solver_cg.h

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// ---------------------------------------------------------------------
//
// Copyright (C) 1998 - 2022 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------
 
#ifndef dealii_solver_cg_h
#define dealii_solver_cg_h
 
 
#include <deal.II/base/config.h>
 
#include <deal.II/base/exceptions.h>
#include <deal.II/base/logstream.h>
#include <deal.II/base/subscriptor.h>
#include <deal.II/base/vectorization.h>
 
#include <deal.II/lac/solver.h>
#include <deal.II/lac/solver_control.h>
#include <deal.II/lac/tridiagonal_matrix.h>
 
#include <cmath>
 
DEAL_II_NAMESPACE_OPEN
 
// forward declaration
#ifndef DOXYGEN
class PreconditionIdentity;
namespace LinearAlgebra
{
  namespace distributed
  {
    template <typename, typename>
    class Vector;
  }
} // namespace LinearAlgebra
#endif
 
 
 
template <typename VectorType = Vector<double>>
class SolverCG : public SolverBase<VectorType>
{
public:
  using size_type = types::global_dof_index;
 
  struct AdditionalData
  {};
 
  SolverCG(SolverControl &           cn,
           VectorMemory<VectorType> &mem,
           const AdditionalData &    data = AdditionalData());
 
  SolverCG(SolverControl &cn, const AdditionalData &data = AdditionalData());
 
  virtual ~SolverCG() override = default;
 
  template <typename MatrixType, typename PreconditionerType>
  void
  solve(const MatrixType &        A,
        VectorType &              x,
        const VectorType &        b,
        const PreconditionerType &preconditioner);
 
  boost::signals2::connection
  connect_coefficients_slot(
    const std::function<void(typename VectorType::value_type,
                             typename VectorType::value_type)> &slot);
 
  boost::signals2::connection
  connect_condition_number_slot(const std::function<void(double)> &slot,
                                const bool every_iteration = false);
 
  boost::signals2::connection
  connect_eigenvalues_slot(
    const std::function<void(const std::vector<double> &)> &slot,
    const bool every_iteration = false);
 
protected:
  virtual void
  print_vectors(const unsigned int step,
                const VectorType & x,
                const VectorType & r,
                const VectorType & d) const;
 
  static void
  compute_eigs_and_cond(
    const std::vector<typename VectorType::value_type> &diagonal,
    const std::vector<typename VectorType::value_type> &offdiagonal,
    const boost::signals2::signal<void(const std::vector<double> &)>
      &                                          eigenvalues_signal,
    const boost::signals2::signal<void(double)> &cond_signal);
 
  AdditionalData additional_data;
 
  boost::signals2::signal<void(typename VectorType::value_type,
                               typename VectorType::value_type)>
    coefficients_signal;
 
  boost::signals2::signal<void(double)> condition_number_signal;
 
  boost::signals2::signal<void(double)> all_condition_numbers_signal;
 
  boost::signals2::signal<void(const std::vector<double> &)> eigenvalues_signal;
 
  boost::signals2::signal<void(const std::vector<double> &)>
    all_eigenvalues_signal;
 
  bool determine_beta_by_flexible_formula;
};
 
 
 
template <typename VectorType = Vector<double>>
class SolverFlexibleCG : public SolverCG<VectorType>
{
public:
  using size_type = types::global_dof_index;
 
  struct AdditionalData
  {};
 
  SolverFlexibleCG(SolverControl &           cn,
                   VectorMemory<VectorType> &mem,
                   const AdditionalData &    data = AdditionalData());
 
  SolverFlexibleCG(SolverControl &       cn,
                   const AdditionalData &data = AdditionalData());
};
 
 
/*------------------------- Implementation ----------------------------*/
 
#ifndef DOXYGEN
 
 
 
template <typename VectorType>
SolverCG<VectorType>::SolverCG(SolverControl &           cn,
                               VectorMemory<VectorType> &mem,
                               const AdditionalData &    data)
  : SolverBase<VectorType>(cn, mem)
  , additional_data(data)
  , determine_beta_by_flexible_formula(false)
{}
 
 
 
template <typename VectorType>
SolverCG<VectorType>::SolverCG(SolverControl &cn, const AdditionalData &data)
  : SolverBase<VectorType>(cn)
  , additional_data(data)
  , determine_beta_by_flexible_formula(false)
{}
 
 
 
template <typename VectorType>
void
SolverCG<VectorType>::print_vectors(const unsigned int,
                                    const VectorType &,
                                    const VectorType &,
                                    const VectorType &) const
{}
 
 
 
template <typename VectorType>
inline void
SolverCG<VectorType>::compute_eigs_and_cond(
  const std::vector<typename VectorType::value_type> &diagonal,
  const std::vector<typename VectorType::value_type> &offdiagonal,
  const boost::signals2::signal<void(const std::vector<double> &)>
    &                                          eigenvalues_signal,
  const boost::signals2::signal<void(double)> &cond_signal)
{
  // Avoid computing eigenvalues unless they are needed.
  if (!cond_signal.empty() || !eigenvalues_signal.empty())
    {
      TridiagonalMatrix<typename VectorType::value_type> T(diagonal.size(),
                                                           true);
      for (size_type i = 0; i < diagonal.size(); ++i)
        {
          T(i, i) = diagonal[i];
          if (i < diagonal.size() - 1)
            T(i, i + 1) = offdiagonal[i];
        }
      T.compute_eigenvalues();
      // Need two eigenvalues to estimate the condition number.
      if (diagonal.size() > 1)
        {
          auto condition_number = T.eigenvalue(T.n() - 1) / T.eigenvalue(0);
          // Condition number is real valued and nonnegative; simply take
          // the absolute value:
          cond_signal(std::abs(condition_number));
        }
      // Avoid copying the eigenvalues of T to a vector unless a signal is
      // connected.
      if (!eigenvalues_signal.empty())
        {
          std::vector<double> eigenvalues(T.n());
          for (unsigned int j = 0; j < T.n(); ++j)
            {
              // for a hermitian matrix, all eigenvalues are real-valued
              // and non-negative, simply return the absolute value:
              eigenvalues[j] = std::abs(T.eigenvalue(j));
            }
          eigenvalues_signal(eigenvalues);
        }
    }
}
 
 
 
namespace internal
{
  namespace SolverCG
  {
    // This base class is used to select different variants of the conjugate
    // gradient solver. The default variant is used for standard matrix and
    // preconditioner arguments, as provided by the derived class
    // IterationWork below, but there is also a specialized variant further
    // down that uses SFINAE to identify whether matrices and preconditioners
    // support special operations on sub-ranges of the vectors.
    template <typename VectorType,
              typename MatrixType,
              typename PreconditionerType>
    struct IterationWorkerBase
    {
      using Number = typename VectorType::value_type;
 
      const MatrixType &        A;
      const PreconditionerType &preconditioner;
      const bool                flexible;
      VectorType &              x;
 
      typename VectorMemory<VectorType>::Pointer r_pointer;
      typename VectorMemory<VectorType>::Pointer p_pointer;
      typename VectorMemory<VectorType>::Pointer v_pointer;
      typename VectorMemory<VectorType>::Pointer z_pointer;
 
      // Define some aliases for simpler access, using the variables 'r' for
      // the residual b - A*x, 'p' for the search direction, and 'v' for the
      // auxiliary vector. This naming convention is used e.g. by the
      // description on
      // https://en.wikipedia.org/wiki/Conjugate_gradient_method. The variable
      // 'z' gets only used for the flexible variant of the CG method.
      VectorType &r;
      VectorType &p;
      VectorType &v;
      VectorType &z;
 
      Number r_dot_preconditioner_dot_r;
      Number alpha;
      Number beta;
      double residual_norm;
      Number previous_alpha;
 
      IterationWorkerBase(const MatrixType &        A,
                          const PreconditionerType &preconditioner,
                          const bool                flexible,
                          VectorMemory<VectorType> &memory,
                          VectorType &              x)
        : A(A)
        , preconditioner(preconditioner)
        , flexible(flexible)
        , x(x)
        , r_pointer(memory)
        , p_pointer(memory)
        , v_pointer(memory)
        , z_pointer(memory)
        , r(*r_pointer)
        , p(*p_pointer)
        , v(*v_pointer)
        , z(*z_pointer)
        , r_dot_preconditioner_dot_r(Number())
        , alpha(Number())
        , beta(Number())
        , residual_norm(0.0)
        , previous_alpha(Number())
      {}
 
      void
      startup(const VectorType &b)
      {
        // Initialize without setting the vector entries, as those would soon
        // be overwritten anyway
        r.reinit(x, true);
        p.reinit(x, true);
        v.reinit(x, true);
        if (flexible)
          z.reinit(x, true);
 
        // compute residual. if vector is zero, then short-circuit the full
        // computation
        if (!x.all_zero())
          {
            A.vmult(r, x);
            r.sadd(-1., 1., b);
          }
        else
          r.equ(1., b);
 
        residual_norm = r.l2_norm();
      }
    };
 
 
 
    // Implementation of a conjugate gradient operation with matrices and
    // preconditioners without special capabilities
    template <typename VectorType,
              typename MatrixType,
              typename PreconditionerType,
              typename = int>
    struct IterationWorker
      : public IterationWorkerBase<VectorType, MatrixType, PreconditionerType>
    {
      using BaseClass =
        IterationWorkerBase<VectorType, MatrixType, PreconditionerType>;
 
      IterationWorker(const MatrixType &        A,
                      const PreconditionerType &preconditioner,
                      const bool                flexible,
                      VectorMemory<VectorType> &memory,
                      VectorType &              x)
        : BaseClass(A, preconditioner, flexible, memory, x)
      {}
 
      using BaseClass::A;
      using BaseClass::alpha;
      using BaseClass::beta;
      using BaseClass::p;
      using BaseClass::preconditioner;
      using BaseClass::r;
      using BaseClass::r_dot_preconditioner_dot_r;
      using BaseClass::residual_norm;
      using BaseClass::v;
      using BaseClass::x;
      using BaseClass::z;
 
      void
      do_iteration(const unsigned int iteration_index)
      {
        using Number = typename VectorType::value_type;
 
        const Number previous_r_dot_preconditioner_dot_r =
          r_dot_preconditioner_dot_r;
 
        if (std::is_same<PreconditionerType, PreconditionIdentity>::value ==
            false)
          {
            preconditioner.vmult(v, r);
            r_dot_preconditioner_dot_r = r * v;
          }
        else
          r_dot_preconditioner_dot_r = residual_norm * residual_norm;
 
        const VectorType &direction =
          std::is_same<PreconditionerType, PreconditionIdentity>::value ? r : v;
 
        if (iteration_index > 1)
          {
            Assert(std::abs(previous_r_dot_preconditioner_dot_r) != 0.,
                   ExcDivideByZero());
            beta =
              r_dot_preconditioner_dot_r / previous_r_dot_preconditioner_dot_r;
            if (this->flexible)
              beta -= (r * z) / previous_r_dot_preconditioner_dot_r;
            p.sadd(beta, 1., direction);
          }
        else
          p.equ(1., direction);
 
        if (this->flexible)
          z.swap(v);
 
        A.vmult(v, p);
 
        const Number p_dot_A_dot_p = p * v;
        Assert(std::abs(p_dot_A_dot_p) != 0., ExcDivideByZero());
 
        this->previous_alpha = alpha;
        alpha                = r_dot_preconditioner_dot_r / p_dot_A_dot_p;
 
        x.add(alpha, p);
        residual_norm = std::sqrt(std::abs(r.add_and_dot(-alpha, v, r)));
      }
 
      void
      finalize_after_convergence(const unsigned int)
      {}
    };
 
 
    // In the following, we provide a specialization of the above
    // IterationWorker class that picks up particular features in the matrix
    // and preconditioners.
 
    // a helper type-trait that leverage SFINAE to figure out if MatrixType has
    // ... MatrixType::vmult(VectorType &, const VectorType&,
    // std::function<...>, std::function<...>) const
    template <typename MatrixType, typename VectorType>
    using vmult_functions_t = decltype(std::declval<MatrixType const>().vmult(
      std::declval<VectorType &>(),
      std::declval<const VectorType &>(),
      std::declval<
        const std::function<void(const unsigned int, const unsigned int)> &>(),
      std::declval<const std::function<void(const unsigned int,
                                            const unsigned int)> &>()));
 
    template <typename MatrixType, typename VectorType>
    constexpr bool has_vmult_functions =
      is_supported_operation<vmult_functions_t, MatrixType, VectorType>;
 
    // a helper type-trait that leverage SFINAE to figure out if
    // PreconditionerType has ... PreconditionerType::apply_to_subrange(const
    // unsigned int, const unsigned int, const Number*, Number*) const
    template <typename PreconditionerType>
    using apply_to_subrange_t =
      decltype(std::declval<PreconditionerType const>()
                 .apply_to_subrange(0U, 0U, nullptr, nullptr));
 
    template <typename PreconditionerType>
    constexpr bool has_apply_to_subrange =
      is_supported_operation<apply_to_subrange_t, PreconditionerType>;
 
    // a helper type-trait that leverage SFINAE to figure out if
    // PreconditionerType has ... PreconditionerType::apply(const
    // unsigned int, const Number) const
    template <typename PreconditionerType>
    using apply_t =
      decltype(std::declval<PreconditionerType const>().apply(0U, 0.0));
 
    template <typename PreconditionerType>
    constexpr bool has_apply =
      is_supported_operation<apply_t, PreconditionerType>;
 
 
    // Internal function to run one iteration of the conjugate gradient solver
    // for matrices and preconditioners that support interleaving the vector
    // updates with the matrix-vector product.
    template <typename VectorType,
              typename MatrixType,
              typename PreconditionerType>
    struct IterationWorker<
      VectorType,
      MatrixType,
      PreconditionerType,
      typename std::enable_if<has_vmult_functions<MatrixType, VectorType> &&
                                (has_apply_to_subrange<PreconditionerType> ||
                                 has_apply<PreconditionerType>)&&std::
                                  is_same<VectorType,
                                          LinearAlgebra::distributed::Vector<
                                            typename VectorType::value_type,
                                            MemorySpace::Host>>::value,
                              int>::type>
      : public IterationWorkerBase<VectorType, MatrixType, PreconditionerType>
    {
      using Number = typename VectorType::value_type;
 
      Number next_r_dot_preconditioner_dot_r;
      Number previous_beta;
 
      IterationWorker(const MatrixType &        A,
                      const PreconditionerType &preconditioner,
                      const bool                flexible,
                      VectorMemory<VectorType> &memory,
                      VectorType &              x)
        : IterationWorkerBase<VectorType, MatrixType, PreconditionerType>(
            A,
            preconditioner,
            flexible,
            memory,
            x)
        , next_r_dot_preconditioner_dot_r(0.)
        , previous_beta(0.)
      {}
 
      // This is the main iteration function, that will use some of the
      // specialized functions below
      void
      do_iteration(const unsigned int iteration_index)
      {
        if (iteration_index > 1)
          {
            previous_beta = this->beta;
            this->beta    = next_r_dot_preconditioner_dot_r /
                         this->r_dot_preconditioner_dot_r;
          }
 
        std::array<VectorizedArray<Number>, 7> vectorized_sums = {};
 
        this->A.vmult(
          this->v,
          this->p,
          [&](const unsigned int begin, const unsigned int end) {
            operation_before_loop(iteration_index, begin, end);
          },
          [&](const unsigned int begin, const unsigned int end) {
            operation_after_loop(begin, end, vectorized_sums);
          });
 
        std::array<Number, 7> scalar_sums;
        for (unsigned int i = 0; i < 7; ++i)
          scalar_sums[i] = vectorized_sums[i][0];
        for (unsigned int l = 1; l < VectorizedArray<Number>::size(); ++l)
          for (unsigned int i = 0; i < 7; ++i)
            scalar_sums[i] += vectorized_sums[i][l];
 
        Utilities::MPI::sum(::ArrayView<const Number>(scalar_sums.data(),
                                                            7),
                            this->r.get_mpi_communicator(),
                            ::ArrayView<Number>(scalar_sums.data(), 7));
 
        this->r_dot_preconditioner_dot_r = scalar_sums[6];
 
        const Number p_dot_A_dot_p = scalar_sums[0];
        Assert(std::abs(p_dot_A_dot_p) != 0., ExcDivideByZero());
 
        this->previous_alpha = this->alpha;
        this->alpha          = this->r_dot_preconditioner_dot_r / p_dot_A_dot_p;
 
        // Round-off errors near zero might yield negative values, so take
        // the absolute value in the next two formulas
        this->residual_norm = std::sqrt(std::abs(
          scalar_sums[3] +
          this->alpha * (-2. * scalar_sums[2] + this->alpha * scalar_sums[1])));
 
        next_r_dot_preconditioner_dot_r = std::abs(
          this->r_dot_preconditioner_dot_r +
          this->alpha * (-2. * scalar_sums[4] + this->alpha * scalar_sums[5]));
      }
 
      // Function that we use if the PreconditionerType implements an apply()
      // function
      template <typename U = void>
      typename std::enable_if<has_apply<PreconditionerType>, U>::type
      operation_before_loop(const unsigned int iteration_index,
                            const unsigned int start_range,
                            const unsigned int end_range) const
      {
        Number *               x       = this->x.begin();
        Number *               r       = this->r.begin();
        Number *               p       = this->p.begin();
        Number *               v       = this->v.begin();
        const Number           alpha   = this->alpha;
        const Number           beta    = this->beta;
        constexpr unsigned int n_lanes = VectorizedArray<Number>::size();
        const unsigned int     end_regular =
          start_range + (end_range - start_range) / n_lanes * n_lanes;
        if (iteration_index == 1)
          {
            // Vectorize by hand since compilers are often pretty bad at
            // doing these steps automatically even with
            // DEAL_II_OPENMP_SIMD_PRAGMA
            for (unsigned int j = start_range; j < end_regular; j += n_lanes)
              {
                VectorizedArray<Number> rj, pj;
                rj.load(r + j);
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int l = 0; l < n_lanes; ++l)
                  pj[l] = this->preconditioner.apply(j + l, rj[l]);
                pj.store(p + j);
                rj = VectorizedArray<Number>();
                rj.store(v + j);
              }
            for (unsigned int j = end_regular; j < end_range; ++j)
              {
                p[j] = this->preconditioner.apply(j, r[j]);
                v[j] = Number();
              }
          }
        else if (iteration_index % 2 == 0)
          {
            for (unsigned int j = start_range; j < end_regular; j += n_lanes)
              {
                VectorizedArray<Number> rj, vj, pj, prec_rj;
                rj.load(r + j);
                vj.load(v + j);
                rj -= alpha * vj;
                rj.store(r + j);
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int l = 0; l < n_lanes; ++l)
                  prec_rj[l] = this->preconditioner.apply(j + l, rj[l]);
                pj.load(p + j);
                pj = beta * pj + prec_rj;
                pj.store(p + j);
                rj = VectorizedArray<Number>();
                rj.store(v + j);
              }
            for (unsigned int j = end_regular; j < end_range; ++j)
              {
                r[j] -= alpha * v[j];
                p[j] = beta * p[j] + this->preconditioner.apply(j, r[j]);
                v[j] = Number();
              }
          }
        else
          {
            const Number alpha_plus_previous_alpha_over_beta =
              alpha + this->previous_alpha / this->previous_beta;
            const Number previous_alpha_over_beta =
              this->previous_alpha / this->previous_beta;
            for (unsigned int j = start_range; j < end_regular; j += n_lanes)
              {
                VectorizedArray<Number> rj, vj, pj, xj, prec_rj, prec_vj;
                xj.load(x + j);
                pj.load(p + j);
                xj += alpha_plus_previous_alpha_over_beta * pj;
                rj.load(r + j);
                vj.load(v + j);
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int l = 0; l < n_lanes; ++l)
                  {
                    prec_rj[l] = this->preconditioner.apply(j + l, rj[l]);
                    prec_vj[l] = this->preconditioner.apply(j + l, vj[l]);
                  }
                xj -= previous_alpha_over_beta * prec_rj;
                xj.store(x + j);
                rj -= alpha * vj;
                rj.store(r + j);
                prec_rj -= alpha * prec_vj;
                pj = beta * pj + prec_rj;
                pj.store(p + j);
                rj = VectorizedArray<Number>();
                rj.store(v + j);
              }
            for (unsigned int j = end_regular; j < end_range; ++j)
              {
                x[j] += alpha_plus_previous_alpha_over_beta * p[j];
                x[j] -= previous_alpha_over_beta *
                        this->preconditioner.apply(j, r[j]);
                r[j] -= alpha * v[j];
                p[j] = beta * p[j] + this->preconditioner.apply(j, r[j]);
                v[j] = Number();
              }
          }
      }
 
      // Function that we use if the PreconditionerType implements an apply()
      // function
      template <typename U = void>
      typename std::enable_if<has_apply<PreconditionerType>, U>::type
      operation_after_loop(
        const unsigned int                      start_range,
        const unsigned int                      end_range,
        std::array<VectorizedArray<Number>, 7> &vectorized_sums) const
      {
        const Number *                         r       = this->r.begin();
        const Number *                         p       = this->p.begin();
        const Number *                         v       = this->v.begin();
        std::array<VectorizedArray<Number>, 7> my_sums = {};
        constexpr unsigned int n_lanes = VectorizedArray<Number>::size();
        const unsigned int     end_regular =
          start_range + (end_range - start_range) / n_lanes * n_lanes;
        for (unsigned int j = start_range; j < end_regular; j += n_lanes)
          {
            VectorizedArray<Number> pj, vj, rj, prec_vj, prec_rj;
            pj.load(p + j);
            vj.load(v + j);
            rj.load(r + j);
            DEAL_II_OPENMP_SIMD_PRAGMA
            for (unsigned int l = 0; l < n_lanes; ++l)
              {
                prec_vj[l] = this->preconditioner.apply(j + l, vj[l]);
                prec_rj[l] = this->preconditioner.apply(j + l, rj[l]);
              }
            my_sums[0] += pj * vj;
            my_sums[1] += vj * vj;
            my_sums[2] += rj * vj;
            my_sums[3] += rj * rj;
            my_sums[4] += rj * prec_vj;
            my_sums[5] += vj * prec_vj;
            my_sums[6] += rj * prec_rj;
          }
        for (unsigned int j = end_regular; j < end_range; ++j)
          {
            const Number prec_v = this->preconditioner.apply(j, v[j]);
            const Number prec_r = this->preconditioner.apply(j, r[j]);
            my_sums[0][0] += p[j] * v[j];
            my_sums[1][0] += v[j] * v[j];
            my_sums[2][0] += r[j] * v[j];
            my_sums[3][0] += r[j] * r[j];
            my_sums[4][0] += r[j] * prec_v;
            my_sums[5][0] += v[j] * prec_v;
            my_sums[6][0] += r[j] * prec_r;
          }
        for (unsigned int i = 0; i < vectorized_sums.size(); ++i)
          vectorized_sums[i] += my_sums[i];
      }
 
      // Function that we use if the PreconditionerType implements an apply()
      // function
      template <typename U = void>
      typename std::enable_if<has_apply<PreconditionerType>, U>::type
      finalize_after_convergence(const unsigned int iteration_index)
      {
        if (iteration_index % 2 == 1 || iteration_index == 2)
          this->x.add(this->alpha, this->p);
        else
          {
            using Number                 = typename VectorType::value_type;
            const unsigned int end_range = this->x.locally_owned_size();
 
            Number *x = this->x.begin();
            Number *r = this->r.begin();
            Number *p = this->p.begin();
 
            // Note that we use 'beta' here rather than 'previous_beta' in the
            // formula above, which is because the shift in beta ->
            // previous_beta has not been applied at this stage, allowing us
            // to recover the previous search direction
            const Number alpha_plus_previous_alpha_over_beta =
              this->alpha + this->previous_alpha / this->previous_beta;
            const Number previous_alpha_over_beta =
              this->previous_alpha / this->previous_beta;
 
            DEAL_II_OPENMP_SIMD_PRAGMA
            for (unsigned int j = 0; j < end_range; ++j)
              {
                x[j] += alpha_plus_previous_alpha_over_beta * p[j] -
                        previous_alpha_over_beta *
                          this->preconditioner.apply(j, r[j]);
              }
          }
      }
 
      // Function that we use if the PreconditionerType does not implement an
      // apply() function, where we instead need to choose the
      // apply_to_subrange function
      template <typename U = void>
      typename std::enable_if<!has_apply<PreconditionerType>, U>::type
      operation_before_loop(const unsigned int iteration_index,
                            const unsigned int start_range,
                            const unsigned int end_range) const
      {
        Number *                       x     = this->x.begin() + start_range;
        Number *                       r     = this->r.begin() + start_range;
        Number *                       p     = this->p.begin() + start_range;
        Number *                       v     = this->v.begin() + start_range;
        const Number                   alpha = this->alpha;
        const Number                   beta  = this->beta;
        constexpr unsigned int         grain_size = 128;
        std::array<Number, grain_size> prec_r;
        if (iteration_index == 1)
          {
            for (unsigned int j = start_range; j < end_range; j += grain_size)
              {
                const unsigned int length = std::min(grain_size, end_range - j);
                this->preconditioner.apply_to_subrange(j,
                                                       j + length,
                                                       r,
                                                       prec_r.data());
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int i = 0; i < length; ++i)
                  {
                    p[i] = prec_r[i];
                    v[i] = Number();
                  }
                p += length;
                r += length;
                v += length;
              }
          }
        else if (iteration_index % 2 == 0)
          {
            for (unsigned int j = start_range; j < end_range; j += grain_size)
              {
                const unsigned int length = std::min(grain_size, end_range - j);
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int i = 0; i < length; ++i)
                  r[i] -= this->alpha * v[i];
                this->preconditioner.apply_to_subrange(j,
                                                       j + length,
                                                       r,
                                                       prec_r.data());
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int i = 0; i < length; ++i)
                  {
                    p[i] = this->beta * p[i] + prec_r[i];
                    v[i] = Number();
                  }
                p += length;
                r += length;
                v += length;
              }
          }
        else
          {
            const Number alpha_plus_previous_alpha_over_beta =
              this->alpha + this->previous_alpha / this->previous_beta;
            const Number previous_alpha_over_beta =
              this->previous_alpha / this->previous_beta;
            for (unsigned int j = start_range; j < end_range; j += grain_size)
              {
                const unsigned int length = std::min(grain_size, end_range - j);
                this->preconditioner.apply_to_subrange(j,
                                                       j + length,
                                                       r,
                                                       prec_r.data());
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int i = 0; i < length; ++i)
                  {
                    x[i] += alpha_plus_previous_alpha_over_beta * p[i] -
                            previous_alpha_over_beta * prec_r[i];
                    r[i] -= this->alpha * v[i];
                  }
                this->preconditioner.apply_to_subrange(j,
                                                       j + length,
                                                       r,
                                                       prec_r.data());
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int i = 0; i < length; ++i)
                  {
                    p[i] = this->beta * p[i] + prec_r[i];
                    v[i] = Number();
                  }
                p += length;
                r += length;
                v += length;
                x += length;
              }
          }
      }
 
      // Function that we use if the PreconditionerType does not implement an
      // apply() function and where we instead need to use the
      // apply_to_subrange function
      template <typename U = void>
      typename std::enable_if<!has_apply<PreconditionerType>, U>::type
      operation_after_loop(
        const unsigned int                      start_range,
        const unsigned int                      end_range,
        std::array<VectorizedArray<Number>, 7> &vectorized_sums) const
      {
        const Number *                         r          = this->r.begin();
        const Number *                         p          = this->p.begin();
        const Number *                         v          = this->v.begin();
        std::array<VectorizedArray<Number>, 7> my_sums    = {};
        constexpr unsigned int                 grain_size = 128;
        Assert(grain_size % VectorizedArray<Number>::size() == 0,
               ExcNotImplemented());
        const unsigned int end_regular =
          start_range + (end_range - start_range) / grain_size * grain_size;
        std::array<Number, grain_size> prec_r;
        std::array<Number, grain_size> prec_v;
        for (unsigned int j = start_range; j < end_regular; j += grain_size)
          {
            this->preconditioner.apply_to_subrange(j,
                                                   j + grain_size,
                                                   r + j,
                                                   prec_r.data());
            this->preconditioner.apply_to_subrange(j,
                                                   j + grain_size,
                                                   v + j,
                                                   prec_v.data());
            VectorizedArray<Number> pj, vj, rj, prec_vj, prec_rj;
            for (unsigned int i = 0; i < grain_size;
                 i += VectorizedArray<Number>::size())
              {
                pj.load(p + j + i);
                vj.load(v + j + i);
                rj.load(r + j + i);
                prec_rj.load(prec_r.data() + i);
                prec_vj.load(prec_v.data() + i);
 
                my_sums[0] += pj * vj;
                my_sums[1] += vj * vj;
                my_sums[2] += rj * vj;
                my_sums[3] += rj * rj;
                my_sums[4] += rj * prec_vj;
                my_sums[5] += vj * prec_vj;
                my_sums[6] += rj * prec_rj;
              }
          }
        const unsigned int length = end_range - end_regular;
        AssertIndexRange(length, grain_size);
        this->preconditioner.apply_to_subrange(end_regular,
                                               end_regular + length,
                                               r + end_regular,
                                               prec_r.data());
        this->preconditioner.apply_to_subrange(end_regular,
                                               end_regular + length,
                                               v + end_regular,
                                               prec_v.data());
        for (unsigned int j = end_regular; j < end_range; ++j)
          {
            my_sums[0][0] += p[j] * v[j];
            my_sums[1][0] += v[j] * v[j];
            my_sums[2][0] += r[j] * v[j];
            my_sums[3][0] += r[j] * r[j];
            my_sums[4][0] += r[j] * prec_v[j - end_regular];
            my_sums[5][0] += v[j] * prec_v[j - end_regular];
            my_sums[6][0] += r[j] * prec_r[j - end_regular];
          }
        for (unsigned int i = 0; i < vectorized_sums.size(); ++i)
          vectorized_sums[i] += my_sums[i];
      }
 
      // Function that we use if the PreconditionerType does not implement an
      // apply() function, where we instead need to choose the
      // apply_to_subrange function
      template <typename U = void>
      typename std::enable_if<!has_apply<PreconditionerType>, U>::type
      finalize_after_convergence(const unsigned int iteration_index)
      {
        if (iteration_index % 2 == 1 || iteration_index == 2)
          this->x.add(this->alpha, this->p);
        else
          {
            const unsigned int end_range = this->x.locally_owned_size();
 
            Number *     x = this->x.begin();
            Number *     r = this->r.begin();
            Number *     p = this->p.begin();
            const Number alpha_plus_previous_alpha_over_beta =
              this->alpha + this->previous_alpha / this->previous_beta;
            const Number previous_alpha_over_beta =
              this->previous_alpha / this->previous_beta;
 
            constexpr unsigned int         grain_size = 128;
            std::array<Number, grain_size> prec_r;
            for (unsigned int j = 0; j < end_range; j += grain_size)
              {
                const unsigned int length = std::min(grain_size, end_range - j);
                this->preconditioner.apply_to_subrange(j,
                                                       j + length,
                                                       r,
                                                       prec_r.data());
                DEAL_II_OPENMP_SIMD_PRAGMA
                for (unsigned int i = 0; i < length; ++i)
                  x[i] += alpha_plus_previous_alpha_over_beta * p[i] -
                          previous_alpha_over_beta * prec_r[i];
 
                x += length;
                r += length;
                p += length;
              }
          }
      }
    };
  } // namespace SolverCG
} // namespace internal
 
 
 
template <typename VectorType>
template <typename MatrixType, typename PreconditionerType>
void
SolverCG<VectorType>::solve(const MatrixType &        A,
                            VectorType &              x,
                            const VectorType &        b,
                            const PreconditionerType &preconditioner)
{
  using number = typename VectorType::value_type;
 
  SolverControl::State solver_state = SolverControl::iterate;
 
  LogStream::Prefix prefix("cg");
 
  // Should we build the matrix for eigenvalue computations?
  const bool do_eigenvalues =
    !condition_number_signal.empty() || !all_condition_numbers_signal.empty() ||
    !eigenvalues_signal.empty() || !all_eigenvalues_signal.empty();
 
  // vectors used for eigenvalue computations
  std::vector<typename VectorType::value_type> diagonal;
  std::vector<typename VectorType::value_type> offdiagonal;
 
  typename VectorType::value_type eigen_beta_alpha = 0;
 
  int it = 0;
 
  internal::SolverCG::
    IterationWorker<VectorType, MatrixType, PreconditionerType>
      worker(
        A, preconditioner, determine_beta_by_flexible_formula, this->memory, x);
 
  worker.startup(b);
 
  solver_state = this->iteration_status(0, worker.residual_norm, x);
  if (solver_state != SolverControl::iterate)
    return;
 
  while (solver_state == SolverControl::iterate)
    {
      it++;
 
      worker.do_iteration(it);
 
      print_vectors(it, x, worker.r, worker.p);
 
      if (it > 1)
        {
          this->coefficients_signal(worker.previous_alpha, worker.beta);
          // set up the vectors containing the diagonal and the off diagonal
          // of the projected matrix.
          if (do_eigenvalues)
            {
              diagonal.push_back(number(1.) / worker.previous_alpha +
                                 eigen_beta_alpha);
              eigen_beta_alpha = worker.beta / worker.previous_alpha;
              offdiagonal.push_back(std::sqrt(worker.beta) /
                                    worker.previous_alpha);
            }
          compute_eigs_and_cond(diagonal,
                                offdiagonal,
                                all_eigenvalues_signal,
                                all_condition_numbers_signal);
        }
 
      solver_state = this->iteration_status(it, worker.residual_norm, x);
    }
 
  worker.finalize_after_convergence(it);
 
  compute_eigs_and_cond(diagonal,
                        offdiagonal,
                        eigenvalues_signal,
                        condition_number_signal);
 
  AssertThrow(solver_state == SolverControl::success,
              SolverControl::NoConvergence(it, worker.residual_norm));
}
 
 
 
template <typename VectorType>
boost::signals2::connection
SolverCG<VectorType>::connect_coefficients_slot(
  const std::function<void(typename VectorType::value_type,
                           typename VectorType::value_type)> &slot)
{
  return coefficients_signal.connect(slot);
}
 
 
 
template <typename VectorType>
boost::signals2::connection
SolverCG<VectorType>::connect_condition_number_slot(
  const std::function<void(double)> &slot,
  const bool                         every_iteration)
{
  if (every_iteration)
    {
      return all_condition_numbers_signal.connect(slot);
    }
  else
    {
      return condition_number_signal.connect(slot);
    }
}
 
 
 
template <typename VectorType>
boost::signals2::connection
SolverCG<VectorType>::connect_eigenvalues_slot(
  const std::function<void(const std::vector<double> &)> &slot,
  const bool                                              every_iteration)
{
  if (every_iteration)
    {
      return all_eigenvalues_signal.connect(slot);
    }
  else
    {
      return eigenvalues_signal.connect(slot);
    }
}
 
 
 
template <typename VectorType>
SolverFlexibleCG<VectorType>::SolverFlexibleCG(SolverControl &           cn,
                                               VectorMemory<VectorType> &mem,
                                               const AdditionalData &)
  : SolverCG<VectorType>(cn, mem)
{
  this->determine_beta_by_flexible_formula = true;
}
 
 
 
template <typename VectorType>
SolverFlexibleCG<VectorType>::SolverFlexibleCG(SolverControl &cn,
                                               const AdditionalData &)
  : SolverCG<VectorType>(cn)
{
  this->determine_beta_by_flexible_formula = true;
}
 
 
 
#endif // DOXYGEN
 
DEAL_II_NAMESPACE_CLOSE
 
#endif