GMRES.h

// 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/.
// SPDX-License-Identifier: MPL-2.0
 
#ifndef EIGEN_GMRES_H
#define EIGEN_GMRES_H
 
// IWYU pragma: private
#include "./InternalHeaderCheck.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, Index& iters,
           const Index& restart_, typename Dest::RealScalar& tol_error) {
  using std::abs;
  using std::sqrt;
 
  using RealScalar = typename Dest::RealScalar;
  using Scalar = typename Dest::Scalar;
  using VectorType = Matrix<Scalar, Dynamic, 1>;
  using FMatrixType = Matrix<Scalar, Dynamic, Dynamic, ColMajor>;
 
  const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
 
  if (rhs.norm() <= considerAsZero) {
    x.setZero();
    tol_error = 0;
    return true;
  }
 
  RealScalar tol = tol_error;
  const Index maxIters = iters;
  iters = 0;
 
  const Index m = mat.rows();
  // A GMRES cycle cannot use more Krylov vectors than the problem dimension, so cap the internal workspace even if
  // the user requests a larger restart value.
  const Index restart = numext::mini(numext::maxi(restart_, Index(1)), m);
 
  // residual and preconditioned residual
  VectorType p0 = rhs - mat * x;
  VectorType r0 = precond.solve(p0);
 
  const RealScalar r0Norm = r0.norm();
  // w(k) tracks the residual norm of the left-preconditioned system. Normalize it by the matching right-hand
  // side, not by the initial residual, so solveWithGuess() uses the same relative-residual criterion as solve().
  const RealScalar rhsNorm = numext::maxi((x.squaredNorm() == 0) ? r0Norm : precond.solve(rhs).norm(), considerAsZero);
  const RealScalar threshold = numext::maxi(tol * rhsNorm, considerAsZero);
 
  // is initial guess already good enough?
  if (r0Norm < threshold) {
    tol_error = r0Norm / rhsNorm;
    return true;
  }
 
  // storage for Hessenberg matrix and Householder data
  FMatrixType H = FMatrixType::Zero(m, restart + 1);
  VectorType w = VectorType::Zero(restart + 1);
  VectorType tau = VectorType::Zero(restart + 1);
 
  // storage for Jacobi rotations
  std::vector<JacobiRotation<Scalar> > G(restart);
 
  // storage for temporaries
  VectorType t(m), v(m), workspace(m), x_new(m);
 
  // generate first Householder vector
  Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
  RealScalar beta;
  r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
  w(0) = Scalar(beta);
 
  for (Index k = 1; k <= restart; ++k) {
    ++iters;
 
    v = VectorType::Unit(m, k - 1);
 
    // apply Householder reflections H_{1} ... H_{k-1} to v
    // TODO: use a HouseholderSequence
    for (Index 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;
    t.noalias() = mat * v;
    v = precond.solve(t);
 
    // apply Householder reflections H_{k-1} ... H_{1} to v
    // TODO: use a HouseholderSequence
    for (Index 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 && k <= restart) {
      // generate new Householder vector
      Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
      v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
 
      // apply Householder reflection H_{k} to v
      v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
    }
 
    // apply old Givens rotations to v
    for (Index i = 0; i < k - 1; ++i) {
      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);
 
    tol_error = abs(w(k)) / rhsNorm;
    bool stop = (k == m || tol_error < tol || iters == maxIters);
 
    if (stop || k == restart) {
      // solve upper triangular system
      Ref<VectorType> y = w.head(k);
      H.topLeftCorner(k, k).template triangularView<Upper>().solveInPlace(y);
 
      // use Horner-like scheme to calculate solution vector
      x_new.setZero();
      for (Index i = k - 1; i >= 0; --i) {
        x_new(i) += y(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 restart
        p0.noalias() = rhs - mat * x;
        r0 = precond.solve(p0);
 
        // clear Hessenberg matrix and Householder data
        H.setZero();
        w.setZero();
        tau.setZero();
 
        // generate first Householder vector
        r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
        w(0) = Scalar(beta);
      }
    }
  }
 
  return false;
}
 
}  // namespace internal
 
template <typename MatrixType_, typename Preconditioner_ = DiagonalPreconditioner<typename MatrixType_::Scalar> >
class GMRES;
 
namespace internal {
 
template <typename MatrixType_, typename Preconditioner_>
struct traits<GMRES<MatrixType_, Preconditioner_> > {
  using MatrixType = MatrixType_;
  using Preconditioner = Preconditioner_;
};
 
}  // namespace internal

template <typename MatrixType_, typename Preconditioner_>
class GMRES : public IterativeSolverBase<GMRES<MatrixType_, Preconditioner_> > {
 protected:
  using Base = IterativeSolverBase<GMRES>;
  using Base::m_error;
  using Base::m_info;
  using Base::m_isInitialized;
  using Base::m_iterations;
  using Base::matrix;
 
 private:
  Index m_restart = 30;
 
 public:
  using Base::_solve_impl;
  using MatrixType = MatrixType_;
  using Scalar = typename MatrixType::Scalar;
  using RealScalar = typename MatrixType::RealScalar;
  using Preconditioner = Preconditioner_;
 
 public:
  GMRES() : Base(), m_restart(30) {}

  template <typename MatrixDerived>
  explicit GMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30) {}

  Index get_restart() const { return m_restart; }

  void set_restart(const Index restart) { m_restart = restart; }

  template <typename Rhs, typename Dest>
  void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const {
    m_iterations = Base::maxIterations();
    m_error = Base::m_tolerance;
    bool ret = internal::gmres(matrix(), b, x, Base::m_preconditioner, m_iterations, m_restart, m_error);
    m_info = (!ret) ? NumericalIssue : m_error <= Base::m_tolerance ? Success : NoConvergence;
  }
 
 protected:
};
 
}  // end namespace Eigen
 
#endif  // EIGEN_GMRES_H