machine catches fire. Another quite important intent is to allow people
to fiddle around with optimization.
+By default, the only documentation that will be built is this tutorial
+in @file{.info} format. To build the GiNaC tutorial and reference manual
+in HTML, DVI, PostScript, or PDF formats, use one of
+
+@example
+$ make html
+$ make dvi
+$ make ps
+$ make pdf
+@end example
+
Generally, the top-level Makefile runs recursively to the
subdirectories. It is therefore safe to go into any subdirectory
(@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
(or @file{@var{INCLUDEDIR}/ginac/}, if specified).
@item
-All documentation (HTML and Postscript) will be stuffed into
+All documentation (info) will be stuffed into
@file{@var{PREFIX}/share/doc/GiNaC/} (or
@file{@var{DATADIR}/doc/GiNaC/}, if @var{DATADIR} was specified).
arbitrary objects. Rather, non-commutativity in GiNaC is a property of the
classes of objects involved, and non-commutative products are formed with
the usual @samp{*} operator, as are ordinary products. GiNaC is capable of
-figuring out by itself which objects commute and will group the factors
+figuring out by itself which objects commutate and will group the factors
by their class. Consider this example:
@example
As can be seen, GiNaC pulls out the overall commutative factor @samp{-16} and
groups the non-commutative factors (the gammas and the su(3) generators)
together while preserving the order of factors within each class (because
-Clifford objects commute with color objects). The resulting expression is a
+Clifford objects commutate with color objects). The resulting expression is a
@emph{commutative} product with two factors that are themselves non-commutative
products (@samp{gamma~mu*gamma~nu} and @samp{T.a*T.b}). For clarification,
parentheses are placed around the non-commutative products in the output.
canonicalize themselves according to rules specified in the implementation
of the non-commutative classes. The drawback is that to work with other than
the built-in algebras you have to implement new classes yourself. Symbols
-always commute and it's not possible to construct non-commutative products
+always commutate and it's not possible to construct non-commutative products
using symbols to represent the algebra elements or generators. User-defined
functions can, however, be specified as being non-commutative.
expressions in GiNaC:
@itemize
-@item @code{return_types::commutative}: Commutes with everything. Most GiNaC
+@item @code{return_types::commutative}: Commutates with everything. Most GiNaC
classes are of this kind.
@item @code{return_types::noncommutative}: Non-commutative, belonging to a
certain class of non-commutative objects which can be determined with the
- @code{return_type_tinfo()} method. Expressions of this category commute
+ @code{return_type_tinfo()} method. Expressions of this category commutate
with everything except @code{noncommutative} expressions of the same
class.
@item @code{return_types::noncommutative_composite}: Non-commutative, composed
of non-commutative objects of different classes. Expressions of this
- category don't commute with any other @code{noncommutative} or
+ category don't commutate with any other @code{noncommutative} or
@code{noncommutative_composite} expressions.
@end itemize
which takes two arguments: the index and a @dfn{representation label} in the
range 0 to 255 which is used to distinguish elements of different Clifford
algebras (this is also called a @dfn{spin line index}). Gammas with different
-labels commute with each other. The dimension of the index can be 4 or (in
+labels commutate with each other. The dimension of the index can be 4 or (in
the framework of dimensional regularization) any symbolic value. Spinor
indices on Dirac gammas are not supported in GiNaC.
GiNaC will complain and/or produce incorrect results.
@cindex @code{dirac_gamma5()}
-There is a special element @samp{gamma5} that commutes with all other
+There is a special element @samp{gamma5} that commutates with all other
gammas, has a unit square, and in 4 dimensions equals
@samp{gamma~0 gamma~1 gamma~2 gamma~3}, provided by
@cindex @code{dirac_trace()}
To calculate the trace of an expression containing strings of Dirac gammas
-you use the function
+you use one of the functions
@example
+ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
+ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
@end example
-This function takes the trace of all gammas with the specified representation
-label; gammas with other labels are left standing. The last argument to
+These functions take the trace over all gammas in the specified set @code{rls}
+or list @code{rll} of representation labels, or the single label @code{rl};
+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
+element, which defaults to 4.
+
+The @code{dirac_trace()} function is a linear functional that is equal to the
+ordinary matrix 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
which takes two arguments: the index and a @dfn{representation label} in the
range 0 to 255 which is used to distinguish elements of different color
-algebras. Objects with different labels commute with each other. The
+algebras. Objects with different labels commutate with each other. The
dimension of the index must be exactly 8 and it should be of class @code{idx},
not @code{varidx}.
@end example
@cindex @code{color_trace()}
-To calculate the trace of an expression containing color objects you use the
-function
+To calculate the trace of an expression containing color objects you use one
+of the functions
@example
+ex color_trace(const ex & e, const std::set<unsigned char> & rls);
+ex color_trace(const ex & e, const lst & rll);
ex color_trace(const ex & e, unsigned char rl = 0);
@end example
-This function takes the trace of all color @samp{T} objects with the
-specified representation label; @samp{T}s with other labels are left
-standing. For example:
+These functions take the trace over all color @samp{T} objects in the
+specified set @code{rls} or list @code{rll} of representation labels, or the
+single label @code{rl}; @samp{T}s with other labels are left standing. For
+example:
@example
...
@end cartouche
To determine whether an expression is commutative or non-commutative and if
-so, with which other expressions it would commute, you use the methods
+so, with which other expressions it would commutate, you use the methods
@code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
for an explanation of these.
@cindex @code{unit()}
@cindex @code{content()}
@cindex @code{primpart()}
+@cindex @code{unitcontprim()}
The methods
ex ex::unit(const ex & x);
ex ex::content(const ex & x);
ex ex::primpart(const ex & x);
+ex ex::primpart(const ex & x, const ex & c);
@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.
+content parts). The second variant of @code{primpart()} expects the previously
+calculated content part of the polynomial in @code{c}, which enables it to
+work faster in the case where the content part has already been computed. The
+product of unit, content, and primitive part is the original polynomial.
+
+Additionally, the method
+
+@example
+void ex::unitcontprim(const ex & x, ex & u, ex & c, ex & p);
+@end example
+
+computes the unit, content, and primitive parts in one go, returning them
+in @code{u}, @code{c}, and @code{p}, respectively.
@subsection GCD, LCM and resultant
option which is explained in the next section.
+@subsection Functions with a variable number of arguments
+
+The @code{DECLARE_FUNCTION} and @code{REGISTER_FUNCTION} macros define
+functions with a fixed number of arguments. Sometimes, though, you may need
+to have a function that accepts a variable number of expressions. One way to
+accomplish this is to pass variable-length lists as arguments. The
+@code{Li()} function uses this method for multiple polylogarithms.
+
+It is also possible to define functions that accept a different number of
+parameters under the same function name, such as the @code{psi()} function
+which can be called either as @code{psi(z)} (the digamma function) or as
+@code{psi(n, z)} (polygamma functions). These are actually two different
+functions in GiNaC that, however, have the same name. Defining such
+functions is not possible with the macros but requires manually fiddling
+with GiNaC internals. If you are interested, please consult the GiNaC source
+code for the @code{psi()} function (@file{inifcns.h} and
+@file{inifcns_gamma.cpp}).
+
@node Printing, Structures, Symbolic functions, Extending GiNaC
@c node-name, next, previous, up
git://www.ginac.de / ginac.git / blobdiff