Provided by: librheolef-dev_7.2-4_amd64 
NAME
gmres - generalized minimum residual algorithm (rheolef-7.2)
SYNOPSIS
template <class Matrix, class Vector, class Preconditioner,
class SmallMatrix, class SmallVector>
int gmres (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M,
SmallMatrix &H, const SmallVector& V, const solver_option& sopt = solver_option())
EXAMPLE
solver_option sopt;
sopt.tol = 1e-7;
sopt.max_iter = 100;
size_t m = sopt.krylov_dimension = 6;
Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> H(m+1,m+1);
Eigen::Matrix<T,Eigen::Dynamic,1> V(m);
int status = gmres (A, x, b, ilut(a), H, V, sopt);
DESCRIPTION
This function solves the unsymmetric linear system A*x=b with the generalized minimum residual algorithm.
The gmres function follows the algorithm described on p. 20 in
R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. Donato,
J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van der Vorst,
Templates for the solution of linear systems: building blocks for iterative methods,
SIAM, 1994.
The fourth argument of gmres is a preconditionner: here, the ilut(5) one is used, for simplicity.
Next, H specifies a matrix to hold the coefficients of the upper Hessenberg matrix constructed by the
gmres iterations, m specifies the number of iterations for each restart. We have used here the eigen
dense matrix and vector types for the H and V vectors, with sizes related to the Krylov space dimension
m. Finally, the solver_option(4) variable sopt transmits the stopping criterion with sopt.tol and
sopt.max_iter.
On return, the sopt.residue and sopt.n_iter indicate the reached residue and the number of iterations
effectively performed. The return status is zero when the prescribed tolerance tol has been obtained, and
non-zero otherwise. Also, the x variable contains the approximate solution. See also the solver_option(4)
for more controls upon the stopping criterion.
REMARKS
gmres requires two matrices as input: A and H. The A matrix, which will typically be sparse, corresponds
to the matrix involved in the linear system A*x=b. Conversely, the H matrix, which will typically be
dense, corresponds to the upper Hessenberg matrix that is constructed during the gmres iterations. Within
gmres, H is used in a different way than A, so its class must supply different functionality. That is, A
is only accessed though its matrix-vector and transpose-matrix-vector multiplication functions. On the
other hand, gmres solves a dense upper triangular linear system of equations on H. Therefore, the class
to which H belongs must provide H(i,j) accessors.
It is important to remember that we use the convention that indices are 0-based. That is H(0,0) is the
first component of the matrix. Also, the type of the matrix must be compatible with the type of single
vector entry. That is, operations such as H(i,j)*x(j) must be able to be carried out.
IMPLEMENTATION
This documentation has been generated from file linalg/lib/gmres.h
The present template implementation is inspired from the IML++ 1.2 iterative method library,
http://math.nist.gov/iml++
template <class Matrix, class Vector, class Preconditioner,
class SmallMatrix, class SmallVector>
int gmres (const Matrix &A, Vector &x, const Vector &b, const Preconditioner &M,
SmallMatrix &H, const SmallVector& V, const solver_option& sopt = solver_option())
{
typedef typename Vector::size_type Size;
typedef typename Vector::float_type Real;
std::string label = (sopt.label != "" ? sopt.label : "gmres");
Size m = sopt.krylov_dimension;
Vector w;
SmallVector s(m+1), cs(m+1), sn(m+1);
Real residue;
Real norm_b = norm(M.solve(b));
Vector r = M.solve(b - A * x);
Real beta = norm(r);
if (sopt.p_err) (*sopt.p_err) << "[" << label << "] # norm_b=" << norm_b << std::endl
<< "[" << label << "] #iteration residue" << std::endl;
if (sopt.absolute_stopping || norm_b == Real(0)) norm_b = 1;
sopt.n_iter = 0;
sopt.residue = norm(r)/norm_b;
if (sopt.residue <= sopt.tol) return 0;
std::vector<Vector> v (m+1);
for (sopt.n_iter = 1; sopt.n_iter <= sopt.max_iter; ) {
v[0] = r/beta;
for (Size i = 0; i < m+1; i++) s(i) = 0; // std::numeric_limits<Float>::max();
s(0) = beta;
for (Size i = 0; i < m && sopt.n_iter <= sopt.max_iter; i++, sopt.n_iter++) {
w = M.solve(A * v[i]);
for (Size k = 0; k <= i; k++) {
H(k, i) = dot(w, v[k]);
w -= H(k, i) * v[k];
}
H(i+1, i) = norm(w);
v[i+1] = w/H(i+1,i);
for (Size k = 0; k < i; k++) {
details::apply_plane_rotation (H(k,i), H(k+1,i), cs(k), sn(k));
}
details::generate_plane_rotation (H(i,i), H(i+1,i), cs(i), sn(i));
details::apply_plane_rotation (H(i,i), H(i+1,i), cs(i), sn(i));
details::apply_plane_rotation (s(i), s(i+1), cs(i), sn(i));
sopt.residue = abs(s(i+1))/norm_b;
if (sopt.p_err) (*sopt.p_err) << "[" << label << "] " << sopt.n_iter << " " << sopt.residue << std::endl;
if (sopt.residue <= sopt.tol) {
details::update (x, i, H, s, v);
return 0;
}
}
details::update (x, m - 1, H, s, v);
r = M.solve(b - A * x);
beta = norm(r);
sopt.residue = beta/norm_b;
if (sopt.p_err) (*sopt.p_err) << "[" << label << "] " << sopt.n_iter << " " << sopt.residue << std::endl;
if (sopt.residue < sopt.tol) return 0;
}
return 1;
}
template <class SmallMatrix, class SmallVector, class Vector, class Vector2, class Size>
void update (Vector& x, Size k, const SmallMatrix& h, const SmallVector& s, Vector2& v) {
SmallVector y = s;
// back solve:
for (int i = k; i >= 0; i--) {
y(i) /= h(i,i);
for (int j = i - 1; j >= 0; j--)
y(j) -= h(j,i) * y(i);
}
for (Size j = 0; j <= k; j++) {
x += v[j] * y(j);
}
}
template<class Real>
void generate_plane_rotation (const Real& dx, const Real& dy, Real& cs, Real& sn) {
if (dy == Real(0)) {
cs = 1.0;
sn = 0.0;
} else if (abs(dy) > abs(dx)) {
Real temp = dx / dy;
sn = 1.0 / sqrt( 1.0 + temp*temp );
cs = temp * sn;
} else {
Real temp = dy / dx;
cs = 1.0 / sqrt( 1.0 + temp*temp );
sn = temp * cs;
}
}
template<class Real>
void apply_plane_rotation (Real& dx, Real& dy, const Real& cs, const Real& sn) {
Real temp = cs * dx + sn * dy;
dy = -sn * dx + cs * dy;
dx = temp;
}
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.
rheolef Version 7.2 gmres(5rheolef)