sparse_matrix_ez.h

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// ------------------------------------------------------------------------
//
// SPDX-License-Identifier: LGPL-2.1-or-later
// Copyright (C) 2002 - 2023 by the deal.II authors
//
// This file is part of the deal.II library.
//
// Part of the source code is dual licensed under Apache-2.0 WITH
// LLVM-exception OR LGPL-2.1-or-later. Detailed license information
// governing the source code and code contributions can be found in
// LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
//
// ------------------------------------------------------------------------
 
#ifndef dealii_sparse_matrix_ez_h
#define dealii_sparse_matrix_ez_h
 
 
#include <deal.II/base/config.h>
 
#include <deal.II/base/enable_observer_pointer.h>
#include <deal.II/base/observer_pointer.h>
 
#include <deal.II/lac/exceptions.h>
 
#include <vector>
 
DEAL_II_NAMESPACE_OPEN
 
// Forward declarations
#ifndef DOXYGEN
template <typename number>
class Vector;
template <typename number>
class FullMatrix;
#endif
 
template <typename number>
class SparseMatrixEZ : public EnableObserverPointer
{
public:
  using size_type = types::global_dof_index;
 
  struct Entry
  {
    Entry();
 
    Entry(const size_type column, const number &value);
 
    size_type column;
 
    number value;
 
    static const size_type invalid = numbers::invalid_size_type;
  };
 
  struct RowInfo
  {
    RowInfo(const size_type start = Entry::invalid);
 
    size_type start;
    unsigned short length;
    unsigned short diagonal;
    static const unsigned short invalid_diagonal =
      static_cast<unsigned short>(-1);
  };
 
public:
  class const_iterator
  {
  private:
    class Accessor
    {
    public:
      Accessor(const SparseMatrixEZ<number> *matrix,
               const size_type               row,
               const unsigned short          index);
 
      size_type
      row() const;
 
      unsigned short
      index() const;
 
      size_type
      column() const;
 
      number
      value() const;
 
    protected:
      const SparseMatrixEZ<number> *matrix;
 
      size_type a_row;
 
      unsigned short a_index;
 
      // Make enclosing class a friend.
      friend class const_iterator;
    };
 
  public:
    const_iterator(const SparseMatrixEZ<number> *matrix,
                   const size_type               row,
                   const unsigned short          index);
 
    const_iterator &
    operator++();
 
    const Accessor &
    operator*() const;
 
    const Accessor *
    operator->() const;
 
    bool
    operator==(const const_iterator &) const;
    bool
    operator!=(const const_iterator &) const;
 
    bool
    operator<(const const_iterator &) const;
 
  private:
    Accessor accessor;
  };
 
  using value_type = number;
 
  SparseMatrixEZ();
 
  SparseMatrixEZ(const SparseMatrixEZ &);
 
  explicit SparseMatrixEZ(const size_type    n_rows,
                          const size_type    n_columns,
                          const size_type    default_row_length = 0,
                          const unsigned int default_increment  = 1);
 
  ~SparseMatrixEZ() override = default;
 
  SparseMatrixEZ<number> &
  operator=(const SparseMatrixEZ<number> &);
 
  SparseMatrixEZ<number> &
  operator=(const double d);
 
  void
  reinit(const size_type n_rows,
         const size_type n_columns,
         size_type       default_row_length = 0,
         unsigned int    default_increment  = 1,
         size_type       reserve            = 0);
 
  void
  clear();
  bool
  empty() const;
 
  size_type
  m() const;
 
  size_type
  n() const;
 
  size_type
  get_row_length(const size_type row) const;
 
  size_type
  n_nonzero_elements() const;
 
  std::size_t
  memory_consumption() const;
 
  template <typename StreamType>
  void
  print_statistics(StreamType &s, bool full = false);
 
  void
  compute_statistics(size_type              &used,
                     size_type              &allocated,
                     size_type              &reserved,
                     std::vector<size_type> &used_by_line,
                     const bool              compute_by_line) const;
  void
  set(const size_type i,
      const size_type j,
      const number    value,
      const bool      elide_zero_values = true);
 
  void
  add(const size_type i, const size_type j, const number value);
 
  template <typename number2>
  void
  add(const std::vector<size_type> &indices,
      const FullMatrix<number2>    &full_matrix,
      const bool                    elide_zero_values = true);
 
  template <typename number2>
  void
  add(const std::vector<size_type> &row_indices,
      const std::vector<size_type> &col_indices,
      const FullMatrix<number2>    &full_matrix,
      const bool                    elide_zero_values = true);
 
  template <typename number2>
  void
  add(const size_type               row,
      const std::vector<size_type> &col_indices,
      const std::vector<number2>   &values,
      const bool                    elide_zero_values = true);
 
  template <typename number2>
  void
  add(const size_type  row,
      const size_type  n_cols,
      const size_type *col_indices,
      const number2   *values,
      const bool       elide_zero_values      = true,
      const bool       col_indices_are_sorted = false);
 
  template <typename MatrixType>
  SparseMatrixEZ<number> &
  copy_from(const MatrixType &source, const bool elide_zero_values = true);
 
  template <typename MatrixType>
  void
  add(const number factor, const MatrixType &matrix);
  number
  operator()(const size_type i, const size_type j) const;
 
  number
  el(const size_type i, const size_type j) const;
  template <typename somenumber>
  void
  vmult(Vector<somenumber> &dst, const Vector<somenumber> &src) const;
 
  template <typename somenumber>
  void
  Tvmult(Vector<somenumber> &dst, const Vector<somenumber> &src) const;
 
  template <typename somenumber>
  void
  vmult_add(Vector<somenumber> &dst, const Vector<somenumber> &src) const;
 
  template <typename somenumber>
  void
  Tvmult_add(Vector<somenumber> &dst, const Vector<somenumber> &src) const;
  number
  l2_norm() const;
  template <typename somenumber>
  void
  precondition_Jacobi(Vector<somenumber>       &dst,
                      const Vector<somenumber> &src,
                      const number              omega = 1.) const;
 
  template <typename somenumber>
  void
  precondition_SSOR(Vector<somenumber>             &dst,
                    const Vector<somenumber>       &src,
                    const number                    om = 1.,
                    const std::vector<std::size_t> &pos_right_of_diagonal =
                      std::vector<std::size_t>()) const;
 
  template <typename somenumber>
  void
  precondition_SOR(Vector<somenumber>       &dst,
                   const Vector<somenumber> &src,
                   const number              om = 1.) const;
 
  template <typename somenumber>
  void
  precondition_TSOR(Vector<somenumber>       &dst,
                    const Vector<somenumber> &src,
                    const number              om = 1.) const;
 
  template <typename MatrixTypeA, typename MatrixTypeB>
  void
  conjugate_add(const MatrixTypeA &A,
                const MatrixTypeB &B,
                const bool         transpose = false);
  const_iterator
  begin() const;
 
  const_iterator
  end() const;
 
  const_iterator
  begin(const size_type r) const;
 
  const_iterator
  end(const size_type r) const;
  void
  print(std::ostream &out) const;
 
  void
  print_formatted(std::ostream      &out,
                  const unsigned int precision   = 3,
                  const bool         scientific  = true,
                  const unsigned int width       = 0,
                  const char        *zero_string = " ",
                  const double       denominator = 1.,
                  const char        *separator   = " ") const;
 
  void
  block_write(std::ostream &out) const;
 
  void
  block_read(std::istream &in);
  DeclException0(ExcNoDiagonal);
 
  DeclException2(ExcInvalidEntry,
                 int,
                 int,
                 << "The entry with index (" << arg1 << ',' << arg2
                 << ") does not exist.");
 
  DeclException2(ExcEntryAllocationFailure,
                 int,
                 int,
                 << "An entry with index (" << arg1 << ',' << arg2
                 << ") cannot be allocated.");
private:
  const Entry *
  locate(const size_type row, const size_type col) const;
 
  Entry *
  locate(const size_type row, const size_type col);
 
  Entry *
  allocate(const size_type row, const size_type col);
 
  template <typename somenumber>
  void
  threaded_vmult(Vector<somenumber>       &dst,
                 const Vector<somenumber> &src,
                 const size_type           begin_row,
                 const size_type           end_row) const;
 
  template <typename somenumber>
  void
  threaded_matrix_norm_square(const Vector<somenumber> &v,
                              const size_type           begin_row,
                              const size_type           end_row,
                              somenumber               *partial_sum) const;
 
  template <typename somenumber>
  void
  threaded_matrix_scalar_product(const Vector<somenumber> &u,
                                 const Vector<somenumber> &v,
                                 const size_type           begin_row,
                                 const size_type           end_row,
                                 somenumber               *partial_sum) const;
 
  size_type n_columns;
 
  std::vector<RowInfo> row_info;
 
  std::vector<Entry> data;
 
  unsigned int increment;
 
  unsigned int saved_default_row_length;
};
 
/*---------------------- Inline functions -----------------------------------*/
 
template <typename number>
inline SparseMatrixEZ<number>::Entry::Entry(const size_type column,
                                            const number   &value)
  : column(column)
  , value(value)
{}
 
 
 
template <typename number>
inline SparseMatrixEZ<number>::Entry::Entry()
  : column(invalid)
  , value(0)
{}
 
 
template <typename number>
inline SparseMatrixEZ<number>::RowInfo::RowInfo(const size_type start)
  : start(start)
  , length(0)
  , diagonal(invalid_diagonal)
{}
 
 
//---------------------------------------------------------------------------
template <typename number>
inline SparseMatrixEZ<number>::const_iterator::Accessor::Accessor(
  const SparseMatrixEZ<number> *matrix,
  const size_type               r,
  const unsigned short          i)
  : matrix(matrix)
  , a_row(r)
  , a_index(i)
{}
 
 
template <typename number>
inline typename SparseMatrixEZ<number>::size_type
SparseMatrixEZ<number>::const_iterator::Accessor::row() const
{
  return a_row;
}
 
 
template <typename number>
inline typename SparseMatrixEZ<number>::size_type
SparseMatrixEZ<number>::const_iterator::Accessor::column() const
{
  return matrix->data[matrix->row_info[a_row].start + a_index].column;
}
 
 
template <typename number>
inline unsigned short
SparseMatrixEZ<number>::const_iterator::Accessor::index() const
{
  return a_index;
}
 
 
 
template <typename number>
inline number
SparseMatrixEZ<number>::const_iterator::Accessor::value() const
{
  return matrix->data[matrix->row_info[a_row].start + a_index].value;
}
 
 
template <typename number>
inline SparseMatrixEZ<number>::const_iterator::const_iterator(
  const SparseMatrixEZ<number> *matrix,
  const size_type               r,
  const unsigned short          i)
  : accessor(matrix, r, i)
{
  // Finish if this is the end()
  if (r == accessor.matrix->m() && i == 0)
    return;
 
  // Make sure we never construct an
  // iterator pointing to a
  // non-existing entry
 
  // If the index points beyond the
  // end of the row, try the next
  // row.
  if (accessor.a_index >= accessor.matrix->row_info[accessor.a_row].length)
    {
      do
        {
          ++accessor.a_row;
        }
      // Beware! If the next row is
      // empty, iterate until a
      // non-empty row is found or we
      // hit the end of the matrix.
      while (accessor.a_row < accessor.matrix->m() &&
             accessor.matrix->row_info[accessor.a_row].length == 0);
    }
}
 
 
template <typename number>
inline typename SparseMatrixEZ<number>::const_iterator &
SparseMatrixEZ<number>::const_iterator::operator++()
{
  Assert(accessor.a_row < accessor.matrix->m(), ExcIteratorPastEnd());
 
  // Increment column index
  ++(accessor.a_index);
  // If index exceeds number of
  // entries in this row, proceed
  // with next row.
  if (accessor.a_index >= accessor.matrix->row_info[accessor.a_row].length)
    {
      accessor.a_index = 0;
      // Do this loop to avoid
      // elements in empty rows
      do
        {
          ++accessor.a_row;
        }
      while (accessor.a_row < accessor.matrix->m() &&
             accessor.matrix->row_info[accessor.a_row].length == 0);
    }
  return *this;
}
 
 
template <typename number>
inline const typename SparseMatrixEZ<number>::const_iterator::Accessor &
SparseMatrixEZ<number>::const_iterator::operator*() const
{
  return accessor;
}
 
 
template <typename number>
inline const typename SparseMatrixEZ<number>::const_iterator::Accessor *
SparseMatrixEZ<number>::const_iterator::operator->() const
{
  return &accessor;
}
 
 
template <typename number>
inline bool
SparseMatrixEZ<number>::const_iterator::operator==(
  const const_iterator &other) const
{
  return (accessor.row() == other.accessor.row() &&
          accessor.index() == other.accessor.index());
}
 
 
template <typename number>
inline bool
SparseMatrixEZ<number>::const_iterator::operator!=(
  const const_iterator &other) const
{
  return !(*this == other);
}
 
 
template <typename number>
inline bool
SparseMatrixEZ<number>::const_iterator::operator<(
  const const_iterator &other) const
{
  return (accessor.row() < other.accessor.row() ||
          (accessor.row() == other.accessor.row() &&
           accessor.index() < other.accessor.index()));
}
 
 
//---------------------------------------------------------------------------
template <typename number>
inline typename SparseMatrixEZ<number>::size_type
SparseMatrixEZ<number>::m() const
{
  return row_info.size();
}
 
 
template <typename number>
inline typename SparseMatrixEZ<number>::size_type
SparseMatrixEZ<number>::n() const
{
  return n_columns;
}
 
 
template <typename number>
inline typename SparseMatrixEZ<number>::Entry *
SparseMatrixEZ<number>::locate(const size_type row, const size_type col)
{
  AssertIndexRange(row, m());
  AssertIndexRange(col, n());
 
  const RowInfo  &r   = row_info[row];
  const size_type end = r.start + r.length;
  for (size_type i = r.start; i < end; ++i)
    {
      Entry *const entry = &data[i];
      if (entry->column == col)
        return entry;
      if (entry->column == Entry::invalid)
        return nullptr;
    }
  return nullptr;
}
 
 
 
template <typename number>
inline const typename SparseMatrixEZ<number>::Entry *
SparseMatrixEZ<number>::locate(const size_type row, const size_type col) const
{
  SparseMatrixEZ<number> *t = const_cast<SparseMatrixEZ<number> *>(this);
  return t->locate(row, col);
}
 
 
template <typename number>
inline typename SparseMatrixEZ<number>::Entry *
SparseMatrixEZ<number>::allocate(const size_type row, const size_type col)
{
  AssertIndexRange(row, m());
  AssertIndexRange(col, n());
 
  RowInfo        &r   = row_info[row];
  const size_type end = r.start + r.length;
 
  size_type i = r.start;
  // If diagonal exists and this
  // column is higher, start only
  // after diagonal.
  if (r.diagonal != RowInfo::invalid_diagonal && col >= row)
    i += r.diagonal;
  // Find position of entry
  while (i < end && data[i].column < col)
    ++i;
 
  // entry found
  if (i != end && data[i].column == col)
    return &data[i];
 
  // Now, we must insert the new
  // entry and move all successive
  // entries back.
 
  // If no more space is available
  // for this row, insert new
  // elements into the vector.
  // TODO:[GK] We should not extend this row if i<end
  if (row != row_info.size() - 1)
    {
      if (end >= row_info[row + 1].start)
        {
          // Failure if increment 0
          Assert(increment != 0, ExcEntryAllocationFailure(row, col));
 
          // Insert new entries
          data.insert(data.begin() + end, increment, Entry());
          // Update starts of
          // following rows
          for (size_type rn = row + 1; rn < row_info.size(); ++rn)
            row_info[rn].start += increment;
        }
    }
  else
    {
      if (end >= data.size())
        {
          // Here, appending a block
          // does not increase
          // performance.
          data.push_back(Entry());
        }
    }
 
  Entry *entry = &data[i];
  // Save original entry
  Entry temp = *entry;
  // Insert new entry here to
  // make sure all entries
  // are ordered by column
  // index
  entry->column = col;
  entry->value  = 0;
  // Update row_info
  ++r.length;
  if (col == row)
    r.diagonal = i - r.start;
  else if (col < row && r.diagonal != RowInfo::invalid_diagonal)
    ++r.diagonal;
 
  if (i == end)
    return entry;
 
  // Move all entries in this
  // row up by one
  for (size_type j = i + 1; j < end; ++j)
    {
      // There should be no invalid
      // entry below end
      Assert(data[j].column != Entry::invalid, ExcInternalError());
 
      // TODO[GK]: This could be done more efficiently by moving starting at the
      // top rather than swapping starting at the bottom
      std::swap(data[j], temp);
    }
  Assert(data[end].column == Entry::invalid, ExcInternalError());
 
  data[end] = temp;
 
  return entry;
}
 
 
 
template <typename number>
inline void
SparseMatrixEZ<number>::set(const size_type i,
                            const size_type j,
                            const number    value,
                            const bool      elide_zero_values)
{
  AssertIsFinite(value);
 
  AssertIndexRange(i, m());
  AssertIndexRange(j, n());
 
  if (elide_zero_values && value == 0.)
    {
      Entry *entry = locate(i, j);
      if (entry != nullptr)
        entry->value = 0.;
    }
  else
    {
      Entry *entry = allocate(i, j);
      entry->value = value;
    }
}
 
 
 
template <typename number>
inline void
SparseMatrixEZ<number>::add(const size_type i,
                            const size_type j,
                            const number    value)
{
  AssertIsFinite(value);
 
  AssertIndexRange(i, m());
  AssertIndexRange(j, n());
 
  // ignore zero additions
  if (std::abs(value) == 0.)
    return;
 
  Entry *entry = allocate(i, j);
  entry->value += value;
}
 
 
template <typename number>
template <typename number2>
void
SparseMatrixEZ<number>::add(const std::vector<size_type> &indices,
                            const FullMatrix<number2>    &full_matrix,
                            const bool                    elide_zero_values)
{
  // TODO: This function can surely be made more efficient
  for (size_type i = 0; i < indices.size(); ++i)
    for (size_type j = 0; j < indices.size(); ++j)
      if ((full_matrix(i, j) != 0) || (elide_zero_values == false))
        add(indices[i], indices[j], full_matrix(i, j));
}
 
 
 
template <typename number>
template <typename number2>
void
SparseMatrixEZ<number>::add(const std::vector<size_type> &row_indices,
                            const std::vector<size_type> &col_indices,
                            const FullMatrix<number2>    &full_matrix,
                            const bool                    elide_zero_values)
{
  // TODO: This function can surely be made more efficient
  for (size_type i = 0; i < row_indices.size(); ++i)
    for (size_type j = 0; j < col_indices.size(); ++j)
      if ((full_matrix(i, j) != 0) || (elide_zero_values == false))
        add(row_indices[i], col_indices[j], full_matrix(i, j));
}
 
 
 
template <typename number>
template <typename number2>
void
SparseMatrixEZ<number>::add(const size_type               row,
                            const std::vector<size_type> &col_indices,
                            const std::vector<number2>   &values,
                            const bool                    elide_zero_values)
{
  // TODO: This function can surely be made more efficient
  for (size_type j = 0; j < col_indices.size(); ++j)
    if ((values[j] != 0) || (elide_zero_values == false))
      add(row, col_indices[j], values[j]);
}
 
 
 
template <typename number>
template <typename number2>
void
SparseMatrixEZ<number>::add(const size_type  row,
                            const size_type  n_cols,
                            const size_type *col_indices,
                            const number2   *values,
                            const bool       elide_zero_values,
                            const bool /*col_indices_are_sorted*/)
{
  // TODO: This function can surely be made more efficient
  for (size_type j = 0; j < n_cols; ++j)
    if ((std::abs(values[j]) != 0) || (elide_zero_values == false))
      add(row, col_indices[j], values[j]);
}
 
 
 
template <typename number>
inline number
SparseMatrixEZ<number>::el(const size_type i, const size_type j) const
{
  const Entry *entry = locate(i, j);
  if (entry)
    return entry->value;
  return 0.;
}
 
 
 
template <typename number>
inline number
SparseMatrixEZ<number>::operator()(const size_type i, const size_type j) const
{
  const Entry *entry = locate(i, j);
  if (entry)
    return entry->value;
  Assert(false, ExcInvalidEntry(i, j));
  return 0.;
}
 
 
template <typename number>
inline typename SparseMatrixEZ<number>::const_iterator
SparseMatrixEZ<number>::begin() const
{
  const_iterator result(this, 0, 0);
  return result;
}
 
template <typename number>
inline typename SparseMatrixEZ<number>::const_iterator
SparseMatrixEZ<number>::end() const
{
  return const_iterator(this, m(), 0);
}
 
template <typename number>
inline typename SparseMatrixEZ<number>::const_iterator
SparseMatrixEZ<number>::begin(const size_type r) const
{
  AssertIndexRange(r, m());
  const_iterator result(this, r, 0);
  return result;
}
 
template <typename number>
inline typename SparseMatrixEZ<number>::const_iterator
SparseMatrixEZ<number>::end(const size_type r) const
{
  AssertIndexRange(r, m());
  const_iterator result(this, r + 1, 0);
  return result;
}
 
template <typename number>
template <typename MatrixType>
inline SparseMatrixEZ<number> &
SparseMatrixEZ<number>::copy_from(const MatrixType &M,
                                  const bool        elide_zero_values)
{
  reinit(M.m(), M.n(), this->saved_default_row_length, this->increment);
 
  // loop over the elements of the argument matrix row by row, as suggested
  // in the documentation of the sparse matrix iterator class, and
  // copy them into the current object
  for (size_type row = 0; row < M.m(); ++row)
    {
      const typename MatrixType::const_iterator end_row = M.end(row);
      for (typename MatrixType::const_iterator entry = M.begin(row);
           entry != end_row;
           ++entry)
        set(row, entry->column(), entry->value(), elide_zero_values);
    }
 
  return *this;
}
 
template <typename number>
template <typename MatrixType>
inline void
SparseMatrixEZ<number>::add(const number factor, const MatrixType &M)
{
  Assert(M.m() == m(), ExcDimensionMismatch(M.m(), m()));
  Assert(M.n() == n(), ExcDimensionMismatch(M.n(), n()));
 
  if (factor == 0.)
    return;
 
  // loop over the elements of the argument matrix row by row, as suggested
  // in the documentation of the sparse matrix iterator class, and
  // add them into the current object
  for (size_type row = 0; row < M.m(); ++row)
    {
      const typename MatrixType::const_iterator end_row = M.end(row);
      for (typename MatrixType::const_iterator entry = M.begin(row);
           entry != end_row;
           ++entry)
        if (entry->value() != 0)
          add(row, entry->column(), factor * entry->value());
    }
}
 
 
 
template <typename number>
template <typename MatrixTypeA, typename MatrixTypeB>
inline void
SparseMatrixEZ<number>::conjugate_add(const MatrixTypeA &A,
                                      const MatrixTypeB &B,
                                      const bool         transpose)
{
  // Compute the result
  // r_ij = \sum_kl b_ik b_jl a_kl
 
  //    Assert (n() == B.m(), ExcDimensionMismatch(n(), B.m()));
  //    Assert (m() == B.m(), ExcDimensionMismatch(m(), B.m()));
  //    Assert (A.n() == B.n(), ExcDimensionMismatch(A.n(), B.n()));
  //    Assert (A.m() == B.n(), ExcDimensionMismatch(A.m(), B.n()));
 
  // Somehow, we have to avoid making
  // this an operation of complexity
  // n^2. For the transpose case, we
  // can go through the non-zero
  // elements of A^-1 and use the
  // corresponding rows of B only.
  // For the non-transpose case, we
  // must find a trick.
  typename MatrixTypeB::const_iterator       b1      = B.begin();
  const typename MatrixTypeB::const_iterator b_final = B.end();
  if (transpose)
    while (b1 != b_final)
      {
        const size_type                      i  = b1->column();
        const size_type                      k  = b1->row();
        typename MatrixTypeB::const_iterator b2 = B.begin();
        while (b2 != b_final)
          {
            const size_type j = b2->column();
            const size_type l = b2->row();
 
            const typename MatrixTypeA::value_type a = A.el(k, l);
 
            if (a != 0.)
              add(i, j, a * b1->value() * b2->value());
            ++b2;
          }
        ++b1;
      }
  else
    {
      // Determine minimal and
      // maximal row for a column in
      // advance.
 
      std::vector<size_type> minrow(B.n(), B.m());
      std::vector<size_type> maxrow(B.n(), 0);
      while (b1 != b_final)
        {
          const size_type r = b1->row();
          if (r < minrow[b1->column()])
            minrow[b1->column()] = r;
          if (r > maxrow[b1->column()])
            maxrow[b1->column()] = r;
          ++b1;
        }
 
      typename MatrixTypeA::const_iterator       ai = A.begin();
      const typename MatrixTypeA::const_iterator ae = A.end();
 
      while (ai != ae)
        {
          const typename MatrixTypeA::value_type a = ai->value();
          // Don't do anything if
          // this entry is zero.
          if (a == 0.)
            continue;
 
          // Now, loop over all rows
          // having possibly a
          // nonzero entry in column
          // ai->row()
          b1 = B.begin(minrow[ai->row()]);
          const typename MatrixTypeB::const_iterator be1 =
            B.end(maxrow[ai->row()]);
          const typename MatrixTypeB::const_iterator be2 =
            B.end(maxrow[ai->column()]);
 
          while (b1 != be1)
            {
              const double b1v = b1->value();
              // We need the product
              // of both. If it is
              // zero, we can save
              // the work
              if (b1->column() == ai->row() && (b1v != 0.))
                {
                  const size_type i = b1->row();
 
                  typename MatrixTypeB::const_iterator b2 =
                    B.begin(minrow[ai->column()]);
                  while (b2 != be2)
                    {
                      if (b2->column() == ai->column())
                        {
                          const size_type j = b2->row();
                          add(i, j, a * b1v * b2->value());
                        }
                      ++b2;
                    }
                }
              ++b1;
            }
          ++ai;
        }
    }
}
 
 
template <typename number>
template <typename StreamType>
inline void
SparseMatrixEZ<number>::print_statistics(StreamType &out, bool full)
{
  size_type              used;
  size_type              allocated;
  size_type              reserved;
  std::vector<size_type> used_by_line;
 
  compute_statistics(used, allocated, reserved, used_by_line, full);
 
  out << "SparseMatrixEZ:used      entries:" << used << std::endl
      << "SparseMatrixEZ:allocated entries:" << allocated << std::endl
      << "SparseMatrixEZ:reserved  entries:" << reserved << std::endl;
 
  if (full)
    {
      for (size_type i = 0; i < used_by_line.size(); ++i)
        if (used_by_line[i] != 0)
          out << "SparseMatrixEZ:entries\t" << i << "\trows\t"
              << used_by_line[i] << std::endl;
    }
}
 
 
DEAL_II_NAMESPACE_CLOSE
 
#endif