This is a tutorial that documents GiNaC @value{VERSION}, an open
framework for symbolic computation within the C++ programming language.
-Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
+Copyright (C) 1999-2024 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
@title GiNaC @value{VERSION}
@subtitle An open framework for symbolic computation within the C++ programming language
@subtitle @value{UPDATED}
-@author @uref{http://www.ginac.de}
+@author @uref{https://www.ginac.de}
@page
@vskip 0pt plus 1filll
-Copyright @copyright{} 1999-2015 Johannes Gutenberg University Mainz, Germany
+Copyright @copyright{} 1999-2024 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
the development, the actual documentation is inside the sources in the
form of comments. That documentation may be parsed by one of the many
Javadoc-like documentation systems. If you fail at generating it you
-may access it from @uref{http://www.ginac.de/reference/, the GiNaC home
+may access it from @uref{https://www.ginac.de/reference/, the GiNaC home
page}. It is an invaluable resource not only for the advanced user who
wishes to extend the system (or chase bugs) but for everybody who wants
to comprehend the inner workings of GiNaC. This little tutorial on the
@section License
The GiNaC framework for symbolic computation within the C++ programming
-language is Copyright @copyright{} 1999-2015 Johannes Gutenberg
+language is Copyright @copyright{} 1999-2024 Johannes Gutenberg
University Mainz, Germany.
This program is free software; you can redistribute it and/or
and run it like this:
@example
-$ c++ hello.cc -o hello -lcln -lginac
+$ c++ hello.cc -o hello -lginac -lcln
$ ./hello
355687428096000*x*y+20922789888000*y^2+6402373705728000*x^2
@end example
@end example
Multivariate polynomials and rational functions may be expanded,
-collected and normalized (i.e. converted to a ratio of two coprime
-polynomials):
+collected, factorized, and normalized (i.e. converted to a ratio of
+two coprime polynomials):
@example
> a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
4*x*y-y^2+x^2
> expand(a*b);
8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
+> factor(%);
+(4*x*y+x^2-y^2)^2*(x^2+3*y^2)
> collect(a+b,x);
4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
> collect(a+b,y);
3*y^2+x^2
@end example
+Here we have made use of the @command{ginsh}-command @code{%} to pop the
+previously evaluated element from @command{ginsh}'s internal stack.
+
You can differentiate functions and expand them as Taylor or Laurent
series in a very natural syntax (the second argument of @code{series} is
a relation defining the evaluation point, the third specifies the
-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
-previously evaluated element from @command{ginsh}'s internal stack.
-
Often, functions don't have roots in closed form. Nevertheless, it's
quite easy to compute a solution numerically, to arbitrary precision:
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 GCC for development
+ISO standard @cite{ISO/IEC 14882:2011(E)}. We used 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. The pkg-config utility is
@uref{http://pkg-config.freedesktop.org}.
Last but not least, the CLN library
is used extensively and needs to be installed on your system.
-Please get it from @uref{ftp://ftpthep.physik.uni-mainz.de/pub/gnu/}
-(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.
+Please get it from @uref{https://www.ginac.de/CLN/} (it is licensed under
+the 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.
@node Configuration, Building GiNaC, Prerequisites, Installation
* Matrices:: Matrices.
* Indexed objects:: Handling indexed quantities.
* Non-commutative objects:: Algebras with non-commutative products.
-* Hash maps:: A faster alternative to std::map<>.
@end menu
Internally, the anonymous evaluator in GiNaC is implemented by the methods
@example
-ex ex::eval(int level = 0) const;
-ex basic::eval(int level = 0) const;
+ex ex::eval() const;
+ex basic::eval() const;
@end example
but unless you are extending GiNaC with your own classes or functions, there
as "@code{\Box}" in LaTeX code (@xref{Input/output}, for more
information about the different output formats of expressions in GiNaC).
GiNaC automatically creates proper LaTeX code for symbols having names of
-greek letters (@samp{alpha}, @samp{mu}, etc.).
+greek letters (@samp{alpha}, @samp{mu}, etc.). You can retrieve the name
+and the LaTeX name of a symbol using the respective methods:
+@cindex @code{get_name()}
+@cindex @code{get_TeX_name()}
+@example
+symbol::get_name() const;
+symbol::get_TeX_name() const;
+@end example
@cindex @code{subs()}
Symbols in GiNaC can't be assigned values. If you need to store results of
The classes therein serve as foundation classes for GiNaC. CLN stands
for Class Library for Numbers or alternatively for Common Lisp Numbers.
In order to find out more about CLN's internals, the reader is referred to
-the documentation of that library. @inforef{Introduction, , cln}, for
+the documentation of that library. @xref{Top,,, cln, The CLN Manual}, for
more information. Suffice to say that it is by itself build on top of
another library, the GNU Multiple Precision library GMP, which is an
extremely fast library for arbitrary long integers and rationals as well
@code{to_int()/to_long()} and @code{to_double()} discard the imaginary
part of complex numbers.
+Note the signature of the above methods, you may need to apply a type
+conversion and call @code{evalf()} as shown in the following example:
+@example
+ ...
+ ex e1 = 1, e2 = sin(Pi/5);
+ cout << ex_to<numeric>(e1).to_int() << endl
+ << ex_to<numeric>(e2.evalf()).to_double() << endl;
+ ...
+@end example
@node Constants, Fundamental containers, Numbers, Basic concepts
@c node-name, next, previous, up
the same type to GiNaC methods such as @code{subs()} and some @code{matrix}
constructors, so you should have a basic understanding of them.
-Lists can be constructed by assigning a comma-separated sequence of
-expressions:
+Lists can be constructed from an initializer list of expressions:
@example
@{
symbol x("x"), y("y");
- lst l;
- l = x, 2, y, x+y;
+ lst l = @{x, 2, y, x+y@};
// now, l is a list holding the expressions 'x', '2', 'y', and 'x+y',
// in that order
...
@end example
-There are also constructors that allow direct creation of lists of up to
-16 expressions, which is often more convenient but slightly less efficient:
-
-@example
- ...
- // This produces the same list 'l' as above:
- // lst l(x, 2, y, x+y);
- // lst l = lst(x, 2, y, x+y);
- ...
-@end example
-
Use the @code{nops()} method to determine the size (number of expressions) of
a list and the @code{op()} method or the @code{[]} operator to access
individual elements:
@example
...
- lst l1, l2;
- l1 = x, 2, y, x+y;
- l2 = 2, x+y, x, y;
+ lst l1 = @{x, 2, y, x+y@};
+ lst l2 = @{2, x+y, x, y@};
l1.sort();
l2.sort();
// l1 and l2 are now equal
@example
...
- lst l3;
- l3 = x, 2, 2, 2, y, x+y, y+x;
+ lst l3 = @{x, 2, 2, 2, y, x+y, y+x@};
l3.unique(); // l3 is now @{x, 2, y, x+y@}
@}
@end example
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
+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.
+Simplifications of
+relationals may be more efficient if preceded by a call to
+@example
+ex relational::canonical() const
+@end example
+which returns an equivalent relation with the zero
+right-hand side. For example:
+@example
+possymbol p("p");
+relational rel = (p >= (p*p-1)/p);
+if (ex_to<relational>(rel.canonical().normal()))
+ cout << "correct inequality" << endl;
+@end example
+However, a user shall not expect that any inequality can be fully
+resolved by GiNaC.
+
@node Integrals, Matrices, Relations, Basic concepts
@c node-name, next, previous, up
@section Integrals
creates a matrix with @samp{r} rows and @samp{c} columns with all elements
set to zero.
-The fastest way to create a matrix with preinitialized elements is to assign
-a list of comma-separated expressions to an empty matrix (see below for an
-example). But you can also specify the elements as a (flat) list with
+The easiest way to create a matrix is using an initializer list of
+initializer lists, all of the same size:
+
+@example
+@{
+ matrix m = @{@{1, -a@},
+ @{a, 1@}@};
+@}
+@end example
+
+You can also specify the elements as a (flat) list with
@example
matrix::matrix(unsigned r, unsigned c, const lst & l);
@cindex @code{symbolic_matrix()}
@example
ex diag_matrix(const lst & l);
+ex diag_matrix(initializer_list<ex> l);
ex unit_matrix(unsigned x);
ex unit_matrix(unsigned r, unsigned c);
ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
const string & tex_base_name);
@end example
-@code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
+@code{diag_matrix()} constructs a square diagonal matrix given the diagonal
elements. @code{unit_matrix()} creates an @samp{x} by @samp{x} (or @samp{r}
by @samp{c}) unit matrix. And finally, @code{symbolic_matrix} constructs a
matrix filled with newly generated symbols made of the specified base name
@example
@{
- matrix m(3,3);
- m = 11, 12, 13,
- 21, 22, 23,
- 31, 32, 33;
+ matrix m = @{@{11, 12, 13@},
+ @{21, 22, 23@},
+ @{31, 32, 33@}@};
cout << reduced_matrix(m, 1, 1) << endl;
// -> [[11,13],[31,33]]
cout << sub_matrix(m, 1, 2, 1, 2) << endl;
@{
symbol a("a"), b("b");
- matrix M(2, 2);
- M = a, 0,
- 0, b;
+ matrix M = @{@{a, 0@},
+ @{0, b@}@};
cout << M << endl;
// -> [[a,0],[0,b]]
cout << M2 << endl;
// -> [[a,0],[0,b]]
- cout << matrix(2, 2, lst(a, 0, 0, b)) << endl;
+ cout << matrix(2, 2, lst@{a, 0, 0, b@}) << endl;
// -> [[a,0],[0,b]]
- cout << lst_to_matrix(lst(lst(a, 0), lst(0, b))) << endl;
+ cout << lst_to_matrix(lst@{lst@{a, 0@}, lst@{0, b@}@}) << endl;
// -> [[a,0],[0,b]]
- cout << diag_matrix(lst(a, b)) << endl;
+ cout << diag_matrix(lst@{a, b@}) << endl;
// -> [[a,0],[0,b]]
cout << unit_matrix(3) << endl;
@example
@{
- matrix A(2, 2), B(2, 2), C(2, 2);
- A = 1, 2,
- 3, 4;
- B = -1, 0,
- 2, 1;
- C = 8, 4,
- 2, 1;
+ matrix A = @{@{ 1, 2@},
+ @{ 3, 4@}@};
+ matrix B = @{@{-1, 0@},
+ @{ 2, 1@}@};
+ matrix C = @{@{ 8, 4@},
+ @{ 2, 1@}@};
matrix result = A.mul(B).sub(C.mul_scalar(2));
cout << result << endl;
ex matrix::determinant(unsigned algo=determinant_algo::automatic) const;
ex matrix::trace() const;
ex matrix::charpoly(const ex & lambda) const;
-unsigned matrix::rank() const;
+unsigned matrix::rank(unsigned algo=solve_algo::automatic) const;
@end example
-The @samp{algo} argument of @code{determinant()} allows to select
-between different algorithms for calculating the determinant. The
-asymptotic speed (as parametrized by the matrix size) can greatly differ
-between those algorithms, depending on the nature of the matrix'
-entries. The possible values are defined in the @file{flags.h} header
-file. By default, GiNaC uses a heuristic to automatically select an
-algorithm that is likely (but not guaranteed) to give the result most
-quickly.
+The optional @samp{algo} argument of @code{determinant()} and @code{rank()}
+functions allows to select between different algorithms for calculating the
+determinant and rank respectively. The asymptotic speed (as parametrized
+by the matrix size) can greatly differ between those algorithms, depending
+on the nature of the matrix' entries. The possible values are defined in
+the @file{flags.h} header file. By default, GiNaC uses a heuristic to
+automatically select an algorithm that is likely (but not guaranteed)
+to give the result most quickly.
-@cindex @code{inverse()} (matrix)
@cindex @code{solve()}
-Matrices may also be inverted using the @code{ex matrix::inverse()}
-method and linear systems may be solved with:
+Linear systems can be solved with:
@example
matrix matrix::solve(const matrix & vars, const matrix & rhs,
contain some of the indeterminates from @code{vars}. If the system is
overdetermined, an exception is thrown.
+@cindex @code{inverse()} (matrix)
+To invert a matrix, use the method:
+
+@example
+matrix matrix::inverse(unsigned algo=solve_algo::automatic) const;
+@end example
+
+The @samp{algo} argument is optional. If given, it must be one of
+@code{solve_algo} defined in @file{flags.h}.
@node Indexed objects, Non-commutative objects, Matrices, Basic concepts
@c node-name, next, previous, up
symbol x("x"), y("y");
// A is a 2x2 matrix, X is a 2x1 vector
- matrix A(2, 2), X(2, 1);
- A = 1, 2,
- 3, 4;
- X = x, y;
+ matrix A = @{@{1, 2@},
+ @{3, 4@}@};
+ matrix X = @{@{x, y@}@};
cout << indexed(A, i, i) << endl;
// -> 5
of the metric tensor.
-@node Non-commutative objects, Hash maps, Indexed objects, Basic concepts
+@node Non-commutative objects, Methods and functions, Indexed objects, Basic concepts
@c node-name, next, previous, up
@section Non-commutative objects
The @code{dirac_trace()} function is a linear functional that is equal to the
ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
functional is not cyclic in
-@tex $D \ne 4$
+@tex
+$D \ne 4$
@end tex
@ifnottex
@math{D != 4}
@cite{The Role of gamma5 in Dimensional Regularization} (@ref{Bibliography}).
The value of the trace itself is also usually different in 4 and in
-@tex $D \ne 4$
+@tex
+$D \ne 4$
@end tex
@ifnottex
@math{D != 4}
dirac_gamma(mu.toggle_variance()) *
(dirac_slash(l, D) + m * dirac_ONE());
e = dirac_trace(e).simplify_indexed(sp);
- e = e.collect(lst(l, ldotq, m));
+ e = e.collect(lst@{l, ldotq, m@});
cout << e << endl;
// -> (8-4*D)*l^2+(8-4*D)*ldotq+4*D*m^2
@}
@subsubsection A generic Clifford algebra
A generic Clifford algebra, i.e. a
-@tex $2^n$
+@tex
+$2^n$
@end tex
@ifnottex
2^n
@end ifnottex
-dimensional algebra with
-generators
-@tex $e_k$
-@end tex
+dimensional algebra with generators
+@tex
+$e_k$
+@end tex
@ifnottex
e_k
@end ifnottex
something very close to @code{dirac_gamma(mu)}, although
@code{dirac_gamma} have more efficient simplification mechanism.
@cindex @code{get_metric()}
+Also, the object created by @code{clifford_unit(mu, minkmetric())} is
+not aware about the symmetry of its metric, see the start of the previous
+paragraph. A more accurate analog of 'dirac_gamma(mu)' should be
+specifies as follows:
+
+@example
+ clifford_unit(mu, indexed(minkmetric(),sy_symm(),varidx(symbol("i"),4),varidx(symbol("j"),4)));
+@end example
+
The method @code{clifford::get_metric()} returns a metric defining this
Clifford number.
...
idx i(symbol("i"), 4);
realsymbol s("s");
- ex M = diag_matrix(lst(1, -1, 0, s));
+ ex M = diag_matrix(lst@{1, -1, 0, s@});
ex e = clifford_unit(i, M);
ex e0 = e.subs(i == 0);
ex e1 = e.subs(i == 1);
...
idx i(symbol("i"), 4);
realsymbol s("s");
- ex M = diag_matrix(lst(1, -1, 0, s));
- ex e0 = lst_to_clifford(lst(1, 0, 0, 0), i, M);
- ex e1 = lst_to_clifford(lst(0, 1, 0, 0), i, M);
- ex e2 = lst_to_clifford(lst(0, 0, 1, 0), i, M);
- ex e3 = lst_to_clifford(lst(0, 0, 0, 1), i, M);
+ ex M = diag_matrix(@{1, -1, 0, s@});
+ ex e0 = lst_to_clifford(lst@{1, 0, 0, 0@}, i, M);
+ ex e1 = lst_to_clifford(lst@{0, 1, 0, 0@}, i, M);
+ ex e2 = lst_to_clifford(lst@{0, 0, 1, 0@}, i, M);
+ ex e3 = lst_to_clifford(lst@{0, 0, 0, 1@}, i, M);
...
@}
@end example
@example
ex clifford_prime(const ex & e)
- inline ex clifford_star(const ex & e) @{ return e.conjugate(); @}
- inline ex clifford_bar(const ex & e) @{ return clifford_prime(e.conjugate()); @}
+ inline ex clifford_star(const ex & e)
+ inline ex clifford_bar(const ex & e)
@end example
The automorphism of a Clifford algebra @code{clifford_prime()} simply
changes signs of all Clifford units in the expression. The reversion
-of a Clifford algebra @code{clifford_star()} coincides with the
-@code{conjugate()} method and effectively reverses the order of Clifford
+of a Clifford algebra @code{clifford_star()} reverses the order of Clifford
units in any product. Finally the main anti-automorphism
of a Clifford algebra @code{clifford_bar()} is the composition of the
previous two, i.e. it makes the reversion and changes signs of all Clifford units
@end example
-@node Hash maps, Methods and functions, Non-commutative objects, Basic concepts
-@c node-name, next, previous, up
-@section Hash Maps
-@cindex hash maps
-@cindex @code{exhashmap} (class)
-
-For your convenience, GiNaC offers the container template @code{exhashmap<T>}
-that can be used as a drop-in replacement for the STL
-@code{std::map<ex, T, ex_is_less>}, using hash tables to provide faster,
-typically constant-time, element look-up than @code{map<>}.
-
-@code{exhashmap<>} supports all @code{map<>} members and operations, with the
-following differences:
-
-@itemize @bullet
-@item
-no @code{lower_bound()} and @code{upper_bound()} methods
-@item
-no reverse iterators, no @code{rbegin()}/@code{rend()}
-@item
-no @code{operator<(exhashmap, exhashmap)}
-@item
-the comparison function object @code{key_compare} is hardcoded to
-@code{ex_is_less}
-@item
-the constructor @code{exhashmap(size_t n)} allows specifying the minimum
-initial hash table size (the actual table size after construction may be
-larger than the specified value)
-@item
-the method @code{size_t bucket_count()} returns the current size of the hash
-table
-@item
-@code{insert()} and @code{erase()} operations invalidate all iterators
-@end itemize
-
-
-@node Methods and functions, Information about expressions, Hash maps, Top
+@node Methods and functions, Information about expressions, Non-commutative objects, Top
@c node-name, next, previous, up
@chapter Methods and functions
@cindex polynomial
* Symmetrization::
* Built-in functions:: List of predefined mathematical functions.
* Multiple polylogarithms::
+* Iterated integrals::
* Complex expressions::
* Solving linear systems of equations::
* Input/output:: Input and output of expressions.
@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
@example
@{
symbol A("A"), B("B"), C("C");
- ex e = lst(lst(A, B), C);
+ ex e = lst@{lst@{A, B@}, C@};
std::copy(e.begin(), e.end(),
std::ostream_iterator<ex>(cout, "\n"));
predicates to the STL:
@example
-class ex_is_less : public std::binary_function<ex, ex, bool> @{
+class ex_is_less @{
public:
bool operator()(const ex &lh, const ex &rh) const;
@};
-class ex_is_equal : public std::binary_function<ex, ex, bool> @{
+class ex_is_equal @{
public:
bool operator()(const ex &lh, const ex &rh) const;
@};
// count the number of expressions equal to '1'
unsigned num_ones = std::count_if(v.begin(), v.end(),
- std::bind2nd(ex_is_equal(), 1));
+ [](const ex& e) @{ return ex_is_equal()(e, 1); @});
@end example
The implementation of @code{ex_is_less} uses the member function
To evaluate them using floating-point arithmetic you need to call
@example
-ex ex::evalf(int level = 0) const;
+ex ex::evalf() const;
@end example
@cindex @code{Digits}
// -> 73
ex e2 = x*y + x;
- cout << "e2(-2, 4) = " << e2.subs(lst(x == -2, y == 4)) << endl;
+ cout << "e2(-2, 4) = " << e2.subs(lst@{x == -2, y == 4@}) << endl;
// -> -10
@}
@end example
If you specify multiple substitutions, they are performed in parallel, so e.g.
-@code{subs(lst(x == y, y == x))} exchanges @samp{x} and @samp{y}.
+@code{subs(lst@{x == y, y == x@})} exchanges @samp{x} and @samp{y}.
The second form of @code{subs()} takes an @code{exmap} object which is a
pair associative container that maps expressions to expressions (currently
symbol x("x"), y("y");
ex e2 = x*y + x;
- cout << "e2(-2, 4) = " << e2.subs(lst(x, y), lst(-2, 4)) << endl;
+ cout << "e2(-2, 4) = " << e2.subs(lst@{x, y@}, lst@{-2, 4@}) << endl;
@}
@end example
One solution to this dilemma is the @dfn{Visitor} design pattern,
which is implemented in GiNaC (actually, Robert Martin's Acyclic Visitor
variation, described in detail in
-@uref{http://objectmentor.com/publications/acv.pdf}). Instead of adding
+@uref{https://condor.depaul.edu/dmumaugh/OOT/Design-Principles/acv.pdf}). Instead of adding
virtual functions to the class hierarchy to implement operations, GiNaC
provides a single "bouncing" method @code{accept()} that takes an instance
of a special @code{visitor} class and redirects execution to the one
@example
(x*y*sin(y)).is_polynomial(x) // Returns true.
-(x*y*sin(y)).is_polynomial(lst(x,y)) // Returns false.
+(x*y*sin(y)).is_polynomial(lst@{x,y@}) // Returns false.
@end example
@subsection Expanding and collecting
@cindex @code{ldegree()}
@cindex @code{coeff()}
-The degree and low degree of a polynomial can be obtained using the two
-methods
+The degree and low degree of a polynomial in expanded form can be obtained
+using the two methods
@example
int ex::degree(const ex & s);
int ex::ldegree(const ex & s);
@end example
-which also work reliably on non-expanded input polynomials (they even work
-on rational functions, returning the asymptotic degree). By definition, the
-degree of zero is zero. To extract a coefficient with a certain power from
-an expanded polynomial you use
+These functions even work on rational functions, returning the asymptotic
+degree. By definition, the degree of zero is zero. To extract a coefficient
+with a certain power from an expanded polynomial you use
@example
ex ex::coeff(const ex & s, int n);
Square-free decomposition is available in GiNaC:
@example
-ex sqrfree(const ex & a, const lst & l = lst());
+ex sqrfree(const ex & a, const lst & l = lst@{@});
@end example
Here is an example that by the way illustrates how the exact form of the
result may slightly depend on the order of differentiation, calling for
symbol x("x"), y("y");
ex BiVarPol = expand(pow(2-2*y,3) * pow(1+x*y,2) * pow(x-2*y,2) * (x+y));
- cout << sqrfree(BiVarPol, lst(x,y)) << endl;
+ cout << sqrfree(BiVarPol, lst@{x,y@}) << endl;
// -> 8*(1-y)^3*(y*x^2-2*y+x*(1-2*y^2))^2*(y+x)
- cout << sqrfree(BiVarPol, lst(y,x)) << endl;
+ cout << sqrfree(BiVarPol, lst@{y,x@}) << endl;
// -> 8*(1-y)^3*(-y*x^2+2*y+x*(-1+2*y^2))^2*(y+x)
cout << sqrfree(BiVarPol) << endl;
Note also, how factors with the same exponents are not fully factorized
with this method.
+@subsection Square-free partial fraction decomposition
+@cindex square-free partial fraction decomposition
+@cindex partial fraction decomposition
+@cindex @code{sqrfree_parfrac()}
+
+GiNaC also supports square-free partial fraction decomposition of
+rational functions:
+@example
+ex sqrfree_parfrac(const ex & a, const symbol & x);
+@end example
+It is called square-free because it assumes a square-free
+factorization of the input's denominator:
+@example
+ ...
+ symbol x("x");
+
+ ex rat = (x-4)/(pow(x,2)*(x+2));
+ cout << sqrfree_parfrac(rat, x) << endl;
+ // -> -2*x^(-2)+3/2*x^(-1)-3/2*(2+x)^(-1)
+@end example
+
@subsection Polynomial factorization
@cindex factorization
@cindex polynomial factorization
These functions will first normalize the expression as described above and
then return the numerator, denominator, or both as a list, respectively.
-If you need both numerator and denominator, calling @code{numer_denom()} is
-faster than using @code{numer()} and @code{denom()} separately.
+If you need both numerator and denominator, call @code{numer_denom()}: it
+is faster than using @code{numer()} and @code{denom()} separately. And even
+more important: a separate evaluation of @code{numer()} and @code{denom()}
+may result in a spurious sign, e.g. for $x/(x^2-1)$ @code{numer()} may
+return $x$ and @code{denom()} $1-x^2$.
@subsection Converting to a polynomial or rational expression
@example
ex ex::to_polynomial(exmap & m);
-ex ex::to_polynomial(lst & l);
@end example
or
@example
ex ex::to_rational(exmap & m);
-ex ex::to_rational(lst & l);
@end example
-on the expression to be converted. The supplied @code{exmap} or @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_polynomial()} or @code{.to_rational()}, so it's
-possible to use it on multiple expressions and get consistent results.
+on the expression to be converted. The supplied @code{exmap} 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_polynomial()} or @code{.to_rational()}, so it's possible to use
+it on multiple expressions and get consistent results.
The difference between @code{.to_polynomial()} and @code{.to_rational()}
is probably best illustrated with an example:
ex a = 2*x/sin(x) - y/(3*sin(x));
cout << a << endl;
- lst lp;
- ex p = a.to_polynomial(lp);
- cout << " = " << p << "\n with " << lp << endl;
+ exmap mp;
+ ex p = a.to_polynomial(mp);
+ cout << " = " << p << "\n with " << mp << endl;
// = symbol3*symbol2*y+2*symbol2*x
// with @{symbol2==sin(x)^(-1),symbol3==-1/3@}
- lst lr;
- ex r = a.to_rational(lr);
- cout << " = " << r << "\n with " << lr << endl;
+ exmap mr;
+ ex r = a.to_rational(mr);
+ cout << " = " << r << "\n with " << mr << endl;
// = -1/3*symbol4^(-1)*y+2*symbol4^(-1)*x
// with @{symbol4==sin(x)@}
@}
ex machin_pi(int degr)
@{
symbol x;
- ex pi_expansion = series_to_poly(atan(x).series(x,degr));
+ ex pi_expansion = series_to_poly(atan(x).series(x==0,degr));
ex pi_approx = 16*pi_expansion.subs(x==numeric(1,5))
-4*pi_expansion.subs(x==numeric(1,239));
return pi_approx;
cout << ex(indexed(A, i, j)).symmetrize() << endl;
// -> 1/2*A.j.i+1/2*A.i.j
- cout << ex(indexed(A, i, j, k)).antisymmetrize(lst(i, j)) << endl;
+ cout << ex(indexed(A, i, j, k)).antisymmetrize(lst@{i, j@}) << endl;
// -> -1/2*A.j.i.k+1/2*A.i.j.k
- cout << ex(lst(a, b, c)).symmetrize_cyclic(lst(a, b, c)) << endl;
+ cout << ex(lst@{a, b, c@}).symmetrize_cyclic(lst@{a, b, c@}) << endl;
// -> 1/3*@{a,b,c@}+1/3*@{b,c,a@}+1/3*@{c,a,b@}
@}
@end example
@cindex @code{zeta()}
@item @code{zetaderiv(n, x)}
@tab derivatives of Riemann's zeta function
+@item @code{iterated_integral(a, y)}
+@tab iterated integral
+@cindex @code{iterated_integral()}
+@item @code{iterated_integral(a, y, N)}
+@tab iterated integral with explicit truncation parameter
+@cindex @code{iterated_integral()}
@item @code{tgamma(x)}
@tab gamma function
@cindex @code{tgamma()}
@cindex @code{psi()}
@item @code{psi(n, x)}
@tab derivatives of psi function (polygamma functions)
+@item @code{EllipticK(x)}
+@tab complete elliptic integral of the first kind
+@cindex @code{EllipticK()}
+@item @code{EllipticE(x)}
+@tab complete elliptic integral of the second kind
+@cindex @code{EllipticE()}
@item @code{factorial(n)}
@tab factorial function @math{n!}
@cindex @code{factorial()}
the maximal expansion. For example, for the exponent GiNaC firstly expands
the argument and then the function. For the logarithm and absolute value,
GiNaC uses the opposite order: firstly expands the function and then its
-argument. Of course, a user can fine-tune this behaviour by sequential
+argument. Of course, a user can fine-tune this behavior by sequential
calls of several @code{expand()} methods with desired flags.
-@node Multiple polylogarithms, Complex expressions, Built-in functions, Methods and functions
+@node Multiple polylogarithms, Iterated integrals, Built-in functions, Methods and functions
@c node-name, next, previous, up
@subsection Multiple polylogarithms
The multiple polylogarithm is the most generic member of a family of functions,
to which others like the harmonic polylogarithm, Nielsen's generalized
polylogarithm and the multiple zeta value belong.
-Everyone of these functions can also be written as a multiple polylogarithm with specific
+Each of these functions can also be written as a multiple polylogarithm with specific
parameters. This whole family of functions is therefore often referred to simply as
multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
will be interpreted as the sequence of signs for the corresponding indices
@code{m} or the sign of the imaginary part for the
corresponding arguments @code{a}, it must contain 1 or -1, e.g.
-@code{zeta(lst(3,4), lst(-1,1))} means
+@code{zeta(lst@{3,4@}, lst@{-1,1@})} means
@tex
$\zeta(\overline{3},4)$
@end tex
@command{zeta(\overline@{3@},4)}
@end ifnottex
and
-@code{G(lst(a,b), lst(-1,1), c)} means
+@code{G(lst@{a,b@}, lst@{-1,1@}, c)} means
@tex
$G(a-0\epsilon,b+0\epsilon;c)$.
@end tex
@end ifnottex
The definition of @code{H} allows indices to be 0, 1 or -1 (in expanded notation) or equally to
be any integer (in compact notation). With GiNaC expanded and compact notation can be mixed,
-e.g. @code{lst(0,0,-1,0,1,0,0)}, @code{lst(0,0,-1,2,0,0)} and @code{lst(-3,2,0,0)} are equivalent as
+e.g. @code{lst@{0,0,-1,0,1,0,0@}}, @code{lst@{0,0,-1,2,0,0@}} and @code{lst@{-3,2,0,0@}} are equivalent as
indices. The anonymous evaluator @code{eval()} tries to reduce the functions, if possible, to
the least-generic multiple polylogarithm. If all arguments are unit, it returns @code{zeta}.
Arguments equal to zero get considered, too. Riemann's zeta function @code{zeta} (with depth one)
@cite{Numerical Evaluation of Multiple Polylogarithms},
J.Vollinga, S.Weinzierl, hep-ph/0410259
-@node Complex expressions, Solving linear systems of equations, Multiple polylogarithms, Methods and functions
+@node Iterated integrals, Complex expressions, Multiple polylogarithms, Methods and functions
+@c node-name, next, previous, up
+@subsection Iterated integrals
+
+Multiple polylogarithms are a particular example of iterated integrals.
+An iterated integral is defined by the function @code{iterated_integral(a,y)}.
+The variable @code{y} gives the upper integration limit for the outermost integration, by convention the lower integration limit is always set to zero.
+The variable @code{a} must be a GiNaC @code{lst} containing sub-classes of @code{integration_kernel} as elements.
+The depth of the iterated integral corresponds to the number of elements of @code{a}.
+The available integrands for iterated integrals are
+(for a more detailed description the user is referred to the publications listed at the end of this section)
+@cartouche
+@multitable @columnfractions .40 .60
+@item @strong{Class} @tab @strong{Description}
+@item @code{integration_kernel()}
+@tab Base class, represents the one-form @math{dy}
+@cindex @code{integration_kernel()}
+@item @code{basic_log_kernel()}
+@tab Logarithmic one-form @math{dy/y}
+@cindex @code{basic_log_kernel()}
+@item @code{multiple_polylog_kernel(z_j)}
+@tab The one-form @math{dy/(y-z_j)}
+@cindex @code{multiple_polylog_kernel()}
+@item @code{ELi_kernel(n, m, x, y)}
+@tab The one form @math{ELi_{n;m}(x;y;q) dq/q}
+@cindex @code{ELi_kernel()}
+@item @code{Ebar_kernel(n, m, x, y)}
+@tab The one form @math{\overline{E}_{n;m}(x;y;q) dq/q}
+@cindex @code{Ebar_kernel()}
+@item @code{Kronecker_dtau_kernel(k, z_j, K, C_k)}
+@tab The one form @math{C_k K (k-1)/(2 \pi i)^k g^{(k)}(z_j,K \tau) dq/q}
+@cindex @code{Kronecker_dtau_kernel()}
+@item @code{Kronecker_dz_kernel(k, z_j, tau, K, C_k)}
+@tab The one form @math{C_k (2 \pi i)^{2-k} g^{(k-1)}(z-z_j,K \tau) dz}
+@cindex @code{Kronecker_dz_kernel()}
+@item @code{Eisenstein_kernel(k, N, a, b, K, C_k)}
+@tab The one form @math{C_k E_{k,N,a,b,K}(\tau) dq/q}
+@cindex @code{Eisenstein_kernel()}
+@item @code{Eisenstein_h_kernel(k, N, r, s, C_k)}
+@tab The one form @math{C_k h_{k,N,r,s}(\tau) dq/q}
+@cindex @code{Eisenstein_h_kernel()}
+@item @code{modular_form_kernel(k, P, C_k)}
+@tab The one form @math{C_k P dq/q}
+@cindex @code{modular_form_kernel()}
+@item @code{user_defined_kernel(f, y)}
+@tab The one form @math{f(y) dy}
+@cindex @code{user_defined_kernel()}
+@end multitable
+@end cartouche
+All parameters are assumed to be such that all integration kernels have a convergent Laurent expansion
+around zero with at most a simple pole at zero.
+The iterated integral may also be called with an optional third parameter
+@code{iterated_integral(a,y,N_trunc)}, in which case the numerical evaluation will truncate the series
+expansion at order @code{N_trunc}.
+
+The classes @code{Eisenstein_kernel()}, @code{Eisenstein_h_kernel()} and @code{modular_form_kernel()}
+provide a method @code{q_expansion_modular_form(q, order)}, which can used to obtain the q-expansion
+of @math{E_{k,N,a,b,K}(\tau)}, @math{h_{k,N,r,s}(\tau)} or @math{P} to the specified order.
+
+Useful publications:
+
+@cite{Numerical evaluation of iterated integrals related to elliptic Feynman integrals},
+M.Walden, S.Weinzierl, arXiv:2010.05271
+
+@node Complex expressions, Solving linear systems of equations, Iterated integrals, Methods and functions
@c node-name, next, previous, up
@section Complex expressions
@c
@example
@{
symbol a("a"), b("b"), x("x"), y("y");
- lst eqns, vars;
- eqns = a*x+b*y==3, x-y==b;
- vars = x, y;
+ lst eqns = @{a*x+b*y==3, x-y==b@};
+ lst vars = @{x, y@};
cout << lsolve(eqns, vars) << endl;
// -> @{x==(3+b^2)/(b+a),y==(3-b*a)/(b+a)@}
@end example
@cindex ginac-excompiler
@code{compile_ex} uses the shell script @code{ginac-excompiler} to start the C
compiler and produce the object files. This shell script comes with GiNaC and
-will be installed together with GiNaC in the configured @code{$PREFIX/bin}
-directory. You can also export additional compiler flags via the $CXXFLAGS
-variable:
+will be installed together with GiNaC in the configured @code{$LIBEXECDIR}
+(typically @code{$PREFIX/libexec} or @code{$PREFIX/lib/ginac}). You can also
+export additional compiler flags via the @env{$CXXFLAGS} variable:
@example
setenv("CXXFLAGS", "-O3 -fomit-frame-pointer -ffast-math", 1);
@example
#include <fstream>
-using namespace std;
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
int main()
@example
// ...
- ofstream out("foobar.gar");
+ ofstream out("foobar.gar", ios::binary);
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}.
+reconstruct the expressions @code{foo} and @code{bar}. The flag
+@code{ios::binary} prevents locales setting of your OS tampers the
+archive file structure.
@cindex @command{viewgar}
The tool @command{viewgar} that comes with GiNaC can be used to view
@example
// ...
archive a2;
- ifstream in("foobar.gar");
+ ifstream in("foobar.gar", ios::binary);
in >> a2;
// ...
@end example
@example
// ...
- lst syms;
- syms = x, y;
+ lst syms = @{x, y@};
ex ex1 = a2.unarchive_ex(syms, "foo");
ex ex2 = a2.unarchive_ex(syms, "the second one");
@example
#include <iostream>
-using namespace std;
-
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
struct sprod_s @{
#include <iostream>
#include <string>
#include <stdexcept>
-using namespace std;
-
#include <ginac/ginac.h>
+using namespace std;
using namespace GiNaC;
@end example
@{
...
public:
- ex eval(int level = 0) const;
+ ex eval() const override;
...
@};
-ex mystring::eval(int level) const
+ex mystring::eval() const
@{
string new_str;
for (size_t i=0; i<str.length(); i++) @{
if (new_str.length() == 0)
return 0;
- else
- return mystring(new_str).hold();
+
+ 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();}.
+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();}.
+
+If our class had subobjects, we would have to evaluate them first (unless
+they are all of type @code{ex}, which are automatically evaluated). We don't
+have any subexpressions in the @code{mystring} class, so we are not concerned
+with this.
Let's confirm that it works:
@cindex @code{calchash()}
@cindex @code{is_equal_same_type()}
@example
-unsigned calchash() const;
-bool is_equal_same_type(const basic & other) const;
+unsigned calchash() const override;
+bool is_equal_same_type(const basic & other) const override;
@end example
The @code{calchash()} method returns an @code{unsigned} hash value for the
might want to provide:
@example
-bool info(unsigned inf) const;
-ex evalf(int level = 0) const;
-ex series(const relational & r, int order, unsigned options = 0) const;
-ex derivative(const symbol & s) const;
+bool info(unsigned inf) const override;
+ex evalf() const override;
+ex series(const relational & r, int order, unsigned options = 0) const override;
+ex derivative(const symbol & s) const override;
@end example
If your class stores sub-expressions (see the scalar product example in the
@cindex @code{let_op()}
@example
-size_t nops() cont;
-ex op(size_t i) const;
-ex & let_op(size_t i);
-ex subs(const lst & ls, const lst & lr, unsigned options = 0) const;
-ex map(map_function & f) const;
+size_t nops() const override;
+ex op(size_t i) const override;
+ex & let_op(size_t i) override;
+ex subs(const lst & ls, const lst & lr, unsigned options = 0) const override;
+ex map(map_function & f) const override;
@end example
@code{let_op()} is a variant of @code{op()} that allows write access. The
multiple interfaces: Though real GiNaC programs have to be written in
some editor, then be compiled, linked and executed, there are more ways
to work with the GiNaC engine. Many people want to play with
-expressions interactively, as in traditional CASs. Currently, two such
-windows into GiNaC have been implemented and many more are possible: the
-tiny @command{ginsh} that is part of the distribution exposes GiNaC's
-types to a command line and second, as a more consistent approach, an
-interactive interface to the Cint C++ interpreter has been put together
-(called GiNaC-cint) that allows an interactive scripting interface
-consistent with the C++ language. It is available from the usual GiNaC
-FTP-site.
+expressions interactively, as in traditional CASs: The tiny
+@command{ginsh} that comes with the distribution exposes many, but not
+all, of GiNaC's types to a command line.
@item
seamless integration: it is somewhere between difficult and impossible
occasionally used other compilers and may be able to give you advice.}
GiNaC uses recent language features like explicit constructors, mutable
members, RTTI, @code{dynamic_cast}s and STL, so ANSI compliance is meant
-literally. Recent GCC versions starting at 2.95.3, although itself not
-yet ANSI compliant, support all needed features.
+literally.
@end itemize
happens. Knowing this will enable you to write much more efficient
code. If you still have an uncertain feeling with copy-on-write
semantics, we recommend you have a look at the
-@uref{http://www.parashift.com/c++-faq-lite/, C++-FAQ lite} by
-Marshall Cline. Chapter 16 covers this issue and presents an
-implementation which is pretty close to the one in GiNaC.
+@uref{https://isocpp.org/faq, C++-FAQ's} chapter on memory management.
+It covers this issue and presents an implementation which is pretty
+close to the one in GiNaC.
@node Internal representation of products and sums, Package tools, Expressions are reference counted, Internal structures
directory @var{prefix} one should set the @var{PKG_CONFIG_PATH}
environment variable to @var{prefix}/lib/pkgconfig for this to work.}
@example
-g++ -o simple `pkg-config --cflags --libs ginac` simple.cpp
+g++ -o simple simple.cpp `pkg-config --cflags --libs ginac`
@end example
This command line might expand to (for example):
@example
-g++ -o simple -lginac -lcln simple.cpp
+g++ -o simple simple.cpp -lginac -lcln
@end example
Not only is the form using @command{pkg-config} easier to type, it will
@itemize @minus{}
@item
-@cite{ISO/IEC 14882:1998: Programming Languages: C++}
+@cite{ISO/IEC 14882:2011: Programming Languages: C++}
@item
@cite{CLN: A Class Library for Numbers}, @email{haible@@ilog.fr, Bruno Haible}
git://www.ginac.de / ginac.git / blobdiff