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) 1999-2002 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{} 1999-2002 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{} 1999-2002 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;
[[1,1],[2,-1]]
> A+2*M;
[[1,1],[2,-1]]+2*[[1,3],[-3,2]]
-> evalm(");
+> 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
+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.
* The Class Hierarchy:: Overview of GiNaC's classes.
+* Error handling:: How the library reports errors.
* Symbols:: Symbolic objects.
* Numbers:: Numerical objects.
* Constants:: Pre-defined constants.
@code{ex}.
-@node The Class Hierarchy, Symbols, Expressions, Basic Concepts
+@node The Class Hierarchy, Error handling, Expressions, Basic Concepts
@c node-name, next, previous, up
@section The Class Hierarchy
@end multitable
@end cartouche
-@node Symbols, Numbers, The Class Hierarchy, Basic Concepts
+
+@node Error handling, Symbols, The Class Hierarchy, 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(void) 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(void)
+@{
+ try @{
+ ...
+ // code using GiNaC
+ ...
+ @} catch (exception &p) @{
+ cerr << p.what() << endl;
+ return 1;
+ @}
+ return 0;
+@}
+@end example
+
+
+@node Symbols, Numbers, Error handling, Basic Concepts
@c node-name, next, previous, up
@section Symbols
@cindex @code{symbol} (class)
@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 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 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 differnt 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
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
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, Matrices, Mathematical functions, Basic Concepts
@c node-name, next, previous, up
A simple example shall illustrate the concepts:
@example
+#include <iostream>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;
@code{j}.
Note the difference between the indices @code{i} and @code{j} which are of
-class @code{idx}, and the index values which are the sybols @code{i_sym}
+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:
@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.
-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:
+Here are some examples of symmetry definitions:
@example
...
- cout << indexed(A, indexed::symmetric, i, j)
- + indexed(A, indexed::symmetric, j, i) << endl;
+ // 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
+
+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, 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;
+ cout << indexed(B, sy_anti(), i, j)
+ + indexed(B, sy_anti(), j, i) << 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, k)
+ + indexed(B, sy_anti(), j, i, k) << endl;
// -> 0
...
@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 dumy indices that appear in different terms of a sum
+@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
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
@end itemize
The @code{clifford} and @code{color} classes are subclasses of
-@code{indexed} because the elements of these algebras ususally carry
+@code{indexed} because the elements of these algebras usually carry
indices. The @code{matrix} class is described in more detail in
@ref{Matrices}.
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 may produce incorrect results.
+
@cindex @code{dirac_gamma5()}
-and there's a special element @samp{gamma5} that commutes with all other
+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
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\}.
In products of dirac gammas, superfluous unity elements are automatically
removed, squares are replaced by their values and @samp{gamma5} is
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
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
@}
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);
* Information About Expressions::
* Substituting Expressions::
* Pattern Matching and Advanced Substitutions::
+* Applying a Function on Subexpressions::
* Polynomial Arithmetic:: Working with polynomials.
* Rational Expressions:: Working with rational functions.
* Symbolic Differentiation::
Actually, if you construct an expression like @code{a == b}, this will be
represented by an object of the @code{relational} class (@pxref{Relations})
-which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
+which is not evaluated until (explicitly or implicitly) cast to a @code{bool}.
There are also two methods
next section.
-@node Pattern Matching and Advanced Substitutions, Polynomial Arithmetic, 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 Pattern matching and advanced substitutions
@cindex @code{wildcard} (class)
In general, having more than one single wildcard as a term of a sum or a
factor of a product (such as @samp{a+$0+$1}) will lead to unpredictable or
-amgiguous results.
+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
contains a linear term you should use the coeff() function instead.)
@end example
+@cindex @code{find()}
+The method
+
+@example
+bool ex::find(const ex & pattern, lst & found);
+@end example
+
+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
+> 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
+
@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
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);
+> 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);
+> 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
@end example
-@node Polynomial Arithmetic, Rational Expressions, Pattern Matching and Advanced Substitutions, Methods and Functions
+@node Applying a Function on Subexpressions, Polynomial Arithmetic, 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()}
+
+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 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 (unsigned i=0; i<e.nops(); i++)
+ sum += calc_trace(e.op(i));
+ return sum;
+ @} else if (is_a<mul>)(e)) @{
+ ...
+ @} else @{
+ ...
+ @}
+@}
+@end example
+
+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.
+
+GiNaC offers the @code{map()} method to aid in the implementation of such
+operations:
+
+@example
+ex ex::map(map_function & f) const;
+ex ex::map(ex (*f)(const ex & e)) const;
+@end example
+
+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.
+
+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
+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
+
+This function object could then be used like this:
+
+@example
+@{
+ ex M = ... // expression with matrices
+ calc_trace do_trace;
+ ex tr = do_trace(M);
+@}
+@end example
+
+Here is another example for you to meditate over. It removes quadratic
+terms in a variable from an expanded polynomial:
+
+@example
+struct map_rem_quad : public map_function @{
+ ex var;
+ map_rem_quad(const ex & var_) : var(var_) @{@}
+
+ ex operator()(const ex & e)
+ @{
+ if (is_a<add>(e) || is_a<mul>(e))
+ return e.map(*this);
+ else if (is_a<power>(e) &&
+ e.op(0).is_equal(var) && e.op(1).info(info_flags::even))
+ return 0;
+ else
+ return e;
+ @}
+@};
+
+...
+
+@{
+ symbol x("x"), y("y");
+
+ ex e;
+ for (int i=0; i<8; i++)
+ e += pow(x, i) * pow(y, 8-i) * (i+1);
+ cout << e << endl;
+ // -> 4*y^5*x^3+5*y^4*x^4+8*y*x^7+7*y^2*x^6+2*y^7*x+6*y^3*x^5+3*y^6*x^2+y^8
+
+ map_rem_quad rem_quad(x);
+ cout << rem_quad(e) << endl;
+ // -> 4*y^5*x^3+8*y*x^7+2*y^7*x+6*y^3*x^5+y^8
+@}
+@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
+> 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
+
+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
+> 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
+
+
+@node Polynomial Arithmetic, Rational Expressions, Applying a Function on Subexpressions, Methods and Functions
@c node-name, next, previous, up
@section Polynomial arithmetic
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
+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}.
in a distributed form with terms like @math{c*x1^e1*...*xn^en}, as specified
by the @code{distributed} flag.
-Note that the original polynomial needs to be in expanded form in order
-for @code{collect()} to be able to find the coefficients properly.
+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()}:
+
+@example
+> 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
@subsection Degree and coefficients
@cindex @code{degree()}
int ex::ldegree(const ex & s);
@end example
-which also work reliably on non-expanded input polynomials (they even work
-on rational functions, returning the asymptotic degree). To extract
-a coefficient with a certain power from an expanded polynomial you use
+These functions only work reliably if the input polynomial is collected in
+terms of the object @samp{s}. Otherwise, they are only guaranteed to return
+the upper/lower bounds of the exponents. If you need accurate results, you
+have to call @code{expand()} and/or @code{collect()} on the input polynomial.
+For example
+
+@example
+> a=(x+1)^2-x^2;
+(1+x)^2-x^2;
+> degree(a,x);
+2
+> degree(expand(a),x);
+1
+@end example
+
+@code{degree()} also works on rational functions, returning the asymptotic
+degree:
+
+@example
+> degree((x+1)/(x^3+1),x);
+-2
+@end example
+
+If the input is not a polynomial or rational function in the variable @samp{s},
+the behavior of @code{degree()} and @code{ldegree()} is undefined.
+
+To extract a coefficient with a certain power from an expanded
+polynomial you use
@example
ex ex::coeff(const ex & s, int n);
polynomial is analyzed:
@example
-#include <ginac/ginac.h>
-using namespace std;
-using namespace GiNaC;
-
-int main()
@{
symbol x("x"), y("y");
ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
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:
+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 result may depend
-on the order of differentiation:
+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(x-2*y*x,3) * pow(x+y,2) * (x-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;
- // -> (y+x)^2*(-1+6*y+8*y^3-12*y^2)*(y-x)*x^3
+ // -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
cout << sqrfree(BiVarPol, lst(y,x)) << endl;
- // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
+ // -> 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
@code{.to_rational()}, described below.
This means that both expressions @code{t1} and @code{t2} are indeed
-simplified in this little program:
+simplified in this little code snippet:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
@{
symbol x("x");
ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
the derivatives of all the monomials:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
@{
symbol x("x"), y("y"), z("z");
ex P = pow(x, 5) + pow(x, 2) + y;
- cout << P.diff(x,2) << endl; // 20*x^3 + 2
+ cout << P.diff(x,2) << endl;
+ // -> 20*x^3 + 2
cout << P.diff(y) << endl; // 1
+ // -> 1
cout << P.diff(z) << endl; // 0
+ // -> 0
@}
@end example
@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
@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 proper but equivalent to @code{pow(x, numeric(1, 2)})
+@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
-@item @code{Derivative(x, l)}
-@tab inert partial differentiation operator (used internally)
+@cindex @code{Order()}
@end multitable
@end cartouche
@{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
@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:
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:
+list of symbols and indices to be used:
@example
@{
- symbol x("x"), y("y");
- ex e("2*x+sin(y)", lst(x, y));
+ symbol x("x"), y("y"), p("p");
+ idx i(symbol("i"), 3);
+ ex e("2*x+sin(y)+p.i", lst(x, y, p, i));
@}
@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.
+output operator @code{<<}. The symbols and indices in the string are matched
+by name to the symbols and indices in the list and if GiNaC encounters a
+symbol or index not specified in the list it will throw an exception. Only
+indices whose values are single symbols can be used (i.e. numeric indices
+or compound indices as in "A.(2*n+1)" are not allowed).
With this constructor, it's also easy to implement interactive GiNaC programs:
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;
- @}
+ 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
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.
+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
switch (p[i].type) @{
case archive_node::PTYPE_BOOL: @{
bool x;
- n.find_bool(name, x);
+ n.find_bool(name, x, j);
cout << (x ? "true" : "false");
break;
@}
case archive_node::PTYPE_UNSIGNED: @{
unsigned x;
- n.find_unsigned(name, x);
+ n.find_unsigned(name, x, j);
cout << x;
break;
@}
case archive_node::PTYPE_STRING: @{
string x;
- n.find_string(name, x);
+ n.find_string(name, x, j);
cout << '\"' << x << '\"';
break;
@}
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
+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.
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.
+provided by CLN are much better suited.
@node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
@example
static ex cos_evalf(const ex & x)
@{
- return cos(ex_to<numeric>(x));
+ if (is_a<numeric>(x))
+ return cos(ex_to<numeric>(x));
+ else
+ return cos(x).hold();
@}
@end example
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
+going to describe how it expands, please consult your preprocessor if you
are curious:
@example
@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
@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
+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
+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
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.
@}
@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).
+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:
@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
+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
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.
@}
@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:
@subsection Automatic evaluation
@cindex @code{hold()}
+@cindex @code{eval()}
@cindex evaluation
When dealing with objects that are just a little more complicated than the
simple string objects we have implemented, chances are that you will want to
@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:
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
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 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
+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");
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]))
@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{The Role of gamma5 in Dimensional Regularization}, Dirk Kreimer, hep-ph/9401354
@end itemize
git://www.ginac.de / ginac.git / blobdiff