This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2010 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2015 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{} 1999-2010 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2015 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
@section License
The GiNaC framework for symbolic computation within the C++ programming
-language is Copyright @copyright{} 1999-2010 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2015 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
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 GCC for development
+ISO standard @cite{ISO/IEC 14882:2011(E)}. We used GCC for development
so if you have a different compiler you are on your own. For the
configuration to succeed you need a Posix compliant shell installed in
@file{/bin/sh}, GNU @command{bash} is fine. The pkg-config utility is
of the non-commutative classes. The drawback is that to work with other than
the built-in algebras you have to implement new classes yourself. Both
symbols and user-defined functions can be specified as being non-commutative.
+For symbols, this is done by subclassing class symbol; for functions,
+by explicitly setting the return type (@pxref{Symbolic functions}).
@cindex @code{return_type()}
@cindex @code{return_type_tinfo()}
Note that the call @code{clifford_unit(mu, minkmetric())} creates
something very close to @code{dirac_gamma(mu)}, although
@code{dirac_gamma} have more efficient simplification mechanism.
-@cindex @code{clifford::get_metric()}
+@cindex @code{get_metric()}
The method @code{clifford::get_metric()} returns a metric defining this
Clifford number.
@{
symbol x("x"), y("y");
- ex e1 = 2*x^2-4*x+3;
+ ex e1 = 2*x*x-4*x+3;
cout << "e1(7) = " << e1.subs(x == 7) << endl;
// -> 73
@code{(x^(-3)*y^(-2)*z).subs(1/(x*y)==c, subs_options::algebraic)} will
return @code{x^(-1)*c^2*z}.
-@strong{Note:} this only works for multiplications
+@strong{Please notice:} this only works for multiplications
and not for locating @code{x+y} within @code{x+y+z}.
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;
+ cout << ex(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;
+ cout << ex(indexed(A, i, j, k)).antisymmetrize(lst(i, j)) << endl;
// -> -1/2*A.j.i.k+1/2*A.i.j.k
- cout << lst(a, b, c).symmetrize_cyclic(lst(a, b, c)) << endl;
+ cout << ex(lst(a, b, c)).symmetrize_cyclic(lst(a, b, c)) << endl;
// -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
@}
@end example
@item @code{log(x)}
@tab natural logarithm
@cindex @code{log()}
+@item @code{eta(x,y)}
+@tab Eta function: @code{eta(x,y) = log(x*y) - log(x) - log(y)}
+@cindex @code{eta()}
@item @code{Li2(x)}
@tab dilogarithm
@cindex @code{Li2()}
@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.
+For functions that have a branch cut in the complex plane, GiNaC
+follows the conventions of C/C++ for systems that do not support a
+signed zero. In particular: the natural logarithm (@code{log}) and
+the square root (@code{sqrt}) both have their branch cuts running
+along the negative real axis. The @code{asin}, @code{acos}, and
+@code{atanh} functions all have two branch cuts starting at +/-1 and
+running away towards infinity along the real axis. The @code{atan} and
+@code{asinh} functions have two branch cuts starting at +/-i and
+running away towards infinity along the imaginary axis. The
+@code{acosh} function has one branch cut starting at +1 and running
+towards -infinity. These functions are continuous as the branch cut
+is approached coming around the finite endpoint of the cut in a
+counter clockwise direction.
+
+@c
+@subsection Expanding functions
+@cindex expand trancedent functions
+@cindex @code{expand_options::expand_transcendental}
+@cindex @code{expand_options::expand_function_args}
+GiNaC knows several expansion laws for trancedent functions, e.g.
+@tex
+$e^{a+b}=e^a e^b$,
+$|zw|=|z|\cdot |w|$
+@end tex
+@ifnottex
+@command{exp(a+b)=exp(a) exp(b), |zw|=|z| |w|}
+@end ifnottex
+or
+@tex
+$\log(c*d)=\log(c)+\log(d)$,
+@end tex
+@ifnottex
+@command{log(cd)=log(c)+log(d)}
+@end ifnottex
+(for positive
+@tex
+$c,\ d$
+@end tex
+@ifnottex
+@command{c, d}
+@end ifnottex
+). In order to use these rules you need to call @code{expand()} method
+with the option @code{expand_options::expand_transcendental}. Another
+relevant option is @code{expand_options::expand_function_args}. Their
+usage and interaction can be seen from the following example:
+@example
+@{
+ symbol x("x"), y("y");
+ ex e=exp(pow(x+y,2));
+ cout << e.expand() << endl;
+ // -> exp((x+y)^2)
+ cout << e.expand(expand_options::expand_transcendental) << endl;
+ // -> exp((x+y)^2)
+ cout << e.expand(expand_options::expand_function_args) << endl;
+ // -> exp(2*x*y+x^2+y^2)
+ cout << e.expand(expand_options::expand_function_args
+ | expand_options::expand_transcendental) << endl;
+ // -> exp(y^2)*exp(2*x*y)*exp(x^2)
+@}
+@end example
+If both flags are set (as in the last call), then GiNaC tries to get
+the maximal expansion. For example, for the exponent GiNaC firstly expands
+the argument and then the function. For the logarithm and absolute value,
+GiNaC uses the opposite order: firstly expands the function and then its
+argument. Of course, a user can fine-tune this behaviour by sequential
+calls of several @code{expand()} methods with desired flags.
@node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
@c node-name, next, previous, up
@}
@end example
-If you declare your own GiNaC functions, then they will conjugate themselves
-by conjugating their arguments. This is the default strategy. If you want to
-change this behavior, you have to supply a specialized conjugation method
-for your function (see @ref{Symbolic functions} and the GiNaC source-code
-for @code{abs} as an example). Also, specialized methods can be provided
-to take real and imaginary parts of user-defined functions.
+If you declare your own GiNaC functions and you want to conjugate them, you
+will have to supply a specialized conjugation method for them (see
+@ref{Symbolic functions} and the GiNaC source-code for @code{abs} as an
+example). GiNaC does not automatically conjugate user-supplied functions
+by conjugating their arguments because this would be incorrect on branch
+cuts. Also, specialized methods can be provided to take real and imaginary
+parts of user-defined functions.
@node Solving linear systems of equations, Input/output, Complex expressions, Methods and functions
@c node-name, next, previous, up
table["x"] = x+log(y)+1;
parser reader(table);
ex e = reader("5*x^3 - x^2");
- // e = 5*(x+log(y)+1)^3 + (x+log(y)+1)^2
+ // e = 5*(x+log(y)+1)^3 - (x+log(y)+1)^2
@}
@end example
parser reader;
ex e = reader("2*x+sin(y)");
symtab table = reader.get_syms();
- symbol x = reader["x"];
- symbol y = reader["y"];
+ symbol x = ex_to<symbol>(table["x"]);
+ symbol y = ex_to<symbol>(table["y"]);
@}
@end example
@}
@end example
-With this parser, it's also easy to implement interactive GiNaC programs:
+With this parser, it's also easy to implement interactive GiNaC programs.
+When running the following program interactively, remember to send an
+EOF marker after the input, e.g. by pressing Ctrl-D on an empty line:
@example
#include <iostream>
@cindex Monte Carlo integration
@code{FUNCP_2P} allows for two variables in the expression. @code{FUNCP_CUBA} is
the correct type to be used with the CUBA library
-(@uref{http://www.feynarts/cuba}) for numerical integrations. The details for the
+(@uref{http://www.feynarts.de/cuba}) for numerical integrations. The details for the
parameters of @code{FUNCP_CUBA} are explained in the CUBA manual.
@cindex compile_ex
@code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
compiler and produce the object files. This shell script comes with GiNaC and
will be installed together with GiNaC in the configured @code{$PREFIX/bin}
-directory.
+directory. You can also export additional compiler flags via the $CXXFLAGS
+variable:
+
+@example
+setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
+compile_ex(...);
+@end example
@subsection Archiving
@cindex @code{archive} (class)
case the function has more than one parameter, and its main application
is for correct handling of the chain rule.
+Derivatives of some functions, for example @code{abs()} and
+@code{Order()}, could not be evaluated through the chain rule. In such
+cases the full derivative may be specified as shown for @code{Order()}:
+
+@example
+static ex Order_expl_derivative(const ex & arg, const symbol & s)
+@{
+ return Order(arg.diff(s));
+@}
+@end example
+
+That is, we need to supply a procedure, which returns the expression of
+derivative with respect to the variable @code{s} for the argument
+@code{arg}. This procedure need to be registered with the function
+through the option @code{expl_derivative_func} (see the next
+Subsection). In contrast, a partial derivative, e.g. as was defined for
+@code{cos()} above, needs to be registered through the option
+@code{derivative_func}.
+
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
eval_func(<C++ function>)
evalf_func(<C++ function>)
derivative_func(<C++ function>)
+expl_derivative_func(<C++ function>)
series_func(<C++ function>)
conjugate_func(<C++ function>)
@end example
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()}.
+numeric evaluation, partial derivatives, explicit derivative, and series
+expansion, respectively. They correspond to the GiNaC methods
+@code{eval()}, @code{evalf()}, @code{diff()} and @code{series()}.
The @code{eval_func()} function needs to use @code{.hold()} if no further
automatic evaluation is desired or possible.
multiple interfaces: Though real GiNaC programs have to be written in
some editor, then be compiled, linked and executed, there are more ways
to work with the GiNaC engine. Many people want to play with
-expressions interactively, as in traditional CASs. Currently, two such
-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 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.
+expressions interactively, as in traditional CASs: The tiny
+@command{ginsh} that comes with the distribution exposes many, but not
+all, of GiNaC's types to a command line.
@item
seamless integration: it is somewhere between difficult and impossible
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.
+literally.
@end itemize
@node Configure script options, Example package, Package tools, Package tools
@c node-name, next, previous, up
-@subsection Configuring a package that uses GiNaC
+@appendixsection Configuring a package that uses GiNaC
The directory where the GiNaC libraries are installed needs
to be found by your system's dynamic linkers (both compile- and run-time
@node Example package, Bibliography, Configure script options, Package tools
@c node-name, next, previous, up
-@subsection Example of a package using GiNaC
+@appendixsection Example of a package using GiNaC
The following shows how to build a simple package using automake
and the @samp{PKG_CHECK_MODULES} macro. The program used here is @file{simple.cpp}:
@itemize @minus{}
@item
-@cite{ISO/IEC 14882:1998: Programming Languages: C++}
+@cite{ISO/IEC 14882:2011: Programming Languages: C++}
@item
@cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
git://www.ginac.de / ginac.git / blobdiff