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Gael Guennebaud
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@ -16,189 +16,191 @@ namespace Eigen {
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namespace internal {
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namespace internal {
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/**
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/**
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* Generalized Minimal Residual Algorithm based on the
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* Generalized Minimal Residual Algorithm based on the
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* Arnoldi algorithm implemented with Householder reflections.
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* Arnoldi algorithm implemented with Householder reflections.
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*
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*
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* Parameters:
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* Parameters:
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* \param mat matrix of linear system of equations
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* \param mat matrix of linear system of equations
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* \param Rhs right hand side vector of linear system of equations
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* \param Rhs right hand side vector of linear system of equations
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* \param x on input: initial guess, on output: solution
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* \param x on input: initial guess, on output: solution
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* \param precond preconditioner used
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* \param precond preconditioner used
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* \param iters on input: maximum number of iterations to perform
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* \param iters on input: maximum number of iterations to perform
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* on output: number of iterations performed
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* on output: number of iterations performed
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* \param restart number of iterations for a restart
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* \param restart number of iterations for a restart
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* \param tol_error on input: relative residual tolerance
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* \param tol_error on input: relative residual tolerance
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* on output: residuum achieved
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* on output: residuum achieved
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*
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*
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* \sa IterativeMethods::bicgstab()
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* \sa IterativeMethods::bicgstab()
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*
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*
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*
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*
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* For references, please see:
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* For references, please see:
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*
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*
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* Saad, Y. and Schultz, M. H.
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* Saad, Y. and Schultz, M. H.
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* GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
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* GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
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* SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869.
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* SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869.
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*
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*
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* Saad, Y.
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* Saad, Y.
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* Iterative Methods for Sparse Linear Systems.
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* Iterative Methods for Sparse Linear Systems.
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* Society for Industrial and Applied Mathematics, Philadelphia, 2003.
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* Society for Industrial and Applied Mathematics, Philadelphia, 2003.
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*
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*
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* Walker, H. F.
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* Walker, H. F.
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* Implementations of the GMRES method.
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* Implementations of the GMRES method.
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* Comput.Phys.Comm. 53, 1989, pp. 311 - 320.
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* Comput.Phys.Comm. 53, 1989, pp. 311 - 320.
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*
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*
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* Walker, H. F.
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* Walker, H. F.
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* Implementation of the GMRES Method using Householder Transformations.
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* Implementation of the GMRES Method using Householder Transformations.
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* SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163.
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* SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163.
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*
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*
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*/
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*/
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template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
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template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
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bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond,
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bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond,
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Index &iters, const Index &restart, typename Dest::RealScalar & tol_error) {
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Index &iters, const Index &restart, typename Dest::RealScalar & tol_error) {
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using std::sqrt;
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using std::sqrt;
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using std::abs;
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using std::abs;
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typedef typename Dest::RealScalar RealScalar;
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typedef typename Dest::RealScalar RealScalar;
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typedef typename Dest::Scalar Scalar;
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typedef typename Dest::Scalar Scalar;
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typedef Matrix < Scalar, Dynamic, 1 > VectorType;
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typedef Matrix < Scalar, Dynamic, 1 > VectorType;
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typedef Matrix < Scalar, Dynamic, Dynamic > FMatrixType;
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typedef Matrix < Scalar, Dynamic, Dynamic > FMatrixType;
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RealScalar tol = tol_error;
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RealScalar tol = tol_error;
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const Index maxIters = iters;
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const Index maxIters = iters;
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iters = 0;
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iters = 0;
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const Index m = mat.rows();
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const Index m = mat.rows();
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// residual and preconditioned residual
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// residual and preconditioned residual
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VectorType p0 = rhs - mat*x;
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VectorType p0 = rhs - mat*x;
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VectorType r0 = precond.solve(p0);
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VectorType r0 = precond.solve(p0);
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VectorType t(m), v(m), workspace(m);
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VectorType t(m), v(m), workspace(m);
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const RealScalar r0Norm = r0.norm();
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const RealScalar r0Norm = r0.norm();
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// is initial guess already good enough?
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// is initial guess already good enough?
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if(r0Norm == 0) {
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if(r0Norm == 0)
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tol_error=0;
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{
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return true;
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tol_error = 0;
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}
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return true;
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}
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// storage for Hessenberg matrix and Householder data
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// storage for Hessenberg matrix and Householder data
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FMatrixType H = FMatrixType::Zero(m, restart + 1);
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FMatrixType H = FMatrixType::Zero(m, restart + 1);
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VectorType w = VectorType::Zero(restart + 1);
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VectorType w = VectorType::Zero(restart + 1);
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VectorType tau = VectorType::Zero(restart + 1);
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VectorType tau = VectorType::Zero(restart + 1);
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// storage for Jacobi rotations
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// storage for Jacobi rotations
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std::vector < JacobiRotation < Scalar > > G(restart);
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std::vector < JacobiRotation < Scalar > > G(restart);
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// generate first Householder vector
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// generate first Householder vector
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Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
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Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
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RealScalar beta;
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RealScalar beta;
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r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
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r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
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w(0)=(Scalar) beta;
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w(0) = Scalar(beta);
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for (Index k = 1; k <= restart; ++k) {
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for (Index k = 1; k <= restart; ++k)
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{
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++iters;
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++iters;
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v = VectorType::Unit(m, k - 1);
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v = VectorType::Unit(m, k - 1);
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// apply Householder reflections H_{1} ... H_{k-1} to v
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for (Index i = k - 1; i >= 0; --i) {
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v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
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}
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// apply Householder reflections H_{1} ... H_{k-1} to v
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// apply matrix M to v: v = mat * v;
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for (Index i = k - 1; i >= 0; --i) {
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t=mat*v;
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v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
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v=precond.solve(t);
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}
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// apply matrix M to v: v = mat * v;
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// apply Householder reflections H_{k-1} ... H_{1} to v
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t=mat*v;
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for (Index i = 0; i < k; ++i) {
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v=precond.solve(t);
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v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
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}
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// apply Householder reflections H_{k-1} ... H_{1} to v
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if (v.tail(m - k).norm() != 0.0)
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for (Index i = 0; i < k; ++i) {
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{
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v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
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if (k <= restart)
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}
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{
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// generate new Householder vector
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if (v.tail(m - k).norm() != 0.0) {
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if (k <= restart) {
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// generate new Householder vector
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Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
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Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
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v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
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v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
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// apply Householder reflection H_{k} to v
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// apply Householder reflection H_{k} to v
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v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
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v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
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}
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}
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}
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if (k > 1)
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}
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{
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for (Index i = 0; i < k - 1; ++i)
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{
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// apply old Givens rotations to v
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v.applyOnTheLeft(i, i + 1, G[i].adjoint());
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}
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}
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if (k > 1) {
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if (k<m && v(k) != (Scalar) 0)
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for (Index i = 0; i < k - 1; ++i) {
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{
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// apply old Givens rotations to v
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// determine next Givens rotation
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v.applyOnTheLeft(i, i + 1, G[i].adjoint());
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G[k - 1].makeGivens(v(k - 1), v(k));
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}
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}
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if (k<m && v(k) != (Scalar) 0) {
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// apply Givens rotation to v and w
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v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
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w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
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}
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// determine next Givens rotation
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// insert coefficients into upper matrix triangle
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G[k - 1].makeGivens(v(k - 1), v(k));
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H.col(k - 1).head(k) = v.head(k);
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// apply Givens rotation to v and w
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bool stop = (k==m || abs(w(k)) < tol * r0Norm || iters == maxIters);
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v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
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w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
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}
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// insert coefficients into upper matrix triangle
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if (stop || k == restart)
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H.col(k - 1).head(k) = v.head(k);
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{
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// solve upper triangular system
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VectorType y = w.head(k);
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H.topLeftCorner(k, k).template triangularView <Upper>().solveInPlace(y);
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bool stop=(k==m || abs(w(k)) < tol * r0Norm || iters == maxIters);
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// use Horner-like scheme to calculate solution vector
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VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1);
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if (stop || k == restart) {
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// apply Householder reflection H_{k} to x_new
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x_new.tail(m - k + 1).applyHouseholderOnTheLeft(H.col(k - 1).tail(m - k), tau.coeffRef(k - 1), workspace.data());
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// solve upper triangular system
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for (Index i = k - 2; i >= 0; --i)
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VectorType y = w.head(k);
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{
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H.topLeftCorner(k, k).template triangularView < Eigen::Upper > ().solveInPlace(y);
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x_new += y(i) * VectorType::Unit(m, i);
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// apply Householder reflection H_{i} to x_new
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x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
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}
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// use Horner-like scheme to calculate solution vector
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x += x_new;
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VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1);
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// apply Householder reflection H_{k} to x_new
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if (stop)
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x_new.tail(m - k + 1).applyHouseholderOnTheLeft(H.col(k - 1).tail(m - k), tau.coeffRef(k - 1), workspace.data());
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{
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return true;
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}
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else
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{
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k=0;
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for (Index i = k - 2; i >= 0; --i) {
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// reset data for restart
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x_new += y(i) * VectorType::Unit(m, i);
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p0 = rhs - mat*x;
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// apply Householder reflection H_{i} to x_new
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r0 = precond.solve(p0);
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x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
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}
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x += x_new;
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// clear Hessenberg matrix and Householder data
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H = FMatrixType::Zero(m, restart + 1);
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w = VectorType::Zero(restart + 1);
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tau = VectorType::Zero(restart + 1);
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if (stop) {
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// generate first Householder vector
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return true;
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r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
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} else {
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w(0)=(Scalar) beta;
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k=0;
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}
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}
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}
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// reset data for restart
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return false;
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p0 = rhs - mat*x;
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r0 = precond.solve(p0);
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// clear Hessenberg matrix and Householder data
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H = FMatrixType::Zero(m, restart + 1);
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w = VectorType::Zero(restart + 1);
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tau = VectorType::Zero(restart + 1);
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// generate first Householder vector
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r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
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w(0)=(Scalar) beta;
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}
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}
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}
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return false;
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}
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}
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@ -314,8 +316,8 @@ public:
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failed = true;
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failed = true;
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}
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}
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m_info = failed ? NumericalIssue
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m_info = failed ? NumericalIssue
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: m_error <= Base::m_tolerance ? Success
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: m_error <= Base::m_tolerance ? Success
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: NoConvergence;
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: NoConvergence;
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m_isInitialized = true;
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m_isInitialized = true;
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}
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}
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