MyGMRES.hpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2012, 2014 Kolja Brix <brix@igpm.rwth-aaachen.de>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
 
#ifndef EIGEN_GMRES_H
#define EIGEN_GMRES_H
 
namespace Eigen { 
 
namespace internal {
 
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond,
        int &iters, const int &restart, typename Dest::RealScalar & tol_error) 
{
 
    using std::sqrt;
    using std::abs;
 
    typedef typename Dest::RealScalar RealScalar;
    typedef typename Dest::Scalar Scalar;
    typedef Matrix < Scalar, Dynamic, 1 > VectorType;
    typedef Matrix < Scalar, Dynamic, Dynamic > FMatrixType;
    
    RealScalar tol0 = precond.solve(rhs).norm();
    RealScalar tol = tol_error * tol0;
    const int maxIters = iters;
    iters = 0;
 
    const int m = mat.rows();
 
    VectorType p0 = rhs - mat*x;
    VectorType r0 = precond.solve(p0);
 
    // is initial guess already good enough?
    if(abs(r0.norm()) < tol) {
        return true; 
    }
 
    VectorType w = VectorType::Zero(restart + 1);
 
    FMatrixType H = FMatrixType::Zero(m, restart + 1); // Hessenberg matrix
    VectorType tau = VectorType::Zero(restart + 1);
    std::vector < JacobiRotation < Scalar > > G(restart);
 
    // generate first Householder vector
    VectorType e(m-1);
    RealScalar beta;
    r0.makeHouseholder(e, tau.coeffRef(0), beta);
    w(0)=(Scalar) beta;
    H.bottomLeftCorner(m - 1, 1) = e;
 
    for (int k = 1; k <= restart; ++k) {
 
        ++iters;
 
        VectorType v = VectorType::Unit(m, k - 1), workspace(m);
 
        // apply Householder reflections H_{1} ... H_{k-1} to v
        for (int i = k - 1; i >= 0; --i) {
            v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
        }
 
        // apply matrix M to v:  v = mat * v;
        VectorType t=mat*v;
        v=precond.solve(t);
 
        // apply Householder reflections H_{k-1} ... H_{1} to v
        for (int i = 0; i < k; ++i) {
            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 (k <= restart) {
 
                // generate new Householder vector
                VectorType e(m - k - 1);
                RealScalar beta;
                v.tail(m - k).makeHouseholder(e, tau.coeffRef(k), beta);
                H.col(k).tail(m - k - 1) = e;
 
                // apply Householder reflection H_{k} to v
                v.tail(m - k).applyHouseholderOnTheLeft(H.col(k).tail(m - k - 1), tau.coeffRef(k), workspace.data());
 
            }
        }
 
        if (k > 1) {
                for (int i = 0; i < k - 1; ++i) {
                        // apply old Givens rotations to v
                        v.applyOnTheLeft(i, i + 1, G[i].adjoint());
                }
        }
 
        if (k<m && v(k) != (Scalar) 0) {
                // determine next Givens rotation
                G[k - 1].makeGivens(v(k - 1), v(k));
 
                // apply Givens rotation to v and w
                v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
                w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
        }
 
        // insert coefficients into upper matrix triangle
        H.col(k - 1).head(k) = v.head(k);
        
        bool stop=(k==m || abs(w(k)) < tol || iters == maxIters);
        
        if (stop || k == restart) {
 
                // solve upper triangular system
                VectorType y = w.head(k);
                H.topLeftCorner(k, k).template triangularView < Eigen::Upper > ().solveInPlace(y);
 
                // use Horner-like scheme to calculate solution vector
                VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1);
 
                // 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());
 
                for (int i = k - 2; i >= 0; --i) {
                        x_new += y(i) * VectorType::Unit(m, i);
                        // 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 += x_new;
 
                if (stop) {
                        return true;
                } else {
                        k=0;
 
                        // reset data for a restart  r0 = rhs - mat * x;
                        VectorType p0=mat*x;
                        VectorType p1=precond.solve(p0);
                        r0 = rhs - p1;
//                                 r0_sqnorm = r0.squaredNorm();
                        w = VectorType::Zero(restart + 1);
                        H = FMatrixType::Zero(m, restart + 1);
                        tau = VectorType::Zero(restart + 1);
 
                        // generate first Householder vector
                        RealScalar beta;
                        r0.makeHouseholder(e, tau.coeffRef(0), beta);
                        w(0)=(Scalar) beta;
                        H.bottomLeftCorner(m - 1, 1) = e;
 
                }
 
        }
    }
    return false;
}
 
}// end namespace internal
}// end namespace Eigen
 
#endif // EIGEN_GMRES_H