#include <ginac/ginac.h>
using namespace GiNaC;
-ex HermitePoly(symbol x, int deg)
+ex HermitePoly(const symbol & x, int n)
@{
- ex HKer=exp(-pow(x,2));
- // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
- return normal(pow(-1,deg) * diff(HKer, x, deg) / HKer);
+ ex HKer=exp(-pow(x, 2));
+ // uses the identity H_n(x) == (-1)^n exp(x^2) (d/dx)^n exp(-x^2)
+ return normal(pow(-1, n) * diff(HKer, x, n) / HKer);
@}
int main()
@c node-name, next, previous, up
@section Prerequisites
-In order to install GiNaC on your system, some prerequisites need
-to be met. First of all, you need to have a C++-compiler adhering to
-the ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
+In order to install GiNaC on your system, some prerequisites need to be
+met. First of all, you need to have a C++-compiler adhering to the
+ANSI-standard @cite{ISO/IEC 14882:1998(E)}. We used @acronym{GCC} for
development so if you have a different compiler you are on your own.
For the configuration to succeed you need a Posix compliant shell
installed in @file{/bin/sh}, GNU @command{bash} is fine. Perl is needed
-by the built process as well, since some of the source files are automatically
-generated by Perl scripts. Last but not least, Bruno Haible's library
-@acronym{CLN} is extensively used and needs to be installed on your system.
-Please get it from @uref{ftp://ftp.santafe.edu/pub/gnu/} or from
-@uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP site}
-(it is covered by GPL) and install it prior to trying to install GiNaC.
-The configure script checks if it can find it and if it cannot
+by the built process as well, since some of the source files are
+automatically generated by Perl scripts. Last but not least, Bruno
+Haible's library @acronym{CLN} is extensively used and needs to be
+installed on your system. Please get it either from
+@uref{ftp://ftp.santafe.edu/pub/gnu/}, from
+@uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/, GiNaC's FTP site} or
+from @uref{ftp://ftp.ilog.fr/pub/Users/haible/gnu/, Bruno Haible's FTP
+site} (it is covered by GPL) and install it prior to trying to install
+GiNaC. The configure script checks if it can find it and if it cannot
it will refuse to continue.
@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:
+
+@cartouche
+@multitable @columnfractions .22 .78
+@item @code{symbol} @tab Algebraic symbols @math{a}, @math{x}, @math{y}@dots{}
+@item @code{constant} @tab Constants like
+@tex
+$\pi$
+@end tex
+@ifnottex
+@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{power} @tab Exponentials such as @math{x^2}, @math{a^b},
+@tex
+$\sqrt{2}$
+@end tex
+@ifnottex
+@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{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
+@end multitable
+@end cartouche
@node Symbols, Numbers, The Class Hierarchy, Basic Concepts
@c node-name, next, previous, up
@section Symbols
-@cindex Symbols (class @code{symbol})
+@cindex @code{symbol} (class)
@cindex hierarchy of classes
@cindex atom
@node Numbers, Constants, Symbols, Basic Concepts
@c node-name, next, previous, up
@section Numbers
-@cindex numbers (class @code{numeric})
+@cindex @code{numeric} (class)
@cindex GMP
@cindex CLN
@node Constants, Fundamental containers, Numbers, Basic Concepts
@c node-name, next, previous, up
@section Constants
-@cindex constants (class @code{constant})
+@cindex @code{constant} (class)
@cindex @code{Pi}
@cindex @code{Catalan}
@node Built-in functions, Relations, Fundamental containers, Basic Concepts
@c node-name, next, previous, up
@section Built-in functions
-@cindex functions (class @code{function})
+@cindex @code{function} (class)
@cindex trigonometric function
@cindex hyperbolic function
@node Relations, Archiving, Built-in functions, Basic Concepts
@c node-name, next, previous, up
@section Relations
-@cindex relations (class @code{relational})
+@cindex @code{relational} (class)
Sometimes, a relation holding between two expressions must be stored
somehow. The class @code{relational} is a convenient container for such
@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.
@node Archiving, Important Algorithms, Relations, Basic Concepts
@c node-name, next, previous, up
@section Archiving Expressions
-@cindex archives (class @code{archive})
+@cindex I/O
+@cindex @code{archive} (class)
GiNaC allows creating @dfn{archives} of expressions which can be stored
to or retrieved from files. To create an archive, you declare an object
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:
@code{A:=x^2+3; coeff(A,x,0);} (GiNaC: @code{A=pow(x,2)+3;
coeff(A,x,0);}) it is clear that MapleV is not trying to be consistent
here. Also, users of MuPAD will in most cases feel more comfortable
-with GiNaC's convention. All function wrappers are always implemented
+with GiNaC's convention. All function wrappers are implemented
as simple inline functions which just call the corresponding method and
are only provided for users uncomfortable with OO who are dead set to
avoid method invocations. Generally, nested function wrappers are much
@code{collect()} accomplishes this task. Here is its declaration:
@example
-ex ex::collect(symbol const & s);
+ex ex::collect(const symbol & s);
@end example
Note that the original polynomial needs to be in expanded form in order
@cindex @code{degree()}
@cindex @code{ldegree()}
@example
-int ex::degree(symbol const & s);
-int ex::ldegree(symbol const & s);
+int ex::degree(const symbol & s);
+int ex::ldegree(const symbol & s);
@end example
where @code{degree()} returns the highest coefficient and
ex EulerNumber(unsigned n)
@{
symbol x;
- ex generator = pow(cosh(x),-1);
+ const ex generator = pow(cosh(x),-1);
return generator.diff(x,n).subs(x==0);
@}
@node Series Expansion, Extending GiNaC, Symbolic Differentiation, Important Algorithms
@c node-name, next, previous, up
@section Series Expansion
-@cindex series expansion
+@cindex @code{series()}
@cindex Taylor expansion
@cindex Laurent expansion
+@cindex @code{pseries} (class)
Expressions know how to expand themselves as a Taylor series or (more
generally) a Laurent series. As in most conventional Computer Algebra
Systems, no distinction is made between those two. There is a class of
-its own for storing such series as well as a class for storing the order
-of the series. As a consequence, if you want to work with series,
-i.e. multiply two series, you need to call the method @code{ex::series}
-again to convert it to a series object with the usual structure
-(expansion plus order term). A sample application from special
-relativity could read:
+its own for storing such series (@code{class pseries}) and a built-in
+function (called @code{Order}) for storing the order term of the series.
+As a consequence, if you want to work with series, i.e. multiply two
+series, you need to call the method @code{ex::series} again to convert
+it to a series object with the usual structure (expansion plus order
+term). A sample application from special relativity could read:
@example
#include <ginac/ginac.h>
look something like this:
@example
-static ex cos_eval_method(ex const & x)
+static ex cos_eval_method(const ex & x)
@{
// if (!x%(2*Pi)) return 1
// if (!x%Pi) return -1
@cindex @code{hold()}
@cindex evaluation
The last line returns @code{cos(x)} if we don't know what else to do and
-stops a potential recursive evaluation by saying @code{.hold()}. We
+stops a potential recursive evaluation by saying @code{.hold()}, which
+sets a flag to the expression signalint that it has been evaluated. We
should also implement a method for numerical evaluation and since we are
lazy we sweep the problem under the rug by calling someone else's
function that does so, in this case the one in class @code{numeric}:
@example
-static ex cos_evalf_method(ex const & x)
+static ex cos_evalf(const ex & x)
@{
- return sin(ex_to_numeric(x));
+ return cos(ex_to_numeric(x));
@}
@end example
-Differentiation will surely turn up and so we need to tell
-@code{sin} how to differentiate itself:
+Differentiation will surely turn up and so we need to tell @code{cos}
+what the first derivative is (higher derivatives (@code{.diff(x,3)} for
+instance are then handled automatically by @code{basic::diff} and
+@code{ex::diff}):
@example
-static ex cos_diff_method(ex const & x, unsigned diff_param)
+static ex cos_derive(const ex & x, unsigned diff_param)
@{
- return cos(x);
+ return -sin(x);
@}
@end example
are curious:
@example
-REGISTER_FUNCTION(cos, cos_eval_method, cos_evalf_method, cos_diff, NULL);
+REGISTER_FUNCTION(cos, cos_eval, cos_evalf, cos_derive, NULL);
@end example
-The first argument is the function's name, the second, third and fourth
-bind the corresponding methods to this objects and the fifth is a slot
-for inserting a method for series expansion. (If set to @code{NULL} it
-defaults to simple Taylor expansion, which is correct if there are no
-poles involved. 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:
+The first argument is the function's name used for calling it and for
+output. The second, third and fourth bind the corresponding methods to
+this objects and the fifth is a slot for inserting a method for series
+expansion. (If set to @code{NULL} it defaults to simple Taylor
+expansion, which is correct if there are no poles involved. 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:
@example
DECLARE_FUNCTION_1P(cos)
However, such an extension is not inherently forbidden by design. In
fact, two interactive interfaces are possible: First, a shell that
exposes GiNaC's types to a command line can readily be written (the tiny
-@command{ginsh} that is part of the distribution being an example) and
-second, as a more consistent approach we are working on an integration
-with the @acronym{CINT} C++ interpreter.
+@command{ginsh} that is part of the distribution being an example).
+Second, as a more consistent approach, an interactive interface to the
+@acronym{CINT} C++ interpreter is under development (called
+@acronym{GiNaC-cint}) that will allow an interactive interface
+consistent with the C++ language.
@item
advanced features: GiNaC cannot compete with a program like
@item
If you move the GiNaC package from its installed location,
-you will need either need to modify @command{ginac-config} script
+you will either need to modify @command{ginac-config} script
manually to point to the new location or rebuild GiNaC.
@end itemize
git://www.ginac.de / ginac.git / blobdiff