@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
ex HermitePoly(const symbol & x, int n)
Linear equation systems can be solved along with basic linear
algebra manipulations over symbolic expressions. In C++ GiNaC offers
a matrix class for this purpose but we can see what it can do using
-@command{ginsh}'s notation of double brackets to type them in:
+@command{ginsh}'s bracket notation to type them in:
@example
> lsolve(a+x*y==z,x);
y^(-1)*(z-a);
-> lsolve([3*x+5*y == 7, -2*x+10*y == -5], [x, y]);
-[x==19/8,y==-1/40]
-> M = [[ [[1, 3]], [[-3, 2]] ]];
-[[ [[1,3]], [[-3,2]] ]]
+> lsolve(@{3*x+5*y == 7, -2*x+10*y == -5@}, @{x, y@});
+@{x==19/8,y==-1/40@}
+> M = [ [1, 3], [-3, 2] ];
+[[1,3],[-3,2]]
> determinant(M);
11
> charpoly(M,lambda);
lambda^2-3*lambda+11
+> A = [ [1, 1], [2, -1] ];
+[[1,1],[2,-1]]
+> A+2*M;
+[[1,1],[2,-1]]+2*[[1,3],[-3,2]]
+> evalm(");
+[[3,7],[-4,3]]
@end example
Multivariate polynomials and rational functions may be expanded,
@example
> a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
--3*y^4+x^4+12*x*y^3+2*x^2*y^2+4*x^3*y
+12*x*y^3+2*x^2*y^2+4*x^3*y-3*y^4+x^4
> b = x^2 + 4*x*y - y^2;
--y^2+x^2+4*x*y
+4*x*y-y^2+x^2
> expand(a*b);
-3*y^6+x^6-24*x*y^5+43*x^2*y^4+16*x^3*y^3+17*x^4*y^2+8*x^5*y
-> collect(a*b,x);
-3*y^6+48*x*y^4+2*x^2*y^2+x^4*(-y^2+x^2+4*x*y)+4*x^3*y*(-y^2+x^2+4*x*y)
+8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
+> collect(a+b,x);
+4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
+> collect(a+b,y);
+12*x*y^3-3*y^4+(-1+2*x^2)*y^2+(4*x+4*x^3)*y+x^2+x^4
> normal(a/b);
3*y^2+x^2
@end example
Generally, the top-level Makefile runs recursively to the
subdirectories. It is therfore safe to go into any subdirectory
-(@code{doc/}, @code{ginsh/}, ...) and simply type @code{make}
+(@code{doc/}, @code{ginsh/}, @dots{}) and simply type @code{make}
@var{target} there in case something went wrong.
* Lists:: Lists of expressions.
* Mathematical functions:: Mathematical functions.
* Relations:: Equality, Inequality and all that.
+* Matrices:: Matrices.
* Indexed objects:: Handling indexed quantities.
+* Non-commutative objects:: Algebras with non-commutative products.
@end menu
The most common class of objects a user deals with is the expression
@code{ex}, representing a mathematical object like a variable, number,
-function, sum, product, etc... Expressions may be put together to form
+function, sum, product, etc@dots{} Expressions may be put together to form
new expressions, passed as arguments to functions, and so on. Here is a
little collection of valid expressions:
@item @code{numeric} @tab All kinds of numbers, @math{42}, @math{7/3*I}, @math{3.14159}@dots{}
@item @code{add} @tab Sums like @math{x+y} or @math{a-(2*b)+3}
@item @code{mul} @tab Products like @math{x*y} or @math{2*a^2*(x+y+z)/b}
+@item @code{ncmul} @tab Products of non-commutative objects
@item @code{power} @tab Exponentials such as @math{x^2}, @math{a^b},
@tex
$\sqrt{2}$
@dots{}
@item @code{pseries} @tab Power Series, e.g. @math{x-1/6*x^3+1/120*x^5+O(x^7)}
@item @code{function} @tab A symbolic function like @math{sin(2*x)}
-@item @code{lst} @tab Lists of expressions [@math{x}, @math{2*y}, @math{3+z}]
-@item @code{matrix} @tab @math{n}x@math{m} matrices of expressions
+@item @code{lst} @tab Lists of expressions @{@math{x}, @math{2*y}, @math{3+z}@}
+@item @code{matrix} @tab @math{m}x@math{n} matrices of expressions
@item @code{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
@item @code{indexed} @tab Indexed object like @math{A_ij}
@item @code{tensor} @tab Special tensor like the delta and metric tensors
@item @code{idx} @tab Index of an indexed object
@item @code{varidx} @tab Index with variance
+@item @code{spinidx} @tab Index with variance and dot (used in Weyl-van-der-Waerden spinor formalism)
+@item @code{wildcard} @tab Wildcard for pattern matching
@end multitable
@end cartouche
Although symbols can be assigned expressions for internal reasons, you
should not do it (and we are not going to tell you how it is done). If
you want to replace a symbol with something else in an expression, you
-can use the expression's @code{.subs()} method (@xref{Substituting Symbols},
-for more information).
+can use the expression's @code{.subs()} method (@pxref{Substituting Expressions}).
@node Numbers, Constants, Symbols, Basic Concepts
// Trott's constant in scientific notation:
numeric trott("1.0841015122311136151E-2");
- cout << two*p << endl; // floating point 6.283...
+ std::cout << two*p << std::endl; // floating point 6.283...
@}
@end example
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
void foo()
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
// some very important constants:
in other CAS or like expressions involving numerical variables in C.
The necessary operators @code{+}, @code{-}, @code{*} and @code{/} have
been overloaded to achieve this goal. When you run the following
-program, the constructor for an object of type @code{mul} is
+code snippet, the constructor for an object of type @code{mul} is
automatically called to hold the product of @code{a} and @code{b} and
then the constructor for an object of type @code{add} is called to hold
the sum of that @code{mul} object and the number one:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
+ ...
symbol a("a"), b("b");
ex MyTerm = 1+a*b;
- // ...
-@}
+ ...
@end example
@cindex @code{pow()}
statement @code{pow(x,2);} to represent @code{x} squared. This direct
construction is necessary since we cannot safely overload the constructor
@code{^} in C++ to construct a @code{power} object. If we did, it would
-have several counterintuitive effects:
+have several counterintuitive and undesired effects:
@itemize @bullet
@item
@cindex @code{append()}
@cindex @code{prepend()}
-The GiNaC class @code{lst} serves for holding a list of arbitrary expressions.
-These are sometimes used to supply a variable number of arguments of the same
-type to GiNaC methods such as @code{subs()} and @code{to_rational()}, so you
-should have a basic understanding about them.
+The GiNaC class @code{lst} serves for holding a @dfn{list} of arbitrary
+expressions. These are sometimes used to supply a variable number of
+arguments of the same type to GiNaC methods such as @code{subs()} and
+@code{to_rational()}, so you should have a basic understanding about them.
-Lists of up to 15 expressions can be directly constructed from single
+Lists of up to 16 expressions can be directly constructed from single
expressions:
@example
@example
// ...
- l.append(4*x); // l is now [x, 2, y, x+y, 4*x]
- l.prepend(0); // l is now [0, x, 2, y, x+y, 4*x]
+ l.append(4*x); // l is now @{x, 2, y, x+y, 4*x@}
+ l.prepend(0); // l is now @{0, x, 2, y, x+y, 4*x@}
@}
@end example
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:
+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:
@cindex Gamma function
@cindex @code{subs()}
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- symbol x("x"), y("y");
-
+ ...
+ symbol x("x"), y("y");
ex foo = x+y/2;
- cout << "tgamma(" << foo << ") -> " << tgamma(foo) << endl;
+ cout << tgamma(foo) << endl;
+ // -> tgamma(x+(1/2)*y)
ex bar = foo.subs(y==1);
- cout << "tgamma(" << bar << ") -> " << tgamma(bar) << endl;
+ cout << tgamma(bar) << endl;
+ // -> tgamma(x+1/2)
ex foobar = bar.subs(x==7);
- cout << "tgamma(" << foobar << ") -> " << tgamma(foobar) << endl;
-@}
-@end example
-
-This program shows how the function returns itself twice and finally an
-expression that may be really useful:
-
-@example
-tgamma(x+(1/2)*y) -> tgamma(x+(1/2)*y)
-tgamma(x+1/2) -> tgamma(x+1/2)
-tgamma(15/2) -> (135135/128)*Pi^(1/2)
+ cout << tgamma(foobar) << endl;
+ // -> (135135/128)*Pi^(1/2)
+ ...
@end example
Besides evaluation most of these functions allow differentiation, series
this.
-@node Relations, Indexed objects, Mathematical functions, Basic Concepts
+@node Relations, Matrices, Mathematical functions, Basic Concepts
@c node-name, next, previous, up
@section Relations
@cindex @code{relational} (class)
@code{expand()} must be called explicitly.
-@node Indexed objects, Methods and Functions, Relations, Basic Concepts
+@node Matrices, Indexed objects, Relations, Basic Concepts
+@c node-name, next, previous, up
+@section Matrices
+@cindex @code{matrix} (class)
+
+A @dfn{matrix} is a two-dimensional array of expressions. The elements of a
+matrix with @math{m} rows and @math{n} columns are accessed with two
+@code{unsigned} indices, the first one in the range 0@dots{}@math{m-1}, the
+second one in the range 0@dots{}@math{n-1}.
+
+There are a couple of ways to construct matrices, with or without preset
+elements:
+
+@example
+matrix::matrix(unsigned r, unsigned c);
+matrix::matrix(unsigned r, unsigned c, const lst & l);
+ex lst_to_matrix(const lst & l);
+ex diag_matrix(const lst & l);
+@end example
+
+The first two functions are @code{matrix} constructors which create a matrix
+with @samp{r} rows and @samp{c} columns. The matrix elements can be
+initialized from a (flat) list of expressions @samp{l}. Otherwise they are
+all set to zero. The @code{lst_to_matrix()} function constructs a matrix
+from a list of lists, each list representing a matrix row. Finally,
+@code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
+elements. Note that the last two functions return expressions, not matrix
+objects.
+
+Matrix elements can be accessed and set using the parenthesis (function call)
+operator:
+
+@example
+const ex & matrix::operator()(unsigned r, unsigned c) const;
+ex & matrix::operator()(unsigned r, unsigned c);
+@end example
+
+It is also possible to access the matrix elements in a linear fashion with
+the @code{op()} method. But C++-style subscripting with square brackets
+@samp{[]} is not available.
+
+Here are a couple of examples that all construct the same 2x2 diagonal
+matrix:
+
+@example
+@{
+ symbol a("a"), b("b");
+ ex e;
+
+ matrix M(2, 2);
+ M(0, 0) = a;
+ M(1, 1) = b;
+ e = M;
+
+ e = matrix(2, 2, lst(a, 0, 0, b));
+
+ e = lst_to_matrix(lst(lst(a, 0), lst(0, b)));
+
+ e = diag_matrix(lst(a, b));
+
+ cout << e << endl;
+ // -> [[a,0],[0,b]]
+@}
+@end example
+
+@cindex @code{transpose()}
+@cindex @code{inverse()}
+There are three ways to do arithmetic with matrices. The first (and most
+efficient one) is to use the methods provided by the @code{matrix} class:
+
+@example
+matrix matrix::add(const matrix & other) const;
+matrix matrix::sub(const matrix & other) const;
+matrix matrix::mul(const matrix & other) const;
+matrix matrix::mul_scalar(const ex & other) const;
+matrix matrix::pow(const ex & expn) const;
+matrix matrix::transpose(void) const;
+matrix matrix::inverse(void) const;
+@end example
+
+All of these methods return the result as a new matrix object. Here is an
+example that calculates @math{A*B-2*C} for three matrices @math{A}, @math{B}
+and @math{C}:
+
+@example
+@{
+ matrix A(2, 2, lst(1, 2, 3, 4));
+ matrix B(2, 2, lst(-1, 0, 2, 1));
+ matrix C(2, 2, lst(8, 4, 2, 1));
+
+ matrix result = A.mul(B).sub(C.mul_scalar(2));
+ cout << result << endl;
+ // -> [[-13,-6],[1,2]]
+ ...
+@}
+@end example
+
+@cindex @code{evalm()}
+The second (and probably the most natural) way is to construct an expression
+containing matrices with the usual arithmetic operators and @code{pow()}.
+For efficiency reasons, expressions with sums, products and powers of
+matrices are not automatically evaluated in GiNaC. You have to call the
+method
+
+@example
+ex ex::evalm() const;
+@end example
+
+to obtain the result:
+
+@example
+@{
+ ...
+ ex e = A*B - 2*C;
+ cout << e << endl;
+ // -> [[1,2],[3,4]]*[[-1,0],[2,1]]-2*[[8,4],[2,1]]
+ cout << e.evalm() << endl;
+ // -> [[-13,-6],[1,2]]
+ ...
+@}
+@end example
+
+The non-commutativity of the product @code{A*B} in this example is
+automatically recognized by GiNaC. There is no need to use a special
+operator here. @xref{Non-commutative objects}, for more information about
+dealing with non-commutative expressions.
+
+Finally, you can work with indexed matrices and call @code{simplify_indexed()}
+to perform the arithmetic:
+
+@example
+@{
+ ...
+ idx i(symbol("i"), 2), j(symbol("j"), 2), k(symbol("k"), 2);
+ e = indexed(A, i, k) * indexed(B, k, j) - 2 * indexed(C, i, j);
+ cout << e << endl;
+ // -> -2*[[8,4],[2,1]].i.j+[[-1,0],[2,1]].k.j*[[1,2],[3,4]].i.k
+ cout << e.simplify_indexed() << endl;
+ // -> [[-13,-6],[1,2]].i.j
+@}
+@end example
+
+Using indices is most useful when working with rectangular matrices and
+one-dimensional vectors because you don't have to worry about having to
+transpose matrices before multiplying them. @xref{Indexed objects}, for
+more information about using matrices with indices, and about indices in
+general.
+
+The @code{matrix} class provides a couple of additional methods for
+computing determinants, traces, and characteristic polynomials:
+
+@example
+ex matrix::determinant(unsigned algo = determinant_algo::automatic) const;
+ex matrix::trace(void) const;
+ex matrix::charpoly(const symbol & lambda) const;
+@end example
+
+The @samp{algo} argument of @code{determinant()} allows to select between
+different algorithms for calculating the determinant. The possible values
+are defined in the @file{flags.h} header file. By default, GiNaC uses a
+heuristic to automatically select an algorithm that is likely to give the
+result most quickly.
+
+
+@node Indexed objects, Non-commutative objects, Matrices, Basic Concepts
@c node-name, next, previous, up
@section Indexed objects
@itemize @bullet
+@cindex contravariant
+@cindex covariant
+@cindex variance
@item Index objects are of class @code{idx} or a subclass. Every index has
a @dfn{value} and a @dfn{dimension} (which is the dimension of the space
the index lives in) which can both be arbitrary expressions but are usually
a number or a simple symbol. In addition, indices of class @code{varidx} have
-a @dfn{variance} (they can be co- or contravariant).
+a @dfn{variance} (they can be co- or contravariant), and indices of class
+@code{spinidx} have a variance and can be @dfn{dotted} or @dfn{undotted}.
@item Indexed objects are of class @code{indexed} or a subclass. They
contain a @dfn{base expression} (which is the expression being indexed), and
@end itemize
@strong{Note:} when printing expressions, covariant indices and indices
-without variance are denoted @samp{.i} while contravariant indices are denoted
-@samp{~i}. In the following, we are going to use that notation in the text
-so instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions
-are not visible in the output.
+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
+instead of @math{A^i_jk} we will write @samp{A~i.j.k}. Index dimensions are
+not visible in the output.
A simple example shall illustrate the concepts:
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
which makes a new index with the same value and dimension but the opposite
variance. By using it you only have to define the index once.
+@cindex @code{spinidx} (class)
+The @code{spinidx} class provides dotted and undotted variant indices, as
+used in the Weyl-van-der-Waerden spinor formalism:
+
+@example
+ ...
+ symbol K("K"), C_sym("C"), D_sym("D");
+ spinidx C(C_sym, 2), D(D_sym); // default is 2-dimensional,
+ // contravariant, undotted
+ spinidx C_co(C_sym, 2, true); // covariant index
+ spinidx D_dot(D_sym, 2, false, true); // contravariant, dotted
+ spinidx D_co_dot(D_sym, 2, true, true); // covariant, dotted
+
+ cout << indexed(K, C, D) << endl;
+ // -> K~C~D
+ cout << indexed(K, C_co, D_dot) << endl;
+ // -> K.C~*D
+ cout << indexed(K, D_co_dot, D) << endl;
+ // -> K.*D~D
+ ...
+@end example
+
+A @code{spinidx} is a @code{varidx} with an additional flag that marks it as
+dotted or undotted. The default is undotted but this can be overridden by
+supplying a fourth argument to the @code{spinidx} constructor. The two
+methods
+
+@example
+bool spinidx::is_dotted(void);
+bool spinidx::is_undotted(void);
+@end example
+
+allow you to check whether or not a @code{spinidx} object is dotted (use
+@code{ex_to_spinidx()} to get the object reference from an expression).
+Finally, the two methods
+
+@example
+ex spinidx::toggle_dot(void);
+ex spinidx::toggle_variance_dot(void);
+@end example
+
+create a new index with the same value and dimension but opposite dottedness
+and the same or opposite variance.
+
@subsection Substituting indices
@cindex @code{subs()}
Sometimes you will want to substitute one symbolic index with another
symbolic or numeric index, for example when calculating one specific element
of a tensor expression. This is done with the @code{.subs()} method, as it
-is done for symbols (see @ref{Substituting Symbols}).
+is done for symbols (see @ref{Substituting Expressions}).
You have two possibilities here. You can either substitute the whole index
by another index or expression:
@end example
@cindex @code{get_free_indices()}
+@cindex Dummy index
@subsection Dummy indices
GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
To be recognized as a dummy index pair, the two indices must be of the same
class and dimension and their value must be the same single symbol (an index
like @samp{2*n+1} is never a dummy index). If the indices are of class
-@code{varidx}, they must also be of opposite variance.
+@code{varidx} they must also be of opposite variance; if they are of class
+@code{spinidx} they must be both dotted or both undotted.
The method @code{.get_free_indices()} returns a vector containing the free
indices of an expression. It also checks that the free indices of the terms
@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
+ 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)
@code{simplify_indexed()} will replace all scalar products of indexed
objects that have the symbols @code{A} and @code{B} as base expressions
with the single value 0. The number, type and dimension of the indices
-doesn't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
+don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
@cindex @code{expand()}
The example above also illustrates a feature of the @code{expand()} method:
@subsubsection Delta tensor
The delta tensor takes two indices, is symmetric and has the matrix
-representation @code{diag(1,1,1,...)}. It is constructed by the function
+representation @code{diag(1, 1, 1, ...)}. It is constructed by the function
@code{delta_tensor()}:
@example
@}
@end example
+@cindex @code{spinor_metric()}
+@subsubsection Spinor metric tensor
+
+The function @code{spinor_metric()} creates an antisymmetric tensor with
+two indices that is used to raise/lower indices of 2-component spinors.
+It is output as @samp{eps}:
+
+@example
+@{
+ symbol psi("psi");
+
+ spinidx A(symbol("A")), B(symbol("B")), C(symbol("C"));
+ ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
+
+ e = spinor_metric(A, B) * indexed(psi, B_co);
+ cout << e.simplify_indexed() << endl;
+ // -> psi~A
+
+ e = spinor_metric(A, B) * indexed(psi, A_co);
+ cout << e.simplify_indexed() << endl;
+ // -> -psi~B
+
+ e = spinor_metric(A_co, B_co) * indexed(psi, B);
+ cout << e.simplify_indexed() << endl;
+ // -> -psi.A
+
+ e = spinor_metric(A_co, B_co) * indexed(psi, A);
+ cout << e.simplify_indexed() << endl;
+ // -> psi.B
+
+ e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
+ cout << e.simplify_indexed() << endl;
+ // -> 2
+
+ e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
+ cout << e.simplify_indexed() << endl;
+ // -> -delta.A~C
+@}
+@end example
+
+The matrix representation of the spinor metric is @code{[[0, 1], [-1, 0]]}.
+
+@cindex @code{epsilon_tensor()}
+@cindex @code{lorentz_eps()}
@subsubsection Epsilon tensor
The epsilon tensor is totally antisymmetric, its number of indices is equal
idx i(symbol("i"), 2), j(symbol("j"), 2);
symbol x("x"), y("y");
+ // A is a 2x2 matrix, X is a 2x1 vector
matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
cout << indexed(A, i, i) << endl;
ex e = indexed(A, i, j) * indexed(X, j);
cout << e.simplify_indexed() << endl;
- // -> [[ [[2*y+x]], [[4*y+3*x]] ]].i
+ // -> [[2*y+x],[4*y+3*x]].i
e = indexed(A, i, j) * indexed(X, i) + indexed(X, j) * 2;
cout << e.simplify_indexed() << endl;
- // -> [[ [[3*y+3*x,6*y+2*x]] ]].j
+ // -> [[3*y+3*x,6*y+2*x]].j
@}
@end example
You can of course obtain the same results with the @code{matrix::add()},
-@code{matrix::mul()} and @code{matrix::trace()} methods but with indices you
-don't have to worry about transposing matrices.
+@code{matrix::mul()} and @code{matrix::trace()} methods (@pxref{Matrices})
+but with indices you don't have to worry about transposing matrices.
Matrix indices always start at 0 and their dimension must match the number
of rows/columns of the matrix. Matrices with one row or one column are
of the metric tensor.
-@node Methods and Functions, Information About Expressions, Indexed objects, Top
+@node Non-commutative objects, Methods and Functions, Indexed objects, Basic Concepts
+@c node-name, next, previous, up
+@section Non-commutative objects
+
+GiNaC is equipped to handle certain non-commutative algebras. Three classes of
+non-commutative objects are built-in which are mostly of use in high energy
+physics:
+
+@itemize
+@item Clifford (Dirac) algebra (class @code{clifford})
+@item su(3) Lie algebra (class @code{color})
+@item Matrices (unindexed) (class @code{matrix})
+@end itemize
+
+The @code{clifford} and @code{color} classes are subclasses of
+@code{indexed} because the elements of these algebras ususally carry
+indices. The @code{matrix} class is described in more detail in
+@ref{Matrices}.
+
+Unlike most computer algebra systems, GiNaC does not primarily provide an
+operator (often denoted @samp{&*}) for representing inert products of
+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
+by their class. Consider this example:
+
+@example
+ ...
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
+ idx a(symbol("a"), 8), b(symbol("b"), 8);
+ ex e = -dirac_gamma(mu) * (2*color_T(a)) * 8 * color_T(b) * dirac_gamma(nu);
+ cout << e << endl;
+ // -> -16*(gamma~mu*gamma~nu)*(T.a*T.b)
+ ...
+@end 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
+@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.
+
+@cindex @code{ncmul} (class)
+Non-commutative products are internally represented by objects of the class
+@code{ncmul}, as opposed to commutative products which are handled by the
+@code{mul} class. You will normally not have to worry about this distinction,
+though.
+
+The advantage of this approach is that you never have to worry about using
+(or forgetting to use) a special operator when constructing non-commutative
+expressions. Also, non-commutative products in GiNaC are more intelligent
+than in other computer algebra systems; they can, for example, automatically
+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
+using symbols to represent the algebra elements or generators. User-defined
+functions can, however, be specified as being non-commutative.
+
+@cindex @code{return_type()}
+@cindex @code{return_type_tinfo()}
+Information about the commutativity of an object or expression can be
+obtained with the two member functions
+
+@example
+unsigned ex::return_type(void) const;
+unsigned ex::return_type_tinfo(void) const;
+@end example
+
+The @code{return_type()} function returns one of three values (defined in
+the header file @file{flags.h}), corresponding to three categories of
+expressions in GiNaC:
+
+@itemize
+@item @code{return_types::commutative}: Commutes 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
+ 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
+ @code{noncommutative_composite} expressions.
+@end itemize
+
+The value returned by the @code{return_type_tinfo()} method is valid only
+when the return type of the expression is @code{noncommutative}. It is a
+value that is unique to the class of the object and usually one of the
+constants in @file{tinfos.h}, or derived therefrom.
+
+Here are a couple of examples:
+
+@cartouche
+@multitable @columnfractions 0.33 0.33 0.34
+@item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
+@item @code{42} @tab @code{commutative} @tab -
+@item @code{2*x-y} @tab @code{commutative} @tab -
+@item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
+@item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
+@item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
+@item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
+@end multitable
+@end cartouche
+
+Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
+@code{TINFO_clifford} for objects with a representation label of zero.
+Other representation labels yield a different @code{return_type_tinfo()},
+but it's the same for any two objects with the same label. This is also true
+for color objects.
+
+A last note: With the exception of matrices, positive integer powers of
+non-commutative objects are automatically expanded in GiNaC. For example,
+@code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
+non-commutative expressions).
+
+
+@cindex @code{clifford} (class)
+@subsection Clifford algebra
+
+@cindex @code{dirac_gamma()}
+Clifford algebra elements (also called Dirac gamma matrices, although GiNaC
+doesn't treat them as matrices) are designated as @samp{gamma~mu} and satisfy
+@samp{gamma~mu*gamma~nu + gamma~nu*gamma~mu = 2*eta~mu~nu} where @samp{eta~mu~nu}
+is the Minkowski metric tensor. Dirac gammas are constructed by the function
+
+@example
+ex dirac_gamma(const ex & mu, unsigned char rl = 0);
+@end example
+
+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
+the framework of dimensional regularization) any symbolic value. Spinor
+indices on Dirac gammas are not supported in GiNaC.
+
+@cindex @code{dirac_ONE()}
+The unity element of a Clifford algebra is constructed by
+
+@example
+ex dirac_ONE(unsigned char rl = 0);
+@end example
+
+@cindex @code{dirac_gamma5()}
+and 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
+
+@example
+ex dirac_gamma5(unsigned char rl = 0);
+@end example
+
+@cindex @code{dirac_gamma6()}
+@cindex @code{dirac_gamma7()}
+The two additional functions
+
+@example
+ex dirac_gamma6(unsigned char rl = 0);
+ex dirac_gamma7(unsigned char rl = 0);
+@end example
+
+return @code{dirac_ONE(rl) + dirac_gamma5(rl)} and @code{dirac_ONE(rl) - dirac_gamma5(rl)},
+respectively.
+
+@cindex @code{dirac_slash()}
+Finally, the function
+
+@example
+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.
+
+In products of dirac gammas, superfluous unity elements are automatically
+removed, squares are replaced by their values and @samp{gamma5} is
+anticommuted to the front. The @code{simplify_indexed()} function performs
+contractions in gamma strings, for example
+
+@example
+@{
+ ...
+ symbol a("a"), b("b"), D("D");
+ varidx mu(symbol("mu"), D);
+ 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
+ e = e.simplify_indexed();
+ cout << e << endl;
+ // -> -gamma~symbol10*a.symbol10*D+2*gamma~symbol10*a.symbol10
+ cout << e.subs(D == 4) << endl;
+ // -> -2*gamma~symbol10*a.symbol10
+ // [ == -2 * dirac_slash(a, D) ]
+ ...
+@}
+@end example
+
+@cindex @code{dirac_trace()}
+To calculate the trace of an expression containing strings of Dirac gammas
+you use the function
+
+@example
+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
+@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
+@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:
+
+@example
+@{
+ // 4 dimensions
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
+ ex e = dirac_gamma(mu) * dirac_gamma(nu) *
+ dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
+ cout << dirac_trace(e).simplify_indexed() << endl;
+ // -> -8*eta~rho~nu
+@}
+...
+@{
+ // D dimensions
+ symbol D("D");
+ varidx mu(symbol("mu"), D), nu(symbol("nu"), D), rho(symbol("rho"), D);
+ ex e = dirac_gamma(mu) * dirac_gamma(nu) *
+ dirac_gamma(mu.toggle_variance()) * dirac_gamma(rho);
+ cout << dirac_trace(e).simplify_indexed() << endl;
+ // -> 8*eta~rho~nu-4*eta~rho~nu*D
+@}
+@end example
+
+Here is an example for using @code{dirac_trace()} to compute a value that
+appears in the calculation of the one-loop vacuum polarization amplitude in
+QED:
+
+@example
+@{
+ symbol q("q"), l("l"), m("m"), ldotq("ldotq"), D("D");
+ varidx mu(symbol("mu"), D), nu(symbol("nu"), D);
+
+ scalar_products sp;
+ sp.add(l, l, pow(l, 2));
+ sp.add(l, q, ldotq);
+
+ ex e = dirac_gamma(mu) *
+ (dirac_slash(l, D) + dirac_slash(q, D) + m * dirac_ONE()) *
+ 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);
+ cout << e << endl;
+ // -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
+@}
+@end example
+
+The @code{canonicalize_clifford()} function reorders all gamma products that
+appear in an expression to a canonical (but not necessarily simple) form.
+You can use this to compare two expressions or for further simplifications:
+
+@example
+@{
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
+ ex e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
+ cout << e << endl;
+ // -> gamma~mu*gamma~nu+gamma~nu*gamma~mu
+
+ e = canonicalize_clifford(e);
+ cout << e << endl;
+ // -> 2*eta~mu~nu
+@}
+@end example
+
+
+@cindex @code{color} (class)
+@subsection Color algebra
+
+@cindex @code{color_T()}
+For computations in quantum chromodynamics, GiNaC implements the base elements
+and structure constants of the su(3) Lie algebra (color algebra). The base
+elements @math{T_a} are constructed by the function
+
+@example
+ex color_T(const ex & a, unsigned char rl = 0);
+@end example
+
+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
+dimension of the index must be exactly 8 and it should be of class @code{idx},
+not @code{varidx}.
+
+@cindex @code{color_ONE()}
+The unity element of a color algebra is constructed by
+
+@example
+ex color_ONE(unsigned char rl = 0);
+@end example
+
+@cindex @code{color_d()}
+@cindex @code{color_f()}
+and the functions
+
+@example
+ex color_d(const ex & a, const ex & b, const ex & c);
+ex color_f(const ex & a, const ex & b, const ex & c);
+@end example
+
+create the symmetric and antisymmetric structure constants @math{d_abc} and
+@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}.
+
+@cindex @code{color_h()}
+There's an additional function
+
+@example
+ex color_h(const ex & a, const ex & b, const ex & c);
+@end example
+
+which returns the linear combination @samp{color_d(a, b, c)+I*color_f(a, b, c)}.
+
+The function @code{simplify_indexed()} performs some simplifications on
+expressions containing color objects:
+
+@example
+@{
+ ...
+ idx a(symbol("a"), 8), b(symbol("b"), 8), c(symbol("c"), 8),
+ k(symbol("k"), 8), l(symbol("l"), 8);
+
+ e = color_d(a, b, l) * color_f(a, b, k);
+ cout << e.simplify_indexed() << endl;
+ // -> 0
+
+ e = color_d(a, b, l) * color_d(a, b, k);
+ cout << e.simplify_indexed() << endl;
+ // -> 5/3*delta.k.l
+
+ e = color_f(l, a, b) * color_f(a, b, k);
+ cout << e.simplify_indexed() << endl;
+ // -> 3*delta.k.l
+
+ e = color_h(a, b, c) * color_h(a, b, c);
+ cout << e.simplify_indexed() << endl;
+ // -> -32/3
+
+ e = color_h(a, b, c) * color_T(b) * color_T(c);
+ cout << e.simplify_indexed() << endl;
+ // -> -2/3*T.a
+
+ e = color_h(a, b, c) * color_T(a) * color_T(b) * color_T(c);
+ cout << e.simplify_indexed() << endl;
+ // -> -8/9*ONE
+
+ e = color_T(k) * color_T(a) * color_T(b) * color_T(k);
+ cout << e.simplify_indexed() << endl;
+ // -> 1/4*delta.b.a*ONE-1/6*T.a*T.b
+ ...
+@end example
+
+@cindex @code{color_trace()}
+To calculate the trace of an expression containing color objects you use the
+function
+
+@example
+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:
+
+@example
+ ...
+ e = color_trace(4 * color_T(a) * color_T(b) * color_T(c));
+ cout << e << endl;
+ // -> -I*f.a.c.b+d.a.c.b
+@}
+@end example
+
+
+@node Methods and Functions, Information About Expressions, Non-commutative objects, Top
@c node-name, next, previous, up
@chapter Methods and Functions
@cindex polynomial
example:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
-
-int main()
-@{
- ex x = numeric(1.0);
-
- cout << "As method: " << sin(x).evalf() << endl;
- cout << "As function: " << evalf(sin(x)) << endl;
-@}
+ ...
+ cout << "As method: " << sin(1).evalf() << endl;
+ cout << "As function: " << evalf(sin(1)) << endl;
+ ...
@end example
@cindex @code{subs()}
@menu
* Information About Expressions::
-* Substituting Symbols::
+* Substituting Expressions::
+* Pattern Matching and Advanced Substitutions::
* Polynomial Arithmetic:: Working with polynomials.
* Rational Expressions:: Working with rational functions.
* Symbolic Differentiation::
* Series Expansion:: Taylor and Laurent expansion.
+* Symmetrization::
* Built-in Functions:: List of predefined mathematical functions.
* Input/Output:: Input and output of expressions.
@end menu
-@node Information About Expressions, Substituting Symbols, Methods and Functions, Methods and Functions
+@node Information About Expressions, Substituting Expressions, Methods and Functions, Methods and Functions
@c node-name, next, previous, up
@section Getting information about expressions
@cindex @code{ex_to_@dots{}}
@cindex @code{Converting ex to other classes}
@cindex @code{info()}
+@cindex @code{return_type()}
+@cindex @code{return_type_tinfo()}
Sometimes it's useful to check whether a given expression is a plain number,
a sum, a polynomial with integer coefficients, or of some other specific type.
-GiNaC provides two functions for this (the first one is actually a macro):
+GiNaC provides a couple of functions for this (the first one is actually a macro):
@example
bool is_ex_of_type(const ex & e, TYPENAME t);
bool ex::info(unsigned flag);
+unsigned ex::return_type(void) const;
+unsigned ex::return_type_tinfo(void) const;
@end example
When the test made by @code{is_ex_of_type()} returns true, it is safe to
@end multitable
@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
+@code{return_type()} and @code{return_type_tinfo()}. @xref{Non-commutative objects},
+for an explanation of these.
+
@subsection Accessing subexpressions
@cindex @code{nops()}
@cindex @code{op()}
-@cindex @code{has()}
@cindex container
@cindex @code{relational} (class)
ex ex::rhs();
@end example
-Finally, the method
-
-@example
-bool ex::has(const ex & other);
-@end example
-
-checks whether an expression contains the given subexpression @code{other}.
-This only works reliably if @code{other} is of an atomic class such as a
-@code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that
-@code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression.
-
@subsection Comparing expressions
@cindex @code{is_equal()}
@code{false}.
Actually, if you construct an expression like @code{a == b}, this will be
-represented by an object of the @code{relational} class (@xref{Relations}.)
+represented by an object of the @code{relational} class (@pxref{Relations})
which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
There are also two methods
expressions will give very surprising results.
-@node Substituting Symbols, Polynomial Arithmetic, Information About Expressions, Methods and Functions
+@node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
@c node-name, next, previous, up
-@section Substituting symbols
+@section Substituting expressions
@cindex @code{subs()}
-Symbols can be replaced with expressions via the @code{.subs()} method:
+Algebraic objects inside expressions can be replaced with arbitrary
+expressions via the @code{.subs()} method:
@example
ex ex::subs(const ex & e);
@end example
In the first form, @code{subs()} accepts a relational of the form
-@samp{symbol == expression} or a @code{lst} of such relationals. E.g.
+@samp{object == expression} or a @code{lst} of such relationals:
@example
@{
symbol x("x"), y("y");
+
ex e1 = 2*x^2-4*x+3;
cout << "e1(7) = " << e1.subs(x == 7) << endl;
+ // -> 73
+
ex e2 = x*y + x;
cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
+ // -> -10
@}
@end example
-will print @samp{73} and @samp{-10}, respectively.
-
If you specify multiple substitutions, they are performed in parallel, so e.g.
@code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
-The second form of @code{subs()} takes two lists, one for the symbols and
-one for the expressions to be substituted (both lists must contain the same
-number of elements). Using this form, you would write @code{subs(lst(x, y), lst(y, x))}
-to exchange @samp{x} and @samp{y}.
+The second form of @code{subs()} takes two lists, one for the objects to be
+replaced and one for the expressions to be substituted (both lists must
+contain the same number of elements). Using this form, you would write
+@code{subs(lst(x, y), lst(y, x))} to exchange @samp{x} and @samp{y}.
+
+@code{subs()} performs syntactic substitution of any complete algebraic
+object; it does not try to match sub-expressions as is demonstrated by the
+following example:
+
+@example
+@{
+ symbol x("x"), y("y"), z("z");
+
+ ex e1 = pow(x+y, 2);
+ cout << e1.subs(x+y == 4) << endl;
+ // -> 16
+
+ ex e2 = sin(x)*sin(y)*cos(x);
+ cout << e2.subs(sin(x) == cos(x)) << endl;
+ // -> cos(x)^2*sin(y)
+
+ ex e3 = x+y+z;
+ cout << e3.subs(x+y == 4) << endl;
+ // -> x+y+z
+ // (and not 4+z as one might expect)
+@}
+@end example
+
+A more powerful form of substitution using wildcards is described in the
+next section.
-@node Polynomial Arithmetic, Rational Expressions, Substituting Symbols, Methods and Functions
+@node Pattern Matching and Advanced Substitutions, Polynomial Arithmetic, Substituting Expressions, Methods and Functions
+@c node-name, next, previous, up
+@section Pattern matching and advanced substitutions
+
+GiNaC allows the use of patterns for checking whether an expression is of a
+certain form or contains subexpressions of a certain form, and for
+substituting expressions in a more general way.
+
+A @dfn{pattern} is an algebraic expression that optionally contains wildcards.
+A @dfn{wildcard} is a special kind of object (of class @code{wildcard}) that
+represents an arbitrary expression. Every wildcard has a @dfn{label} which is
+an unsigned integer number to allow having multiple different wildcards in a
+pattern. Wildcards are printed as @samp{$label} (this is also the way they
+are specified in @command{ginsh}. In C++ code, wildcard objects are created
+with the call
+
+@example
+ex wild(unsigned label = 0);
+@end example
+
+which is simply a wrapper for the @code{wildcard()} constructor with a shorter
+name.
+
+Some examples for patterns:
+
+@multitable @columnfractions .5 .5
+@item @strong{Constructed as} @tab @strong{Output as}
+@item @code{wild()} @tab @samp{$0}
+@item @code{pow(x,wild())} @tab @samp{x^$0}
+@item @code{atan2(wild(1),wild(2))} @tab @samp{atan2($1,$2)}
+@item @code{indexed(A,idx(wild(),3))} @tab @samp{A.$0}
+@end multitable
+
+Notes:
+
+@itemize
+@item Wildcards behave like symbols and are subject to the same algebraic
+ rules. E.g., @samp{$0+2*$0} is automatically transformed to @samp{3*$0}.
+@item As shown in the last example, to use wildcards for indices you have to
+ use them as the value of an @code{idx} object. This is because indices must
+ always be of class @code{idx} (or a subclass).
+@item Wildcards only represent expressions or subexpressions. It is not
+ possible to use them as placeholders for other properties like index
+ dimension or variance, representation labels, symmetry of indexed objects
+ etc.
+@item Because wildcards are commutative, it is not possible to use wildcards
+ as part of noncommutative products.
+@item A pattern does not have to contain wildcards. @samp{x} and @samp{x+y}
+ are also valid patterns.
+@end itemize
+
+@cindex @code{match()}
+The most basic application of patterns is to check whether an expression
+matches a given pattern. This is done by the function
+
+@example
+bool ex::match(const ex & pattern);
+bool ex::match(const ex & pattern, lst & repls);
+@end example
+
+This function returns @code{true} when the expression matches the pattern
+and @code{false} if it doesn't. If used in the second form, the actual
+subexpressions matched by the wildcards get returned in the @code{repls}
+object as a list of relations of the form @samp{wildcard == expression}.
+If @code{match()} returns false, the state of @code{repls} is undefined.
+For reproducible results, the list should be empty when passed to
+@code{match()}, but it is also possible to find similarities in multiple
+expressions by passing in the result of a previous match.
+
+The matching algorithm works as follows:
+
+@itemize
+@item A single wildcard matches any expression. If one wildcard appears
+ multiple times in a pattern, it must match the same expression in all
+ places (e.g. @samp{$0} matches anything, and @samp{$0*($0+1)} matches
+ @samp{x*(x+1)} but not @samp{x*(y+1)}).
+@item If the expression is not of the same class as the pattern, the match
+ fails (i.e. a sum only matches a sum, a function only matches a function,
+ etc.).
+@item If the pattern is a function, it only matches the same function
+ (i.e. @samp{sin($0)} matches @samp{sin(x)} but doesn't match @samp{exp(x)}).
+@item Except for sums and products, the match fails if the number of
+ subexpressions (@code{nops()}) is not equal to the number of subexpressions
+ of the pattern.
+@item If there are no subexpressions, the expressions and the pattern must
+ be equal (in the sense of @code{is_equal()}).
+@item Except for sums and products, each subexpression (@code{op()}) must
+ match the corresponding subexpression of the pattern.
+@end itemize
+
+Sums (@code{add}) and products (@code{mul}) are treated in a special way to
+account for their commutativity and associativity:
+
+@itemize
+@item If the pattern contains a term or factor that is a single wildcard,
+ this one is used as the @dfn{global wildcard}. If there is more than one
+ such wildcard, one of them is chosen as the global wildcard in a random
+ way.
+@item Every term/factor of the pattern, except the global wildcard, is
+ matched against every term of the expression in sequence. If no match is
+ found, the whole match fails. Terms that did match are not considered in
+ further matches.
+@item If there are no unmatched terms left, the match succeeds. Otherwise
+ the match fails unless there is a global wildcard in the pattern, in
+ which case this wildcard matches the remaining terms.
+@end itemize
+
+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.
+
+Here are some examples in @command{ginsh} to demonstrate how it works (the
+@code{match()} function in @command{ginsh} returns @samp{FAIL} if the
+match fails, and the list of wildcard replacements otherwise):
+
+@example
+> match((x+y)^a,(x+y)^a);
+@{@}
+> match((x+y)^a,(x+y)^b);
+FAIL
+> match((x+y)^a,$1^$2);
+@{$1==x+y,$2==a@}
+> match((x+y)^a,$1^$1);
+FAIL
+> match((x+y)^(x+y),$1^$1);
+@{$1==x+y@}
+> match((x+y)^(x+y),$1^$2);
+@{$1==x+y,$2==x+y@}
+> match((a+b)*(a+c),($1+b)*($1+c));
+@{$1==a@}
+> match((a+b)*(a+c),(a+$1)*(a+$2));
+@{$1==c,$2==b@}
+ (Unpredictable. The result might also be [$1==c,$2==b].)
+> match((a+b)*(a+c),($1+$2)*($1+$3));
+ (The result is undefined. Due to the sequential nature of the algorithm
+ and the re-ordering of terms in GiNaC, the match for the first factor
+ may be @{$1==a,$2==b@} in which case the match for the second factor
+ succeeds, or it may be @{$1==b,$2==a@} which causes the second match to
+ fail.)
+> match(a*(x+y)+a*z+b,a*$1+$2);
+ (This is also ambiguous and may return either @{$1==z,$2==a*(x+y)+b@} or
+ @{$1=x+y,$2=a*z+b@}.)
+> match(a+b+c+d+e+f,c);
+FAIL
+> match(a+b+c+d+e+f,c+$0);
+@{$0==a+e+b+f+d@}
+> match(a+b+c+d+e+f,c+e+$0);
+@{$0==a+b+f+d@}
+> match(a+b,a+b+$0);
+@{$0==0@}
+> match(a*b^2,a^$1*b^$2);
+FAIL
+ (The matching is syntactic, not algebraic, and "a" doesn't match "a^$1"
+ even if a==a^1.)
+> match(x*atan2(x,x^2),$0*atan2($0,$0^2));
+@{$0==x@}
+> match(atan2(y,x^2),atan2(y,$0));
+@{$0==x^2@}
+@end example
+
+@cindex @code{has()}
+A more general way to look for patterns in expressions is provided by the
+member function
+
+@example
+bool ex::has(const ex & pattern);
+@end example
+
+This function checks whether a pattern is matched by an expression itself or
+by any of its subexpressions.
+
+Again some examples in @command{ginsh} for illustration (in @command{ginsh},
+@code{has()} returns @samp{1} for @code{true} and @samp{0} for @code{false}):
+
+@example
+> has(x*sin(x+y+2*a),y);
+1
+> has(x*sin(x+y+2*a),x+y);
+0
+ (This is because in GiNaC, "x+y" is not a subexpression of "x+y+2*a" (which
+ has the subexpressions "x", "y" and "2*a".)
+> has(x*sin(x+y+2*a),x+y+$1);
+1
+ (But this is possible.)
+> has(x*sin(2*(x+y)+2*a),x+y);
+0
+ (This fails because "2*(x+y)" automatically gets converted to "2*x+2*y" of
+ which "x+y" is not a subexpression.)
+> has(x+1,x^$1);
+0
+ (Although x^1==x and x^0==1, neither "x" nor "1" are actually of the form
+ "x^something".)
+> has(4*x^2-x+3,$1*x);
+1
+> has(4*x^2+x+3,$1*x);
+0
+ (Another possible pitfall. The first expression matches because the term
+ "-x" has the form "(-1)*x" in GiNaC. To check whether a polynomial
+ contains a linear term you should use the coeff() function instead.)
+@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
+used in the search patterns as well as in the replacement expressions, where
+they get replaced by the expressions matched by them. @code{subs()} doesn't
+know anything about algebra; it performs purely syntactic substitutions.
+
+Some examples:
+
+@example
+> subs(a^2+b^2+(x+y)^2,$1^2==$1^3);
+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);
+(a+b+c)^2
+> subs((a+b+c)^2,a+b+$1==x+$1);
+(x+c)^2
+> 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
+> subs(4*x^3-2*x^2+5*x-1,x^$0==a^$0);
+-1+5*x-2*a^2+4*a^3
+> subs(sin(1+sin(x)),sin($1)==cos($1));
+cos(1+cos(x))
+> expand(subs(a*sin(x+y)^2+a*cos(x+y)^2+b,cos($1)^2==1-sin($1)^2));
+a+b
+@end example
+
+The last example would be written in C++ in this way:
+
+@example
+@{
+ symbol a("a"), b("b"), x("x"), y("y");
+ e = a*pow(sin(x+y), 2) + a*pow(cos(x+y), 2) + b;
+ e = e.subs(pow(cos(wild()), 2) == 1-pow(sin(wild()), 2));
+ cout << e.expand() << endl;
+ // -> a+b
+@}
+@end example
+
+
+@node Polynomial Arithmetic, Rational Expressions, Pattern Matching and Advanced Substitutions, Methods and Functions
@c node-name, next, previous, up
@section Polynomial arithmetic
@code{collect()} accomplishes this task:
@example
-ex ex::collect(const symbol & s);
+ex ex::collect(const ex & s, bool distributed = false);
@end example
+The first argument to @code{collect()} can also be a list of objects in which
+case the result is either a recursively collected polynomial, or a polynomial
+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
-to be able to find the coefficients properly.
+for @code{collect()} to be able to find the coefficients properly.
@subsection Degree and coefficients
@cindex @code{degree()}
methods
@example
-int ex::degree(const symbol & s);
-int ex::ldegree(const symbol & s);
+int ex::degree(const ex & s);
+int ex::ldegree(const ex & s);
@end example
which also work reliably on non-expanded input polynomials (they even work
a coefficient with a certain power from an expanded polynomial you use
@example
-ex ex::coeff(const symbol & s, int n);
+ex ex::coeff(const ex & s, int n);
@end example
You can also obtain the leading and trailing coefficients with the methods
@example
-ex ex::lcoeff(const symbol & s);
-ex ex::tcoeff(const symbol & s);
+ex ex::lcoeff(const ex & s);
+ex ex::tcoeff(const ex & s);
@end example
which are equivalent to @code{coeff(s, degree(s))} and @code{coeff(s, ldegree(s))},
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
or even from run to run since the internal canonical ordering is not
within the user's sphere of influence.
+@code{degree()}, @code{ldegree()}, @code{coeff()}, @code{lcoeff()},
+@code{tcoeff()} and @code{collect()} can also be used to a certain degree
+with non-polynomial expressions as they not only work with symbols but with
+constants, functions and indexed objects as well:
+
+@example
+@{
+ symbol a("a"), b("b"), c("c");
+ idx i(symbol("i"), 3);
+
+ ex e = pow(sin(x) - cos(x), 4);
+ cout << e.degree(cos(x)) << endl;
+ // -> 4
+ cout << e.expand().coeff(sin(x), 3) << endl;
+ // -> -4*cos(x)
+
+ e = indexed(a+b, i) * indexed(b+c, i);
+ e = e.expand(expand_options::expand_indexed);
+ cout << e.collect(indexed(b, i)) << endl;
+ // -> a.i*c.i+(a.i+c.i)*b.i+b.i^2
+@}
+@end example
+
@subsection Polynomial division
@cindex polynomial division
@end example
+@subsection Square-free decomposition
+@cindex square-free decomposition
+@cindex factorization
+@cindex @code{sqrfree()}
+
+GiNaC still lacks proper factorization support. Some form of
+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:
+@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:
+@example
+ ...
+ symbol x("x"), y("y");
+ ex BiVarPol = expand(pow(x-2*y*x,3) * pow(x+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
+
+ cout << sqrfree(BiVarPol, lst(y,x)) << endl;
+ // -> (1-2*y)^3*(y+x)^2*(-y+x)*x^3
+
+ cout << sqrfree(BiVarPol) << endl;
+ // -> depending on luck, any of the above
+ ...
+@end example
+
+
@node Rational Expressions, Symbolic Differentiation, Polynomial Arithmetic, Methods and Functions
@c node-name, next, previous, up
@section Rational expressions
symbol x("x");
ex t1 = (pow(x,2) + 2*x + 1)/(x + 1);
ex t2 = (pow(sin(x),2) + 2*sin(x) + 1)/(sin(x) + 1);
- cout << "t1 is " << t1.normal() << endl;
- cout << "t2 is " << t2.normal() << endl;
+ std::cout << "t1 is " << t1.normal() << std::endl;
+ std::cout << "t2 is " << t2.normal() << std::endl;
@}
@end example
@cindex denominator
@cindex @code{numer()}
@cindex @code{denom()}
+@cindex @code{numer_denom()}
The numerator and denominator of an expression can be obtained with
@example
ex ex::numer();
ex ex::denom();
+ex ex::numer_denom();
@end example
These functions will first normalize the expression as described above and
-then return the numerator or denominator, respectively.
+then return the numerator, denominator, or both as a list, respectively.
+If you need both numerator and denominator, calling @code{numer_denom()} is
+faster than using @code{numer()} and @code{denom()} separately.
@subsection Converting to a rational expression
int main()
@{
for (unsigned i=0; i<11; i+=2)
- cout << EulerNumber(i) << endl;
+ std::cout << EulerNumber(i) << std::endl;
return 0;
@}
@end example
@code{i} by two since all odd Euler numbers vanish anyways.
-@node Series Expansion, Built-in Functions, Symbolic Differentiation, Methods and Functions
+@node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
@c node-name, next, previous, up
@section Series expansion
@cindex @code{series()}
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
int main()
@{
+ using std::cout; // just for fun, another way of...
+ using std::endl; // ...dealing with this namespace std.
ex pi_frac;
for (int i=2; i<12; i+=2) @{
pi_frac = mechain_pi(i);
@end example
-@node Built-in Functions, Input/Output, Series Expansion, Methods and Functions
+@node Symmetrization, Built-in Functions, Series Expansion, Methods and Functions
+@c node-name, next, previous, up
+@section Symmetrization
+@cindex @code{symmetrize()}
+@cindex @code{antisymmetrize()}
+
+The two methods
+
+@example
+ex ex::symmetrize(const lst & l);
+ex ex::antisymmetrize(const lst & l);
+@end example
+
+symmetrize an expression by returning the symmetric or antisymmetric sum
+over all permutations of the specified list of objects, weighted by the
+number of permutations.
+
+The two additional methods
+
+@example
+ex ex::symmetrize();
+ex ex::antisymmetrize();
+@end example
+
+symmetrize or antisymmetrize an expression over its free indices.
+
+Symmetrization is most useful with indexed expressions but can be used with
+almost any kind of object (anything that is @code{subs()}able):
+
+@example
+@{
+ 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;
+ // -> 1/2*A.j.i+1/2*A.i.j
+ cout << 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(lst(a, b, c)) << endl;
+ // -> 1/6*@{a,b,c@}+1/6*@{c,a,b@}+1/6*@{b,a,c@}+1/6*@{c,b,a@}+1/6*@{b,c,a@}+1/6*@{a,c,b@}
+@}
+@end example
+
+
+@node Built-in Functions, Input/Output, Symmetrization, Methods and Functions
@c node-name, next, previous, up
@section Predefined mathematical functions
@{
symbol x("x");
ex e = 4.5+pow(x,2)*3/2;
- cout << e << endl; // prints '4.5+3/2*x^2'
+ cout << e << endl; // prints '(4.5)+3/2*x^2'
// ...
@end example
into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
is printed as @samp{x^2}).
-To print an expression in a way that can be directly used in a C or C++
-program, you use the method
+It is possible to print expressions in a number of different formats with
+the method
@example
-void ex::printcsrc(ostream & os, unsigned type, const char *name);
+void ex::print(const print_context & c, unsigned level = 0);
@end example
-This outputs a line in the form of a variable definition @code{<type> <name> = <expression>}.
-The possible types are defined in @file{ginac/flags.h} (@code{csrc_types})
-and mostly affect the way in which floating point numbers are written:
+@cindex @code{print_context} (class)
+The type of @code{print_context} object passed in determines the format
+of the output. The possible types are defined in @file{ginac/print.h}.
+All constructors of @code{print_context} and derived classes take an
+@code{ostream &} as their first argument.
+
+To print an expression in a way that can be directly used in a C or C++
+program, you pass a @code{print_csrc} object like this:
@example
// ...
- e.printcsrc(cout, csrc_types::ctype_float, "f");
- e.printcsrc(cout, csrc_types::ctype_double, "d");
- e.printcsrc(cout, csrc_types::ctype_cl_N, "n");
+ cout << "float f = ";
+ e.print(print_csrc_float(cout));
+ cout << ";\n";
+
+ cout << "double d = ";
+ e.print(print_csrc_double(cout));
+ cout << ";\n";
+
+ cout << "cl_N n = ";
+ e.print(print_csrc_cl_N(cout));
+ cout << ";\n";
// ...
@end example
+The three possible types mostly affect the way in which floating point
+numbers are written.
+
The above example will produce (note the @code{x^2} being converted to @code{x*x}):
@example
float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00;
-cl_N n = (cl_F("3.0")/cl_F("2.0"))*(x*x)+cl_F("4.5");
+cl_N n = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
@end example
-Finally, there are the two methods @code{printraw()} and @code{printtree()} intended for GiNaC
-developers, that provide a dump of the internal structure of an expression for
-debugging purposes:
+The @code{print_context} type @code{print_tree} provides a dump of the
+internal structure of an expression for debugging purposes:
@example
// ...
- e.printraw(cout); cout << endl << endl;
- e.printtree(cout);
+ e.print(print_tree(cout));
@}
@end example
produces
@example
-ex(+((power(ex(symbol(name=x,serial=1,hash=150875740,flags=11)),ex(numeric(2)),hash=2,flags=3),numeric(3/2)),,hash=0,flags=3))
-
-type=Q25GiNaC3add, hash=0 (0x0), flags=3, nops=2
- power: hash=2 (0x2), flags=3
- x (symbol): serial=1, hash=150875740 (0x8fe2e5c), flags=11
- 2 (numeric): hash=2147483714 (0x80000042), flags=11
- 3/2 (numeric): hash=2147483745 (0x80000061), flags=11
+add, hash=0x0, flags=0x3, nops=2
+ power, hash=0x9, flags=0x3, nops=2
+ x (symbol), serial=3, hash=0x44a113a6, flags=0xf
+ 2 (numeric), hash=0x80000042, flags=0xf
+ 3/2 (numeric), hash=0x80000061, flags=0xf
-----
overall_coeff
- 4.5L0 (numeric): hash=2147483723 (0x8000004b), flags=11
+ 4.5L0 (numeric), hash=0x8000004b, flags=0xf
=====
@end example
-The @code{printtree()} method is also available in @command{ginsh} as the
-@code{print()} function.
+This kind of output is also available in @command{ginsh} as the @code{print()}
+function.
+
+Another useful output format is for LaTeX parsing in mathematical mode.
+It is rather similar to the default @code{print_context} but provides
+some braces needed by LaTeX for delimiting boxes and also converts some
+common objects to conventional LaTeX names. It is possible to give symbols
+a special name for LaTeX output by supplying it as a second argument to
+the @code{symbol} constructor.
+
+For example, the code snippet
+
+@example
+ // ...
+ symbol x("x");
+ ex foo = lgamma(x).series(x==0,3);
+ foo.print(print_latex(std::cout));
+@end example
+
+will print out:
+
+@example
+ @{(-\ln(x))@}+@{(-\gamma_E)@} x+@{(1/12 \pi^2)@} x^@{2@}+\mathcal@{O@}(x^3)
+@end example
+
+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:
+
+@example
+static void my_print(const ex & e)
+@{
+ if (is_ex_of_type(e, function))
+ cout << ex_to_function(e).get_name();
+ else
+ cout << e.bp->class_name();
+ cout << "(";
+ unsigned n = e.nops();
+ if (n)
+ for (unsigned i=0; i<n; i++) @{
+ my_print(e.op(i));
+ if (i != n-1)
+ cout << ",";
+ @}
+ else
+ cout << e;
+ cout << ")";
+@}
+
+int main(void)
+@{
+ my_print(pow(3, x) - 2 * sin(y / Pi)); cout << endl;
+ return 0;
+@}
+@end example
+
+This will produce
+
+@example
+add(power(numeric(3),symbol(x)),mul(sin(mul(power(constant(Pi),numeric(-1)),
+symbol(y))),numeric(-2)))
+@end example
+
+If you need an output format that makes it possible to accurately
+reconstruct an expression by feeding the output to a suitable parser or
+object factory, you should consider storing the expression in an
+@code{archive} object and reading the object properties from there.
+See the section on archiving for more information.
@subsection Expression input
#include <string>
#include <stdexcept>
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
expression a unique name:
@example
-#include <ginac/ginac.h>
#include <fstream>
+using namespace std;
+#include <ginac/ginac.h>
using namespace GiNaC;
int main()
different symbol than the @code{x} which was defined at the beginning of
the program, altough 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
+@code{archive} and @code{archive_node} classes have a couple of member
+functions that let you access the stored properties:
+
+@example
+static void my_print2(const archive_node & n)
+@{
+ string class_name;
+ n.find_string("class", class_name);
+ cout << class_name << "(";
+
+ archive_node::propinfovector p;
+ n.get_properties(p);
+
+ unsigned num = p.size();
+ for (unsigned i=0; i<num; i++) @{
+ const string &name = p[i].name;
+ if (name == "class")
+ continue;
+ cout << name << "=";
+
+ unsigned count = p[i].count;
+ if (count > 1)
+ cout << "@{";
+
+ for (unsigned j=0; j<count; j++) @{
+ switch (p[i].type) @{
+ case archive_node::PTYPE_BOOL: @{
+ bool x;
+ n.find_bool(name, x);
+ cout << (x ? "true" : "false");
+ break;
+ @}
+ case archive_node::PTYPE_UNSIGNED: @{
+ unsigned x;
+ n.find_unsigned(name, x);
+ cout << x;
+ break;
+ @}
+ case archive_node::PTYPE_STRING: @{
+ string x;
+ n.find_string(name, x);
+ cout << '\"' << x << '\"';
+ break;
+ @}
+ case archive_node::PTYPE_NODE: @{
+ const archive_node &x = n.find_ex_node(name, j);
+ my_print2(x);
+ break;
+ @}
+ @}
+
+ if (j != count-1)
+ cout << ",";
+ @}
+
+ if (count > 1)
+ cout << "@}";
+
+ if (i != num-1)
+ cout << ",";
+ @}
+
+ cout << ")";
+@}
+
+int main(void)
+@{
+ ex e = pow(2, x) - y;
+ archive ar(e, "e");
+ my_print2(ar.get_top_node(0)); cout << endl;
+ return 0;
+@}
+@end example
+
+This will produce:
+
+@example
+add(rest=@{power(basis=numeric(number="2"),exponent=symbol(name="x")),
+symbol(name="y")@},coeff=@{numeric(number="1"),numeric(number="-1")@},
+overall_coeff=numeric(number="0"))
+@end example
+
+Be warned, however, that the set of properties and their meaning for each
+class may change between GiNaC versions.
@node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
@{
...
public:
- void print(ostream &os, unsigned upper_precedence) const;
+ void print(const print_context &c, unsigned level = 0) const;
...
@};
-void mystring::print(ostream &os, unsigned upper_precedence) const
+void mystring::print(const print_context &c, unsigned level) const
@{
- os << '\"' << str << '\"';
+ // print_context::s is a reference to an ostream
+ c.s << '\"' << str << '\"';
@}
@end example
-The @code{upper_precedence} argument is only required for container classes
-to correctly parenthesize the output. Let's try again to print the expression:
+The @code{level} argument is only required for container classes to
+correctly parenthesize the output. Let's try again to print the expression:
@example
cout << e << endl;
@}
if (new_str.length() == 0)
- return _ex0();
+ return 0;
else
return mystring(new_str).hold();
@}
some more functions that you will want to re-implement, such as
@code{evalf()}, @code{series()} or @code{op()}. Have a look at @file{basic.h}
or the header file of the class you want to make a subclass of to see
-what's there. You can, of course, also add your own new member functions.
-In this case you will probably want to define a little helper function like
+what's there. One member function that you will most likely want to
+implement for terminal classes like the described string class is
+@code{calcchash()} that returns an @code{unsigned} hash value for the object
+which will allow GiNaC to compare and canonicalize expressions much more
+efficiently.
+
+You can, of course, also add your own new member functions. In this case you
+will probably want to define a little helper function like
@example
inline const mystring &ex_to_mystring(const ex &e)
@}
@end example
-that let's you get at the object inside an expression (after you have verified
-that the type is correct) so you can call member functions that are specific
-to the class.
+that let's you get at the object inside an expression (after you have
+verified that the type is correct) so you can call member functions that are
+specific to the class.
That's it. May the source be with you!
@item
development tools: powerful development tools exist for C++, like fancy
editors (e.g. with automatic indentation and syntax highlighting),
-debuggers, visualization tools, documentation generators...
+debuggers, visualization tools, documentation generators@dots{}
@item
modularization: C++ programs can easily be split into modules by
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@example
#include <ginac/ginac.h>
-using namespace GiNaC;
int main(void)
@{
- symbol x("x");
- ex a = sin(x);
- cout << "Derivative of " << a << " is " << a.diff(x) << endl;
+ GiNaC::symbol x("x");
+ GiNaC::ex a = GiNaC::sin(x);
+ std::cout << "Derivative of " << a
+ << " is " << a.diff(x) << std::endl;
return 0;
@}
@end example
J.H. Davenport, Y. Siret, and E. Tournier, ISBN 0-12-204230-1, 1988,
Academic Press, London
+@item
+@cite{The Role of gamma5 in Dimensional Regularization}, D. Kreimer, hep-ph/9401354
+
@end itemize
git://www.ginac.de / ginac.git / blobdiff