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gmres

generalized minimum residual algorithm

# 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 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 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 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;
}