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Gael Guennebaud 2015-06-09 15:23:08 +02:00
parent 04665ef9e1
commit cacbc5679d
1 changed files with 150 additions and 148 deletions
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124
unsupported/Eigen/src/IterativeSolvers/GMRES.h
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@ -16,42 +16,42 @@ namespace Eigen {
namespace internal { namespace internal {
/** /**
* Generalized Minimal Residual Algorithm based on the * Generalized Minimal Residual Algorithm based on the
* Arnoldi algorithm implemented with Householder reflections. * Arnoldi algorithm implemented with Householder reflections.
* *
* Parameters: * Parameters:
* \param mat matrix of linear system of equations * \param mat matrix of linear system of equations
* \param Rhs right hand side vector of linear system of equations * \param Rhs right hand side vector of linear system of equations
* \param x on input: initial guess, on output: solution * \param x on input: initial guess, on output: solution
* \param precond preconditioner used * \param precond preconditioner used
* \param iters on input: maximum number of iterations to perform * \param iters on input: maximum number of iterations to perform
* on output: number of iterations performed * on output: number of iterations performed
* \param restart number of iterations for a restart * \param restart number of iterations for a restart
* \param tol_error on input: relative residual tolerance * \param tol_error on input: relative residual tolerance
* on output: residuum achieved * on output: residuum achieved
* *
* \sa IterativeMethods::bicgstab() * \sa IterativeMethods::bicgstab()
* *
* *
* For references, please see: * For references, please see:
* *
* Saad, Y. and Schultz, M. H. * Saad, Y. and Schultz, M. H.
* GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems. * GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
* SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869. * SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869.
* *
* Saad, Y. * Saad, Y.
* Iterative Methods for Sparse Linear Systems. * Iterative Methods for Sparse Linear Systems.
* Society for Industrial and Applied Mathematics, Philadelphia, 2003. * Society for Industrial and Applied Mathematics, Philadelphia, 2003.
* *
* Walker, H. F. * Walker, H. F.
* Implementations of the GMRES method. * Implementations of the GMRES method.
* Comput.Phys.Comm. 53, 1989, pp. 311 - 320. * Comput.Phys.Comm. 53, 1989, pp. 311 - 320.
* *
* Walker, H. F. * Walker, H. F.
* Implementation of the GMRES Method using Householder Transformations. * Implementation of the GMRES Method using Householder Transformations.
* SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163. * SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163.
* *
*/ */
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner> template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond, bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond,
Index &iters, const Index &restart, typename Dest::RealScalar & tol_error) { Index &iters, const Index &restart, typename Dest::RealScalar & tol_error) {
@ -78,8 +78,9 @@ bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Precondition
const RealScalar r0Norm = r0.norm(); const RealScalar r0Norm = r0.norm();
// is initial guess already good enough? // is initial guess already good enough?
if(r0Norm == 0) { if(r0Norm == 0)
tol_error=0; {
tol_error = 0;
return true; return true;
} }
@ -95,10 +96,10 @@ bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Precondition
Ref<VectorType> H0_tail = H.col(0).tail(m - 1); Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
RealScalar beta; RealScalar beta;
r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta); r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
w(0)=(Scalar) beta; w(0) = Scalar(beta);
for (Index k = 1; k <= restart; ++k) {
for (Index k = 1; k <= restart; ++k)
{
++iters; ++iters;
v = VectorType::Unit(m, k - 1); v = VectorType::Unit(m, k - 1);
@ -117,29 +118,30 @@ bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Precondition
v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data()); v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
} }
if (v.tail(m - k).norm() != 0.0) { if (v.tail(m - k).norm() != 0.0)
if (k <= restart) { {
if (k <= restart)
{
// generate new Householder vector // generate new Householder vector
Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1); Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta); v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
// apply Householder reflection H_{k} to v // apply Householder reflection H_{k} to v
v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data()); v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
} }
} }
if (k > 1) { if (k > 1)
for (Index i = 0; i < k - 1; ++i) { {
for (Index i = 0; i < k - 1; ++i)
{
// apply old Givens rotations to v // apply old Givens rotations to v
v.applyOnTheLeft(i, i + 1, G[i].adjoint()); v.applyOnTheLeft(i, i + 1, G[i].adjoint());
} }
} }
if (k<m && v(k) != (Scalar) 0) { if (k<m && v(k) != (Scalar) 0)
{
// determine next Givens rotation // determine next Givens rotation
G[k - 1].makeGivens(v(k - 1), v(k)); G[k - 1].makeGivens(v(k - 1), v(k));
@ -151,13 +153,13 @@ bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Precondition
// insert coefficients into upper matrix triangle // insert coefficients into upper matrix triangle
H.col(k - 1).head(k) = v.head(k); H.col(k - 1).head(k) = v.head(k);
bool stop=(k==m || abs(w(k)) < tol * r0Norm || iters == maxIters); bool stop = (k==m || abs(w(k)) < tol * r0Norm || iters == maxIters);
if (stop || k == restart) {
if (stop || k == restart)
{
// solve upper triangular system // solve upper triangular system
VectorType y = w.head(k); VectorType y = w.head(k);
H.topLeftCorner(k, k).template triangularView < Eigen::Upper > ().solveInPlace(y); H.topLeftCorner(k, k).template triangularView <Upper>().solveInPlace(y);
// use Horner-like scheme to calculate solution vector // use Horner-like scheme to calculate solution vector
VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1); VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1);
@ -165,7 +167,8 @@ bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Precondition
// apply Householder reflection H_{k} to x_new // apply Householder reflection H_{k} to x_new
x_new.tail(m - k + 1).applyHouseholderOnTheLeft(H.col(k - 1).tail(m - k), tau.coeffRef(k - 1), workspace.data()); x_new.tail(m - k + 1).applyHouseholderOnTheLeft(H.col(k - 1).tail(m - k), tau.coeffRef(k - 1), workspace.data());
for (Index i = k - 2; i >= 0; --i) { for (Index i = k - 2; i >= 0; --i)
{
x_new += y(i) * VectorType::Unit(m, i); x_new += y(i) * VectorType::Unit(m, i);
// apply Householder reflection H_{i} to x_new // apply Householder reflection H_{i} to x_new
x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data()); x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
@ -173,9 +176,12 @@ bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Precondition
x += x_new; x += x_new;
if (stop) { if (stop)
{
return true; return true;
} else { }
else
{
k=0; k=0;
// reset data for restart // reset data for restart
@ -190,12 +196,8 @@ bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Precondition
// generate first Householder vector // generate first Householder vector
r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta); r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
w(0)=(Scalar) beta; w(0)=(Scalar) beta;
} }
} }
} }
return false; return false;