MINRES.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2012 Giacomo Po <gpo@ucla.edu>
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2018 David Hyde <dabh@stanford.edu>
//
// 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_MINRES_H
#define EIGEN_MINRES_H

// IWYU pragma: private
#include "./InternalHeaderCheck.h"

namespace Eigen {

namespace internal {

template <typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
EIGEN_DONT_INLINE void minres(const MatrixType& mat, const Rhs& rhs, Dest& x, const Preconditioner& precond,
                              Index& iters, typename Dest::RealScalar& tol_error) {
  using std::sqrt;
  typedef typename Dest::RealScalar RealScalar;
  typedef typename Dest::Scalar Scalar;
  typedef Matrix<Scalar, Dynamic, 1> VectorType;

  // Check for zero rhs
  const RealScalar rhsNorm2(rhs.squaredNorm());
  if (rhsNorm2 == 0) {
    x.setZero();
    iters = 0;
    tol_error = 0;
    return;
  }

  // initialize
  const Index maxIters(iters);                                    // initialize maxIters to iters
  const Index N(mat.cols());                                      // the size of the matrix
  const RealScalar threshold2(tol_error * tol_error * rhsNorm2);  // convergence threshold (compared to residualNorm2)

  // Initialize preconditioned Lanczos
  VectorType v_old(N);                // will be initialized inside loop
  VectorType v(VectorType::Zero(N));  // initialize v
  VectorType v_new(rhs - mat * x);    // initialize v_new
  RealScalar residualNorm2(v_new.squaredNorm());
  VectorType w(N);                         // will be initialized inside loop
  VectorType w_new(precond.solve(v_new));  // initialize w_new
                                           //            RealScalar beta; // will be initialized inside loop
  RealScalar beta_new2(v_new.dot(w_new));
  eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
  RealScalar beta_new(sqrt(beta_new2));
  const RealScalar beta_one(beta_new);
  // Initialize other variables
  RealScalar c(1.0);  // the cosine of the Givens rotation
  RealScalar c_old(1.0);
  RealScalar s(0.0);                      // the sine of the Givens rotation
  RealScalar s_old(0.0);                  // the sine of the Givens rotation
  VectorType p_oold(N);                   // will be initialized in loop
  VectorType p_old(VectorType::Zero(N));  // initialize p_old=0
  VectorType p(p_old);                    // initialize p=0
  RealScalar eta(1.0);

  iters = 0;  // reset iters
  while (iters < maxIters) {
    // Preconditioned Lanczos
    /* Note that there are 4 variants on the Lanczos algorithm. These are
     * described in Paige, C. C. (1972). Computational variants of
     * the Lanczos method for the eigenproblem. IMA Journal of Applied
     * Mathematics, 10(3), 373-381. The current implementation corresponds
     * to the case A(2,7) in the paper. It also corresponds to
     * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear
     * Systems, 2003 p.173. For the preconditioned version see
     * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987).
     */
    const RealScalar beta(beta_new);
    v_old = v;          // update: at first time step, this makes v_old = 0 so value of beta doesn't matter
    v_new /= beta_new;  // overwrite v_new for next iteration
    w_new /= beta_new;  // overwrite w_new for next iteration
    v = v_new;          // update
    w = w_new;          // update
    v_new.noalias() = mat * w - beta * v_old;  // compute v_new
    const RealScalar alpha = v_new.dot(w);
    v_new -= alpha * v;            // overwrite v_new
    w_new = precond.solve(v_new);  // overwrite w_new
    beta_new2 = v_new.dot(w_new);  // compute beta_new
    eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
    beta_new = sqrt(beta_new2);  // compute beta_new

    // Givens rotation
    const RealScalar r2 = s * alpha + c * c_old * beta;  // s, s_old, c and c_old are still from previous iteration
    const RealScalar r3 = s_old * beta;                  // s, s_old, c and c_old are still from previous iteration
    const RealScalar r1_hat = c * alpha - c_old * s * beta;
    const RealScalar r1 = sqrt(std::pow(r1_hat, 2) + std::pow(beta_new, 2));
    c_old = c;          // store for next iteration
    s_old = s;          // store for next iteration
    c = r1_hat / r1;    // new cosine
    s = beta_new / r1;  // new sine

    // Update solution
    p_oold = p_old;
    p_old = p;
    p.noalias() = (w - r2 * p_old - r3 * p_oold) / r1;  // IS NOALIAS REQUIRED?
    x += beta_one * c * eta * p;

    /* Update the squared residual. Note that this is the estimated residual.
    The real residual |Ax-b|^2 may be slightly larger */
    residualNorm2 *= s * s;

    if (residualNorm2 < threshold2) {
      break;
    }

    eta = -s * eta;  // update eta
    iters++;         // increment iteration number (for output purposes)
  }

  /* Compute error. Note that this is the estimated error. The real
   error |Ax-b|/|b| may be slightly larger */
  tol_error = std::sqrt(residualNorm2 / rhsNorm2);
}

}  // namespace internal

template <typename MatrixType_, int UpLo_ = Lower, typename Preconditioner_ = IdentityPreconditioner>
class MINRES;

namespace internal {

template <typename MatrixType_, int UpLo_, typename Preconditioner_>
struct traits<MINRES<MatrixType_, UpLo_, Preconditioner_> > {
  typedef MatrixType_ MatrixType;
  typedef Preconditioner_ Preconditioner;
};

}  // namespace internal

template <typename MatrixType_, int UpLo_, typename Preconditioner_>
class MINRES : public IterativeSolverBase<MINRES<MatrixType_, UpLo_, Preconditioner_> > {
  typedef IterativeSolverBase<MINRES> Base;
  using Base::m_error;
  using Base::m_info;
  using Base::m_isInitialized;
  using Base::m_iterations;
  using Base::matrix;

 public:
  using Base::_solve_impl;
  typedef MatrixType_ MatrixType;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef Preconditioner_ Preconditioner;

  enum { UpLo = UpLo_ };

 public:
  MINRES() : Base() {}

  template <typename MatrixDerived>
  explicit MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}

  ~MINRES() {}

  template <typename Rhs, typename Dest>
  void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const {
    typedef typename Base::MatrixWrapper MatrixWrapper;
    typedef typename Base::ActualMatrixType ActualMatrixType;
    enum {
      TransposeInput = (!MatrixWrapper::MatrixFree) && (UpLo == (Lower | Upper)) && (!MatrixType::IsRowMajor) &&
                       (!NumTraits<Scalar>::IsComplex)
    };
    typedef std::conditional_t<TransposeInput, Transpose<const ActualMatrixType>, ActualMatrixType const&>
        RowMajorWrapper;
    EIGEN_STATIC_ASSERT(internal::check_implication(MatrixWrapper::MatrixFree, UpLo == (Lower | Upper)),
                        MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
    typedef std::conditional_t<UpLo == (Lower | Upper), RowMajorWrapper,
                               typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type>
        SelfAdjointWrapper;

    m_iterations = Base::maxIterations();
    m_error = Base::m_tolerance;
    RowMajorWrapper row_mat(matrix());
    internal::minres(SelfAdjointWrapper(row_mat), b, x, Base::m_preconditioner, m_iterations, m_error);
    m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
  }

 protected:
};

}  // end namespace Eigen

#endif  // EIGEN_MINRES_H