This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2000 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2001 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{} 1999-2000 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2001 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
* A Tour of GiNaC:: A quick tour of the library.
* Installation:: How to install the package.
* Basic Concepts:: Description of fundamental classes.
-* Important Algorithms:: Algorithms for symbolic manipulations.
+* Methods and Functions:: Algorithms for symbolic manipulations.
* Extending GiNaC:: How to extend the library.
* A Comparison With Other CAS:: Compares GiNaC to traditional CAS.
* Internal Structures:: Description of some internal structures.
@section License
The GiNaC framework for symbolic computation within the C++ programming
-language is Copyright @copyright{} 1999-2000 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2001 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
@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)
@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
x-1/6*x^3+Order(x^4)
> series(1/tan(x),x==0,4);
x^(-1)-1/3*x+Order(x^2)
-> series(Gamma(x),x==0,3);
-x^(-1)-gamma+(1/12*Pi^2+1/2*gamma^2)*x+
-(-1/3*zeta(3)-1/12*Pi^2*gamma-1/6*gamma^3)*x^2+Order(x^3)
+> series(tgamma(x),x==0,3);
+x^(-1)-Euler+(1/12*Pi^2+1/2*Euler^2)*x+
+(-1/3*zeta(3)-1/12*Pi^2*Euler-1/6*Euler^3)*x^2+Order(x^3)
> evalf(");
x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
-(0.90747907608088628905)*x^2+Order(x^3)
-> series(Gamma(2*sin(x)-2),x==Pi/2,6);
--(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*gamma^2-1/240)*(x-1/2*Pi)^2
--gamma-1/12+Order((x-1/2*Pi)^3)
+> series(tgamma(2*sin(x)-2),x==Pi/2,6);
+-(x-1/2*Pi)^(-2)+(-1/12*Pi^2-1/2*Euler^2-1/240)*(x-1/2*Pi)^2
+-Euler-1/12+Order((x-1/2*Pi)^3)
@end example
Here we have made use of the @command{ginsh}-command @code{"} to pop the
benchmark some predefined problems with different sizes and display the
CPU time used in seconds. Each individual test should return a message
@samp{passed}. This is mostly intended to be a QA-check if something
-was broken during development, not a sanity check of your system.
-Another intent is to allow people to fiddle around with optimization.
+was broken during development, not a sanity check of your system. Some
+of the tests in sections @emph{checks} and @emph{timings} may require
+insane amounts of memory and CPU time. Feel free to kill them if your
+machine catches fire. Another quite important intent is to allow people
+to fiddle around with optimization.
Generally, the top-level Makefile runs recursively to the
subdirectories. It is therfore safe to go into any subdirectory
* Numbers:: Numerical objects.
* Constants:: Pre-defined constants.
* Fundamental containers:: The power, add and mul classes.
-* Built-in functions:: Mathematical functions.
+* Lists:: Lists of expressions.
+* Mathematical functions:: Mathematical functions.
* Relations:: Equality, Inequality and all that.
-* Archiving:: Storing expression libraries in files.
+* Indexed objects:: Handling indexed quantities.
+* Non-commutative objects:: Algebras with non-commutative products.
@end menu
ex MyEx5 = MyEx4 + 1; // similar to above
@end example
-Expressions are handles to other more fundamental objects, that many
-times contain other expressions thus creating a tree of expressions
+Expressions are handles to other more fundamental objects, that often
+contain other expressions thus creating a tree of expressions
(@xref{Internal Structures}, for particular examples). Most methods on
@code{ex} therefore run top-down through such an expression tree. For
example, the method @code{has()} scans recursively for occurrences of
helpers) are internally derived from one abstract base class called
@code{basic}. You do not have to deal with objects of class
@code{basic}, instead you'll be dealing with symbols, numbers,
-containers of expressions and so on. You'll soon learn in this chapter
-how many of the functions on symbols are really classes. This is
-because simple symbolic arithmetic is not supported by languages like
-C++ so in a certain way GiNaC has to implement its own arithmetic.
+containers of expressions and so on.
@cindex container
@cindex atom
To get an idea about what kinds of symbolic composits may be built we
-have a look at the most important classes in the class hierarchy. The
-oval classes are atomic ones and the squared classes are containers.
-The dashed line symbolizes a `points to' or `handles' relationship while
-the solid lines stand for `inherits from' relationship in the class
-hierarchy:
+have a look at the most important classes in the class hierarchy and
+some of the relations among the classes:
@image{classhierarchy}
-Some of the classes shown here (the ones sitting in white boxes) are
-abstract base classes that are of no interest at all for the user. They
-are used internally in order to avoid code duplication if two or more
-classes derived from them share certain features. An example would be
-@code{expairseq}, which is a container for a sequence of pairs each
-consisting of one expression and a number (@code{numeric}). What
-@emph{is} visible to the user are the derived classes @code{add} and
-@code{mul}, representing sums of terms and products, respectively.
-@xref{Internal Structures}, where these two classes are described in
-more detail.
-
-At this point, we only summarize what kind of mathematical objects are
-stored in the different classes in above diagram in order to give you a
-overview:
+The abstract classes shown here (the ones without drop-shadow) are of no
+interest for the user. They are used internally in order to avoid code
+duplication if two or more classes derived from them share certain
+features. An example is @code{expairseq}, a container for a sequence of
+pairs each consisting of one expression and a number (@code{numeric}).
+What @emph{is} visible to the user are the derived classes @code{add}
+and @code{mul}, representing sums and products. @xref{Internal
+Structures}, where these two classes are described in more detail. The
+following table shortly summarizes what kinds of mathematical objects
+are stored in the different classes:
@cartouche
@multitable @columnfractions .22 .78
@math{Pi}
@end ifnottex
@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{a*(x+y+z)*b*2}
+@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
$\sqrt{2}$
@code{sqrt(}@math{2}@code{)}
@end ifnottex
@dots{}
-@item @code{pseries} @tab Power Series, e.g. @math{x+1/6*x^3+1/120*x^5+O(x^7)}
+@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{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{relational} @tab A relation like the identity @math{x}@code{==}@math{y}
-@item @code{color} @tab Element of the @math{SU(3)} Lie-algebra
-@item @code{isospin} @tab Element of the @math{SU(2)} Lie-algebra
-@item @code{idx} @tab Index of a tensor object
-@item @code{coloridx} @tab Index of a @math{SU(3)} tensor
+@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.
+can use the expression's @code{.subs()} method (@xref{Substituting Expressions},
+for more information).
@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:
cout << answer.is_integer() << endl; // false, it's 21/5
answer *= ten;
cout << answer.is_integer() << endl; // true, it's 42 now!
- // ...
@}
@end example
@item @code{.is_real()}
@tab @dots{}a real integer, rational or float (i.e. is not complex)
@item @code{.is_cinteger()}
-@tab @dots{}a (complex) integer, such as @math{2-3*I}
+@tab @dots{}a (complex) integer (such as @math{2-3*I})
@item @code{.is_crational()}
@tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
@end multitable
@cindex @code{Pi}
@cindex @code{Catalan}
-@cindex @code{gamma}
+@cindex @code{Euler}
@cindex @code{evalf()}
Constants behave pretty much like symbols except that they return some
specific number when the method @code{.evalf()} is called.
@item @code{Catalan}
@tab Catalan's constant
@tab 0.91596559417721901505460351493238411
-@item @code{gamma}
+@item @code{Euler}
@tab Euler's (or Euler-Mascheroni) constant
@tab 0.57721566490153286060651209008240243
@end multitable
@end cartouche
-@node Fundamental containers, Built-in functions, Constants, Basic Concepts
+@node Fundamental containers, Lists, Constants, Basic Concepts
@c node-name, next, previous, up
@section Fundamental containers: the @code{power}, @code{add} and @code{mul} classes
@cindex polynomial
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
canonical form.
-@node Built-in functions, Relations, Fundamental containers, Basic Concepts
+@node Lists, Mathematical functions, Fundamental containers, Basic Concepts
+@c node-name, next, previous, up
+@section Lists of expressions
+@cindex @code{lst} (class)
+@cindex lists
+@cindex @code{nops()}
+@cindex @code{op()}
+@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.
+
+Lists of up to 15 expressions can be directly constructed from single
+expressions:
+
+@example
+@{
+ symbol x("x"), y("y");
+ lst l(x, 2, y, x+y);
+ // now, l is a list holding the expressions 'x', '2', 'y', and 'x+y'
+ // ...
+@end example
+
+Use the @code{nops()} method to determine the size (number of expressions) of
+a list and the @code{op()} method to access individual elements:
+
+@example
+ // ...
+ cout << l.nops() << endl; // prints '4'
+ cout << l.op(2) << " " << l.op(0) << endl; // prints 'y x'
+ // ...
+@end example
+
+Finally you can append or prepend an expression to a list with the
+@code{append()} and @code{prepend()} methods:
+
+@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@}
+@}
+@end example
+
+
+@node Mathematical functions, Relations, Lists, Basic Concepts
@c node-name, next, previous, up
-@section Built-in functions
+@section Mathematical functions
@cindex @code{function} (class)
@cindex trigonometric function
@cindex hyperbolic function
There are quite a number of useful functions hard-wired into GiNaC. For
-instance, all trigonometric and hyperbolic functions are implemented.
-They 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:
+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, 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 << "Gamma(" << foo << ") -> " << Gamma(foo) << endl;
+ cout << tgamma(foo) << endl;
+ // -> tgamma(x+(1/2)*y)
ex bar = foo.subs(y==1);
- cout << "Gamma(" << bar << ") -> " << Gamma(bar) << endl;
+ cout << tgamma(bar) << endl;
+ // -> tgamma(x+1/2)
ex foobar = bar.subs(x==7);
- cout << "Gamma(" << foobar << ") -> " << Gamma(foobar) << endl;
- // ...
-@}
+ cout << tgamma(foobar) << endl;
+ // -> (135135/128)*Pi^(1/2)
+ ...
@end example
-This program shows how the function returns itself twice and finally an
-expression that may be really useful:
-
-@example
-Gamma(x+(1/2)*y) -> Gamma(x+(1/2)*y)
-Gamma(x+1/2) -> Gamma(x+1/2)
-Gamma(15/2) -> (135135/128)*Pi^(1/2)
-@end example
-
-@cindex branch cut
-For functions that have a branch cut in the complex plane GiNaC follows
-the conventions for C++ as defined in the ANSI standard. In particular:
-the natural logarithm (@code{log}) and the square root (@code{sqrt})
-both have their branch cuts running along the negative real axis where
-the points on the axis itself belong to the upper part.
-
Besides evaluation most of these functions allow differentiation, series
expansion and so on. Read the next chapter in order to learn more about
this.
-@node Relations, Archiving, Built-in functions, Basic Concepts
+@node Relations, Indexed objects, Mathematical functions, Basic Concepts
@c node-name, next, previous, up
@section Relations
@cindex @code{relational} (class)
They are created by simply using the C++ operators @code{==}, @code{!=},
@code{<}, @code{<=}, @code{>} and @code{>=} between two expressions.
-@xref{Built-in functions}, for examples where various applications of
-the @code{.subs()} method show how objects of class relational are used
-as arguments. There they provide an intuitive syntax for substitutions.
-They can also used for creating systems of equations that are to be
-solved for unknown variables. More applications of this class will
-appear throughout the next chapters.
+@xref{Mathematical functions}, for examples where various applications
+of the @code{.subs()} method show how objects of class relational are
+used as arguments. There they provide an intuitive syntax for
+substitutions. They are also used as arguments to the @code{ex::series}
+method, where the left hand side of the relation specifies the variable
+to expand in and the right hand side the expansion point. They can also
+be used for creating systems of equations that are to be solved for
+unknown variables. But the most common usage of objects of this class
+is rather inconspicuous in statements of the form @code{if
+(expand(pow(a+b,2))==a*a+2*a*b+b*b) @{...@}}. Here, an implicit
+conversion from @code{relational} to @code{bool} takes place. Note,
+however, that @code{==} here does not perform any simplifications, hence
+@code{expand()} must be called explicitly.
+
+
+@node Indexed objects, Non-commutative objects, Relations, Basic Concepts
+@c node-name, next, previous, up
+@section Indexed objects
+
+GiNaC allows you to handle expressions containing general indexed objects in
+arbitrary spaces. It is also able to canonicalize and simplify such
+expressions and perform symbolic dummy index summations. There are a number
+of predefined indexed objects provided, like delta and metric tensors.
+There are few restrictions placed on indexed objects and their indices and
+it is easy to construct nonsense expressions, but our intention is to
+provide a general framework that allows you to implement algorithms with
+indexed quantities, getting in the way as little as possible.
-@node Archiving, Important Algorithms, Relations, Basic Concepts
-@c node-name, next, previous, up
-@section Archiving Expressions
-@cindex I/O
-@cindex @code{archive} (class)
+@cindex @code{idx} (class)
+@cindex @code{indexed} (class)
+@subsection Indexed quantities and their indices
-GiNaC allows creating @dfn{archives} of expressions which can be stored
-to or retrieved from files. To create an archive, you declare an object
-of class @code{archive} and archive expressions in it, giving each
-expressions a unique name:
+Indexed expressions in GiNaC are constructed of two special types of objects,
+@dfn{index objects} and @dfn{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), 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
+one or more indices.
+
+@end itemize
+
+@strong{Note:} 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
+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>
-#include <fstream>
+using namespace std;
using namespace GiNaC;
int main()
@{
- symbol x("x"), y("y"), z("z");
+ symbol i_sym("i"), j_sym("j");
+ idx i(i_sym, 3), j(j_sym, 3);
- ex foo = sin(x + 2*y) + 3*z + 41;
- ex bar = foo + 1;
+ symbol A("A");
+ cout << indexed(A, i, j) << endl;
+ // -> A.i.j
+ ...
+@end example
- archive a;
- a.archive_ex(foo, "foo");
- a.archive_ex(bar, "the second one");
- // ...
+The @code{idx} constructor takes two arguments, the index value and the
+index dimension. First we define two index objects, @code{i} and @code{j},
+both with the numeric dimension 3. The value of the index @code{i} is the
+symbol @code{i_sym} (which prints as @samp{i}) and the value of the index
+@code{j} is the symbol @code{j_sym} (which prints as @samp{j}). Next we
+construct an expression containing one indexed object, @samp{A.i.j}. It has
+the symbol @code{A} as its base expression and the two indices @code{i} and
+@code{j}.
+
+Note the difference between the indices @code{i} and @code{j} which are of
+class @code{idx}, and the index values which are the sybols @code{i_sym}
+and @code{j_sym}. The indices of indexed objects cannot directly be symbols
+or numbers but must be index objects. For example, the following is not
+correct and will raise an exception:
+
+@example
+symbol i("i"), j("j");
+e = indexed(A, i, j); // ERROR: indices must be of type idx
@end example
-The archive can then be written to a file:
+You can have multiple indexed objects in an expression, index values can
+be numeric, and index dimensions symbolic:
@example
- // ...
- ofstream out("foobar.gar");
- out << a;
- out.close();
- // ...
+ ...
+ symbol B("B"), dim("dim");
+ cout << 4 * indexed(A, i)
+ + indexed(B, idx(j_sym, 4), idx(2, 3), idx(i_sym, dim)) << endl;
+ // -> B.j.2.i+4*A.i
+ ...
@end example
-The file @file{foobar.gar} contains all information that is needed to
-reconstruct the expressions @code{foo} and @code{bar}.
+@code{B} has a 4-dimensional symbolic index @samp{k}, a 3-dimensional numeric
+index of value 2, and a symbolic index @samp{i} with the symbolic dimension
+@samp{dim}. Note that GiNaC doesn't automatically notify you that the free
+indices of @samp{A} and @samp{B} in the sum don't match (you have to call
+@code{simplify_indexed()} for that, see below).
-@cindex @command{viewgar}
-The tool @command{viewgar} that comes with GiNaC can be used to view
-the contents of GiNaC archive files:
+In fact, base expressions, index values and index dimensions can be
+arbitrary expressions:
@example
-$ viewgar foobar.gar
-foo = 41+sin(x+2*y)+3*z
-the second one = 42+sin(x+2*y)+3*z
+ ...
+ cout << indexed(A+B, idx(2*i_sym+1, dim/2)) << endl;
+ // -> (B+A).(1+2*i)
+ ...
@end example
-The point of writing archive files is of course that they can later be
-read in again:
+It's also possible to construct nonsense like @samp{Pi.sin(x)}. You will not
+get an error message from this but you will probably not be able to do
+anything useful with it.
+
+@cindex @code{get_value()}
+@cindex @code{get_dimension()}
+The methods
@example
- // ...
- archive a2;
- ifstream in("foobar.gar");
- in >> a2;
- // ...
+ex idx::get_value(void);
+ex idx::get_dimension(void);
@end example
-And the stored expressions can be retrieved by their name:
+return the value and dimension of an @code{idx} object. If you have an index
+in an expression, such as returned by calling @code{.op()} on an indexed
+object, you can get a reference to the @code{idx} object with the function
+@code{ex_to_idx()} on the expression.
+
+There are also the methods
@example
- // ...
- lst syms;
- syms.append(x); syms.append(y);
+bool idx::is_numeric(void);
+bool idx::is_symbolic(void);
+bool idx::is_dim_numeric(void);
+bool idx::is_dim_symbolic(void);
+@end example
- ex ex1 = a2.unarchive_ex(syms, "foo");
- ex ex2 = a2.unarchive_ex(syms, "the second one");
+for checking whether the value and dimension are numeric or symbolic
+(non-numeric). Using the @code{info()} method of an index (see @ref{Information
+About Expressions}) returns information about the index value.
- cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
- cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
- cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
- // ...
-@}
+@cindex @code{varidx} (class)
+If you need co- and contravariant indices, use the @code{varidx} class:
+
+@example
+ ...
+ symbol mu_sym("mu"), nu_sym("nu");
+ varidx mu(mu_sym, 4), nu(nu_sym, 4); // default is contravariant ~mu, ~nu
+ varidx mu_co(mu_sym, 4, true); // covariant index .mu
+
+ cout << indexed(A, mu, nu) << endl;
+ // -> A~mu~nu
+ cout << indexed(A, mu_co, nu) << endl;
+ // -> A.mu~nu
+ cout << indexed(A, mu.toggle_variance(), nu) << endl;
+ // -> A.mu~nu
+ ...
@end example
-Note that you have to supply a list of the symbols which are to be inserted
-in the expressions. Symbols in archives are stored by their name only and
-if you don't specify which symbols you have, unarchiving the expression will
-create new symbols with that name. E.g. if you hadn't included @code{x} in
-the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
-have had no effect because the @code{x} in @code{ex1} would have been a
-different symbol than the @code{x} which was defined at the beginning of
-the program, altough both would appear as @samp{x} when printed.
+A @code{varidx} is an @code{idx} with an additional flag that marks it as
+co- or contravariant. The default is a contravariant (upper) index, but
+this can be overridden by supplying a third argument to the @code{varidx}
+constructor. The two methods
+@example
+bool varidx::is_covariant(void);
+bool varidx::is_contravariant(void);
+@end example
+allow you to check the variance of a @code{varidx} object (use @code{ex_to_varidx()}
+to get the object reference from an expression). There's also the very useful
+method
-@node Important Algorithms, Polynomial Expansion, Archiving, Top
-@c node-name, next, previous, up
-@chapter Important Algorithms
-@cindex polynomial
+@example
+ex varidx::toggle_variance(void);
+@end example
-In this chapter the most important algorithms provided by GiNaC will be
-described. Some of them are implemented as functions on expressions,
-others are implemented as methods provided by expression objects. If
-they are methods, there exists a wrapper function around it, so you can
-alternatively call it in a functional way as shown in the simple
-example:
+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
-#include <ginac/ginac.h>
-using namespace GiNaC;
+ ...
+ 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
-int main()
-@{
- ex x = numeric(1.0);
-
- cout << "As method: " << sin(x).evalf() << endl;
- cout << "As function: " << evalf(sin(x)) << endl;
- // ...
-@}
+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
-@cindex @code{subs()}
-The general rule is that wherever methods accept one or more parameters
-(@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
-wrapper accepts is the same but preceded by the object to act on
-(@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
-most natural one in an OO model but it may lead to confusion for MapleV
-users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
-would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
-@code{A} and @code{x}). On the other hand, since MapleV returns 3 on
-@code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
-coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
-here. Also, users of MuPAD will in most cases feel more comfortable
-with GiNaC's convention. All function wrappers are implemented
-as simple inline functions which just call the corresponding method and
-are only provided for users uncomfortable with OO who are dead set to
-avoid method invocations. Generally, nested function wrappers are much
-harder to read than a sequence of methods and should therefore be
-avoided if possible. On the other hand, not everything in GiNaC is a
-method on class @code{ex} and sometimes calling a function cannot be
-avoided.
+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
-@menu
-* Polynomial Expansion::
-* Collecting expressions::
-* Polynomial Arithmetic::
-* Symbolic Differentiation::
-* Series Expansion::
-@end menu
+@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.
-@node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms
-@c node-name, next, previous, up
-@section Polynomial Expansion
-@cindex @code{expand()}
+@subsection Substituting indices
-A polynomial in one or more variables has many equivalent
-representations. Some useful ones serve a specific purpose. Consider
-for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
-21*y*z + 4*z^2} (written down here in output-style). It is equivalent
-to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
-representations are the recursive ones where one collects for exponents
-in one of the three variable. Since the factors are themselves
-polynomials in the remaining two variables the procedure can be
-repeated. In our expample, two possibilities would be @math{(4*y + z)*x
-+ 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
-x*z}.
+@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 Expressions}).
-To bring an expression into expanded form, its method @code{.expand()}
-may be called. In our example above, this corresponds to @math{4*x*y +
-x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
-GiNaC is not easily guessable you should be prepared to see different
-orderings of terms in such sums!
+You have two possibilities here. You can either substitute the whole index
+by another index or expression:
+@example
+ ...
+ ex e = indexed(A, mu_co);
+ cout << e << " becomes " << e.subs(mu_co == nu) << endl;
+ // -> A.mu becomes A~nu
+ cout << e << " becomes " << e.subs(mu_co == varidx(0, 4)) << endl;
+ // -> A.mu becomes A~0
+ cout << e << " becomes " << e.subs(mu_co == 0) << endl;
+ // -> A.mu becomes A.0
+ ...
+@end example
-@node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms
-@c node-name, next, previous, up
-@section Collecting expressions
-@cindex @code{collect()}
-@cindex @code{coeff()}
+The third example shows that trying to replace an index with something that
+is not an index will substitute the index value instead.
-Another useful representation of multivariate polynomials is as a
-univariate polynomial in one of the variables with the coefficients
-being polynomials in the remaining variables. The method
-@code{collect()} accomplishes this task. Here is its declaration:
+Alternatively, you can substitute the @emph{symbol} of a symbolic index by
+another expression:
@example
-ex ex::collect(const symbol & s);
+ ...
+ ex e = indexed(A, mu_co);
+ cout << e << " becomes " << e.subs(mu_sym == nu_sym) << endl;
+ // -> A.mu becomes A.nu
+ cout << e << " becomes " << e.subs(mu_sym == 0) << endl;
+ // -> A.mu becomes A.0
+ ...
@end example
-Note that the original polynomial needs to be in expanded form in order
-to be able to find the coefficients properly. The range of occuring
-coefficients can be checked using the two methods
+As you see, with the second method only the value of the index will get
+substituted. Its other properties, including its dimension, remain unchanged.
+If you want to change the dimension of an index you have to substitute the
+whole index by another one with the new dimension.
+
+Finally, substituting the base expression of an indexed object works as
+expected:
-@cindex @code{degree()}
-@cindex @code{ldegree()}
@example
-int ex::degree(const symbol & s);
-int ex::ldegree(const symbol & s);
+ ...
+ ex e = indexed(A, mu_co);
+ cout << e << " becomes " << e.subs(A == A+B) << endl;
+ // -> A.mu becomes (B+A).mu
+ ...
@end example
-where @code{degree()} returns the highest coefficient and
-@code{ldegree()} the lowest one. (These two methods work also reliably
-on non-expanded input polynomials). An application is illustrated in
-the next example, where a multivariate polynomial is analyzed:
+@subsection Symmetries
+
+Indexed objects can be declared as being totally symmetric or antisymmetric
+with respect to their indices. In this case, GiNaC will automatically bring
+the indices into a canonical order which allows for some immediate
+simplifications:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+ ...
+ cout << indexed(A, indexed::symmetric, i, j)
+ + indexed(A, indexed::symmetric, j, i) << endl;
+ // -> 2*A.j.i
+ cout << indexed(B, indexed::antisymmetric, i, j)
+ + indexed(B, indexed::antisymmetric, j, j) << endl;
+ // -> -B.j.i
+ cout << indexed(B, indexed::antisymmetric, i, j)
+ + indexed(B, indexed::antisymmetric, j, i) << endl;
+ // -> 0
+ ...
+@end example
-int main()
+@cindex @code{get_free_indices()}
+@cindex Dummy index
+@subsection Dummy indices
+
+GiNaC treats certain symbolic index pairs as @dfn{dummy indices} meaning
+that a summation over the index range is implied. Symbolic indices which are
+not dummy indices are called @dfn{free indices}. Numeric indices are neither
+dummy nor free indices.
+
+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; 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
+of a sum are consistent:
+
+@example
@{
- symbol x("x"), y("y");
- ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
- - pow(x+y,2) + 2*pow(y+2,2) - 8;
- ex Poly = PolyInp.expand();
-
- for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
- cout << "The x^" << i << "-coefficient is "
- << Poly.coeff(x,i) << endl;
- @}
- cout << "As polynomial in y: "
- << Poly.collect(y) << endl;
- // ...
+ symbol A("A"), B("B"), C("C");
+
+ symbol i_sym("i"), j_sym("j"), k_sym("k"), l_sym("l");
+ idx i(i_sym, 3), j(j_sym, 3), k(k_sym, 3), l(l_sym, 3);
+
+ ex e = indexed(A, i, j) * indexed(B, j, k) + indexed(C, k, l, i, l);
+ cout << exprseq(e.get_free_indices()) << endl;
+ // -> (.i,.k)
+ // 'j' and 'l' are dummy indices
+
+ symbol mu_sym("mu"), nu_sym("nu"), rho_sym("rho"), sigma_sym("sigma");
+ varidx mu(mu_sym, 4), nu(nu_sym, 4), rho(rho_sym, 4), sigma(sigma_sym, 4);
+
+ e = indexed(A, mu, nu) * indexed(B, nu.toggle_variance(), rho)
+ + indexed(C, mu, sigma, rho, sigma.toggle_variance());
+ cout << exprseq(e.get_free_indices()) << endl;
+ // -> (~mu,~rho)
+ // 'nu' is a dummy index, but 'sigma' is not
+
+ e = indexed(A, mu, mu);
+ cout << exprseq(e.get_free_indices()) << endl;
+ // -> (~mu)
+ // 'mu' is not a dummy index because it appears twice with the same
+ // variance
+
+ e = indexed(A, mu, nu) + 42;
+ cout << exprseq(e.get_free_indices()) << endl; // ERROR
+ // this will throw an exception:
+ // "add::get_free_indices: inconsistent indices in sum"
@}
@end example
-When run, it returns an output in the following fashion:
+@cindex @code{simplify_indexed()}
+@subsection Simplifying indexed expressions
+
+In addition to the few automatic simplifications that GiNaC performs on
+indexed expressions (such as re-ordering the indices of symmetric tensors
+and calculating traces and convolutions of matrices and predefined tensors)
+there is the method
@example
-The x^0-coefficient is y^2+11*y
-The x^1-coefficient is 5*y^2-2*y
-The x^2-coefficient is -1
-The x^3-coefficient is 4*y
-As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
+ex ex::simplify_indexed(void);
+ex ex::simplify_indexed(const scalar_products & sp);
@end example
-As always, the exact output may vary between different versions of GiNaC
-or even from run to run since the internal canonical ordering is not
-within the user's sphere of influence.
+that performs some more expensive operations:
+
+@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)
+@item as a special case of dummy index summation, it can replace scalar products
+ of two tensors with a user-defined value
+@end itemize
+The last point is done with the help of the @code{scalar_products} class
+which is used to store scalar products with known values (this is not an
+arithmetic class, you just pass it to @code{simplify_indexed()}):
-@node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms
-@c node-name, next, previous, up
-@section Polynomial Arithmetic
+@example
+@{
+ symbol A("A"), B("B"), C("C"), i_sym("i");
+ idx i(i_sym, 3);
-@subsection GCD and LCM
-@cindex GCD
-@cindex LCM
+ scalar_products sp;
+ sp.add(A, B, 0); // A and B are orthogonal
+ sp.add(A, C, 0); // A and C are orthogonal
+ sp.add(A, A, 4); // A^2 = 4 (A has length 2)
-The functions for polynomial greatest common divisor and least common
-multiple have the synopsis:
+ e = indexed(A + B, i) * indexed(A + C, i);
+ cout << e << endl;
+ // -> (B+A).i*(A+C).i
-@example
-ex gcd(const ex & a, const ex & b);
-ex lcm(const ex & a, const ex & b);
+ cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp)
+ << endl;
+ // -> 4+C.i*B.i
+@}
@end example
-The functions @code{gcd()} and @code{lcm()} accept two expressions
-@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}.
+The @code{scalar_products} object @code{sp} acts as a storage for the
+scalar products added to it with the @code{.add()} method. This method
+takes three arguments: the two expressions of which the scalar product is
+taken, and the expression to replace it with. After @code{sp.add(A, B, 0)},
+@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
+don't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0.
-@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+@cindex @code{expand()}
+The example above also illustrates a feature of the @code{expand()} method:
+if passed the @code{expand_indexed} option it will distribute indices
+over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}.
-int main()
-@{
- symbol x("x"), y("y"), z("z");
- ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
- ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
+@cindex @code{tensor} (class)
+@subsection Predefined tensors
- ex P_gcd = gcd(P_a, P_b);
- // x + 5*y + 4*z
- ex P_lcm = lcm(P_a, P_b);
+Some frequently used special tensors such as the delta, epsilon and metric
+tensors are predefined in GiNaC. They have special properties when
+contracted with other tensor expressions and some of them have constant
+matrix representations (they will evaluate to a number when numeric
+indices are specified).
+
+@cindex @code{delta_tensor()}
+@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
+@code{delta_tensor()}:
+
+@example
+@{
+ symbol A("A"), B("B");
+
+ idx i(symbol("i"), 3), j(symbol("j"), 3),
+ 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;
+ cout << e.simplify_indexed() << endl;
+ // -> B.i.j*A.i.j
+
+ cout << delta_tensor(i, i) << endl;
+ // -> 3
+@}
+@end example
+
+@cindex @code{metric_tensor()}
+@subsubsection General metric tensor
+
+The function @code{metric_tensor()} creates a general symmetric metric
+tensor with two indices that can be used to raise/lower tensor indices. The
+metric tensor is denoted as @samp{g} in the output and if its indices are of
+mixed variance it is automatically replaced by a delta tensor:
+
+@example
+@{
+ symbol A("A");
+
+ varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4), rho(symbol("rho"), 4);
+
+ ex e = metric_tensor(mu, nu) * indexed(A, nu.toggle_variance(), rho);
+ cout << e.simplify_indexed() << endl;
+ // -> A~mu~rho
+
+ e = delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho);
+ cout << e.simplify_indexed() << endl;
+ // -> g~mu~rho
+
+ e = metric_tensor(mu.toggle_variance(), nu.toggle_variance())
+ * metric_tensor(nu, rho);
+ cout << e.simplify_indexed() << endl;
+ // -> delta.mu~rho
+
+ e = metric_tensor(nu.toggle_variance(), rho.toggle_variance())
+ * metric_tensor(mu, nu) * (delta_tensor(mu.toggle_variance(), rho)
+ + indexed(A, mu.toggle_variance(), rho));
+ cout << e.simplify_indexed() << endl;
+ // -> 4+A.rho~rho
+@}
+@end example
+
+@cindex @code{lorentz_g()}
+@subsubsection Minkowski metric tensor
+
+The Minkowski metric tensor is a special metric tensor with a constant
+matrix representation which is either @code{diag(1, -1, -1, ...)} (negative
+signature, the default) or @code{diag(-1, 1, 1, ...)} (positive signature).
+It is created with the function @code{lorentz_g()} (although it is output as
+@samp{eta}):
+
+@example
+@{
+ varidx mu(symbol("mu"), 4);
+
+ e = delta_tensor(varidx(0, 4), mu.toggle_variance())
+ * lorentz_g(mu, varidx(0, 4)); // negative signature
+ cout << e.simplify_indexed() << endl;
+ // -> 1
+
+ e = delta_tensor(varidx(0, 4), mu.toggle_variance())
+ * lorentz_g(mu, varidx(0, 4), true); // positive signature
+ cout << e.simplify_indexed() << endl;
+ // -> -1
+@}
+@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
+to the dimension of the index space (the indices must all be of the same
+numeric dimension), and @samp{eps.1.2.3...} (resp. @samp{eps~0~1~2...}) is
+defined to be 1. Its behaviour with indices that have a variance also
+depends on the signature of the metric. Epsilon tensors are output as
+@samp{eps}.
+
+There are three functions defined to create epsilon tensors in 2, 3 and 4
+dimensions:
+
+@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);
+@end example
+
+The first two functions create an epsilon tensor in 2 or 3 Euclidean
+dimensions, the last function creates an epsilon tensor in a 4-dimensional
+Minkowski space (the last @code{bool} argument specifies whether the metric
+has negative or positive signature, as in the case of the Minkowski metric
+tensor).
+
+@subsection Linear algebra
+
+The @code{matrix} class can be used with indices to do some simple linear
+algebra (linear combinations and products of vectors and matrices, traces
+and scalar products):
+
+@example
+@{
+ idx i(symbol("i"), 2), j(symbol("j"), 2);
+ symbol x("x"), y("y");
+
+ matrix A(2, 2, lst(1, 2, 3, 4)), X(2, 1, lst(x, y));
+
+ cout << indexed(A, i, i) << endl;
+ // -> 5
+
+ ex e = indexed(A, i, j) * indexed(X, j);
+ cout << e.simplify_indexed() << endl;
+ // -> [[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
+@}
+@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.
+
+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
+vectors and can have one or two indices (it doesn't matter whether it's a
+row or a column vector). Other matrices must have two indices.
+
+You should be careful when using indices with variance on matrices. GiNaC
+doesn't look at the variance and doesn't know that @samp{F~mu~nu} and
+@samp{F.mu.nu} are different matrices. In this case you should use only
+one form for @samp{F} and explicitly multiply it with a matrix representation
+of the metric tensor.
+
+
+@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.
+
+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.
+
+As a last note, 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
+
+In this chapter the most important algorithms provided by GiNaC will be
+described. Some of them are implemented as functions on expressions,
+others are implemented as methods provided by expression objects. If
+they are methods, there exists a wrapper function around it, so you can
+alternatively call it in a functional way as shown in the simple
+example:
+
+@example
+ ...
+ cout << "As method: " << sin(1).evalf() << endl;
+ cout << "As function: " << evalf(sin(1)) << endl;
+ ...
+@end example
+
+@cindex @code{subs()}
+The general rule is that wherever methods accept one or more parameters
+(@var{arg1}, @var{arg2}, @dots{}) the order of arguments the function
+wrapper accepts is the same but preceded by the object to act on
+(@var{object}, @var{arg1}, @var{arg2}, @dots{}). This approach is the
+most natural one in an OO model but it may lead to confusion for MapleV
+users because where they would type @code{A:=x+1; subs(x=2,A);} GiNaC
+would require @code{A=x+1; subs(A,x==2);} (after proper declaration of
+@code{A} and @code{x}). On the other hand, since MapleV returns 3 on
+@code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
+coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
+here. Also, users of MuPAD will in most cases feel more comfortable
+with GiNaC's convention. All function wrappers are implemented
+as simple inline functions which just call the corresponding method and
+are only provided for users uncomfortable with OO who are dead set to
+avoid method invocations. Generally, nested function wrappers are much
+harder to read than a sequence of methods and should therefore be
+avoided if possible. On the other hand, not everything in GiNaC is a
+method on class @code{ex} and sometimes calling a function cannot be
+avoided.
+
+@menu
+* Information About Expressions::
+* 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 Expressions, Methods and Functions, Methods and Functions
+@c node-name, next, previous, up
+@section Getting information about expressions
+
+@subsection Checking expression types
+@cindex @code{is_ex_of_type()}
+@cindex @code{ex_to_numeric()}
+@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 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
+call one of the functions @code{ex_to_@dots{}}, where @code{@dots{}} is
+one of the class names (@xref{The Class Hierarchy}, for a list of all
+classes). For example, assuming @code{e} is an @code{ex}:
+
+@example
+@{
+ @dots{}
+ if (is_ex_of_type(e, numeric))
+ numeric n = ex_to_numeric(e);
+ @dots{}
+@}
+@end example
+
+@code{is_ex_of_type()} allows you to check whether the top-level object of
+an expression @samp{e} is an instance of the GiNaC class @samp{t}
+(@xref{The Class Hierarchy}, for a list of all classes). This is most useful,
+e.g., for checking whether an expression is a number, a sum, or a product:
+
+@example
+@{
+ symbol x("x");
+ ex e1 = 42;
+ ex e2 = 4*x - 3;
+ is_ex_of_type(e1, numeric); // true
+ is_ex_of_type(e2, numeric); // false
+ is_ex_of_type(e1, add); // false
+ is_ex_of_type(e2, add); // true
+ is_ex_of_type(e1, mul); // false
+ is_ex_of_type(e2, mul); // false
+@}
+@end example
+
+The @code{info()} method is used for checking certain attributes of
+expressions. The possible values for the @code{flag} argument are defined
+in @file{ginac/flags.h}, the most important being explained in the following
+table:
+
+@cartouche
+@multitable @columnfractions .30 .70
+@item @strong{Flag} @tab @strong{Returns true if the object is@dots{}}
+@item @code{numeric}
+@tab @dots{}a number (same as @code{is_ex_of_type(..., numeric)})
+@item @code{real}
+@tab @dots{}a real integer, rational or float (i.e. is not complex)
+@item @code{rational}
+@tab @dots{}an exact rational number (integers are rational, too)
+@item @code{integer}
+@tab @dots{}a (non-complex) integer
+@item @code{crational}
+@tab @dots{}an exact (complex) rational number (such as @math{2/3+7/2*I})
+@item @code{cinteger}
+@tab @dots{}a (complex) integer (such as @math{2-3*I})
+@item @code{positive}
+@tab @dots{}not complex and greater than 0
+@item @code{negative}
+@tab @dots{}not complex and less than 0
+@item @code{nonnegative}
+@tab @dots{}not complex and greater than or equal to 0
+@item @code{posint}
+@tab @dots{}an integer greater than 0
+@item @code{negint}
+@tab @dots{}an integer less than 0
+@item @code{nonnegint}
+@tab @dots{}an integer greater than or equal to 0
+@item @code{even}
+@tab @dots{}an even integer
+@item @code{odd}
+@tab @dots{}an odd integer
+@item @code{prime}
+@tab @dots{}a prime integer (probabilistic primality test)
+@item @code{relation}
+@tab @dots{}a relation (same as @code{is_ex_of_type(..., relational)})
+@item @code{relation_equal}
+@tab @dots{}a @code{==} relation
+@item @code{relation_not_equal}
+@tab @dots{}a @code{!=} relation
+@item @code{relation_less}
+@tab @dots{}a @code{<} relation
+@item @code{relation_less_or_equal}
+@tab @dots{}a @code{<=} relation
+@item @code{relation_greater}
+@tab @dots{}a @code{>} relation
+@item @code{relation_greater_or_equal}
+@tab @dots{}a @code{>=} relation
+@item @code{symbol}
+@tab @dots{}a symbol (same as @code{is_ex_of_type(..., symbol)})
+@item @code{list}
+@tab @dots{}a list (same as @code{is_ex_of_type(..., lst)})
+@item @code{polynomial}
+@tab @dots{}a polynomial (i.e. only consists of sums and products of numbers and symbols with positive integer powers)
+@item @code{integer_polynomial}
+@tab @dots{}a polynomial with (non-complex) integer coefficients
+@item @code{cinteger_polynomial}
+@tab @dots{}a polynomial with (possibly complex) integer coefficients (such as @math{2-3*I})
+@item @code{rational_polynomial}
+@tab @dots{}a polynomial with (non-complex) rational coefficients
+@item @code{crational_polynomial}
+@tab @dots{}a polynomial with (possibly complex) rational coefficients (such as @math{2/3+7/2*I})
+@item @code{rational_function}
+@tab @dots{}a rational function (@math{x+y}, @math{z/(x+y)})
+@item @code{algebraic}
+@tab @dots{}an algebraic object (@math{sqrt(2)}, @math{sqrt(x)-1})
+@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 container
+@cindex @code{relational} (class)
+
+GiNaC provides the two methods
+
+@example
+unsigned ex::nops();
+ex ex::op(unsigned i);
+@end example
+
+for accessing the subexpressions in the container-like GiNaC classes like
+@code{add}, @code{mul}, @code{lst}, and @code{function}. @code{nops()}
+determines the number of subexpressions (@samp{operands}) contained, while
+@code{op()} returns the @code{i}-th (0..@code{nops()-1}) subexpression.
+In the case of a @code{power} object, @code{op(0)} will return the basis
+and @code{op(1)} the exponent. For @code{indexed} objects, @code{op(0)}
+is the base expression and @code{op(i)}, @math{i>0} are the indices.
+
+The left-hand and right-hand side expressions of objects of class
+@code{relational} (and only of these) can also be accessed with the methods
+
+@example
+ex ex::lhs();
+ex ex::rhs();
+@end example
+
+
+@subsection Comparing expressions
+@cindex @code{is_equal()}
+@cindex @code{is_zero()}
+
+Expressions can be compared with the usual C++ relational operators like
+@code{==}, @code{>}, and @code{<} but if the expressions contain symbols,
+the result is usually not determinable and the result will be @code{false},
+except in the case of the @code{!=} operator. You should also be aware that
+GiNaC will only do the most trivial test for equality (subtracting both
+expressions), so something like @code{(pow(x,2)+x)/x==x+1} will return
+@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}.)
+which is not evaluated until (explicitly or implicitely) cast to a @code{bool}.
+
+There are also two methods
+
+@example
+bool ex::is_equal(const ex & other);
+bool ex::is_zero();
+@end example
+
+for checking whether one expression is equal to another, or equal to zero,
+respectively.
+
+@strong{Warning:} You will also find an @code{ex::compare()} method in the
+GiNaC header files. This method is however only to be used internally by
+GiNaC to establish a canonical sort order for terms, and using it to compare
+expressions will give very surprising results.
+
+
+@node Substituting Expressions, Pattern Matching and Advanced Substitutions, Information About Expressions, Methods and Functions
+@c node-name, next, previous, up
+@section Substituting expressions
+@cindex @code{subs()}
+
+Algebraic objects inside expressions can be replaced with arbitrary
+expressions via the @code{.subs()} method:
+
+@example
+ex ex::subs(const ex & e);
+ex ex::subs(const lst & syms, const lst & repls);
+@end example
+
+In the first form, @code{subs()} accepts a relational of the form
+@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
+
+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 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 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+y),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+y),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
+
+@subsection Expanding and collecting
+@cindex @code{expand()}
+@cindex @code{collect()}
+
+A polynomial in one or more variables has many equivalent
+representations. Some useful ones serve a specific purpose. Consider
+for example the trivariate polynomial @math{4*x*y + x*z + 20*y^2 +
+21*y*z + 4*z^2} (written down here in output-style). It is equivalent
+to the factorized polynomial @math{(x + 5*y + 4*z)*(4*y + z)}. Other
+representations are the recursive ones where one collects for exponents
+in one of the three variable. Since the factors are themselves
+polynomials in the remaining two variables the procedure can be
+repeated. In our expample, two possibilities would be @math{(4*y + z)*x
++ 20*y^2 + 21*y*z + 4*z^2} and @math{20*y^2 + (21*z + 4*x)*y + 4*z^2 +
+x*z}.
+
+To bring an expression into expanded form, its method
+
+@example
+ex ex::expand();
+@end example
+
+may be called. In our example above, this corresponds to @math{4*x*y +
+x*z + 20*y^2 + 21*y*z + 4*z^2}. Again, since the canonical form in
+GiNaC is not easily guessable you should be prepared to see different
+orderings of terms in such sums!
+
+Another useful representation of multivariate polynomials is as a
+univariate polynomial in one of the variables with the coefficients
+being polynomials in the remaining variables. The method
+@code{collect()} accomplishes this task:
+
+@example
+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
+for @code{collect()} to be able to find the coefficients properly.
+
+@subsection Degree and coefficients
+@cindex @code{degree()}
+@cindex @code{ldegree()}
+@cindex @code{coeff()}
+
+The degree and low degree of a polynomial can be obtained using the two
+methods
+
+@example
+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
+on rational functions, returning the asymptotic degree). To extract
+a coefficient with a certain power from an expanded polynomial you use
+
+@example
+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 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))},
+respectively.
+
+An application is illustrated in the next example, where a multivariate
+polynomial is analyzed:
+
+@example
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x"), y("y");
+ ex PolyInp = 4*pow(x,3)*y + 5*x*pow(y,2) + 3*y
+ - pow(x+y,2) + 2*pow(y+2,2) - 8;
+ ex Poly = PolyInp.expand();
+
+ for (int i=Poly.ldegree(x); i<=Poly.degree(x); ++i) @{
+ cout << "The x^" << i << "-coefficient is "
+ << Poly.coeff(x,i) << endl;
+ @}
+ cout << "As polynomial in y: "
+ << Poly.collect(y) << endl;
+@}
+@end example
+
+When run, it returns an output in the following fashion:
+
+@example
+The x^0-coefficient is y^2+11*y
+The x^1-coefficient is 5*y^2-2*y
+The x^2-coefficient is -1
+The x^3-coefficient is 4*y
+As polynomial in y: -x^2+(5*x+1)*y^2+(-2*x+4*x^3+11)*y
+@end example
+
+As always, the exact output may vary between different versions of GiNaC
+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
+@cindex quotient
+@cindex remainder
+@cindex pseudo-remainder
+@cindex @code{quo()}
+@cindex @code{rem()}
+@cindex @code{prem()}
+@cindex @code{divide()}
+
+The two functions
+
+@example
+ex quo(const ex & a, const ex & b, const symbol & x);
+ex rem(const ex & a, const ex & b, const symbol & x);
+@end example
+
+compute the quotient and remainder of univariate polynomials in the variable
+@samp{x}. The results satisfy @math{a = b*quo(a, b, x) + rem(a, b, x)}.
+
+The additional function
+
+@example
+ex prem(const ex & a, const ex & b, const symbol & x);
+@end example
+
+computes the pseudo-remainder of @samp{a} and @samp{b} which satisfies
+@math{c*a = b*q + prem(a, b, x)}, where @math{c = b.lcoeff(x) ^ (a.degree(x) - b.degree(x) + 1)}.
+
+Exact division of multivariate polynomials is performed by the function
+
+@example
+bool divide(const ex & a, const ex & b, ex & q);
+@end example
+
+If @samp{b} divides @samp{a} over the rationals, this function returns @code{true}
+and returns the quotient in the variable @code{q}. Otherwise it returns @code{false}
+in which case the value of @code{q} is undefined.
+
+
+@subsection Unit, content and primitive part
+@cindex @code{unit()}
+@cindex @code{content()}
+@cindex @code{primpart()}
+
+The methods
+
+@example
+ex ex::unit(const symbol & x);
+ex ex::content(const symbol & x);
+ex ex::primpart(const symbol & x);
+@end example
+
+return the unit part, content part, and primitive polynomial of a multivariate
+polynomial with respect to the variable @samp{x} (the unit part being the sign
+of the leading coefficient, the content part being the GCD of the coefficients,
+and the primitive polynomial being the input polynomial divided by the unit and
+content parts). The product of unit, content, and primitive part is the
+original polynomial.
+
+
+@subsection GCD and LCM
+@cindex GCD
+@cindex LCM
+@cindex @code{gcd()}
+@cindex @code{lcm()}
+
+The functions for polynomial greatest common divisor and least common
+multiple have the synopsis
+
+@example
+ex gcd(const ex & a, const ex & b);
+ex lcm(const ex & a, const ex & b);
+@end example
+
+The functions @code{gcd()} and @code{lcm()} accept two expressions
+@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}.
+
+@example
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x"), y("y"), z("z");
+ ex P_a = 4*x*y + x*z + 20*pow(y, 2) + 21*y*z + 4*pow(z, 2);
+ ex P_b = x*y + 3*x*z + 5*pow(y, 2) + 19*y*z + 12*pow(z, 2);
+
+ ex P_gcd = gcd(P_a, P_b);
+ // x + 5*y + 4*z
+ ex P_lcm = lcm(P_a, P_b);
// 4*x*y^2 + 13*y*x*z + 20*y^3 + 81*y^2*z + 67*y*z^2 + 3*x*z^2 + 12*z^3
- // ...
@}
@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
+
@subsection The @code{normal} method
@cindex @code{normal()}
+@cindex simplification
@cindex temporary replacement
-While in common symbolic code @code{gcd()} and @code{lcm()} are not too
-heavily used, simplification is called for frequently. Therefore
-@code{.normal()}, which provides some basic form of simplification, has
-become a method of class @code{ex}, just like @code{.expand()}. It
-converts a rational function into an equivalent rational function where
-numerator and denominator are coprime. This means, it finds the GCD of
-numerator and denominator and cancels it. If it encounters some object
-which does not belong to the domain of rationals (a function for
-instance), that object is replaced by a temporary symbol. This means
-that both expressions @code{t1} and @code{t2} are indeed simplified in
-this little program:
+Some basic form of simplification of expressions is called for frequently.
+GiNaC provides the method @code{.normal()}, which converts a rational function
+into an equivalent rational function of the form @samp{numerator/denominator}
+where numerator and denominator are coprime. If the input expression is already
+a fraction, it just finds the GCD of numerator and denominator and cancels it,
+otherwise it performs fraction addition and multiplication.
+
+@code{.normal()} can also be used on expressions which are not rational functions
+as it will replace all non-rational objects (like functions or non-integer
+powers) by temporary symbols to bring the expression to the domain of rational
+functions before performing the normalization, and re-substituting these
+symbols afterwards. This algorithm is also available as a separate method
+@code{.to_rational()}, described below.
+
+This means that both expressions @code{t1} and @code{t2} are indeed
+simplified in this little program:
@example
#include <ginac/ginac.h>
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
normalized to @code{P_a/P_b} = @code{(4*y+z)/(y+3*z)}.
-@node Symbolic Differentiation, Series Expansion, Polynomial Arithmetic, Important Algorithms
+@subsection Numerator and denominator
+@cindex numerator
+@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, 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
+@cindex @code{to_rational()}
+
+Some of the methods described so far only work on polynomials or rational
+functions. GiNaC provides a way to extend the domain of these functions to
+general expressions by using the temporary replacement algorithm described
+above. You do this by calling
+
+@example
+ex ex::to_rational(lst &l);
+@end example
+
+on the expression to be converted. The supplied @code{lst} will be filled
+with the generated temporary symbols and their replacement expressions in
+a format that can be used directly for the @code{subs()} method. It can also
+already contain a list of replacements from an earlier application of
+@code{.to_rational()}, so it's possible to use it on multiple expressions
+and get consistent results.
+
+For example,
+
+@example
+@{
+ symbol x("x");
+ ex a = pow(sin(x), 2) - pow(cos(x), 2);
+ ex b = sin(x) + cos(x);
+ ex q;
+ lst l;
+ divide(a.to_rational(l), b.to_rational(l), q);
+ cout << q.subs(l) << endl;
+@}
+@end example
+
+will print @samp{sin(x)-cos(x)}.
+
+
+@node Symbolic Differentiation, Series Expansion, Rational Expressions, Methods and Functions
@c node-name, next, previous, up
-@section Symbolic Differentiation
+@section Symbolic differentiation
@cindex differentiation
@cindex @code{diff()}
@cindex chain rule
cout << P.diff(x,2) << endl; // 20*x^3 + 2
cout << P.diff(y) << endl; // 1
cout << P.diff(z) << endl; // 0
- // ...
@}
@end example
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, Extending GiNaC, Symbolic Differentiation, Important Algorithms
+@node Series Expansion, Symmetrization, Symbolic Differentiation, Methods and Functions
@c node-name, next, previous, up
-@section Series Expansion
+@section Series expansion
@cindex @code{series()}
@cindex Taylor expansion
@cindex Laurent expansion
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
cout << "the inverse square of this series is " << endl
<< pow(mass_nonrel,-2).series(v==0, 10) << endl;
-
- // ...
@}
@end example
the order term off:
@example
-#include <ginac/ginac.h>
-using namespace GiNaC;
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+ex mechain_pi(int degr)
+@{
+ symbol x;
+ ex pi_expansion = series_to_poly(atan(x).series(x,degr));
+ ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
+ -4*pi_expansion.subs(x==numeric(1,239));
+ return pi_approx;
+@}
+
+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);
+ cout << i << ":\t" << pi_frac << endl
+ << "\t" << pi_frac.evalf() << endl;
+ @}
+ return 0;
+@}
+@end example
+
+Note how we just called @code{.series(x,degr)} instead of
+@code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
+method @code{series()}: if the first argument is a symbol the expression
+is expanded in that symbol around point @code{0}. When you run this
+program, it will type out:
+
+@example
+2: 3804/1195
+ 3.1832635983263598326
+4: 5359397032/1706489875
+ 3.1405970293260603143
+6: 38279241713339684/12184551018734375
+ 3.141621029325034425
+8: 76528487109180192540976/24359780855939418203125
+ 3.141591772182177295
+10: 327853873402258685803048818236/104359128170408663038552734375
+ 3.1415926824043995174
+@end example
+
+
+@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
+
+GiNaC contains the following predefined mathematical functions:
+
+@cartouche
+@multitable @columnfractions .30 .70
+@item @strong{Name} @tab @strong{Function}
+@item @code{abs(x)}
+@tab absolute value
+@item @code{csgn(x)}
+@tab complex sign
+@item @code{sqrt(x)}
+@tab square root (not a GiNaC function proper but equivalent to @code{pow(x, numeric(1, 2)})
+@item @code{sin(x)}
+@tab sine
+@item @code{cos(x)}
+@tab cosine
+@item @code{tan(x)}
+@tab tangent
+@item @code{asin(x)}
+@tab inverse sine
+@item @code{acos(x)}
+@tab inverse cosine
+@item @code{atan(x)}
+@tab inverse tangent
+@item @code{atan2(y, x)}
+@tab inverse tangent with two arguments
+@item @code{sinh(x)}
+@tab hyperbolic sine
+@item @code{cosh(x)}
+@tab hyperbolic cosine
+@item @code{tanh(x)}
+@tab hyperbolic tangent
+@item @code{asinh(x)}
+@tab inverse hyperbolic sine
+@item @code{acosh(x)}
+@tab inverse hyperbolic cosine
+@item @code{atanh(x)}
+@tab inverse hyperbolic tangent
+@item @code{exp(x)}
+@tab exponential function
+@item @code{log(x)}
+@tab natural logarithm
+@item @code{Li2(x)}
+@tab Dilogarithm
+@item @code{zeta(x)}
+@tab Riemann's zeta function
+@item @code{zeta(n, x)}
+@tab derivatives of Riemann's zeta function
+@item @code{tgamma(x)}
+@tab Gamma function
+@item @code{lgamma(x)}
+@tab logarithm of Gamma function
+@item @code{beta(x, y)}
+@tab Beta function (@code{tgamma(x)*tgamma(y)/tgamma(x+y)})
+@item @code{psi(x)}
+@tab psi (digamma) function
+@item @code{psi(n, x)}
+@tab derivatives of psi function (polygamma functions)
+@item @code{factorial(n)}
+@tab factorial function
+@item @code{binomial(n, m)}
+@tab binomial coefficients
+@item @code{Order(x)}
+@tab order term function in truncated power series
+@item @code{Derivative(x, l)}
+@tab inert partial differentiation operator (used internally)
+@end multitable
+@end cartouche
+
+@cindex branch cut
+For functions that have a branch cut in the complex plane GiNaC follows
+the conventions for C++ as defined in the ANSI standard as far as
+possible. In particular: the natural logarithm (@code{log}) and the
+square root (@code{sqrt}) both have their branch cuts running along the
+negative real axis where the points on the axis itself belong to the
+upper part (i.e. continuous with quadrant II). The inverse
+trigonometric and hyperbolic functions are not defined for complex
+arguments by the C++ standard, however. In GiNaC we follow the
+conventions used by CLN, which in turn follow the carefully designed
+definitions in the Common Lisp standard. It should be noted that this
+convention is identical to the one used by the C99 standard and by most
+serious CAS. It is to be expected that future revisions of the C++
+standard incorporate these functions in the complex domain in a manner
+compatible with C99.
+
+
+@node Input/Output, Extending GiNaC, Built-in Functions, Methods and Functions
+@c node-name, next, previous, up
+@section Input and output of expressions
+@cindex I/O
+
+@subsection Expression output
+@cindex printing
+@cindex output of expressions
+
+The easiest way to print an expression is to write it to a stream:
+
+@example
+@{
+ symbol x("x");
+ ex e = 4.5+pow(x,2)*3/2;
+ cout << e << endl; // prints '(4.5)+3/2*x^2'
+ // ...
+@end example
+
+The output format is identical to the @command{ginsh} input syntax and
+to that used by most computer algebra systems, but not directly pastable
+into a GiNaC C++ program (note that in the above example, @code{pow(x,2)}
+is printed as @samp{x^2}).
+
+It is possible to print expressions in a number of different formats with
+the method
+
+@example
+void ex::print(const print_context & c, unsigned level = 0);
+@end example
+
+@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
+ // ...
+ 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 = (cln::cl_F("3.0")/cln::cl_F("2.0"))*(x*x)+cln::cl_F("4.5");
+@end example
+
+The @code{print_context} type @code{print_tree} provides a dump of the
+internal structure of an expression for debugging purposes:
+
+@example
+ // ...
+ e.print(print_tree(cout));
+@}
+@end example
+
+produces
+
+@example
+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=0x8000004b, flags=0xf
+ =====
+@end example
+
+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
+@cindex input of expressions
+
+GiNaC provides no way to directly read an expression from a stream because
+you will usually want the user to be able to enter something like @samp{2*x+sin(y)}
+and have the @samp{x} and @samp{y} correspond to the symbols @code{x} and
+@code{y} you defined in your program and there is no way to specify the
+desired symbols to the @code{>>} stream input operator.
+
+Instead, GiNaC lets you construct an expression from a string, specifying the
+list of symbols to be used:
+
+@example
+@{
+ symbol x("x"), y("y");
+ ex e("2*x+sin(y)", lst(x, y));
+@}
+@end example
+
+The input syntax is the same as that used by @command{ginsh} and the stream
+output operator @code{<<}. The symbols in the string are matched by name to
+the symbols in the list and if GiNaC encounters a symbol not specified in
+the list it will throw an exception.
+
+With this constructor, it's also easy to implement interactive GiNaC programs:
+
+@example
+#include <iostream>
+#include <string>
+#include <stdexcept>
+#include <ginac/ginac.h>
+using namespace std;
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x");
+ string s;
+
+ cout << "Enter an expression containing 'x': ";
+ getline(cin, s);
+
+ try @{
+ ex e(s, lst(x));
+ cout << "The derivative of " << e << " with respect to x is ";
+ cout << e.diff(x) << ".\n";
+ @} catch (exception &p) @{
+ cerr << p.what() << endl;
+ @}
+@}
+@end example
+
+
+@subsection Archiving
+@cindex @code{archive} (class)
+@cindex archiving
+
+GiNaC allows creating @dfn{archives} of expressions which can be stored
+to or retrieved from files. To create an archive, you declare an object
+of class @code{archive} and archive expressions in it, giving each
+expression a unique name:
+
+@example
+#include <fstream>
+using namespace std;
+#include <ginac/ginac.h>
+using namespace GiNaC;
+
+int main()
+@{
+ symbol x("x"), y("y"), z("z");
+
+ ex foo = sin(x + 2*y) + 3*z + 41;
+ ex bar = foo + 1;
+
+ archive a;
+ a.archive_ex(foo, "foo");
+ a.archive_ex(bar, "the second one");
+ // ...
+@end example
+
+The archive can then be written to a file:
+
+@example
+ // ...
+ ofstream out("foobar.gar");
+ out << a;
+ out.close();
+ // ...
+@end example
+
+The file @file{foobar.gar} contains all information that is needed to
+reconstruct the expressions @code{foo} and @code{bar}.
+
+@cindex @command{viewgar}
+The tool @command{viewgar} that comes with GiNaC can be used to view
+the contents of GiNaC archive files:
+
+@example
+$ viewgar foobar.gar
+foo = 41+sin(x+2*y)+3*z
+the second one = 42+sin(x+2*y)+3*z
+@end example
+
+The point of writing archive files is of course that they can later be
+read in again:
+
+@example
+ // ...
+ archive a2;
+ ifstream in("foobar.gar");
+ in >> a2;
+ // ...
+@end example
-ex mechain_pi(int degr)
-@{
- symbol x;
- ex pi_expansion = series_to_poly(atan(x).series(x,degr));
- ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
- -4*pi_expansion.subs(x==numeric(1,239));
- return pi_approx;
+And the stored expressions can be retrieved by their name:
+
+@example
+ // ...
+ lst syms(x, y);
+
+ ex ex1 = a2.unarchive_ex(syms, "foo");
+ ex ex2 = a2.unarchive_ex(syms, "the second one");
+
+ cout << ex1 << endl; // prints "41+sin(x+2*y)+3*z"
+ cout << ex2 << endl; // prints "42+sin(x+2*y)+3*z"
+ cout << ex1.subs(x == 2) << endl; // prints "41+sin(2+2*y)+3*z"
@}
+@end example
-int main()
+Note that you have to supply a list of the symbols which are to be inserted
+in the expressions. Symbols in archives are stored by their name only and
+if you don't specify which symbols you have, unarchiving the expression will
+create new symbols with that name. E.g. if you hadn't included @code{x} in
+the @code{syms} list above, the @code{ex1.subs(x == 2)} statement would
+have had no effect because the @code{x} in @code{ex1} would have been a
+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)
@{
- ex pi_frac;
- for (int i=2; i<12; i+=2) @{
- pi_frac = mechain_pi(i);
- cout << i << ":\t" << pi_frac << endl
- << "\t" << pi_frac.evalf() << endl;
+ 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
-Note how we just called @code{.series(x,degr)} instead of
-@code{.series(x==0,degr)}. This is a simple shortcut for @code{ex}'s
-method @code{series()}: if the first argument is a symbol the expression
-is expanded in that symbol around point @code{0}. When you run this
-program, it will type out:
+This will produce:
@example
-2: 3804/1195
- 3.1832635983263598326
-4: 5359397032/1706489875
- 3.1405970293260603143
-6: 38279241713339684/12184551018734375
- 3.141621029325034425
-8: 76528487109180192540976/24359780855939418203125
- 3.141591772182177295
-10: 327853873402258685803048818236/104359128170408663038552734375
- 3.1415926824043995174
+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, Series Expansion, Top
+
+@node Extending GiNaC, What does not belong into GiNaC, Input/Output, Top
@c node-name, next, previous, up
@chapter Extending GiNaC
@menu
* What does not belong into GiNaC:: What to avoid.
* Symbolic functions:: Implementing symbolic functions.
+* Adding classes:: Defining new algebraic classes.
@end menu
provided by @acronym{CLN} are much better suited.
-@node Symbolic functions, A Comparison With Other CAS, What does not belong into GiNaC, Extending GiNaC
+@node Symbolic functions, Adding classes, What does not belong into GiNaC, Extending GiNaC
@c node-name, next, previous, up
@section Symbolic functions
The easiest and most instructive way to start with is probably to
-implement your own function. Objects of class @code{function} are
-inserted into the system via a kind of `registry'. They get a serial
-number that is used internally to identify them but you usually need not
-worry about this. What you have to care for are functions that are
-called when the user invokes certain methods. These are usual
+implement your own function. GiNaC's functions are objects of class
+@code{function}. The preprocessor is then used to convert the function
+names to objects with a corresponding serial number that is used
+internally to identify them. You usually need not worry about this
+number. New functions may be inserted into the system via a kind of
+`registry'. It is your responsibility to care for some functions that
+are called when the user invokes certain methods. These are usual
C++-functions accepting a number of @code{ex} as arguments and returning
one @code{ex}. As an example, if we have a look at a simplified
implementation of the cosine trigonometric function, we first need a
implement does have a pole somewhere in the complex plane, you need to
write another method for Laurent expansion around that point.
-Now that all the ingrediences for @code{cos} have been set up, we need
+Now that all the ingredients for @code{cos} have been set up, we need
to tell the system about it. This is done by a macro and we are not
going to descibe how it expands, please consult your preprocessor if you
are curious:
which is correct if there are no poles involved as is the case for the
@code{cos} function. The way GiNaC handles poles in case there are any
is best understood by studying one of the examples, like the Gamma
-function for instance. (In essence the function first checks if there
-is a pole at the evaluation point and falls back to Taylor expansion if
-there isn't. Then, the pole is regularized by some suitable
-transformation.) Also, the new function needs to be declared somewhere.
-This may also be done by a convenient preprocessor macro:
+(@code{tgamma}) function for instance. (In essence the function first
+checks if there is a pole at the evaluation point and falls back to
+Taylor expansion if there isn't. Then, the pole is regularized by some
+suitable transformation.) Also, the new function needs to be declared
+somewhere. This may also be done by a convenient preprocessor macro:
@example
DECLARE_FUNCTION_1P(cos)
assure you that functions are GiNaC's most macro-intense classes. We
have done our best to avoid macros where we can.)
+
+@node Adding classes, A Comparison With Other CAS, Symbolic functions, Extending GiNaC
+@c node-name, next, previous, up
+@section Adding classes
+
+If you are doing some very specialized things with GiNaC you may find that
+you have to implement your own algebraic classes to fit your needs. This
+section will explain how to do this by giving the example of a simple
+'string' class. After reading this section you will know how to properly
+declare a GiNaC class and what the minimum required member functions are
+that you have to implement. We only cover the implementation of a 'leaf'
+class here (i.e. one that doesn't contain subexpressions). Creating a
+container class like, for example, a class representing tensor products is
+more involved but this section should give you enough information so you can
+consult the source to GiNaC's predefined classes if you want to implement
+something more complicated.
+
+@subsection GiNaC's run-time type information system
+
+@cindex hierarchy of classes
+@cindex RTTI
+All algebraic classes (that is, all classes that can appear in expressions)
+in GiNaC are direct or indirect subclasses of the class @code{basic}. So a
+@code{basic *} (which is essentially what an @code{ex} is) represents a
+generic pointer to an algebraic class. Occasionally it is necessary to find
+out what the class of an object pointed to by a @code{basic *} really is.
+Also, for the unarchiving of expressions it must be possible to find the
+@code{unarchive()} function of a class given the class name (as a string). A
+system that provides this kind of information is called a run-time type
+information (RTTI) system. The C++ language provides such a thing (see the
+standard header file @file{<typeinfo>}) but for efficiency reasons GiNaC
+implements its own, simpler RTTI.
+
+The RTTI in GiNaC is based on two mechanisms:
+
+@itemize @bullet
+
+@item
+The @code{basic} class declares a member variable @code{tinfo_key} which
+holds an unsigned integer that identifies the object's class. These numbers
+are defined in the @file{tinfos.h} header file for the built-in GiNaC
+classes. They all start with @code{TINFO_}.
+
+@item
+By means of some clever tricks with static members, GiNaC maintains a list
+of information for all classes derived from @code{basic}. The information
+available includes the class names, the @code{tinfo_key}s, and pointers
+to the unarchiving functions. This class registry is defined in the
+@file{registrar.h} header file.
+
+@end itemize
+
+The disadvantage of this proprietary RTTI implementation is that there's
+a little more to do when implementing new classes (C++'s RTTI works more
+or less automatic) but don't worry, most of the work is simplified by
+macros.
+
+@subsection A minimalistic example
+
+Now we will start implementing a new class @code{mystring} that allows
+placing character strings in algebraic expressions (this is not very useful,
+but it's just an example). This class will be a direct subclass of
+@code{basic}. You can use this sample implementation as a starting point
+for your own classes.
+
+The code snippets given here assume that you have included some header files
+as follows:
+
+@example
+#include <iostream>
+#include <string>
+#include <stdexcept>
+using namespace std;
+
+#include <ginac/ginac.h>
+using namespace GiNaC;
+@end example
+
+The first thing we have to do is to define a @code{tinfo_key} for our new
+class. This can be any arbitrary unsigned number that is not already taken
+by one of the existing classes but it's better to come up with something
+that is unlikely to clash with keys that might be added in the future. The
+numbers in @file{tinfos.h} are modeled somewhat after the class hierarchy
+which is not a requirement but we are going to stick with this scheme:
+
+@example
+const unsigned TINFO_mystring = 0x42420001U;
+@end example
+
+Now we can write down the class declaration. The class stores a C++
+@code{string} and the user shall be able to construct a @code{mystring}
+object from a C or C++ string:
+
+@example
+class mystring : public basic
+@{
+ GINAC_DECLARE_REGISTERED_CLASS(mystring, basic)
+
+public:
+ mystring(const string &s);
+ mystring(const char *s);
+
+private:
+ string str;
+@};
+
+GIANC_IMPLEMENT_REGISTERED_CLASS(mystring, basic)
+@end example
+
+The @code{GINAC_DECLARE_REGISTERED_CLASS} and @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
+macros are defined in @file{registrar.h}. They take the name of the class
+and its direct superclass as arguments and insert all required declarations
+for the RTTI system. The @code{GINAC_DECLARE_REGISTERED_CLASS} should be
+the first line after the opening brace of the class definition. The
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in the
+source (at global scope, of course, not inside a function).
+
+@code{GINAC_DECLARE_REGISTERED_CLASS} contains, among other things the
+declarations of the default and copy constructor, the destructor, the
+assignment operator and a couple of other functions that are required. It
+also defines a type @code{inherited} which refers to the superclass so you
+don't have to modify your code every time you shuffle around the class
+hierarchy. @code{GINAC_IMPLEMENT_REGISTERED_CLASS} implements the copy
+constructor, the destructor and the assignment operator.
+
+Now there are nine member functions we have to implement to get a working
+class:
+
+@itemize
+
+@item
+@code{mystring()}, the default constructor.
+
+@item
+@code{void destroy(bool call_parent)}, which is used in the destructor and the
+assignment operator to free dynamically allocated members. The @code{call_parent}
+specifies whether the @code{destroy()} function of the superclass is to be
+called also.
+
+@item
+@code{void copy(const mystring &other)}, which is used in the copy constructor
+and assignment operator to copy the member variables over from another
+object of the same class.
+
+@item
+@code{void archive(archive_node &n)}, the archiving function. This stores all
+information needed to reconstruct an object of this class inside an
+@code{archive_node}.
+
+@item
+@code{mystring(const archive_node &n, const lst &sym_lst)}, the unarchiving
+constructor. This constructs an instance of the class from the information
+found in an @code{archive_node}.
+
+@item
+@code{ex unarchive(const archive_node &n, const lst &sym_lst)}, the static
+unarchiving function. It constructs a new instance by calling the unarchiving
+constructor.
+
+@item
+@code{int compare_same_type(const basic &other)}, which is used internally
+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
+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
+must provide a @code{compare_same_type()} function, even those representing
+objects for which no reasonable algebraic ordering relationship can be
+defined.
+
+@item
+And, of course, @code{mystring(const string &s)} and @code{mystring(const char *s)}
+which are the two constructors we declared.
+
+@end itemize
+
+Let's proceed step-by-step. The default constructor looks like this:
+
+@example
+mystring::mystring() : inherited(TINFO_mystring)
+@{
+ // dynamically allocate resources here if required
+@}
+@end example
+
+The golden rule is that in all constructors you have to set the
+@code{tinfo_key} member to the @code{TINFO_*} value of your class. Otherwise
+it will be set by the constructor of the superclass and all hell will break
+loose in the RTTI. For your convenience, the @code{basic} class provides
+a constructor that takes a @code{tinfo_key} value, which we are using here
+(remember that in our case @code{inherited = basic}). If the superclass
+didn't have such a constructor, we would have to set the @code{tinfo_key}
+to the right value manually.
+
+In the default constructor you should set all other member variables to
+reasonable default values (we don't need that here since our @code{str}
+member gets set to an empty string automatically). The constructor(s) are of
+course also the right place to allocate any dynamic resources you require.
+
+Next, the @code{destroy()} function:
+
+@example
+void mystring::destroy(bool call_parent)
+@{
+ // free dynamically allocated resources here if required
+ if (call_parent)
+ inherited::destroy(call_parent);
+@}
+@end example
+
+This function is where we free all dynamically allocated resources. We don't
+have any so we're not doing anything here, but if we had, for example, used
+a C-style @code{char *} to store our string, this would be the place to
+@code{delete[]} the string storage. If @code{call_parent} is true, we have
+to call the @code{destroy()} function of the superclass after we're done
+(to mimic C++'s automatic invocation of superclass destructors where
+@code{destroy()} is called from outside a destructor).
+
+The @code{copy()} function just copies over the member variables from
+another object:
+
+@example
+void mystring::copy(const mystring &other)
+@{
+ inherited::copy(other);
+ str = other.str;
+@}
+@end example
+
+We can simply overwrite the member variables here. There's no need to worry
+about dynamically allocated storage. The assignment operator (which is
+automatically defined by @code{GINAC_IMPLEMENT_REGISTERED_CLASS}, as you
+recall) calls @code{destroy()} before it calls @code{copy()}. You have to
+explicitly call the @code{copy()} function of the superclass here so
+all the member variables will get copied.
+
+Next are the three functions for archiving. You have to implement them even
+if you don't plan to use archives, but the minimum required implementation
+is really simple. First, the archiving function:
+
+@example
+void mystring::archive(archive_node &n) const
+@{
+ inherited::archive(n);
+ n.add_string("string", str);
+@}
+@end example
+
+The only thing that is really required is calling the @code{archive()}
+function of the superclass. Optionally, you can store all information you
+deem necessary for representing the object into the passed
+@code{archive_node}. We are just storing our string here. For more
+information on how the archiving works, consult the @file{archive.h} header
+file.
+
+The unarchiving constructor is basically the inverse of the archiving
+function:
+
+@example
+mystring::mystring(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+@{
+ n.find_string("string", str);
+@}
+@end example
+
+If you don't need archiving, just leave this function empty (but you must
+invoke the unarchiving constructor of the superclass). Note that we don't
+have to set the @code{tinfo_key} here because it is done automatically
+by the unarchiving constructor of the @code{basic} class.
+
+Finally, the unarchiving function:
+
+@example
+ex mystring::unarchive(const archive_node &n, const lst &sym_lst)
+@{
+ return (new mystring(n, sym_lst))->setflag(status_flags::dynallocated);
+@}
+@end example
+
+You don't have to understand how exactly this works. Just copy these four
+lines into your code literally (replacing the class name, of course). It
+calls the unarchiving constructor of the class and unless you are doing
+something very special (like matching @code{archive_node}s to global
+objects) you don't need a different implementation. For those who are
+interested: setting the @code{dynallocated} flag puts the object under
+the control of GiNaC's garbage collection. It will get deleted automatically
+once it is no longer referenced.
+
+Our @code{compare_same_type()} function uses a provided function to compare
+the string members:
+
+@example
+int mystring::compare_same_type(const basic &other) const
+@{
+ const mystring &o = static_cast<const mystring &>(other);
+ int cmpval = str.compare(o.str);
+ if (cmpval == 0)
+ return 0;
+ else if (cmpval < 0)
+ return -1;
+ else
+ return 1;
+@}
+@end example
+
+Although this function takes a @code{basic &}, it will always be a reference
+to an object of exactly the same class (objects of different classes are not
+comparable), so the cast is safe. If this function returns 0, the two objects
+are considered equal (in the sense that @math{A-B=0}), so you should compare
+all relevant member variables.
+
+Now the only thing missing is our two new constructors:
+
+@example
+mystring::mystring(const string &s) : inherited(TINFO_mystring), str(s)
+@{
+ // dynamically allocate resources here if required
+@}
+
+mystring::mystring(const char *s) : inherited(TINFO_mystring), str(s)
+@{
+ // dynamically allocate resources here if required
+@}
+@end example
+
+No surprises here. We set the @code{str} member from the argument and
+remember to pass the right @code{tinfo_key} to the @code{basic} constructor.
+
+That's it! We now have a minimal working GiNaC class that can store
+strings in algebraic expressions. Let's confirm that the RTTI works:
+
+@example
+ex e = mystring("Hello, world!");
+cout << is_ex_of_type(e, mystring) << endl;
+ // -> 1 (true)
+
+cout << e.bp->class_name() << endl;
+ // -> mystring
+@end example
+
+Obviously it does. Let's see what the expression @code{e} looks like:
+
+@example
+cout << e << endl;
+ // -> [mystring object]
+@end example
+
+Hm, not exactly what we expect, but of course the @code{mystring} class
+doesn't yet know how to print itself. This is done in the @code{print()}
+member function. Let's say that we wanted to print the string surrounded
+by double quotes:
+
+@example
+class mystring : public basic
+@{
+ ...
+public:
+ void print(const print_context &c, unsigned level = 0) const;
+ ...
+@};
+
+void mystring::print(const print_context &c, unsigned level) const
+@{
+ // print_context::s is a reference to an ostream
+ c.s << '\"' << str << '\"';
+@}
+@end example
+
+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;
+ // -> "Hello, world!"
+@end example
+
+Much better. The @code{mystring} class can be used in arbitrary expressions:
+
+@example
+e += mystring("GiNaC rulez");
+cout << e << endl;
+ // -> "GiNaC rulez"+"Hello, world!"
+@end example
+
+(note that GiNaC's automatic term reordering is in effect here), or even
+
+@example
+e = pow(mystring("One string"), 2*sin(Pi-mystring("Another string")));
+cout << e << endl;
+ // -> "One string"^(2*sin(-"Another string"+Pi))
+@end example
+
+Whether this makes sense is debatable but remember that this is only an
+example. At least it allows you to implement your own symbolic algorithms
+for your objects.
+
+Note that GiNaC's algebraic rules remain unchanged:
+
+@example
+e = mystring("Wow") * mystring("Wow");
+cout << e << endl;
+ // -> "Wow"^2
+
+e = pow(mystring("First")-mystring("Second"), 2);
+cout << e.expand() << endl;
+ // -> -2*"First"*"Second"+"First"^2+"Second"^2
+@end example
+
+There's no way to, for example, make GiNaC's @code{add} class perform string
+concatenation. You would have to implement this yourself.
+
+@subsection Automatic evaluation
+
+@cindex @code{hold()}
+@cindex evaluation
+When dealing with objects that are just a little more complicated than the
+simple string objects we have implemented, chances are that you will want to
+have some automatic simplifications or canonicalizations performed on them.
+This is done in the evaluation member function @code{eval()}. Let's say that
+we wanted all strings automatically converted to lowercase with
+non-alphabetic characters stripped, and empty strings removed:
+
+@example
+class mystring : public basic
+@{
+ ...
+public:
+ ex eval(int level = 0) const;
+ ...
+@};
+
+ex mystring::eval(int level) const
+@{
+ string new_str;
+ for (int i=0; i<str.length(); i++) @{
+ char c = str[i];
+ if (c >= 'A' && c <= 'Z')
+ new_str += tolower(c);
+ else if (c >= 'a' && c <= 'z')
+ new_str += c;
+ @}
+
+ if (new_str.length() == 0)
+ return 0;
+ else
+ return mystring(new_str).hold();
+@}
+@end example
+
+The @code{level} argument is used to limit the recursion depth of the
+evaluation. We don't have any subexpressions in the @code{mystring} class
+so we are not concerned with this. If we had, we would call the @code{eval()}
+functions of the subexpressions with @code{level - 1} as the argument if
+@code{level != 1}. The @code{hold()} member function sets a flag in the
+object that prevents further evaluation. Otherwise we might end up in an
+endless loop. When you want to return the object unmodified, use
+@code{return this->hold();}.
+
+Let's confirm that it works:
+
+@example
+ex e = mystring("Hello, world!") + mystring("!?#");
+cout << e << endl;
+ // -> "helloworld"
+
+e = mystring("Wow!") + mystring("WOW") + mystring(" W ** o ** W");
+cout << e << endl;
+ // -> 3*"wow"
+@end example
+
+@subsection Other member functions
+
+We have implemented only a small set of member functions to make the class
+work in the GiNaC framework. For a real algebraic class, there are probably
+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. 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)
+@{
+ return static_cast<const mystring &>(*e.bp);
+@}
+@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's it. May the source be with you!
-@node A Comparison With Other CAS, Advantages, Symbolic functions, Top
+@node A Comparison With Other CAS, Advantages, Adding classes, Top
@c node-name, next, previous, up
@chapter A Comparison With Other CAS
@cindex advocacy
@item
development tools: powerful development tools exist for C++, like fancy
editors (e.g. with automatic indentation and syntax highlighting),
-debuggers, visualization tools, documentation tools...
+debuggers, visualization tools, documentation generators...
@item
modularization: C++ programs can easily be split into modules by
@item
efficiency: often large parts of a program do not need symbolic
calculations at all. Why use large integers for loop variables or
-arbitrary precision arithmetics where double accuracy is sufficient?
-For pure symbolic applications, GiNaC is comparable in speed with other
-CAS.
+arbitrary precision arithmetics where @code{int} and @code{double} are
+sufficient? For pure symbolic applications, GiNaC is comparable in
+speed with other CAS.
@end itemize
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
cout << e2 << endl; // prints sin(x+2*y)+3*z+41
e2 += 1; // e2 is copied into a new object
cout << e2 << endl; // prints sin(x+2*y)+3*z+42
- // ...
@}
@end example
@example
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
cout << e1 << endl // prints x+3*y
<< e2 << endl // prints (x+3*y)^3
<< e3 << endl; // prints 3*(x+3*y)^2*cos((x+3*y)^3)
- // ...
@}
@end example
@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
AC_PROG_INSTALL
AC_LANG_CPLUSPLUS
-AM_PATH_GINAC(0.4.0, [
+AM_PATH_GINAC(0.7.0, [
LIBS="$LIBS $GINACLIB_LIBS"
- CPPFLAGS="$CFLAGS $GINACLIB_CPPFLAGS"
+ CPPFLAGS="$CPPFLAGS $GINACLIB_CPPFLAGS"
], AC_MSG_ERROR([need to have GiNaC installed]))
AC_OUTPUT(Makefile)
The only command in this which is not standard for automake
is the @samp{AM_PATH_GINAC} macro.
-That command does the following:
-
-@display
-If a GiNaC version greater than 0.4.0 is found, adds @env{$GINACLIB_LIBS} to
-@env{$LIBS} and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, dies
-with the error message `need to have GiNaC installed'
-@end display
+That command does the following: If a GiNaC version greater or equal
+than 0.7.0 is found, then it adds @env{$GINACLIB_LIBS} to @env{$LIBS}
+and @env{$GINACLIB_CPPFLAGS} to @env{$CPPFLAGS}. Otherwise, it dies with
+the error message `need to have GiNaC installed'
And the @file{Makefile.am}, which will be used to build the Makefile.
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
@printindex cp
@bye
-
git://www.ginac.de / ginac.git / blobdiff