text: #ifndef AMGCL_RELAXATION_GAUSS_SEIDEL_HPP
  #define AMGCL_RELAXATION_GAUSS_SEIDEL_HPP
  
  /*
  The MIT License
  
  Copyright (c) 2012-2021 Denis Demidov <dennis.demidov@gmail.com>
  
  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  copies of the Software, and to permit persons to whom the Software is
  furnished to do so, subject to the following conditions:
  
  The above copyright notice and this permission notice shall be included in
  all copies or substantial portions of the Software.
  
  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
  THE SOFTWARE.
  */
  
  /**
   * \file   amgcl/relaxation/gauss_seidel.hpp
   * \author Denis Demidov <dennis.demidov@gmail.com>
   * \brief  Gauss-Seidel relaxation scheme.
   */
  
  #include <numeric>
  
  #include <memory>
  
  #include <amgcl/backend/interface.hpp>
  #include <amgcl/util.hpp>
  
  #ifdef _OPENMP
  #  include <omp.h>
  #endif
  
  namespace amgcl {
  namespace relaxation {
  
  /// Gauss-Seidel relaxation.
  /**
   * \note This is a serial relaxation and is only applicable to backends that
   * support matrix row iteration (e.g. amgcl::backend::builtin or
   * amgcl::backend::eigen).
   *
   * \param Backend Backend for temporary structures allocation.
   * \ingroup relaxation
   */
  template <class Backend>
  struct gauss_seidel {
      /// Relaxation parameters.
      struct params {
          /// Use serial version of the algorithm
          bool serial;
  
          params() : serial(false) {}
  
  #ifndef AMGCL_NO_BOOST
          params(const boost::property_tree::ptree &p)
              : AMGCL_PARAMS_IMPORT_VALUE(p, serial)
          {
              check_params(p, {"serial"});
          }
  
          void get(boost::property_tree::ptree &p, const std::string &path) const {
              AMGCL_PARAMS_EXPORT_VALUE(p, path, serial);
          }
  #endif
      };
  
      bool is_serial;
  
      /// \copydoc amgcl::relaxation::damped_jacobi::damped_jacobi
      template <class Matrix>
      gauss_seidel( const Matrix &A, const params &prm, const typename Backend::params&)
          : is_serial(prm.serial || num_threads() < 4)
      {
          if(!is_serial) {
              forward  = std::make_shared< parallel_sweep<true>  >(A);
              backward = std::make_shared< parallel_sweep<false> >(A);
          }
      }
  
      /// \copydoc amgcl::relaxation::damped_jacobi::apply_pre
      template <class Matrix, class VectorRHS, class VectorX, class VectorTMP>
      void apply_pre(
              const Matrix &A, const VectorRHS &rhs, VectorX &x, VectorTMP&
              ) const
      {
          if (is_serial)
              serial_sweep(A, rhs, x, true);
          else
              forward->sweep(rhs, x);
      }
  
      /// \copydoc amgcl::relaxation::damped_jacobi::apply_post
      template <class Matrix, class VectorRHS, class VectorX, class VectorTMP>
      void apply_post(
              const Matrix &A, const VectorRHS &rhs, VectorX &x, VectorTMP&
              ) const
      {
          if (is_serial)
              serial_sweep(A, rhs, x, false);
          else
              backward->sweep(rhs, x);
      }
  
      template <class Matrix, class VectorRHS, class VectorX>
      void apply(const Matrix &A, const VectorRHS &rhs, VectorX &x) const
      {
          backend::clear(x);
          if (is_serial) {
              serial_sweep(A, rhs, x, true);
              serial_sweep(A, rhs, x, false);
          } else {
              forward->sweep(rhs, x);
              backward->sweep(rhs, x);
          }
      }
  
      size_t bytes() const {
          size_t b = 0;
          if (forward)  b += forward->bytes();
          if (backward) b += backward->bytes();
          return b;
      }
  
      private:
          static int num_threads() {
  #ifdef _OPENMP
              return omp_get_max_threads();
  #else
              return 1;
  #endif
          }
  
          static int thread_id() {
  #ifdef _OPENMP
              return omp_get_thread_num();
  #else
              return 0;
  #endif
          }
  
          template <class Matrix, class VectorRHS, class VectorX>
          static void serial_sweep(
                  const Matrix &A, const VectorRHS &rhs, VectorX &x, bool forward)
          {
              typedef typename backend::value_type<Matrix>::type val_type;
              typedef typename math::rhs_of<val_type>::type rhs_type;
  
              const ptrdiff_t n = backend::rows(A);
  
              const ptrdiff_t beg = forward ? 0 : n-1;
              const ptrdiff_t end = forward ? n : -1;
              const ptrdiff_t inc = forward ? 1 : -1;
  
              for(ptrdiff_t i = beg; i != end; i += inc) {
                  val_type D = math::identity<val_type>();
                  rhs_type X;
                  X = rhs[i];
  
                  for (auto a = backend::row_begin(A, i); a; ++a) {
                      ptrdiff_t c = a.col();
                      val_type  v = a.value();
  
                      if (c == i)
                          D = v;
                      else
                          X -= v * x[c];
                  }
  
                  x[i] = math::inverse(D) * X;
              }
          }
  
          template <bool forward>
          struct parallel_sweep {
              typedef typename Backend::value_type value_type;
              typedef typename math::rhs_of<value_type>::type rhs_type;
  
              struct task {
                  ptrdiff_t beg, end;
                  task(ptrdiff_t beg, ptrdiff_t end) : beg(beg), end(end) {}
              };
  
              int nthreads;
  
              // thread-specific storage:
              std::vector< std::vector<task>       > tasks;
              std::vector< std::vector<ptrdiff_t>  > ptr;
              std::vector< std::vector<ptrdiff_t>  > col;
              std::vector< std::vector<value_type> > val;
              std::vector< std::vector<ptrdiff_t>  > ord;
  
              template <class Matrix>
              parallel_sweep(const Matrix &A)
                  : nthreads(num_threads()), tasks(nthreads),
                    ptr(nthreads), col(nthreads), val(nthreads), ord(nthreads)
              {
                  ptrdiff_t n    = backend::rows(A);
                  ptrdiff_t nlev = 0;
  
                  std::vector<ptrdiff_t> level(n, 0);
                  std::vector<ptrdiff_t> order(n, 0);
  
                  // 1. split rows into levels.
                  ptrdiff_t beg = forward ? 0 : n-1;
                  ptrdiff_t end = forward ? n :  -1;
                  ptrdiff_t inc = forward ? 1 :  -1;
  
                  for(ptrdiff_t i = beg; i != end; i += inc) {
                      ptrdiff_t l = level[i];
  
                      for(auto a = row_begin(A, i); a; ++a) {
                          ptrdiff_t c = a.col();
  
                          if (forward) {
                              if (c >= i) continue;
                          } else {
                              if (c <= i) continue;
                          }
  
                          l = std::max(l, level[c]+1);
                      }
  
                      level[i] = l;
                      nlev = std::max(nlev, l+1);
                  }
  
  
                  // 2. reorder matrix rows.
                  std::vector<ptrdiff_t> start(nlev+1, 0);
  
                  for(ptrdiff_t i = 0; i < n; ++i)
                      ++start[level[i]+1];
  
                  std::partial_sum(start.begin(), start.end(), start.begin());
  
                  for(ptrdiff_t i = 0; i < n; ++i)
                      order[start[level[i]]++] = i;
  
                  std::rotate(start.begin(), start.end() - 1, start.end());
                  start[0] = 0;
  
  
                  // 3. Organize matrix rows into tasks.
                  //    Each level is split into nthreads tasks.
                  std::vector<ptrdiff_t> thread_rows(nthreads, 0);
                  std::vector<ptrdiff_t> thread_cols(nthreads, 0);
  
  #pragma omp parallel
                  {
                      int tid = thread_id();
                      tasks[tid].reserve(nlev);
  
                      for(ptrdiff_t lev = 0; lev < nlev; ++lev) {
                          // split each level into tasks.
                          ptrdiff_t lev_size = start[lev+1] - start[lev];
                          ptrdiff_t chunk_size = (lev_size + nthreads - 1) / nthreads;
  
                          ptrdiff_t beg = std::min(tid * chunk_size, lev_size);
                          ptrdiff_t end = std::min(beg + chunk_size, lev_size);
  
                          beg += start[lev];
                          end += start[lev];
  
                          tasks[tid].push_back(task(beg, end));
  
                          // count rows and nonzeros in the current task
                          thread_rows[tid] += end - beg;
                          for(ptrdiff_t i = beg; i < end; ++i) {
                              ptrdiff_t j = order[i];
                              thread_cols[tid] += row_nonzeros(A, j);
                          }
                      }
                  }
  
                  // 4. reorganize matrix data for better cache and NUMA locality.
  #pragma omp parallel
                  {
                      int tid = thread_id();
  
                      col[tid].reserve(thread_cols[tid]);
                      val[tid].reserve(thread_cols[tid]);
                      ord[tid].reserve(thread_rows[tid]);
                      ptr[tid].reserve(thread_rows[tid] + 1);
                      ptr[tid].push_back(0);
  
                      for(task &t : tasks[tid]) {
                          ptrdiff_t loc_beg = ptr[tid].size() - 1;
                          ptrdiff_t loc_end = loc_beg;
  
                          for(ptrdiff_t r = t.beg; r < t.end; ++r, ++loc_end) {
                              ptrdiff_t i = order[r];
  
                              ord[tid].push_back(i);
  
                              for(auto a = row_begin(A, i); a; ++a) {
                                  col[tid].push_back(a.col());
                                  val[tid].push_back(a.value());
                              }
  
                              ptr[tid].push_back(col[tid].size());
                          }
  
                          t.beg = loc_beg;
                          t.end = loc_end;
                      }
                  }
              }
  
              template <class Vector1, class Vector2>
              void sweep(const Vector1 &rhs, Vector2 &x) const {
  #pragma omp parallel
                  {
                      int tid = thread_id();
  
                      for(const task &t : tasks[tid]) {
                          for(ptrdiff_t r = t.beg; r < t.end; ++r) {
                              ptrdiff_t i   = ord[tid][r];
                              ptrdiff_t beg = ptr[tid][r];
                              ptrdiff_t end = ptr[tid][r+1];
  
                              value_type D = math::identity<value_type>();
                              rhs_type X;
                              X = rhs[i];
  
                              for(ptrdiff_t j = beg; j < end; ++j) {
                                  ptrdiff_t  c = col[tid][j];
                                  value_type v = val[tid][j];
  
                                  if (c == i)
                                      D = v;
                                  else
                                      X -= v * x[c];
                              }
  
                              x[i] = math::inverse(D) * X;
                          }
  
                          // each task corresponds to a level, so we need
                          // to synchronize across threads at this point:
  #pragma omp barrier
                          ;
                      }
                  }
              }
  
              size_t bytes() const {
                  size_t b = 0;
  
                  for(int i = 0; i < nthreads; ++i) {
                      b += sizeof(task) * tasks[i].size();
                      b += backend::bytes(ptr[i]);
                      b += backend::bytes(col[i]);
                      b += backend::bytes(val[i]);
                      b += backend::bytes(ord[i]);
                  }
  
                  return b;
              }
          };
  
          std::shared_ptr< parallel_sweep<true>  > forward;
          std::shared_ptr< parallel_sweep<false> > backward;
  };
  
  } // namespace relaxation
  
  namespace backend {
  
  template <class Backend>
  struct relaxation_is_supported<
      Backend,
      relaxation::gauss_seidel,
      typename std::enable_if<
          !Backend::provides_row_iterator::value
          >::type
      > : std::false_type
  {};
  
  } // namespace backend
  } // namespace amgcl
  
  
  #endif
  