BiCGSTAB.h

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@inria.fr>
//
// 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_BICGSTAB_H
#define EIGEN_BICGSTAB_H

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

namespace Eigen {

namespace internal {

template <typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
bool bicgstab(const MatrixType& mat, const Rhs& rhs, Dest& x, const Preconditioner& precond, Index& iters,
              typename Dest::RealScalar& tol_error) {
  typedef typename Dest::RealScalar RealScalar;
  typedef typename Dest::Scalar Scalar;
  typedef Matrix<Scalar, Dynamic, 1> VectorType;
  RealScalar tol = tol_error;
  Index maxIters = iters;

  Index n = mat.cols();
  VectorType r = rhs - mat * x;
  VectorType r0 = r;

  RealScalar r0_norm = r0.stableNorm();
  RealScalar r_norm = r0_norm;
  RealScalar rhs_norm = rhs.stableNorm();
  if (rhs_norm == 0) {
    x.setZero();
    return true;
  }
  Scalar rho(1);
  Scalar alpha(0);
  Scalar w(1);

  VectorType v = VectorType::Zero(n), p = VectorType::Zero(n);
  VectorType y(n), z(n);
  VectorType kt(n), ks(n);

  VectorType s(n), t(n);

  RealScalar eps = NumTraits<Scalar>::epsilon();
  Index i = 0;
  Index restarts = 0;

  while (r_norm > tol && i < maxIters) {
    Scalar rho_old = rho;
    rho = r0.dot(r);
    if (Eigen::numext::abs(rho) / Eigen::numext::maxi(r0_norm, r_norm) < eps * Eigen::numext::mini(r0_norm, r_norm)) {
      // The new residual vector became too orthogonal to the arbitrarily chosen direction r0
      // Let's restart with a new r0:
      r = rhs - mat * x;
      r0 = r;
      rho = r.squaredNorm();
      r0_norm = r.stableNorm();
      alpha = Scalar(0);
      w = Scalar(1);
      if (restarts++ == 0) i = 0;
    }
    Scalar beta = (rho / rho_old) * (alpha / w);
    p = r + beta * (p - w * v);

    y = precond.solve(p);

    v.noalias() = mat * y;
    Scalar theta = r0.dot(v);
    // For small angles ∠(r0, v) < eps, random restart.
    RealScalar v_norm = v.stableNorm();
    if (Eigen::numext::abs(theta) / Eigen::numext::maxi(r0_norm, v_norm) < eps * Eigen::numext::mini(r0_norm, v_norm)) {
      r = rhs - mat * x;
      r0.setRandom();
      r0_norm = r0.stableNorm();
      rho = Scalar(1);
      alpha = Scalar(0);
      w = Scalar(1);
      if (restarts++ == 0) i = 0;
      continue;
    }
    alpha = rho / theta;
    s = r - alpha * v;

    z = precond.solve(s);
    t.noalias() = mat * z;

    RealScalar tmp = t.squaredNorm();
    if (tmp > RealScalar(0)) {
      w = t.dot(s) / tmp;
    } else {
      w = Scalar(0);
    }
    x += alpha * y + w * z;
    r = s - w * t;
    r_norm = r.stableNorm();
    ++i;
  }

  tol_error = r_norm / rhs_norm;
  iters = i;
  return true;
}

}  // namespace internal

template <typename MatrixType_, typename Preconditioner_ = DiagonalPreconditioner<typename MatrixType_::Scalar> >
class BiCGSTAB;

namespace internal {

template <typename MatrixType_, typename Preconditioner_>
struct traits<BiCGSTAB<MatrixType_, Preconditioner_> > {
  typedef MatrixType_ MatrixType;
  typedef Preconditioner_ Preconditioner;
};

}  // namespace internal

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

 public:
  typedef MatrixType_ MatrixType;
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef Preconditioner_ Preconditioner;

 public:
  BiCGSTAB() : Base() {}

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

  ~BiCGSTAB() {}

  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::bicgstab(matrix(), b, x, Base::m_preconditioner, m_iterations, m_error);

    m_info = (!ret) ? NumericalIssue : m_error <= Base::m_tolerance ? Success : NoConvergence;
  }

 protected:
};

}  // end namespace Eigen

#endif  // EIGEN_BICGSTAB_H