index d8970b5bc3ed41b11d49801227f159720a3a3254..ce36a1b5e93a630be2657e294b13831cbaf8a505 100644 (file)
@@ -23,7 +23,7 @@
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
@@ -52,7 +52,7 @@ notice identical to this one.

@page
@vskip 0pt plus 1filll
@sp 2
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
@@ -82,7 +82,7 @@ framework for symbolic computation within the C++ programming language.
* 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.
@@ -135,7 +135,7 @@ the near future.

The GiNaC framework for symbolic computation within the C++ programming
University Mainz, Germany.

This program is free software; you can redistribute it and/or
@@ -388,15 +388,15 @@ tan(x)^2+1
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
@@ -658,9 +658,10 @@ meta-class for storing all mathematical objects.
* 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.

@@ -684,8 +685,8 @@ ex MyEx4 = sin(x + 2*y) + 3*z + 41; // containing a function
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
@@ -707,36 +708,26 @@ mathematical objects, all of which (except for @code{ex} and some
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
@@ -749,8 +740,8 @@ $\pi$
@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{power} @tab Exponentials such as @math{x^2}, @math{a^b},
@tex
$\sqrt{2}$
@@ -759,15 +750,15 @@ $\sqrt{2}$
@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{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
@end multitable
@end cartouche

@@ -807,7 +798,8 @@ for instance) will always reveal their difference.  Watch out, please.
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 Symbols},
+for more information).

@node Numbers, Constants, Symbols, Basic Concepts
@@ -858,7 +850,6 @@ int main()
numeric trott("1.0841015122311136151E-2");

cout << two*p << endl;  // floating point 6.283...
-    // ...
@}
@end example

@@ -962,7 +953,6 @@ int main()
cout << answer.is_integer() << endl;  // false, it's 21/5
cout << answer.is_integer() << endl;  // true, it's 42 now!
-    // ...
@}
@end example

@@ -1003,7 +993,7 @@ following table.
@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
@@ -1017,7 +1007,7 @@ following table.

@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.
@@ -1033,14 +1023,14 @@ The predefined known constants are:
@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
@@ -1125,17 +1115,66 @@ expression twice, either implicitly or explicitly, results in the same
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
+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:

@@ -1150,12 +1189,11 @@ int main()
symbol x("x"), y("y");

ex foo = x+y/2;
-    cout << "Gamma(" << foo << ") -> " << Gamma(foo) << endl;
+    cout << "tgamma(" << foo << ") -> " << tgamma(foo) << endl;
ex bar = foo.subs(y==1);
-    cout << "Gamma(" << bar << ") -> " << Gamma(bar) << endl;
+    cout << "tgamma(" << bar << ") -> " << tgamma(bar) << endl;
ex foobar = bar.subs(x==7);
-    cout << "Gamma(" << foobar << ") -> " << Gamma(foobar) << endl;
-    // ...
+    cout << "tgamma(" << foobar << ") -> " << tgamma(foobar) << endl;
@}
@end example

@@ -1163,24 +1201,17 @@ 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)
+tgamma(x+(1/2)*y) -> tgamma(x+(1/2)*y)
+tgamma(x+1/2) -> tgamma(x+1/2)
+tgamma(15/2) -> (135135/128)*Pi^(1/2)
@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)
a relation between them that signals equality, inequality and so on.
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, Methods and Functions, 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.

-@node Archiving, Important Algorithms, Relations, Basic Concepts
-@c    node-name, next, previous, up
-@section Archiving Expressions
-@cindex I/O
-@cindex @code{archive} (class)
+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.

-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:
+@cindex @code{idx} (class)
+@cindex @code{indexed} (class)
+@subsection Indexed quantities and their indices
+
+Indexed expressions in GiNaC are constructed of two special types of objects,
+@dfn{index objects} and @dfn{indexed objects}.
+
+@itemize @bullet
+
+@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).
+
+@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}. 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 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 + +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 + +@example +ex varidx::toggle_variance(void); +@end 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. + +@subsection Substituting indices + +@cindex @code{subs()} +Sometimes you will want to substitute one symbolic index with another +symbolic or numeric index, for example when calculating one specific element +of a tensor expression. This is done with the @code{.subs()} method, as it +is done for symbols (see @ref{Substituting Symbols}). + +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 + +The third example shows that trying to replace an index with something that +is not an index will substitute the index value instead. + +Alternatively, you can substitute the @emph{symbol} of a symbolic index by +another expression: + +@example + ... + 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 + +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: + +@example + ... + ex e = indexed(A, mu_co); + cout << e << " becomes " << e.subs(A == A+B) << endl; + // -> A.mu becomes (B+A).mu + ... +@end example + +@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 + ... + 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 + +@cindex @code{get_free_indices()} +@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. + +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 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 -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. +@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 +ex ex::simplify_indexed(void); +ex ex::simplify_indexed(const scalar_products & sp); +@end example + +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 (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()}): + +@example +@{ + symbol A("A"), B("B"), C("C"), i_sym("i"); + idx i(i_sym, 3); + + 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) + + e = indexed(A + B, i) * indexed(A + C, i); + cout << e << endl; + // -> (B+A).i*(A+C).i + + cout << e.expand(expand_options::expand_indexed).simplify_indexed(sp) + << endl; + // -> 4+C.i*B.i +@} +@end example + +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 +doesn't matter; @samp{A~mu~nu*B.mu.nu} would also be replaced by 0. + +@cindex @code{expand()} +The example above also illustrates a feature of the @code{expand()} method: +if passed the @code{expand_indexed} option it will distribute indices +over sums, so @samp{(A+B).i} becomes @samp{A.i+B.i}. + +@cindex @code{tensor} (class) +@subsection Predefined tensors + +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 + +@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 (sums 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), X(2, 1); + A.set(0, 0, 1); A.set(0, 1, 2); + A.set(1, 0, 3); A.set(1, 1, 4); + X.set(0, 0, x); X.set(1, 0, 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); + cout << e.simplify_indexed() << endl; + // -> [[ [[3*y+2*x,5*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 Important Algorithms, Polynomial Expansion, Archiving, Top + +@node Methods and Functions, Information About Expressions, Indexed objects, Top @c node-name, next, previous, up -@chapter Important Algorithms +@chapter Methods and Functions @cindex polynomial In this chapter the most important algorithms provided by GiNaC will be @@ -1313,7 +1788,6 @@ int main() cout << "As method: " << sin(x).evalf() << endl; cout << "As function: " << evalf(sin(x)) << endl; - // ... @} @end example method on class @code{ex} and sometimes calling a function cannot be avoided. @menu -* Polynomial Expansion:: -* Collecting expressions:: -* Polynomial Arithmetic:: +* Information About Expressions:: +* Substituting Symbols:: +* Polynomial Arithmetic:: Working with polynomials. +* Rational Expressions:: Working with rational functions. * Symbolic Differentiation:: -* Series Expansion:: +* Series Expansion:: Taylor and Laurent expansion. +* Built-in Functions:: List of predefined mathematical functions. +* Input/Output:: Input and output of expressions. @end menu -@node Polynomial Expansion, Collecting expressions, Important Algorithms, Important Algorithms +@node Information About Expressions, Substituting Symbols, Methods and Functions, Methods and Functions @c node-name, next, previous, up -@section Polynomial Expansion -@cindex @code{expand()} - -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 @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! +@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()} -@node Collecting expressions, Polynomial Arithmetic, Polynomial Expansion, Important Algorithms -@c node-name, next, previous, up -@section Collecting expressions -@cindex @code{collect()} -@cindex @code{coeff()} - -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: +Sometimes it's useful to check whether a given expression is a plain number, +a sum, a polynomial with integer coefficients, or of some other specific type. +GiNaC provides two functions for this (the first one is actually a macro): @example -ex ex::collect(const symbol & s); +bool is_ex_of_type(const ex & e, TYPENAME t); +bool ex::info(unsigned flag); @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 +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}: -@cindex @code{degree()} -@cindex @code{ldegree()} @example -int ex::degree(const symbol & s); -int ex::ldegree(const symbol & s); +@{ + @dots{} + if (is_ex_of_type(e, numeric)) + numeric n = ex_to_numeric(e); + @dots{} +@} @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: +@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 -#include <ginac/ginac.h> -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; - @} + 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 + + +@subsection Accessing subexpressions +@cindex @code{nops()} +@cindex @code{op()} +@cindex @code{has()} +@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 + +Finally, the method + +@example +bool ex::has(const ex & other); +@end example + +checks whether an expression contains the given subexpression @code{other}. +This only works reliably if @code{other} is of an atomic class such as a +@code{numeric} or a @code{symbol}. It is, e.g., not possible to verify that +@code{a+b+c} contains @code{a+c} (or @code{a+b}) as a subexpression. + + +@subsection Comparing expressions +@cindex @code{is_equal()} +@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 Symbols, Polynomial Arithmetic, Information About Expressions, Methods and Functions +@c node-name, next, previous, up +@section Substituting symbols +@cindex @code{subs()} + +Symbols can be replaced with 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{symbol == expression} or a @code{lst} of such relationals. E.g. + +@example +@{ + symbol x("x"), y("y"); + ex e1 = 2*x^2-4*x+3; + cout << "e1(7) = " << e1.subs(x == 7) << endl; + ex e2 = x*y + x; + cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl; +@} +@end example + +will print @samp{73} and @samp{-10}, respectively. + +If you specify multiple substitutions, they are performed in parallel, so e.g. +@code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}. + +The second form of @code{subs()} takes two lists, one for the symbols and +one for the expressions to be substituted (both lists must contain the same +number of elements). Using this form, you would write @code{subs(lst(x, y), lst(y, x))} +to exchange @samp{x} and @samp{y}. + + +@node Polynomial Arithmetic, Rational Expressions, Substituting Symbols, 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 symbol & s); +@end example + +Note that the original polynomial needs to be in expanded form in order +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 symbol & s); +int ex::ldegree(const symbol & 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 symbol & s, int n); +@end example + +You can also obtain the leading and trailing coefficients with the methods + +@example +ex ex::lcoeff(const symbol & s); +ex ex::tcoeff(const symbol & s); +@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 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 @@ -1438,16 +2173,75 @@ or even from run to run since the internal canonical ordering is not within the user's sphere of influence. -@node Polynomial Arithmetic, Symbolic Differentiation, Collecting expressions, Important Algorithms -@c node-name, next, previous, up -@section Polynomial Arithmetic +@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: +multiple have the synopsis @example ex gcd(const ex & a, const ex & b); @@ -1474,25 +2268,35 @@ int main() // 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 + +@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> @@ -1505,7 +2309,6 @@ int main() 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; - // ... @} @end example @@ -1514,9 +2317,62 @@ the sample-polynomials from the section about GCD and LCM above would be 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()} + +The numerator and denominator of an expression can be obtained with + +@example +ex ex::numer(); +ex ex::denom(); +@end example + +These functions will first normalize the expression as described above and +then return the numerator or denominator, respectively. + + +@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 @@ -1538,7 +2394,6 @@ int main() cout << P.diff(x,2) << endl; // 20*x^3 + 2 cout << P.diff(y) << endl; // 1 cout << P.diff(z) << endl; // 0 - // ... @} @end example @@ -1582,9 +2437,9 @@ When you run it, it produces the sequence @code{1}, @code{-1}, @code{5}, @code{i} by two since all odd Euler numbers vanish anyways. -@node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms +@node Series Expansion, Built-in Functions, 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 @@ -1616,8 +2471,6 @@ int main() cout << "the inverse square of this series is " << endl << pow(mass_nonrel,-2).series(v==0, 10) << endl; - - // ... @} @end example solution is to use the function @code{series_to_poly()} to simply strip 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() +@{ + 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 Built-in Functions, Input/Output, Series Expansion, 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}). + +To print an expression in a way that can be directly used in a C or C++ +program, you use the method + +@example +void ex::printcsrc(ostream & os, unsigned type, const char *name); +@end example + +This outputs a line in the form of a variable definition @code{<type> <name> = <expression>}. +The possible types are defined in @file{ginac/flags.h} (@code{csrc_types}) +and mostly affect the way in which floating point numbers are written: + +@example + // ... + e.printcsrc(cout, csrc_types::ctype_float, "f"); + e.printcsrc(cout, csrc_types::ctype_double, "d"); + e.printcsrc(cout, csrc_types::ctype_cl_N, "n"); + // ... +@end example + +The above example will produce (note the @code{x^2} being converted to @code{x*x}): + +@example +float f = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00; +double d = (3.000000e+00/2.000000e+00)*(x*x)+4.500000e+00; +cl_N n = (cl_F("3.0")/cl_F("2.0"))*(x*x)+cl_F("4.5"); +@end example + +Finally, there are the two methods @code{printraw()} and @code{printtree()} intended for GiNaC +developers, that provide a dump of the internal structure of an expression for +debugging purposes: + +@example + // ... + e.printraw(cout); cout << endl << endl; + e.printtree(cout); +@} +@end example + +produces + +@example +ex(+((power(ex(symbol(name=x,serial=1,hash=150875740,flags=11)),ex(numeric(2)),hash=2,flags=3),numeric(3/2)),,hash=0,flags=3)) + +type=Q25GiNaC3add, hash=0 (0x0), flags=3, nops=2 + power: hash=2 (0x2), flags=3 + x (symbol): serial=1, hash=150875740 (0x8fe2e5c), flags=11 + 2 (numeric): hash=2147483714 (0x80000042), flags=11 + 3/2 (numeric): hash=2147483745 (0x80000061), flags=11 + ----- + overall_coeff + 4.5L0 (numeric): hash=2147483723 (0x8000004b), flags=11 + ===== +@end example + +The @code{printtree()} method is also available in @command{ginsh} as the +@code{print()} function. + + +@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 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 <ginac/ginac.h> +#include <fstream> +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

-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;
-@}
+The point of writing archive files is of course that they can later be

-int main()
-@{
-    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;
-@}
+@example
+    // ...
+    archive a2;
+    ifstream in("foobar.gar");
+    in >> a2;
+    // ...
@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:
+And the stored expressions can be retrieved by their name:

@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
+    // ...
+    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

+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.
+
+

-@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

@@ -1705,6 +2868,7 @@ authors---they will happily incorporate them into future versions.
* What does not belong into GiNaC::  What to avoid.
* Symbolic functions::               Implementing symbolic functions.
+* Adding classes::                   Defining new algebraic classes.

@@ -1732,16 +2896,18 @@ inefficient.  For this purpose, the underlying foundation classes
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
@@ -1796,7 +2962,7 @@ enough to know how to differentiate.  But if the function you want to
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:
@@ -1817,11 +2983,11 @@ expansion method is given, GiNaC defaults to simple Taylor expansion,
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)
mechanisms.  Please, have a look at the real implementation in GiNaC.
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
+
+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
+
+@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);
+@}
+@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(ostream &os, unsigned upper_precedence) const;
+    ...
+@};
+
+void mystring::print(ostream &os, unsigned upper_precedence) const
+@{
+    os << '\"' << str << '\"';
+@}
+@end example
+
+The @code{upper_precedence} argument is only required for container classes
+to correctly parenthesize the output. Let's try again to print the expression:
+
+@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 _ex0();
+    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. 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
@c    node-name, next, previous, up
@chapter A Comparison With Other CAS
@@ -2019,7 +3677,6 @@ 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

@@ -2051,7 +3708,6 @@ 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

@@ -2359,9 +4015,9 @@ AC_PROG_CXX
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)
@@ -2370,13 +4026,10 @@ 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.

@@ -2457,4 +4110,3 @@ Academic Press, London
@printindex cp

@bye
-