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
You should have received a copy of the GNU General Public License
along with this program; see the file COPYING. If not, write to the
-Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
-MA 02111-1307, USA.
+Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
+MA 02110-1301, USA.
@node A Tour of GiNaC, How to use it from within C++, Introduction, Top
Here we have made use of the @command{ginsh}-command @code{%} to pop the
previously evaluated element from @command{ginsh}'s internal stack.
+Often, functions don't have roots in closed form. Nevertheless, it's
+quite easy to compute a solution numerically, to arbitrary precision:
+
+@cindex fsolve
+@example
+> Digits=50:
+> fsolve(cos(x)==x,x,0,2);
+0.7390851332151606416553120876738734040134117589007574649658
+> f=exp(sin(x))-x:
+> X=fsolve(f,x,-10,10);
+2.2191071489137460325957851882042901681753665565320678854155
+> subs(f,x==X);
+-6.372367644529809108115521591070847222364418220770475144296E-58
+@end example
+
+Notice how the final result above differs slightly from zero by about
+@math{6*10^(-58)}. This is because with 50 decimal digits precision the
+root cannot be represented more accurately than @code{X}. Such
+inaccuracies are to be expected when computing with finite floating
+point values.
+
If you ever wanted to convert units in C or C++ and found this is
cumbersome, here is the solution. Symbolic types can always be used as
tags for different types of objects. Converting from wrong units to 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
+generated by Perl scripts. Last but not least, the CLN library
+is used extensively and needs to be installed on your system.
+Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
+(it is covered by GPL) and install it prior to trying to install
GiNaC. The configure script checks if it can find it and if it cannot
it will refuse to continue.
a lookup table is used.
If you know that an expression holds an integral, you can get the
-integration variable, the left boundary, right boundary and integrant by
+integration variable, the left boundary, right boundary and integrand by
respectively calling @code{.op(0)}, @code{.op(1)}, @code{.op(2)}, and
@code{.op(3)}. Differentiating integrals with respect to variables works
as expected. Note that it makes no sense to differentiate an integral
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);
+ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
+ const string & tex_base_name);
@end example
@code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
matrix filled with newly generated symbols made of the specified base name
and the position of each element in the matrix.
+Matrices often arise by omitting elements of another matrix. For
+instance, the submatrix @code{S} of a matrix @code{M} takes a
+rectangular block from @code{M}. The reduced matrix @code{R} is defined
+by removing one row and one column from a matrix @code{M}. (The
+determinant of a reduced matrix is called a @emph{Minor} of @code{M} and
+can be used for computing the inverse using Cramer's rule.)
+
+@cindex @code{sub_matrix()}
+@cindex @code{reduced_matrix()}
+@example
+ex sub_matrix(const matrix&m, unsigned r, unsigned nr, unsigned c, unsigned nc);
+ex reduced_matrix(const matrix& m, unsigned r, unsigned c);
+@end example
+
+The function @code{sub_matrix()} takes a row offset @code{r} and a
+column offset @code{c} and takes a block of @code{nr} rows and @code{nc}
+columns. The function @code{reduced_matrix()} has two integer arguments
+that specify which row and column to remove:
+
+@example
+@{
+ matrix m(3,3);
+ m = 11, 12, 13,
+ 21, 22, 23,
+ 31, 32, 33;
+ cout << reduced_matrix(m, 1, 1) << endl;
+ // -> [[11,13],[31,33]]
+ cout << sub_matrix(m, 1, 2, 1, 2) << endl;
+ // -> [[22,23],[32,33]]
+@}
+@end example
+
Matrix elements can be accessed and set using the parenthesis (function call)
operator:
method and linear systems may be solved with:
@example
-matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
+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}
@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
@}
@end example
+@cindex @code{expand_dummy_sum()}
+A dummy index summation like
+@tex
+$ a_i b^i$
+@end tex
+@ifnottex
+a.i b~i
+@end ifnottex
+can be expanded for indices with numeric
+dimensions (e.g. 3) into the explicit sum like
+@tex
+$a_1b^1+a_2b^2+a_3b^3 $.
+@end tex
+@ifnottex
+a.1 b~1 + a.2 b~2 + a.3 b~3.
+@end ifnottex
+This is performed by the function
+
+@example
+ ex expand_dummy_sum(const ex & e, bool subs_idx = false);
+@end example
+
+which takes an expression @code{e} and returns the expanded sum for all
+dummy indices with numeric dimensions. If the parameter @code{subs_idx}
+is set to @code{true} then all substitutions are made by @code{idx} class
+indices, i.e. without variance. In this case the above sum
+@tex
+$ a_i b^i$
+@end tex
+@ifnottex
+a.i b~i
+@end ifnottex
+will be expanded to
+@tex
+$a_1b_1+a_2b_2+a_3b_3 $.
+@end tex
+@ifnottex
+a.1 b.1 + a.2 b.2 + a.3 b.3.
+@end ifnottex
+
+
@cindex @code{simplify_indexed()}
@subsection Simplifying indexed expressions
k(symbol("k"), 3), l(symbol("l"), 3);
ex e = indexed(A, i, j) * indexed(B, k, l)
- * delta_tensor(i, k) * delta_tensor(j, l) << endl;
+ * delta_tensor(i, k) * delta_tensor(j, l);
cout << e.simplify_indexed() << endl;
// -> B.i.j*A.i.j
@example
ex epsilon_tensor(const ex & i1, const ex & i2);
ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
-ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
+ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
+ bool pos_sig = false);
@end example
The first two functions create an epsilon tensor in 2 or 3 Euclidean
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,
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 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
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
+functional is not cyclic in
+@tex $D \ne 4$
+@end tex
+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:
+@tex $D \ne 4$
+@end tex
+dimensions:
@example
@{
$2^n$
@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
+generators
+@tex $e_k$
+@end tex
+satisfying the identities
+@tex
+$e_i e_j + e_j e_i = M(i, j) + M(j, i) $
+@end tex
+@ifnottex
+e~i e~j + e~j e~i = M(i, j) + M(j, i)
+@end ifnottex
+for some bilinear form (@code{metric})
+@math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180)
+and contain symbolic entries. Such generators are created by the
+function
@example
- ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
+ ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0,
+ bool anticommuting = false);
@end example
where @code{mu} should be a @code{varidx} class object indexing the
-generators, @code{metr} defines the metric @math{B(i, j)} and can be
+generators, an index @code{mu} with a numeric value may be of type
+@code{idx} as well.
+Parameter @code{metr} defines the metric @math{M(i, j)} and can be
represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
-object, optional parameter @code{rl} allows to distinguish different
-Clifford algebras (which will commute with each other). Note that the call
-@code{clifford_unit(mu, minkmetric())} creates something very close to
-@code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
-metric defining this Clifford number.
+object. Optional parameter @code{rl} allows to distinguish different
+Clifford algebras, which will commute with each other. The last
+optional parameter @code{anticommuting} defines if the anticommuting
+assumption (i.e.
+@tex
+$e_i e_j + e_j e_i = 0$)
+@end tex
+@ifnottex
+e~i e~j + e~j e~i = 0)
+@end ifnottex
+will be used for contraction of Clifford units. If the @code{metric} is
+supplied by a @code{matrix} object, then the value of
+@code{anticommuting} is calculated automatically and the supplied one
+will be ignored. One can overcome this by giving @code{metric} through
+matrix wrapped into an @code{indexed} object.
+
+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()}
+The method @code{clifford::get_metric()} returns a metric defining this
+Clifford number.
+@cindex @code{clifford::is_anticommuting()}
+The method @code{clifford::is_anticommuting()} returns the
+@code{anticommuting} property of a unit.
+
+If the matrix @math{M(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(M, sy_symm(), i, j));
+@end example
+
+since this may yield some further automatic simplifications. Again, for a
+metric defined through a @code{matrix} such a symmetry is detected
+automatically.
Individual generators of a Clifford algebra can be accessed in several
ways. For example
@example
@{
...
- varidx nu(symbol("nu"), 3);
- matrix M(3, 3) = 1, 0, 0,
- 0,-1, 0,
- 0, 0, 0;
+ varidx nu(symbol("nu"), 4);
+ realsymbol s("s");
+ ex M = diag_matrix(lst(1, -1, 0, s));
ex e = clifford_unit(nu, M);
ex e0 = e.subs(nu == 0);
ex e1 = e.subs(nu == 1);
ex e2 = e.subs(nu == 2);
+ ex e3 = e.subs(nu == 3);
...
@}
@end example
-will produce three generators of a Clifford algebra with properties
-@code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1} and @code{pow(e2, 2) = 0}.
+will produce four anti-commuting generators of a Clifford algebra with properties
+@tex
+$e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
+@end tex
+@ifnottex
+@code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and
+@code{pow(e3, 2) = s}.
+@end ifnottex
@cindex @code{lst_to_clifford()}
A similar effect can be achieved from the function
@example
ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
- unsigned char rl = 0);
+ unsigned char rl = 0, bool anticommuting = false);
+ ex lst_to_clifford(const ex & v, const ex & e);
@end example
-which converts a list or vector @samp{v = (v~0, v~1, ..., v~n)} into
-the Clifford number @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n} with @samp{e.k}
-being created by @code{clifford_unit(mu, metr, rl)}. The previous code
-may be rewritten with the help of @code{lst_to_clifford()} as follows
+which converts a list or vector
+@tex
+$v = (v^0, v^1, ..., v^n)$
+@end tex
+@ifnottex
+@samp{v = (v~0, v~1, ..., v~n)}
+@end ifnottex
+into the
+Clifford number
+@tex
+$v^0 e_0 + v^1 e_1 + ... + v^n e_n$
+@end tex
+@ifnottex
+@samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
+@end ifnottex
+with @samp{e.k}
+directly supplied in the second form of the procedure. In the first form
+the Clifford unit @samp{e.k} is generated by the call of
+@code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten
+with the help of @code{lst_to_clifford()} as follows
@example
@{
...
- varidx nu(symbol("nu"), 3);
- matrix M(3, 3) = 1, 0, 0,
- 0,-1, 0,
- 0, 0, 0;
- ex e0 = lst_to_clifford(lst(1, 0, 0), nu, M);
- ex e1 = lst_to_clifford(lst(0, 1, 0), nu, M);
- ex e2 = lst_to_clifford(lst(0, 0, 1), nu, M);
+ varidx nu(symbol("nu"), 4);
+ realsymbol s("s");
+ ex M = diag_matrix(lst(1, -1, 0, s));
+ ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
+ ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
+ ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
+ ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
...
@}
@end example
@end example
which takes an expression @code{e} and tries to find a list
-@samp{v = (v~0, v~1, ..., v~n)} such that @samp{e = v~0 c.0 + v~1 c.1 + ...
-+ v~n c.n} with respect to the given Clifford units @code{c} and none of
-@samp{v~k} contains the Clifford units @code{c} (of course, this
+@tex
+$v = (v^0, v^1, ..., v^n)$
+@end tex
+@ifnottex
+@samp{v = (v~0, v~1, ..., v~n)}
+@end ifnottex
+such that
+@tex
+$e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
+@end tex
+@ifnottex
+@samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
+@end ifnottex
+with respect to the given Clifford units @code{c} and with none of the
+@samp{v~k} containing Clifford units @code{c} (of course, this
may be impossible). This function can use an @code{algebraic} method
-(default) or a symbolic one. With the @code{algebraic} method @samp{v~k} are calculated as
-@samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2) = 0} for some @samp{k}
+(default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
+@tex
+$(e c_k + c_k e)/c_k^2$. If $c_k^2$
+@end tex
+@ifnottex
+@samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
+@end ifnottex
+is zero or is not @code{numeric} for some @samp{k}
then the method will be automatically changed to symbolic. The same effect
is obtained by the assignment (@code{algebraic = false}) in the procedure call.
@tex
$e^*$
@end tex
+@ifnottex
+e*
+@end ifnottex
and
@tex
$\overline{e}$
@end tex
+@ifnottex
+@code{\bar@{e@}}
+@end ifnottex
used in Clifford algebra textbooks.
@cindex @code{clifford_norm()}
@cindex @code{clifford_inverse()}
calculates the norm of a Clifford number from the expression
@tex
-$||e||^2 = e\overline{e}$
+$||e||^2 = e\overline{e}$.
@end tex
-. The inverse of a Clifford expression is returned
-by the function
+@ifnottex
+@code{||e||^2 = e \bar@{e@}}
+@end ifnottex
+ The inverse of a Clifford expression is returned by the function
@example
ex clifford_inverse(const ex & e);
which calculates it as
@tex
-$e^{-1} = e/||e||^2$
+$e^{-1} = \overline{e}/||e||^2$.
@end tex
-. If
+@ifnottex
+@math{e^@{-1@} = \bar@{e@}/||e||^2}
+@end ifnottex
+ If
@tex
$||e|| = 0$
@end tex
+@ifnottex
+@math{||e||=0}
+@end ifnottex
then an exception is raised.
@cindex @code{remove_dirac_ONE()}
The function @code{canonicalize_clifford()} works for a
generic Clifford algebra in a similar way as for Dirac gammas.
-The last provided function is
+The next provided function is
@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, bool anticommuting = false);
+ ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
+ unsigned char rl = 0, bool anticommuting = false);
@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
-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.
+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{M = [[a, b], [c, d]]}. The parameter @code{G} defines
+the metric of the surrounding (pseudo-)Euclidean space. This can be an
+indexed object, tensormetric, matrix or a Clifford unit, in the later
+case the optional parameters @code{rl} and @code{anticommuting} are ignored
+even if supplied. The returned value of this function is a list of
+components of the resulting vector.
+
+@cindex @code{clifford_max_label()}
+Finally the function
+
+@example
+char clifford_max_label(const ex & e, bool ignore_ONE = false);
+@end example
+can detect a presence of Clifford objects in the expression @code{e}: if
+such objects are found it returns the maximal
+@code{representation_label} of them, otherwise @code{-1}. The optional
+parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should
+be ignored during the search.
+
+LaTeX output for Clifford units looks like
+@code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the
+@code{representation_label} and @code{\nu} is the index of the
+corresponding unit. This provides a flexible typesetting with a suitable
+defintion of the @code{\clifford} command. For example, the definition
+@example
+ \newcommand@{\clifford@}[1][]@{@}
+@end example
+typesets all Clifford units identically, while the alternative definition
+@example
+ \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
+@end example
+prints units with @code{representation_label=0} as
+@tex
+$e$,
+@end tex
+@ifnottex
+@code{e},
+@end ifnottex
+with @code{representation_label=1} as
+@tex
+$\tilde{e}$
+@end tex
+@ifnottex
+@code{\tilde@{e@}}
+@end ifnottex
+ and with @code{representation_label=2} as
+@tex
+$\breve{e}$.
+@end tex
+@ifnottex
+@code{\breve@{e@}}.
+@end ifnottex
@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
+@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,
@math{f_abc} which satisfy @math{@{T_a, T_b@} = 1/3 delta_ab + d_abc T_c}
and @math{[T_a, T_b] = i f_abc T_c}.
+These functions evaluate to their numerical values,
+if you supply numeric indices to them. The index values should be in
+the range from 1 to 8, not from 0 to 7. This departure from usual conventions
+goes along better with the notations used in physical literature.
+
@cindex @code{color_h()}
There's an additional function
* Built-in Functions:: List of predefined mathematical functions.
* Multiple polylogarithms::
* Complex Conjugation::
-* Built-in Functions:: List of predefined mathematical functions.
* Solving Linear Systems of Equations::
* Input/Output:: Input and output of expressions.
@end menu
@end example
The optional last argument to @code{subs()} is a combination of
-@code{subs_options} flags. There are two options available:
+@code{subs_options} flags. There are three 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.
+about patterns and algebraic substitutions. The third option,
+@code{subs_options::no_index_renaming} disables the feature that dummy
+indices are renamed if the subsitution could give a result in which a
+dummy index occurs more than two times. This is sometimes necessary if
+you want to use @code{subs()} to rename your dummy indices.
@code{subs()} performs syntactic substitution of any complete algebraic
object; it does not try to match sub-expressions as is demonstrated by the
@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
+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)
+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@});
@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}.
+and @code{lcm(a,b)} returns the product of @code{a} and @code{b}. Note that all
+the coefficients must be rationals.
@example
#include <ginac/ginac.h>
@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
needs to be solved:
@example
-ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
+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
@example
// ...
- cout << latex; // all output to cout will be in LaTeX format from now on
+ 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
will print
@example
- @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
+ @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
+ +\mathcal@{O@}(\circ^@{3@})
@end example
@cindex @code{index_dimensions}
// our own one
set_print_func<power, print_latex>(my_print_power_as_latex);
- // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
+ // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
+ // \uparrow@{2@}@}"
cout << e << endl;
@}
@end example
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;
+ return lhs.left.compare(rhs.left) < 0
+ ? true : lhs.right.compare(rhs.right) < 0;
@}
@end example
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
a macro to automate the process of checking for GiNaC.
@example
-AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
+AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
+ [, @var{ACTION-IF-NOT-FOUND}]]])
@end example
This macro:
git://www.ginac.de / ginac.git / blobdiff