e.bp->class_name();
- cout << "(";
- unsigned n = e.nops();
- if (n)
- for (unsigned i=0; i<n; i++) @{
- my_print(e.op(i));
- if (i != n-1)
- cout << ",";
- @}
+ if (is_a<numeric>(x))
+ return cos(ex_to<numeric>(x));
else
- cout << e;
- cout << ")";
+ return cos(x).hold();
@}
+@end example
+
+Since we are lazy we defer the problem of numeric evaluation to somebody else,
+in this case the @code{cos()} function for @code{numeric} objects, which in
+turn hands it over to the @code{cos()} function in CLN. The @code{.hold()}
+isn't really needed here, but reminds us that the corresponding @code{eval()}
+function would require it in this place.
+
+Differentiation will surely turn up and so we need to tell @code{cos}
+what its first derivative is (higher derivatives, @code{.diff(x,3)} for
+instance, are then handled automatically by @code{basic::diff} and
+@code{ex::diff}):
-int main(void)
+@example
+static ex cos_deriv(const ex & x, unsigned diff_param)
@{
- my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
- return 0;
+ return -sin(x);
@}
@end example
-This will produce
+@cindex product rule
+The second parameter is obligatory but uninteresting at this point. It
+specifies which parameter to differentiate in a partial derivative in
+case the function has more than one parameter, and its main application
+is for correct handling of the chain rule.
+
+An implementation of the series expansion is not needed for @code{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 @code{cos()} is). @code{tan()}, on
+the other hand, does have poles and may need to do Laurent expansion:
@example
-add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
-symbol(y))),numeric(-2)))
+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);
+@}
@end example
-If you need an output format that makes it possible to accurately
-reconstruct an expression by feeding the output to a suitable parser or
-object factory, you should consider storing the expression in an
-@code{archive} object and reading the object properties from there.
-See the section on archiving for more information.
+The @code{series()} implementation of a function @emph{must} return a
+@code{pseries} object, otherwise your code will crash.
+Now that all the ingredients have been set up, the @code{REGISTER_FUNCTION}
+macro is used to tell the system how the @code{cos()} function behaves:
-@subsection Expression input
-@cindex input of expressions
+@example
+REGISTER_FUNCTION(cos, eval_func(cos_eval).
+ evalf_func(cos_evalf).
+ derivative_func(cos_deriv).
+ latex_name("\\cos"));
+@end example
-GiNaC provides no way to directly read an expression from a stream because
-you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
-and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
-@code{y} you defined in your program and there is no way to specify the
-desired symbols to the @code{>>} stream input operator.
+This registers the @code{cos_eval()}, @code{cos_evalf()} and
+@code{cos_deriv()} C++ functions with the @code{cos()} function, and also
+gives it a proper LaTeX name.
-Instead, GiNaC lets you construct an expression from a string, specifying the
-list of symbols to be used:
+@subsection Function options
+
+GiNaC functions understand several more options which are always
+specified as @code{.option(params)}. None of them are required, but you
+need to specify at least one option to @code{REGISTER_FUNCTION()} (usually
+the @code{eval()} method).
@example
-@{
- symbol x("x"), y("y");
- ex e("2*x+sin(y)", lst(x, y));
-@}
+eval_func(<C++ function>)
+evalf_func(<C++ function>)
+derivative_func(<C++ function>)
+series_func(<C++ function>)
@end example
-The input syntax is the same as that used by @command{ginsh} and the stream
-output operator @code{<<}. The symbols in the string are matched by name to
-the symbols in the list and if GiNaC encounters a symbol not specified in
-the list it will throw an exception.
+These specify the C++ functions that implement symbolic evaluation,
+numeric evaluation, partial derivatives, and series expansion, respectively.
+They correspond to the GiNaC methods @code{eval()}, @code{evalf()},
+@code{diff()} and @code{series()}.
-With this constructor, it's also easy to implement interactive GiNaC programs:
+The @code{eval_func()} function needs to use @code{.hold()} if no further
+automatic evaluation is desired or possible.
+
+If no @code{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 @code{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.
@example
-#include <iostream>
-#include <string>
-#include <stdexcept>
-#include <ginac/ginac.h>
-using namespace std;
-using namespace GiNaC;
+latex_name(const string & n)
+@end example
-int main()
-@{
- symbol x("x");
- string s;
-
- cout << "Enter an expression containing 'x': ";
- getline(cin, s);
-
- try @{
- ex e(s, lst(x));
- cout << "The derivative of " << e << " with respect to x is ";
- cout << e.diff(x) << ".\n";
- @} catch (exception &p) @{
- cerr << p.what() << endl;
- @}
-@}
+specifies the LaTeX code that represents the name of the function in LaTeX
+output. The default is to put the function name in an @code{\mbox@{@}}.
+
+@example
+do_not_evalf_params()
@end example
+This tells @code{evalf()} to not recursively evaluate the parameters of the
+function before calling the @code{evalf_func()}.
-@subsection Archiving
-@cindex @code{archive} (class)
-@cindex archiving
+@example
+set_return_type(unsigned return_type, unsigned return_type_tinfo)
+@end example
-GiNaC allows creating @dfn{archives} of expressions which can be stored
-to or retrieved from files. To create an archive, you declare an object
-of class @code{archive} and archive expressions in it, giving each
-expression a unique name:
+This allows you to explicitly specify the commutation properties of the
+function (@xref{Non-commutative objects}, for an explanation of
+(non)commutativity in GiNaC). For example, you can use
+@code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
+GiNaC treat your function like a matrix. By default, functions inherit the
+commutation properties of their first argument.
@example
-#include <fstream>
+set_symmetry(const symmetry & s)
+@end example
+
+specifies the symmetry properties of the function with respect to its
+arguments. @xref{Indexed objects}, for an explanation of symmetry
+specifications. GiNaC will automatically rearrange the arguments of
+symmetric functions into a canonical order.
+
+
+@node Structures, Adding classes, Symbolic functions, Extending GiNaC
+@c node-name, next, previous, up
+@section 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
+@code{basic} that can be used in GiNaC expressions).
+
+GiNaC offers two ways of accomplishing this: either by using the
+@code{structure<T>} template class, or by rolling your own class from
+scratch. This section will discuss the @code{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 @code{structure<T>} template, defined in the GiNaC header file
+@file{structure.h}, wraps a type that you supply (usually a C++ @code{struct}
+or @code{class}) into a GiNaC object that can be used in expressions.
+
+@subsection Example: scalar products
+
+Let's suppose that we need a way to handle some kind of abstract scalar
+product of the form @samp{<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++ @code{struct}:
+
+@example
+#include <iostream>
using namespace std;
+
#include <ginac/ginac.h>
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;
+struct sprod_s @{
+ ex left, right;
- archive a;
- a.archive_ex(foo, "foo");
- a.archive_ex(bar, "the second one");
- // ...
+ sprod_s() @{@}
+ sprod_s(ex l, ex r) : left(l), right(r) @{@}
+@};
@end example
-The archive can then be written to a file:
+The default constructor is required. Now, to make a GiNaC class out of this
+data structure, we need only one line:
@example
- // ...
- ofstream out("foobar.gar");
- out << a;
- out.close();
- // ...
+typedef structure<sprod_s> sprod;
@end example
-The file @file{foobar.gar} contains all information that is needed to
-reconstruct the expressions @code{foo} and @code{bar}.
-
-@cindex @command{viewgar}
-The tool @command{viewgar} that comes with GiNaC can be used to view
-the contents of GiNaC archive files:
+That's it. This line constructs an algebraic class @code{sprod} which
+contains objects of type @code{sprod_s}. We can now use @code{sprod} in
+expressions like any other GiNaC class:
@example
-$ viewgar foobar.gar
-foo = 41+sin(x+2*y)+3*z
-the second one = 42+sin(x+2*y)+3*z
+...
+ symbol a("a"), b("b");
+ ex e = sprod(sprod_s(a, b));
+...
@end example
-The point of writing archive files is of course that they can later be
-read in again:
+Note the difference between @code{sprod} which is the algebraic class, and
+@code{sprod_s} which is the unadorned C++ structure containing the @code{left}
+and @code{right} data members. As shown above, an @code{sprod} can be
+constructed from an @code{sprod_s} object.
+
+If you find the nested @code{sprod(sprod_s())} constructor too unwieldy,
+you could define a little wrapper function like this:
@example
- // ...
- archive a2;
- ifstream in("foobar.gar");
- in >> a2;
- // ...
+inline ex make_sprod(ex left, ex right)
+@{
+ return sprod(sprod_s(left, right));
+@}
@end example
-And the stored expressions can be retrieved by their name:
+The @code{sprod_s} object contained in @code{sprod} can be accessed with
+the GiNaC @code{ex_to<>()} function followed by the @code{->} operator or
+@code{get_struct()}:
@example
- // ...
- lst syms(x, y);
-
- ex ex1 = a2.unarchive_ex(syms, "foo");
- ex ex2 = a2.unarchive_ex(syms, "the second one");
-
- cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
- cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
- cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
-@}
+...
+ cout << ex_to<sprod>(e)->left << endl;
+ // -> a
+ cout << ex_to<sprod>(e).get_struct().right << endl;
+ // -> b
+...
@end example
-Note that you have to supply a list of the symbols which are to be inserted
-in the expressions. Symbols in archives are stored by their name only and
-if you don't specify which symbols you have, unarchiving the expression will
-create new symbols with that name. E.g. if you hadn't included @code{x} in
-the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
-have had no effect because the @code{x} in @code{ex1} would have been a
-different symbol than the @code{x} which was defined at the beginning of
-the program, altough both would appear as @samp{x} when printed.
+You only have read access to the members of @code{sprod_s}.
-You can also use the information stored in an @code{archive} object to
-output expressions in a format suitable for exact reconstruction. The
-@code{archive} and @code{archive_node} classes have a couple of member
-functions that let you access the stored properties:
+The type definition of @code{sprod} is enough to write your own algorithms
+that deal with scalar products, for example:
@example
-static void my_print2(const archive_node & n)
+ex swap_sprod(ex p)
@{
- string class_name;
- n.find_string("class", class_name);
- cout << class_name << "(";
+ 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;
+@}
- archive_node::propinfovector p;
- n.get_properties(p);
+...
+ f = swap_sprod(e);
+ // f is now <b|a>
+...
+@end example
- unsigned num = p.size();
- for (unsigned i=0; i<num; i++) @{
- const string &name = p[i].name;
- if (name == "class")
- continue;
- cout << name << "=";
+@subsection Structure output
- unsigned count = p[i].count;
- if (count > 1)
- cout << "@{";
+While the @code{sprod} type is useable it still leaves something to be
+desired, most notably proper output:
- for (unsigned j=0; j<count; j++) @{
- switch (p[i].type) @{
- case archive_node::PTYPE_BOOL: @{
- bool x;
- n.find_bool(name, x);
- cout << (x ? "true" : "false");
- break;
- @}
- case archive_node::PTYPE_UNSIGNED: @{
- unsigned x;
- n.find_unsigned(name, x);
- cout << x;
- break;
- @}
- case archive_node::PTYPE_STRING: @{
- string x;
- n.find_string(name, x);
- cout << '\"' << x << '\"';
- break;
- @}
- case archive_node::PTYPE_NODE: @{
- const archive_node &x = n.find_ex_node(name, j);
- my_print2(x);
- break;
- @}
- @}
+@example
+...
+ cout << e << endl;
+ // -> [structure object]
+...
+@end example
- if (j != count-1)
- cout << ",";
- @}
+By default, any structure types you define will be printed as
+@samp{[structure object]}. To override this, you can specialize the
+template's @code{print()} member function. 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:
- if (count > 1)
- cout << "@}";
+@example
+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);
- if (i != num-1)
- cout << ",";
- @}
+ // get the contained sprod_s object
+ const sprod_s & sp = get_struct();
- cout << ")";
+ // print_context::s is a reference to an ostream
+ c.s << "<" << sp.left << "|" << sp.right << ">";
@}
+@end example
-int main(void)
-@{
- ex e = pow(2, x) - y;
- archive ar(e, "e");
- my_print2(ar.get_top_node(0)); cout << endl;
- return 0;
-@}
+Now we can print expressions containing scalar products:
+
+@example
+...
+ cout << e << endl;
+ // -> <a|b>
+ cout << swap_sprod(e) << endl;
+ // -> <b|a>
+...
@end example
-This will produce:
+@subsection Comparing structures
+
+The @code{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 @code{sprod_s}. This can lead to some confusing
+and undesired behavior:
@example
-add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
-symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
-overall_coeff=numeric(number="0"))
+...
+ 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)
+...
@end example
-Be warned, however, that the set of properties and their meaning for each
-class may change between GiNaC versions.
+To remedy this, we first need to define the operators @code{==} and @code{<}
+for objects of type @code{sprod_s}:
+@example
+inline bool operator==(const sprod_s & lhs, const sprod_s & rhs)
+@{
+ return lhs.left.is_equal(rhs.left) && lhs.right.is_equal(rhs.right);
+@}
-@node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
-@c node-name, next, previous, up
-@chapter Extending GiNaC
+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;
+@}
+@end example
-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.
+The ordering established by the @code{<} operator doesn't have to make any
+algebraic sense, but it needs to be well defined. Note that we can't use
+expressions like @code{lhs.left == rhs.left} or @code{lhs.left < rhs.left}
+in the implementation of these operators because they would construct
+GiNaC @code{relational} objects which in the case of @code{<} do not
+establish a well defined ordering (for arbitrary expressions, GiNaC can't
+decide which one is algebraically 'less').
-@menu
-* What does not belong into GiNaC:: What to avoid.
-* Symbolic functions:: Implementing symbolic functions.
-* Adding classes:: Defining new algebraic classes.
-@end menu
+Next, we need to change our definition of the @code{sprod} type to let
+GiNaC know that an ordering relation exists for the embedded objects:
+@example
+typedef structure<sprod_s, compare_std_less> sprod;
+@end example
-@node What does not belong into GiNaC, Symbolic functions, Extending GiNaC, Extending GiNaC
-@c node-name, next, previous, up
-@section What doesn't belong into GiNaC
+@code{sprod} objects then behave as expected:
-@cindex @command{ginsh}
-First of all, GiNaC's name must be read literally. It is designed to be
-a library for use within C++. The tiny @command{ginsh} accompanying
-GiNaC makes this even more clear: it doesn't even attempt to provide a
-language. There are no loops or conditional expressions in
-@command{ginsh}, it is merely a window into the library for the
-programmer to test stuff (or to show off). Still, the design of a
-complete CAS with a language of its own, graphical capabilites and all
-this on top of GiNaC is possible and is without doubt a nice project for
-the future.
+@example
+...
+ 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>
+...
+@end example
-There are many built-in functions in GiNaC that do not know how to
-evaluate themselves numerically to a precision declared at runtime
-(using @code{Digits}). Some may be evaluated at certain points, but not
-generally. This ought to be fixed. However, doing numerical
-computations with GiNaC's quite abstract classes is doomed to be
-inefficient. For this purpose, the underlying foundation classes
-provided by @acronym{CLN} are much better suited.
+The @code{compare_std_less} policy parameter tells GiNaC to use the
+@code{std::less} and @code{std::equal_to} functors to compare objects of
+type @code{sprod_s}. By default, these functors forward their work to the
+standard @code{<} and @code{==} operators, which we have overloaded.
+Alternatively, we could have specialized @code{std::less} and
+@code{std::equal_to} for class @code{sprod_s}.
+GiNaC provides two other comparison policies for @code{structure<T>}
+objects: the default @code{compare_all_equal}, and @code{compare_bitwise}
+which does a bit-wise comparison of the contained @code{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 @code{T} class might have.
-@node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
-@c node-name, next, previous, up
-@section Symbolic functions
+@subsection 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
+@code{nops()} and @code{op()} methods:
-The easiest and most instructive way to start with is probably to
-implement your own function. GiNaC's functions are objects of class
-@code{function}. The preprocessor is then used to convert the function
-names to objects with a corresponding serial number that is used
-internally to identify them. You usually need not worry about this
-number. New functions may be inserted into the system via a kind of
-`registry'. It is your responsibility to care for some functions that
-are called when the user invokes certain methods. These are usual
-C++-functions accepting a number of @code{ex} as arguments and returning
-one @code{ex}. As an example, if we have a look at a simplified
-implementation of the cosine trigonometric function, we first need a
-function that is called when one wishes to @code{eval} it. It could
-look something like this:
-
-@example
-static ex cos_eval_method(const ex & x)
+@example
+size_t sprod::nops() const
@{
- // if (!x%(2*Pi)) return 1
- // if (!x%Pi) return -1
- // if (!x%Pi/2) return 0
- // care for other cases...
- return cos(x).hold();
+ return 2;
@}
-@end example
-
-@cindex @code{hold()}
-@cindex evaluation
-The last line returns @code{cos(x)} if we don't know what else to do and
-stops a potential recursive evaluation by saying @code{.hold()}, which
-sets a flag to the expression signaling that it has been evaluated. We
-should also implement a method for numerical evaluation and since we are
-lazy we sweep the problem under the rug by calling someone else's
-function that does so, in this case the one in class @code{numeric}:
-@example
-static ex cos_evalf(const ex & x)
+ex sprod::op(size_t i) const
@{
- return cos(ex_to_numeric(x));
+ switch (i) @{
+ case 0:
+ return get_struct().left;
+ case 1:
+ return get_struct().right;
+ default:
+ throw std::range_error("sprod::op(): no such operand");
+ @}
@}
@end example
-Differentiation will surely turn up and so we need to tell @code{cos}
-what the first derivative is (higher derivatives (@code{.diff(x,3)} for
-instance are then handled automatically by @code{basic::diff} and
-@code{ex::diff}):
+Implementing @code{nops()} and @code{op()} for container types such as
+@code{sprod} has two other nice side effects:
+
+@itemize @bullet
+@item
+@code{has()} works as expected
+@item
+GiNaC generates better hash keys for the objects (the default implementation
+of @code{calchash()} takes subexpressions into account)
+@end itemize
+
+@cindex @code{let_op()}
+There is a non-const variant of @code{op()} called @code{let_op()} that
+allows replacing subexpressions:
@example
-static ex cos_deriv(const ex & x, unsigned diff_param)
+ex & sprod::let_op(size_t i)
@{
- return -sin(x);
+ // 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");
+ @}
@}
@end example
-@cindex product rule
-The second parameter is obligatory but uninteresting at this point. It
-specifies which parameter to differentiate in a partial derivative in
-case the function has more than one parameter and its main application
-is for correct handling of the chain rule. For Taylor expansion, it is
-enough to know how to differentiate. But if the function you want to
-implement does have a pole somewhere in the complex plane, you need to
-write another method for Laurent expansion around that point.
+Once we have provided @code{let_op()} we also get @code{subs()} and
+@code{map()} for free. In fact, every container class that returns a non-null
+@code{nops()} value must either implement @code{let_op()} or provide custom
+implementations of @code{subs()} and @code{map()}.
-Now that all the ingredients for @code{cos} have been set up, we need
-to tell the system about it. This is done by a macro and we are not
-going to descibe how it expands, please consult your preprocessor if you
-are curious:
+In turn, the availability of @code{map()} enables the recursive behavior of a
+couple of other default method implementations, in particular @code{evalf()},
+@code{evalm()}, @code{normal()}, @code{diff()} and @code{expand()}. Although
+we probably want to provide our own version of @code{expand()} for scalar
+products that turns expressions like @samp{<a+b|c>} into @samp{<a|c>+<b|c>}.
+This is left as an exercise for the reader.
-@example
-REGISTER_FUNCTION(cos, eval_func(cos_eval).
- evalf_func(cos_evalf).
- derivative_func(cos_deriv));
-@end example
-
-The first argument is the function's name used for calling it and for
-output. The second binds the corresponding methods as options to this
-object. Options are separated by a dot and can be given in an arbitrary
-order. GiNaC functions understand several more options which are always
-specified as @code{.option(params)}, for example a method for series
-expansion @code{.series_func(cos_series)}. Again, if no series
-expansion method is given, GiNaC defaults to simple Taylor expansion,
-which is correct if there are no poles involved as is the case for the
-@code{cos} function. The way GiNaC handles poles in case there are any
-is best understood by studying one of the examples, like the Gamma
-(@code{tgamma}) function for instance. (In essence the function first
-checks if there is a pole at the evaluation point and falls back to
-Taylor expansion if there isn't. Then, the pole is regularized by some
-suitable transformation.) Also, the new function needs to be declared
-somewhere. This may also be done by a convenient preprocessor macro:
+The @code{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 @code{eval()}). There is, however, one topic
+that shall be addressed here, as it demonstrates one peculiarity of the
+@code{structure<T>} template: archiving.
+
+@subsection 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 @code{archive()} member function and the
+unarchiving constructor (the @code{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 @code{class_name()} member
+function) is @samp{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
+@code{get_class_name()} member function. Here is a sample implementation
+for enabling archiving of the scalar product type defined above:
@example
-DECLARE_FUNCTION_1P(cos)
+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);
+@}
@end example
-The suffix @code{_1P} stands for @emph{one parameter}. Of course, this
-implementation of @code{cos} is very incomplete and lacks several safety
-mechanisms. Please, have a look at the real implementation in GiNaC.
-(By the way: in case you are worrying about all the macros above we can
-assure you that functions are GiNaC's most macro-intense classes. We
-have done our best to avoid macros where we can.)
+Note that the unarchiving constructor is @code{sprod::structure} and not
+@code{sprod::sprod}, and that we don't need to supply an
+@code{sprod::unarchive()} function.
-@node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
+@node Adding classes, A Comparison With Other CAS, Structures, Extending GiNaC
@c node-name, next, previous, up
@section Adding classes
-If you are doing some very specialized things with GiNaC you may find that
-you have to implement your own algebraic classes to fit your needs. This
-section will explain how to do this by giving the example of a simple
-'string' class. After reading this section you will know how to properly
-declare a GiNaC class and what the minimum required member functions are
-that you have to implement. We only cover the implementation of a 'leaf'
-class here (i.e. one that doesn't contain subexpressions). Creating a
-container class like, for example, a class representing tensor products is
-more involved but this section should give you enough information so you can
-consult the source to GiNaC's predefined classes if you want to implement
-something more complicated.
+The @code{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.
@subsection GiNaC's run-time type information system
string str;
@};
-GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
+GINAC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
@end example
The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
-macros are defined in @file{registrar.h}. They take the name of the class
+macros are defined in @file{registrar.h}. They take the name of the class
and its direct superclass as arguments and insert all required declarations
for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
the first line after the opening brace of the class definition. The
source (at global scope, of course, not inside a function).
@code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
-declarations of the default and copy constructor, the destructor, the
-assignment operator and a couple of other functions that are required. It
-also defines a type @code{inherited} which refers to the superclass so you
-don't have to modify your code every time you shuffle around the class
-hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
-constructor, the destructor and the assignment operator.
-
-Now there are nine member functions we have to implement to get a working
+declarations of the default constructor and a couple of other functions that
+are required. It also defines a type @code{inherited} which refers to the
+superclass so you don't have to modify your code every time you shuffle around
+the class hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} registers the
+class with the GiNaC RTTI.
+
+Now there are seven member functions we have to implement to get a working
class:
@itemize
@item
@code{mystring()}, the default constructor.
-@item
-@code{void destroy(bool call_parent)}, which is used in the destructor and the
-assignment operator to free dynamically allocated members. The @code{call_parent}
-specifies whether the @code{destroy()} function of the superclass is to be
-called also.
-
-@item
-@code{void copy(const mystring &other)}, which is used in the copy constructor
-and assignment operator to copy the member variables over from another
-object of the same class.
-
@item
@code{void archive(archive_node &n)}, the archiving function. This stores all
information needed to reconstruct an object of this class inside an
@code{archive_node}.
@item
-@code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
+@code{mystring(const archive_node &n, lst &sym_lst)}, the unarchiving
constructor. This constructs an instance of the class from the information
found in an @code{archive_node}.
@item
-@code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
+@code{ex unarchive(const archive_node &n, lst &sym_lst)}, the static
unarchiving function. It constructs a new instance by calling the unarchiving
constructor.
@item
+@cindex @code{compare_same_type()}
@code{int compare_same_type(const basic &other)}, which is used internally
by GiNaC to establish a canonical sort order for terms. It returns 0, +1 or
-1, depending on the relative order of this object and the @code{other}
Let's proceed step-by-step. The default constructor looks like this:
@example
-mystring::mystring() : inherited(TINFO_mystring)
-@{
- // dynamically allocate resources here if required
-@}
+mystring::mystring() : inherited(TINFO_mystring) @{@}
@end example
The golden rule is that in all constructors you have to set the
it will be set by the constructor of the superclass and all hell will break
loose in the RTTI. For your convenience, the @code{basic} class provides
a constructor that takes a @code{tinfo_key} value, which we are using here
-(remember that in our case @code{inherited = basic}). If the superclass
+(remember that in our case @code{inherited == basic}). If the superclass
didn't have such a constructor, we would have to set the @code{tinfo_key}
to the right value manually.
In the default constructor you should set all other member variables to
reasonable default values (we don't need that here since our @code{str}
-member gets set to an empty string automatically). The constructor(s) are of
-course also the right place to allocate any dynamic resources you require.
-
-Next, the @code{destroy()} function:
-
-@example
-void mystring::destroy(bool call_parent)
-@{
- // free dynamically allocated resources here if required
- if (call_parent)
- inherited::destroy(call_parent);
-@}
-@end example
-
-This function is where we free all dynamically allocated resources. We don't
-have any so we're not doing anything here, but if we had, for example, used
-a C-style @code{char *} to store our string, this would be the place to
-@code{delete[]} the string storage. If @code{call_parent} is true, we have
-to call the @code{destroy()} function of the superclass after we're done
-(to mimic C++'s automatic invocation of superclass destructors where
-@code{destroy()} is called from outside a destructor).
-
-The @code{copy()} function just copies over the member variables from
-another object:
-
-@example
-void mystring::copy(const mystring &other)
-@{
- inherited::copy(other);
- str = other.str;
-@}
-@end example
-
-We can simply overwrite the member variables here. There's no need to worry
-about dynamically allocated storage. The assignment operator (which is
-automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
-recall) calls @code{destroy()} before it calls @code{copy()}. You have to
-explicitly call the @code{copy()} function of the superclass here so
-all the member variables will get copied.
+member gets set to an empty string automatically).
Next are the three functions for archiving. You have to implement them even
if you don't plan to use archives, but the minimum required implementation
-is really simple. First, the archiving function:
+is really simple. First, the archiving function:
@example
void mystring::archive(archive_node &n) const
The only thing that is really required is calling the @code{archive()}
function of the superclass. Optionally, you can store all information you
deem necessary for representing the object into the passed
-@code{archive_node}. We are just storing our string here. For more
+@code{archive_node}. We are just storing our string here. For more
information on how the archiving works, consult the @file{archive.h} header
file.
function:
@example
-mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+mystring::mystring(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
@{
n.find_string("string", str);
@}
Finally, the unarchiving function:
@example
-ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
+ex mystring::unarchive(const archive_node &n, lst &sym_lst)
@{
return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
@}
@end example
-You don't have to understand how exactly this works. Just copy these four
-lines into your code literally (replacing the class name, of course). It
-calls the unarchiving constructor of the class and unless you are doing
-something very special (like matching @code{archive_node}s to global
-objects) you don't need a different implementation. For those who are
-interested: setting the @code{dynallocated} flag puts the object under
-the control of GiNaC's garbage collection. It will get deleted automatically
-once it is no longer referenced.
+You don't have to understand how exactly this works. Just copy these
+four lines into your code literally (replacing the class name, of
+course). It calls the unarchiving constructor of the class and unless
+you are doing something very special (like matching @code{archive_node}s
+to global objects) you don't need a different implementation. For those
+who are interested: setting the @code{dynallocated} flag puts the object
+under the control of GiNaC's garbage collection. It will get deleted
+automatically once it is no longer referenced.
Our @code{compare_same_type()} function uses a provided function to compare
the string members:
Now the only thing missing is our two new constructors:
@example
-mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
-@{
- // dynamically allocate resources here if required
-@}
-
-mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
-@{
- // dynamically allocate resources here if required
-@}
+mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s) @{@}
+mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s) @{@}
@end example
No surprises here. We set the @code{str} member from the argument and
@example
ex e = mystring("Hello, world!");
-cout << is_ex_of_type(e, mystring) << endl;
+cout << is_a<mystring>(e) << endl;
// -> 1 (true)
cout << e.bp->class_name() << endl;
// -> "GiNaC rulez"+"Hello, world!"
@end example
-(note that GiNaC's automatic term reordering is in effect here), or even
+(GiNaC's automatic term reordering is in effect here), or even
@example
e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
@subsection Automatic evaluation
-@cindex @code{hold()}
@cindex evaluation
+@cindex @code{eval()}
+@cindex @code{hold()}
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.
@end example
The @code{level} argument is used to limit the recursion depth of the
-evaluation. We don't have any subexpressions in the @code{mystring} class
-so we are not concerned with this. If we had, we would call the @code{eval()}
-functions of the subexpressions with @code{level - 1} as the argument if
-@code{level != 1}. The @code{hold()} member function sets a flag in the
-object that prevents further evaluation. Otherwise we might end up in an
-endless loop. When you want to return the object unmodified, use
-@code{return this->hold();}.
+evaluation. We don't have any subexpressions in the @code{mystring}
+class so we are not concerned with this. If we had, we would call the
+@code{eval()} functions of the subexpressions with @code{level - 1} as
+the argument if @code{level != 1}. The @code{hold()} member function
+sets a flag in the object that prevents further evaluation. Otherwise
+we might end up in an endless loop. When you want to return the object
+unmodified, use @code{return this->hold();}.
Let's confirm that it works:
// -> 3*"wow"
@end example
-@subsection Other member functions
+@subsection Optional member functions
We have implemented only a small set of member functions to make the class
-work in the GiNaC framework. For a real algebraic class, there are probably
-some more functions that you will want to re-implement, such as
-@code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
-or the header file of the class you want to make a subclass of to see
-what's there. You can, of course, also add your own new member functions.
-In this case you will probably want to define a little helper function like
+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:
+@cindex @code{calchash()}
+@cindex @code{is_equal_same_type()}
@example
-inline const mystring &ex_to_mystring(const ex &e)
-@{
- return static_cast<const mystring &>(*e.bp);
-@}
+unsigned calchash() const;
+bool is_equal_same_type(const basic &other) const;
+@end example
+
+The @code{calchash()} method returns an @code{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
+@code{calchash()} calculates a hash value out of the @code{tinfo_key} of the
+class and all subexpressions that are accessible via @code{op()}.
+
+@code{is_equal_same_type()} works like @code{compare_same_type()} but only
+tests for equality without establishing an ordering relation, which is often
+faster. The default implementation of @code{is_equal_same_type()} just calls
+@code{compare_same_type()} and tests its result for zero.
+
+@subsection Other member functions
+
+For a real algebraic class, there are probably some more functions that you
+might want to provide:
+
+@example
+bool info(unsigned inf) const;
+ex evalf(int level = 0) const;
+ex series(const relational & r, int order, unsigned options = 0) const;
+ex derivative(const symbol & s) const;
+@end example
+
+If your class stores sub-expressions (see the scalar product example in the
+previous section) you will probably want to override
+
+@cindex @code{let_op()}
+@example
+size_t nops() cont;
+ex op(size_t i) const;
+ex & let_op(size_t i);
+ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
+ex map(map_function & f) const;
@end example
-that let's you get at the object inside an expression (after you have verified
-that the type is correct) so you can call member functions that are specific
-to the class.
+@code{let_op()} is a variant of @code{op()} that allows write access. The
+default implementations of @code{subs()} and @code{map()} use it, so you have
+to implement either @code{let_op()}, or @code{subs()} and @code{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 @code{ex} by stripping the
+@code{ex} off using the @code{ex_to<mystring>(e)} function when that
+should become a need.
That's it. May the source be with you!
disadvantages over these systems.
@menu
-* Advantages:: Stengths of the GiNaC approach.
+* Advantages:: Strengths of the GiNaC approach.
* Disadvantages:: Weaknesses of the GiNaC approach.
* Why C++?:: Attractiveness of C++.
@end menu
@item
development tools: powerful development tools exist for C++, like fancy
editors (e.g. with automatic indentation and syntax highlighting),
-debuggers, visualization tools, documentation generators...
+debuggers, visualization tools, documentation generators@dots{}
@item
modularization: C++ programs can easily be split into modules by
windows into GiNaC have been implemented and many more are possible: the
tiny @command{ginsh} that is part of the distribution exposes GiNaC's
types to a command line and second, as a more consistent approach, an
-interactive interface to the @acronym{Cint} C++ interpreter has been put
-together (called @acronym{GiNaC-cint}) that allows an interactive
-scripting interface consistent with the C++ language.
+interactive interface to the Cint C++ interpreter has been put together
+(called GiNaC-cint) that allows an interactive scripting interface
+consistent with the C++ language. It is available from the usual GiNaC
+FTP-site.
@item
-seemless integration: it is somewhere between difficult and impossible
+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,
portability: While the GiNaC library itself is designed to avoid any
platform dependent features (it should compile on any ANSI compliant C++
compiler), the currently used version of the CLN library (fast large
-integer and arbitrary precision arithmetics) can be compiled only on
-systems with a recently new C++ compiler from the GNU Compiler
-Collection (@acronym{GCC}).@footnote{This is because CLN uses
-PROVIDE/REQUIRE like macros to let the compiler gather all static
-initializations, which works for GNU C++ only.} GiNaC uses recent
-language features like explicit constructors, mutable members, RTTI,
-@code{dynamic_cast}s and STL, so ANSI compliance is meant literally.
-Recent @acronym{GCC} versions starting at 2.95, although itself not yet
-ANSI compliant, support all needed features.
+integer and arbitrary precision arithmetics) can only by compiled
+without hassle on systems with the C++ compiler from the GNU Compiler
+Collection (GCC).@footnote{This is because CLN uses PROVIDE/REQUIRE like
+macros to let the compiler gather all static initializations, which
+works for GNU C++ only. 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.}
+GiNaC uses recent language features like explicit constructors, mutable
+members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
+literally. Recent GCC versions starting at 2.95.3, although itself not
+yet ANSI compliant, support all needed features.
@end itemize
@cindex reference counting
@cindex copy-on-write
@cindex garbage collection
-An expression is extremely light-weight since internally it works like a
-handle to the actual representation and really holds nothing more than a
-pointer to some other object. What this means in practice is that
-whenever you create two @code{ex} and set the second equal to the first
-no copying process is involved. Instead, the copying takes place as soon
-as you try to change the second. Consider the simple sequence of code:
+In GiNaC, there is an @emph{intrusive reference-counting} mechanism at work
+where the counter belongs to the algebraic objects derived from class
+@code{basic} but is maintained by the smart pointer class @code{ptr}, of
+which @code{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 @code{ex} and set the second equal to the
+first no copying process is involved. Instead, the copying takes place
+as soon as you try to change the second. Consider the simple sequence
+of code:
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
can be:
@example
-#include <ginac/ginac.h>
-using namespace std;
-using namespace GiNaC;
-
-int main()
@{
symbol x("x"), y("y");
@example
#include <ginac/ginac.h>
-int main(void)
+int main()
@{
GiNaC::symbol x("x");
GiNaC::ex a = GiNaC::sin(x);
AC_PROG_INSTALL
AC_LANG_CPLUSPLUS
-AM_PATH_GINAC(0.7.0, [
+AM_PATH_GINAC(0.9.0, [
LIBS="$LIBS $GINACLIB_LIBS"
CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
], AC_MSG_ERROR([need to have GiNaC installed]))
@end example
This @file{Makefile.am}, says that we are building a single executable,
-from a single sourcefile @file{simple.cpp}. Since every program
+from a single source file @file{simple.cpp}. Since every program
we are building uses GiNaC we simply added the GiNaC options
to @env{$LIBS} and @env{$CPPFLAGS}, but in other circumstances, we might
want to specify them on a per-program basis: for instance by
@item
@cite{Computer Algebra: Systems and Algorithms for Algebraic Computation},
-J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
+James H. Davenport, Yvon Siret and Evelyne Tournier, ISBN 0-12-204230-1, 1988,
Academic Press, London
@item
-@cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
+@cite{Computer Algebra Systems - A Practical Guide},
+Michael J. Wester (editor), ISBN 0-471-98353-5, 1999, Wiley, Chichester
+
+@item
+@cite{The Art of Computer Programming, Vol 2: Seminumerical Algorithms},
+Donald E. Knuth, ISBN 0-201-89684-2, 1998, Addison Wesley
+
+@item
+@cite{Pi Unleashed}, J@"org Arndt and Christoph Haenel,
+ISBN 3-540-66572-2, 2001, Springer, Heidelberg
+
+@item
+@cite{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
@end itemize
git://www.ginac.de / ginac.git / blobdiff