Provided by: librheolef-dev_7.2-2_amd64 
NAME
solver - direct and iterative solver (rheolef-7.2)
DESCRIPTION
The class implements the numerical resolution of a linear system. Let a be a square and
invertible matrix in the csr(4) sparse format. The construction of a solver writes:
solver sa (a);
and the resolution of a*x = b expresses simply:
vec<Float> x = sa.solve(b);
When the matrix is modified in a computation loop, the solver could be re-initialized by:
sa.update_values (new_a);
vec<Float> x = sa.solve(b);
DIRECT VERSUS ITERATIVE
The choice between a direct or an iterative method for solving the linear system is by
default performed automatically: it depends upon the sparsity pattern of the matrix, in
order to achieve the best performances. The solver_option(4) class allows one to change
this default behavior.
solver_option sopt;
sopt.iterative = true;
solver sa (a, sopt);
vec<Float> x = sa.solve(b);
The direct approach bases on the Choleski factorization for a symmetric definite positive
matrix, and on the LU one otherwise. Conversely, the iterative approach bases on the cg(5)
conjugate gradient algorithm for a symmetric definite positive matrix, and on the gmres(5)
algorithm otherwise.
COMPUTING THE DETERMINANT
This feature is useful e.g. when tracking a change of sign in the determinant of a matrix,
e.g. during the continuation(3) algorithm. When using a direct method, the determinant of
the matrix can be computed as:
solver_option sopt;
sopt.iterative = false;
solver sa (a, sopt);
cout << sa.det().mantissa << "*"
<< sa.det().base << "^"
<< sa.det().exponant << endl;
The sa.det() method returns an object of type solver::determinant_type that contains a
mantissa and an exponent in a given base (generally 2 or 10). For some rare direct
solvers, the computation of the determinant is not yet fully supported: it is the case for
the Cholesky factorization from the eigen library. In you find such a problem, please
switch to another solver library, see the solver_option(4) class.
AUTOMATIC CHOICE AND CUSTOMIZATION
When the matrix is obtained from the finite element discretization of 3D partial
differential problems, the iterative solver is the default choice. Otherwise, the direct
solver is selected.
More precisely, the choice between direct or iterative solver depends upon the
a.pattern_dimension() property of the csr(4) sparse matrix. When this pattern dimension is
3, an iterative method is faster and less memory consuming than a direct one. See
usersguide for a discussion on this subject.
The symmetry-positive-definiteness of the matrix is tested via the a.is_symmetric() and
a.is_definite_positive() properties of the csr(4) sparse matrix. These properties
determine the choices between Cholesky/LU methods for the direct case, and between
cg/gmres algorithms for the iterative one. Most of the time, these properties are
automatically well initialized by the finite element assembly procedure, via the
integrate(3) function.
Nevertheless, in some special cases, e.g. a linear combination of matrix, or when the
matrix has been read from a file, it could be necessary to force either the symmetry or
the positive-definiteness property by the appropriate csr(4) member function before to
send the matrix to the solver.
PRECONDITIONNERS FOR ITERATIVE SOLVERS
When using an iterative method, the default is to perform no preconditionning. Several
preconditionners are available: the mic(5) modified incomplete Cholesky for symmetric
matrix and the ilut(5) incomplete LU one for unsymmetric matrix and the do-nothing eye(5)
identity preconditionner. A preconditionner can be supplied via:
solver_option sopt;
sopt.iterative = true;
solver sa (a, sopt);
sa.set_preconditionner (ilut(a));
vec<Float> x = sa.solve(b);
Note also the eye(5) that returns the solver for the identity matrix: it could be be used
for specifying that we do not use a preconditionner. This is the default behavior. The
set_preconditionner member function should be called before the first call to the solve
method: if no preconditioner has been defined at the first call to solve, the default
eye(5) preconditionner is selected.
IMPLEMENTATION
This documentation has been generated from file linalg/lib/solver.h
The solver class is simply an alias to the solver_basic class
typedef solver_basic<Float> solver;
The solver_basic class provides an interface to a data container:
template <class T, class M = rheo_default_memory_model>
class solver_basic: public smart_pointer_clone<solver_abstract_rep<T,M> > {
public:
// typedefs:
typedef solver_abstract_rep<T,M> rep;
typedef smart_pointer_clone<rep> base;
typedef typename rep::size_type size_type;
typedef typename rep::determinant_type determinant_type;
// allocators:
solver_basic ();
explicit solver_basic (const csr<T,M>& a, const solver_option& opt = solver_option());
void update_values (const csr<T,M>& a);
// accessors:
vec<T,M> trans_solve (const vec<T,M>& b) const;
vec<T,M> solve (const vec<T,M>& b) const;
determinant_type det() const;
const solver_option& option() const;
void set_preconditioner (const solver_basic<T,M>&);
bool initialized() const;
std::string name() const;
};
AUTHOR
Pierre Saramito <Pierre.Saramito@imag.fr>
COPYRIGHT
Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL
version 3 or later <http://gnu.org/licenses/gpl.html>. This is free software: you
are free to change and redistribute it. There is NO WARRANTY, to the extent permitted by
law.