This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2005 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{} 1999-2004 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2005 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{} 1999-2004 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2005 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
@end itemize
-@strong{Note:} when printing expressions, covariant indices and indices
+@strong{Please notice:} when printing expressions, covariant indices and indices
without variance are denoted @samp{.i} while contravariant indices are
denoted @samp{~i}. Dotted indices have a @samp{*} in front of the index
value. In the following, we are going to use that notation in the text so
ex dirac_ONE(unsigned char rl = 0);
@end example
-@strong{Note:} You must always use @code{dirac_ONE()} when referring to
+@strong{Please notice:} 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,
@end tex
dimensional algebra with
generators @samp{e~k} satisfying the identities
-@samp{e~i e~j + e~j e~i = B(i, j)} for some symmetric matrix (@code{metric})
-@math{B(i, j)}. Such generators are created by the function
+@samp{e~i e~j + e~j e~i = B(i, j)} for some matrix (@code{metric})
+@math{B(i, j)}, which may be non-symmetric. Such generators are created
+by the function
@example
ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
@code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
metric defining this Clifford number.
+If the matrix @math{B(i, j)} is in fact symmetric you may prefer to create
+the Clifford algebra units with a call like that
+
+@example
+ ex e = clifford_unit(mu, indexed(B, sy_symm(), i, j));
+@end example
+
+since this may yield some further automatic simplifications.
+
Individual generators of a Clifford algebra can be accessed in several
ways. For example
@cindex @code{clifford_moebius_map()}
@example
ex clifford_moebius_map(const ex & a, const ex & b, const ex & c,
- const ex & d, const ex & v, const ex & G);
+ const ex & d, const ex & v, const ex & G, unsigned char rl = 0);
+ ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
@end example
It takes a list or vector @code{v} and makes the Moebius
(conformal or linear-fractional) transformation @samp{v ->
-(av+b)/(cv+d)} defined by the matrix @samp{[[a, b], [c, d]]}. The last
+(av+b)/(cv+d)} defined by the matrix @samp{M = [[a, b], [c, d]]}. The
parameter @code{G} defines the metric of the surrounding
(pseudo-)Euclidean space. The returned value of this function is a list
of components of the resulting vector.
ex color_ONE(unsigned char rl = 0);
@end example
-@strong{Note:} You must always use @code{color_ONE()} when referring to
+@strong{Please notice:} 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,
@cite{Special Values of Multiple Polylogarithms},
J.Borwein, D.Bradley, D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. 907-941
+@cite{Numerical Evaluation of Multiple Polylogarithms},
+J.Vollinga, S.Weinzierl, hep-ph/0410259
+
@node Complex Conjugation, Solving Linear Systems of Equations, Multiple polylogarithms, Methods and Functions
@c node-name, next, previous, up
@section Complex Conjugation
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}
object. If it returns 0, the objects are considered equal.
-@strong{Note:} This has nothing to do with the (numeric) ordering
+@strong{Please notice:} This has nothing to do with the (numeric) ordering
relationship expressed by @code{<}, @code{>=} etc (which cannot be defined
for non-numeric classes). For example, @code{numeric(1).compare_same_type(numeric(2))}
may return +1 even though 1 is clearly smaller than 2. Every GiNaC class
git://www.ginac.de / ginac.git / blobdiff