This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2009 Johannes Gutenberg University Mainz, Germany
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
@page
@vskip 0pt plus 1filll
-Copyright @copyright{} 1999-2007 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2009 Johannes Gutenberg University Mainz, Germany
@sp 2
Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
@section License
The GiNaC framework for symbolic computation within the C++ programming
-language is Copyright @copyright{} 1999-2007 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2009 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
warning levels. If omitted, it defaults to @option{-g
-O2}.@footnote{The @command{configure} script is itself generated from
the file @file{configure.ac}. It is only distributed in packaged
-releases of GiNaC. If you got the naked sources, e.g. from CVS, you
+releases of GiNaC. If you got the naked sources, e.g. from git, you
must generate @command{configure} along with the various
@file{Makefile.in} by using the @command{autoreconf} utility. This will
require a fair amount of support from your local toolchain, though.}
@tab modulus in positive representation (in the range @code{[0, abs(b)-1]} with the sign of b, or zero)
@cindex @code{mod()}
@item @code{smod(a, b)}
-@tab modulus in symmetric representation (in the range @code{[-iquo(abs(b)-1, 2), iquo(abs(b), 2)]})
+@tab modulus in symmetric representation (in the range @code{[-iquo(abs(b), 2), iquo(abs(b), 2)]})
@cindex @code{smod()}
@item @code{irem(a, b)}
@tab integer remainder (has the sign of @math{a}, or is zero)
anything useful with it.
@cindex @code{get_value()}
-@cindex @code{get_dimension()}
+@cindex @code{get_dim()}
The methods
@example
ex idx::get_value();
-ex idx::get_dimension();
+ex idx::get_dim();
@end example
return the value and dimension of an @code{idx} object. If you have an index
obtained with the two member functions
@example
-unsigned ex::return_type() const;
-unsigned ex::return_type_tinfo() const;
+unsigned ex::return_type() const;
+return_type_t ex::return_type_tinfo() const;
@end example
The @code{return_type()} function returns one of three values (defined in
@code{noncommutative_composite} expressions.
@end itemize
-The value returned by the @code{return_type_tinfo()} method is valid only
-when the return type of the expression is @code{noncommutative}. It is a
-value that is unique to the class of the object, but may vary every time a
-GiNaC program is being run (it is dynamically assigned on start-up).
+The @code{return_type_tinfo()} method returns an object of type
+@code{return_type_t} that contains information about the type of the expression
+and, if given, its representation label (see section on dirac gamma matrices for
+more details). The objects of type @code{return_type_t} can be tested for
+equality to test whether two expressions belong to the same category and
+therefore may not commute.
Here are a couple of examples:
@cartouche
-@multitable @columnfractions 0.33 0.33 0.34
-@item @strong{Expression} @tab @strong{@code{return_type()}} @tab @strong{@code{return_type_tinfo()}}
-@item @code{42} @tab @code{commutative} @tab -
-@item @code{2*x-y} @tab @code{commutative} @tab -
-@item @code{dirac_ONE()} @tab @code{noncommutative} @tab @code{TINFO_clifford}
-@item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative} @tab @code{TINFO_clifford}
-@item @code{2*color_T(a)} @tab @code{noncommutative} @tab @code{TINFO_color}
-@item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite} @tab -
+@multitable @columnfractions .6 .4
+@item @strong{Expression} @tab @strong{@code{return_type()}}
+@item @code{42} @tab @code{commutative}
+@item @code{2*x-y} @tab @code{commutative}
+@item @code{dirac_ONE()} @tab @code{noncommutative}
+@item @code{dirac_gamma(mu)*dirac_gamma(nu)} @tab @code{noncommutative}
+@item @code{2*color_T(a)} @tab @code{noncommutative}
+@item @code{dirac_ONE()*color_T(a)} @tab @code{noncommutative_composite}
@end multitable
@end cartouche
-Note: the @code{return_type_tinfo()} of Clifford objects is only equal to
-@code{TINFO_clifford} for objects with a representation label of zero.
-Other representation labels yield a different @code{return_type_tinfo()},
-but it's the same for any two objects with the same label. This is also true
-for color objects.
-
A last note: With the exception of matrices, positive integer powers of
non-commutative objects are automatically expanded in GiNaC. For example,
@code{pow(a*b, 2)} becomes @samp{a*b*a*b} if @samp{a} and @samp{b} are
bool is_exactly_a<T>(const ex & e);
bool ex::info(unsigned flag);
unsigned ex::return_type() const;
-unsigned ex::return_type_tinfo() const;
+return_type_t ex::return_type_tinfo() const;
@end example
When the test made by @code{is_a<T>()} returns true, it is safe to call
@example
bool ex::match(const ex & pattern);
-bool ex::match(const ex & pattern, lst & repls);
+bool ex::match(const ex & pattern, exmap& repls);
@end example
This function returns @code{true} when the expression matches the pattern
and @code{false} if it doesn't. If used in the second form, the actual
-subexpressions matched by the wildcards get returned in the @code{repls}
-object as a list of relations of the form @samp{wildcard == expression}.
-If @code{match()} returns false, the state of @code{repls} is undefined.
-For reproducible results, the list should be empty when passed to
-@code{match()}, but it is also possible to find similarities in multiple
-expressions by passing in the result of a previous match.
+subexpressions matched by the wildcards get returned in the associative
+array @code{repls} with @samp{wildcard} as a key. If @code{match()}
+returns false, @code{repls} remains unmodified.
The matching algorithm works as follows:
The method
@example
-bool ex::find(const ex & pattern, lst & found);
+bool ex::find(const ex & pattern, exset& found);
@end example
works a bit like @code{has()} but it doesn't stop upon finding the first
@cindex factorization
@cindex @code{sqrfree()}
-GiNaC still lacks proper factorization support. Some form of
-factorization is, however, easily implemented by noting that factors
-appearing in a polynomial with power two or more also appear in the
-derivative and hence can easily be found by computing the GCD of the
-original polynomial and its derivatives. Any decent system has an
-interface for this so called square-free factorization. So we provide
-one, too:
+Square-free decomposition is available in GiNaC:
@example
ex sqrfree(const ex & a, const lst & l = lst());
@end example
Note also, how factors with the same exponents are not fully factorized
with this method.
+@subsection Polynomial factorization
+@cindex factorization
+@cindex polynomial factorization
+@cindex @code{factor()}
+
+Polynomials can also be fully factored with a call to the function
+@example
+ex factor(const ex & a, unsigned int options = 0);
+@end example
+The factorization works for univariate and multivariate polynomials with
+rational coefficients. The following code snippet shows its capabilities:
+@example
+ ...
+ cout << factor(pow(x,2)-1) << endl;
+ // -> (1+x)*(-1+x)
+ cout << factor(expand((x-y*z)*(x-pow(y,2)-pow(z,3))*(x+y+z))) << endl;
+ // -> (y+z+x)*(y*z-x)*(y^2-x+z^3)
+ cout << factor(pow(x,2)-1+sin(pow(x,2)-1)) << endl;
+ // -> -1+sin(-1+x^2)+x^2
+ ...
+@end example
+The results are as expected except for the last one where no factorization
+seems to have been done. This is due to the default option
+@command{factor_options::polynomial} (equals zero) to @command{factor()}, which
+tells GiNaC to try a factorization only if the expression is a valid polynomial.
+In the shown example this is not the case, because one term is a function.
+
+There exists a second option @command{factor_options::all}, which tells GiNaC to
+ignore non-polynomial parts of an expression and also to look inside function
+arguments. With this option the example gives:
+@example
+ ...
+ cout << factor(pow(x,2)-1+sin(pow(x,2)-1), factor_options::all)
+ << endl;
+ // -> (-1+x)*(1+x)+sin((-1+x)*(1+x))
+ ...
+@end example
+GiNaC's factorization functions cannot handle algebraic extensions. Therefore
+the following example does not factor:
+@example
+ ...
+ cout << factor(pow(x,2)-2) << endl;
+ // -> -2+x^2 and not (x-sqrt(2))*(x+sqrt(2))
+ ...
+@end example
+Factorization is useful in many applications. A lot of algorithms in computer
+algebra depend on the ability to factor a polynomial. Of course, factorization
+can also be used to simplify expressions, but it is costly and applying it to
+complicated expressions (high degrees or many terms) may consume far too much
+time. So usually, looking for a GCD at strategic points in a calculation is the
+cheaper and more appropriate alternative.
@node Rational expressions, Symbolic differentiation, Polynomial arithmetic, Methods and functions
@c node-name, next, previous, up
@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:
+Instead, GiNaC lets you read an expression from a stream or a string,
+specifying the mapping between the input strings and symbols to be used:
@example
@{
- symbol x("x"), y("y");
- ex e("2*x+sin(y)", lst(x, y));
+ symbol x, y;
+ symtab table;
+ table["x"] = x;
+ table["y"] = y;
+ parser reader(table);
+ ex e = reader("2*x+sin(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.
+output operator @code{<<}. Matching between the input strings and expressions
+is given by @samp{table}. The @samp{table} in this example instructs GiNaC
+to substitute any input substring ``x'' with symbol @code{x}. Likewise,
+the substring ``y'' will be replaced with symbol @code{y}. It's also possible
+to map input (sub)strings to arbitrary expressions:
-With this constructor, it's also easy to implement interactive GiNaC programs:
+@example
+@{
+ symbol x, y;
+ symtab table;
+ table["x"] = x+log(y)+1;
+ parser reader(table);
+ ex e = reader("5*x^3 - x^2");
+ // e = 5*(x+log(y)+1)^3 + (x+log(y)+1)^2
+@}
+@end example
+
+If no mapping is specified for a particular string GiNaC will create a symbol
+with corresponding name. Later on you can obtain all parser generated symbols
+with @code{get_syms()} method:
+
+@example
+@{
+ parser reader;
+ ex e = reader("2*x+sin(y)");
+ symtab table = reader.get_syms();
+ symbol x = reader["x"];
+ symbol y = reader["y"];
+@}
+@end example
+
+Sometimes you might want to prevent GiNaC from inserting these extra symbols
+(for example, you want treat an unexpected string in the input as an error).
+
+@example
+@{
+ symtab table;
+ table["x"] = symbol();
+ parser reader(table);
+ parser.strict = true;
+ ex e;
+ try @{
+ e = reader("2*x+sin(y)");
+ @} catch (parse_error& err) @{
+ cerr << err.what() << endl;
+ // prints "unknown symbol "y" in the input"
+ @}
+@}
+@end example
+
+With this parser, it's also easy to implement interactive GiNaC programs:
@example
#include <iostream>
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;
- @}
+ cout << "Enter an expression containing 'x': " << flush;
+ parser reader;
+
+ try @{
+ ex e = reader(cin);
+ symtab table = reader.get_syms();
+ symbol x = table.find("x") != table.end() ?
+ ex_to<symbol>(table["x"]) : symbol("x");
+ cout << "The derivative of " << e << " with respect to x is ";
+ cout << e.diff(x) << "." << endl;
+ @} catch (exception &p) @{
+ cerr << p.what() << endl;
+ @}
@}
@end example
function before calling the @code{evalf_func()}.
@example
-set_return_type(unsigned return_type, unsigned return_type_tinfo)
+set_return_type(unsigned return_type, const return_type_t * return_type_tinfo)
@end example
This allows you to explicitly specify the commutation properties of the
function (@xref{Non-commutative objects}, for an explanation of
-(non)commutativity in GiNaC). For example, you can use
-@code{set_return_type(return_types::noncommutative, TINFO_matrix)} to make
-GiNaC treat your function like a matrix. By default, functions inherit the
-commutation properties of their first argument.
+(non)commutativity in GiNaC). For example, with an object of type
+@code{return_type_t} created like
+
+@example
+return_type_t my_type = make_return_type_t<matrix>();
+@end example
+
+you can use @code{set_return_type(return_types::noncommutative, &my_type)} to
+make GiNaC treat your function like a matrix. By default, functions inherit the
+commutation properties of their first argument. The utilized template function
+@code{make_return_type_t<>()}
+
+@example
+template<typename T> inline return_type_t make_return_type_t(const unsigned rl = 0)
+@end example
+
+can also be called with an argument specifying the representation label of the
+non-commutative function (see section on dirac gamma matrices for more
+details).
@example
set_symmetry(const symmetry & s)
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
+@subsection Hierarchy of algebraic classes.
@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:
-
+@code{basic *} represents a generic pointer to an algebraic class. Working
+with such pointers directly is cumbersome (think of memory management), hence
+GiNaC wraps them into @code{ex} (@pxref{Expressions are reference counted}).
+To make such wrapping possible every algebraic class has to implement several
+methods. Visitors (@pxref{Visitors and tree traversal}), printing, and
+(un)archiving (@pxref{Input/output}) require helper methods too. But don't
+worry, most of the work is simplified by the following macros (defined
+in @file{registrar.h}):
@itemize @bullet
+@item @code{GINAC_DECLARE_REGISTERED_CLASS}
+@item @code{GINAC_IMPLEMENT_REGISTERED_CLASS}
+@item @code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT}
+@end itemize
-@item
-The @code{basic} class declares a member variable @code{tinfo_key} which
-holds a variable of @code{tinfo_t} type (which is actually just
-@code{const void*}) that identifies the object's class.
-
-@item
-By means of some clever tricks with static members, GiNaC maintains a list
-of information for all classes derived from @code{basic}. The information
-available includes the class names, the @code{tinfo_key}s, and pointers
-to the unarchiving functions. This class registry is defined in the
-@file{registrar.h} header file.
+The @code{GINAC_DECLARE_REGISTERED_CLASS} macro inserts declarations
+required for memory management, visitors, printing, and (un)archiving.
+It takes the name of the class and its direct superclass as arguments.
+The @code{GINAC_DECLARE_REGISTERED_CLASS} should be the first line after
+the opening brace of the class definition.
-@end itemize
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS} takes the same arguments as
+@code{GINAC_DECLARE_REGISTERED_CLASS}. It initializes certain static
+members of a class so that printing and (un)archiving works. The
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS} may appear anywhere else in
+the source (at global scope, of course, not inside a function).
-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 automatically) but don't worry, most of the work is simplified by
-macros.
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} is a variant of
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS}. It allows specifying additional
+options, such as custom printing functions.
@subsection A minimalistic 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:
+object from a string:
@example
class mystring : public basic
public:
mystring(const string & s);
- mystring(const char * s);
private:
string str;
GINAC_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 constructor 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} registers the
-class with the GiNaC RTTI (there is also a
-@code{GINAC_IMPLEMENT_REGISTERED_CLASS_OPT} which allows specifying additional
-options for the class, and which we will be using instead in a few minutes).
-
-Now there are seven member functions we have to implement to get a working
+The @code{GINAC_DECLARE_REGISTERED_CLASS} macro insert declarations required
+for memory management, visitors, printing, and (un)archiving.
+@code{GINAC_IMPLEMENT_REGISTERED_CLASS} initializes certain static members
+of a class so that printing and (un)archiving works.
+
+Now there are three member functions we have to implement to get a working
class:
@itemize
@item
@code{mystring()}, the default constructor.
-@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, 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, lst & sym_lst)}, the static
-unarchiving function. It constructs a new instance by calling the unarchiving
-constructor.
-
@item
@cindex @code{compare_same_type()}
@code{int compare_same_type(const basic & other)}, which is used internally
defined.
@item
-And, of course, @code{mystring(const string & s)} and @code{mystring(const char * s)}
-which are the two constructors we declared.
+And, of course, @code{mystring(const string& s)} which is the constructor
+we declared.
@end itemize
Let's proceed step-by-step. The default constructor looks like this:
@example
-mystring::mystring() : inherited(&mystring::tinfo_static) @{@}
+mystring::mystring() @{ @}
@end example
-The golden rule is that in all constructors you have to set the
-@code{tinfo_key} member to the @code{&your_class_name::tinfo_static}
-@footnote{Each GiNaC class has a static member called tinfo_static.
-This member is declared by the GINAC_DECLARE_REGISTERED_CLASS macros
-and defined by the GINAC_IMPLEMENT_REGISTERED_CLASS macros.}. 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).
-Next are the three functions for archiving. You have to implement them even
-if you don't plan to use archives, but the minimum required implementation
-is really simple. First, the archiving function:
-
-@example
-void mystring::archive(archive_node & n) const
-@{
- inherited::archive(n);
- n.add_string("string", str);
-@}
-@end example
-
-The only thing that is really required is calling the @code{archive()}
-function of the superclass. Optionally, you can store all information you
-deem necessary for representing the object into the passed
-@code{archive_node}. We are just storing our string here. For more
-information on how the archiving works, consult the @file{archive.h} header
-file.
-
-The unarchiving constructor is basically the inverse of the archiving
-function:
-
-@example
-mystring::mystring(const archive_node & n, 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, 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:
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:
+Now the only thing missing is our constructor:
@example
-mystring::mystring(const string & s)
- : inherited(&mystring::tinfo_static), str(s) @{@}
-mystring::mystring(const char * s)
- : inherited(&mystring::tinfo_static), str(s) @{@}
+mystring::mystring(const string& s) : str(s) @{ @}
@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.
+No surprises here. We set the @code{str} member from the argument.
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:
@subsection Upgrading extension classes from older version of GiNaC
-If you got some extension classes for GiNaC 1.3.X some changes are
-necessary in order to make your code work with GiNaC 1.4.
-
-@itemize @bullet
-@item constructors which set @code{tinfo_key} such as
+GiNaC used to use a custom run time type information system (RTTI). It was
+removed from GiNaC. Thus, one needs to rewrite constructors which set
+@code{tinfo_key} (which does not exist any more). For example,
@example
-myclass::myclass() : inherited(TINFO_myclass) @{@}
+myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
@end example
-need to be rewritten as
+needs to be rewritten as
@example
-myclass::myclass() : inherited(&myclass::tinfo_static) @{@}
+myclass::myclass() @{@}
@end example
-@item TINO_myclass is not necessary any more and can be removed.
-
-@end itemize
-
-
@node A comparison with other CAS, Advantages, Adding classes, Top
@c node-name, next, previous, up
@chapter A Comparison With Other CAS
advanced features: GiNaC cannot compete with a program like
@emph{Reduce} which exists for more than 30 years now or @emph{Maple}
which grows since 1981 by the work of dozens of programmers, with
-respect to mathematical features. Integration, factorization,
+respect to mathematical features. Integration,
non-trivial simplifications, limits etc. are missing in GiNaC (and are
not planned for the near future).
git://www.ginac.de / ginac.git / blobdiff