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Activate FSAL property in dorpri timestepper * scripts/ode/private/runge_kutta_45_dorpri.m: don't compute first stage if values from previous iteration are passed. * scripts/ode/private/integrate_adaptive.m: do not update cmputed stages if timestep is rejected.
author Carlo de Falco <carlo.defalco@polimi.it>
date Sat, 03 Oct 2015 07:32:50 +0200
parents a9574e3c6e9e
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/*

Copyright (C) 1993-2015 John W. Eaton
Copyright (C) 2008 Jaroslav Hajek
Copyright (C) 2009-2010 VZLU Prague

This file is part of Octave.

Octave is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License as published by the
Free Software Foundation; either version 3 of the License, or (at your
option) any later version.

Octave is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
for more details.

You should have received a copy of the GNU General Public License
along with Octave; see the file COPYING.  If not, see
<http://www.gnu.org/licenses/>.

*/

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif

#include <cassert>

#include "Array-util.h"
#include "CMatrix.h"
#include "dMatrix.h"
#include "CNDArray.h"
#include "dNDArray.h"
#include "fCMatrix.h"
#include "fMatrix.h"
#include "fCNDArray.h"
#include "fNDArray.h"
#include "oct-cmplx.h"
#include "dDiagMatrix.h"
#include "fDiagMatrix.h"
#include "CDiagMatrix.h"
#include "fCDiagMatrix.h"
#include "lo-array-gripes.h"
#include "quit.h"

#include "error.h"
#include "xdiv.h"

static inline bool
result_ok (octave_idx_type info)
{
  assert (info != -1);

  return (info != -2);
}

static void
solve_singularity_warning (double rcond)
{
  gripe_singular_matrix (rcond);
}

template <class T1, class T2>
bool
mx_leftdiv_conform (const T1& a, const T2& b, blas_trans_type blas_trans)
{
  octave_idx_type a_nr = blas_trans == blas_no_trans ? a.rows () : a.cols ();
  octave_idx_type b_nr = b.rows ();

  if (a_nr != b_nr)
    {
      octave_idx_type a_nc = blas_trans == blas_no_trans ? a.cols ()
                                                         : a.rows ();
      octave_idx_type b_nc = b.cols ();

      gripe_nonconformant ("operator \\", a_nr, a_nc, b_nr, b_nc);
      return false;
    }

  return true;
}

#define INSTANTIATE_MX_LEFTDIV_CONFORM(T1, T2) \
  template bool mx_leftdiv_conform (const T1&, const T2&, blas_trans_type)

INSTANTIATE_MX_LEFTDIV_CONFORM (Matrix, Matrix);
INSTANTIATE_MX_LEFTDIV_CONFORM (Matrix, ComplexMatrix);
INSTANTIATE_MX_LEFTDIV_CONFORM (ComplexMatrix, Matrix);
INSTANTIATE_MX_LEFTDIV_CONFORM (ComplexMatrix, ComplexMatrix);

template <class T1, class T2>
bool
mx_div_conform (const T1& a, const T2& b)
{
  octave_idx_type a_nc = a.cols ();
  octave_idx_type b_nc = b.cols ();

  if (a_nc != b_nc)
    {
      octave_idx_type a_nr = a.rows ();
      octave_idx_type b_nr = b.rows ();

      gripe_nonconformant ("operator /", a_nr, a_nc, b_nr, b_nc);
      return false;
    }

  return true;
}

#define INSTANTIATE_MX_DIV_CONFORM(T1, T2) \
  template bool mx_div_conform (const T1&, const T2&)

INSTANTIATE_MX_DIV_CONFORM (Matrix, Matrix);
INSTANTIATE_MX_DIV_CONFORM (Matrix, ComplexMatrix);
INSTANTIATE_MX_DIV_CONFORM (ComplexMatrix, Matrix);
INSTANTIATE_MX_DIV_CONFORM (ComplexMatrix, ComplexMatrix);

// Right division functions.
//
//       op2 / op1:   m   cm
//            +--   +---+----+
//   matrix         | 1 |  3 |
//                  +---+----+
//   complex_matrix | 2 |  4 |
//                  +---+----+

// -*- 1 -*-
Matrix
xdiv (const Matrix& a, const Matrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return Matrix ();

  octave_idx_type info;
  double rcond = 0.0;

  Matrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// -*- 2 -*-
ComplexMatrix
xdiv (const Matrix& a, const ComplexMatrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return ComplexMatrix ();

  octave_idx_type info;
  double rcond = 0.0;

  ComplexMatrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// -*- 3 -*-
ComplexMatrix
xdiv (const ComplexMatrix& a, const Matrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return ComplexMatrix ();

  octave_idx_type info;
  double rcond = 0.0;

  ComplexMatrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// -*- 4 -*-
ComplexMatrix
xdiv (const ComplexMatrix& a, const ComplexMatrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return ComplexMatrix ();

  octave_idx_type info;
  double rcond = 0.0;

  ComplexMatrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// Funny element by element division operations.
//
//       op2 \ op1:   s   cs
//            +--   +---+----+
//   matrix         | 1 |  3 |
//                  +---+----+
//   complex_matrix | 2 |  4 |
//                  +---+----+

Matrix
x_el_div (double a, const Matrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  Matrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

ComplexMatrix
x_el_div (double a, const ComplexMatrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  ComplexMatrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

ComplexMatrix
x_el_div (const Complex a, const Matrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  ComplexMatrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

ComplexMatrix
x_el_div (const Complex a, const ComplexMatrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  ComplexMatrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

// Funny element by element division operations.
//
//          op2 \ op1:   s   cs
//               +--   +---+----+
//   N-d array         | 1 |  3 |
//                     +---+----+
//   complex N-d array | 2 |  4 |
//                     +---+----+

NDArray
x_el_div (double a, const NDArray& b)
{
  NDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

ComplexNDArray
x_el_div (double a, const ComplexNDArray& b)
{
  ComplexNDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

ComplexNDArray
x_el_div (const Complex a, const NDArray& b)
{
  ComplexNDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

ComplexNDArray
x_el_div (const Complex a, const ComplexNDArray& b)
{
  ComplexNDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

// Left division functions.
//
//       op2 \ op1:   m   cm
//            +--   +---+----+
//   matrix         | 1 |  3 |
//                  +---+----+
//   complex_matrix | 2 |  4 |
//                  +---+----+

// -*- 1 -*-
Matrix
xleftdiv (const Matrix& a, const Matrix& b, MatrixType &typ,
          blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return Matrix ();

  octave_idx_type info;
  double rcond = 0.0;
  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

// -*- 2 -*-
ComplexMatrix
xleftdiv (const Matrix& a, const ComplexMatrix& b, MatrixType &typ,
          blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return ComplexMatrix ();

  octave_idx_type info;
  double rcond = 0.0;

  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

// -*- 3 -*-
ComplexMatrix
xleftdiv (const ComplexMatrix& a, const Matrix& b, MatrixType &typ,
          blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return ComplexMatrix ();

  octave_idx_type info;
  double rcond = 0.0;
  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

// -*- 4 -*-
ComplexMatrix
xleftdiv (const ComplexMatrix& a, const ComplexMatrix& b, MatrixType &typ,
          blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return ComplexMatrix ();

  octave_idx_type info;
  double rcond = 0.0;
  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

static void
solve_singularity_warning (float rcond)
{
  gripe_singular_matrix (rcond);
}

INSTANTIATE_MX_LEFTDIV_CONFORM (FloatMatrix, FloatMatrix);
INSTANTIATE_MX_LEFTDIV_CONFORM (FloatMatrix, FloatComplexMatrix);
INSTANTIATE_MX_LEFTDIV_CONFORM (FloatComplexMatrix, FloatMatrix);
INSTANTIATE_MX_LEFTDIV_CONFORM (FloatComplexMatrix, FloatComplexMatrix);

INSTANTIATE_MX_DIV_CONFORM (FloatMatrix, FloatMatrix);
INSTANTIATE_MX_DIV_CONFORM (FloatMatrix, FloatComplexMatrix);
INSTANTIATE_MX_DIV_CONFORM (FloatComplexMatrix, FloatMatrix);
INSTANTIATE_MX_DIV_CONFORM (FloatComplexMatrix, FloatComplexMatrix);

// Right division functions.
//
//       op2 / op1:   m   cm
//            +--   +---+----+
//   matrix         | 1 |  3 |
//                  +---+----+
//   complex_matrix | 2 |  4 |
//                  +---+----+

// -*- 1 -*-
FloatMatrix
xdiv (const FloatMatrix& a, const FloatMatrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return FloatMatrix ();

  octave_idx_type info;
  float rcond = 0.0;

  FloatMatrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// -*- 2 -*-
FloatComplexMatrix
xdiv (const FloatMatrix& a, const FloatComplexMatrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return FloatComplexMatrix ();

  octave_idx_type info;
  float rcond = 0.0;

  FloatComplexMatrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// -*- 3 -*-
FloatComplexMatrix
xdiv (const FloatComplexMatrix& a, const FloatMatrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return FloatComplexMatrix ();

  octave_idx_type info;
  float rcond = 0.0;

  FloatComplexMatrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// -*- 4 -*-
FloatComplexMatrix
xdiv (const FloatComplexMatrix& a, const FloatComplexMatrix& b, MatrixType &typ)
{
  if (! mx_div_conform (a, b))
    return FloatComplexMatrix ();

  octave_idx_type info;
  float rcond = 0.0;

  FloatComplexMatrix result
    = b.solve (typ, a.transpose (), info, rcond,
               solve_singularity_warning, true, blas_trans);

  return result.transpose ();
}

// Funny element by element division operations.
//
//       op2 \ op1:   s   cs
//            +--   +---+----+
//   matrix         | 1 |  3 |
//                  +---+----+
//   complex_matrix | 2 |  4 |
//                  +---+----+

FloatMatrix
x_el_div (float a, const FloatMatrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  FloatMatrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

FloatComplexMatrix
x_el_div (float a, const FloatComplexMatrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  FloatComplexMatrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

FloatComplexMatrix
x_el_div (const FloatComplex a, const FloatMatrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  FloatComplexMatrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

FloatComplexMatrix
x_el_div (const FloatComplex a, const FloatComplexMatrix& b)
{
  octave_idx_type nr = b.rows ();
  octave_idx_type nc = b.columns ();

  FloatComplexMatrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      {
        octave_quit ();
        result (i, j) = a / b (i, j);
      }

  return result;
}

// Funny element by element division operations.
//
//          op2 \ op1:   s   cs
//               +--   +---+----+
//   N-d array         | 1 |  3 |
//                     +---+----+
//   complex N-d array | 2 |  4 |
//                     +---+----+

FloatNDArray
x_el_div (float a, const FloatNDArray& b)
{
  FloatNDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

FloatComplexNDArray
x_el_div (float a, const FloatComplexNDArray& b)
{
  FloatComplexNDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

FloatComplexNDArray
x_el_div (const FloatComplex a, const FloatNDArray& b)
{
  FloatComplexNDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

FloatComplexNDArray
x_el_div (const FloatComplex a, const FloatComplexNDArray& b)
{
  FloatComplexNDArray result (b.dims ());

  for (octave_idx_type i = 0; i < b.numel (); i++)
    {
      octave_quit ();
      result (i) = a / b (i);
    }

  return result;
}

// Left division functions.
//
//       op2 \ op1:   m   cm
//            +--   +---+----+
//   matrix         | 1 |  3 |
//                  +---+----+
//   complex_matrix | 2 |  4 |
//                  +---+----+

// -*- 1 -*-
FloatMatrix
xleftdiv (const FloatMatrix& a, const FloatMatrix& b, MatrixType &typ,
          blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return FloatMatrix ();

  octave_idx_type info;
  float rcond = 0.0;
  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

// -*- 2 -*-
FloatComplexMatrix
xleftdiv (const FloatMatrix& a, const FloatComplexMatrix& b, MatrixType &typ,
          blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return FloatComplexMatrix ();

  octave_idx_type info;
  float rcond = 0.0;

  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

// -*- 3 -*-
FloatComplexMatrix
xleftdiv (const FloatComplexMatrix& a, const FloatMatrix& b, MatrixType &typ,
          blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return FloatComplexMatrix ();

  octave_idx_type info;
  float rcond = 0.0;
  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

// -*- 4 -*-
FloatComplexMatrix
xleftdiv (const FloatComplexMatrix& a, const FloatComplexMatrix& b,
          MatrixType &typ, blas_trans_type transt)
{
  if (! mx_leftdiv_conform (a, b, transt))
    return FloatComplexMatrix ();

  octave_idx_type info;
  float rcond = 0.0;
  return a.solve (typ, b, info, rcond, solve_singularity_warning, true, transt);
}

// Diagonal matrix division.

template <class MT, class DMT>
MT
mdm_div_impl (const MT& a, const DMT& d)
{
  if (! mx_div_conform (a, d))
    return MT ();

  octave_idx_type m = a.rows ();
  octave_idx_type n = d.rows ();
  octave_idx_type l = d.length ();
  MT x (m, n);
  typedef typename DMT::element_type S;
  typedef typename MT::element_type T;
  const T *aa = a.data ();
  const S *dd = d.data ();
  T *xx = x.fortran_vec ();

  for (octave_idx_type j = 0; j < l; j++)
    {
      const S del = dd[j];
      if (del != S ())
        for (octave_idx_type i = 0; i < m; i++)
          xx[i] = aa[i] / del;
      else
        for (octave_idx_type i = 0; i < m; i++)
          xx[i] = T ();
      aa += m; xx += m;
    }

  for (octave_idx_type i = l*m; i < n*m; i++)
    xx[i] = T ();

  return x;
}

// Right division functions.
//
//       op2 / op1:   dm  cdm
//            +--   +---+----+
//   matrix         | 1 |    |
//                  +---+----+
//   complex_matrix | 2 |  3 |
//                  +---+----+

// -*- 1 -*-
Matrix
xdiv (const Matrix& a, const DiagMatrix& b)
{ return mdm_div_impl (a, b); }

// -*- 2 -*-
ComplexMatrix
xdiv (const ComplexMatrix& a, const DiagMatrix& b)
{ return mdm_div_impl (a, b); }

// -*- 3 -*-
ComplexMatrix
xdiv (const ComplexMatrix& a, const ComplexDiagMatrix& b)
{ return mdm_div_impl (a, b); }

// Right division functions, float type.
//
//       op2 / op1:   dm  cdm
//            +--   +---+----+
//   matrix         | 1 |    |
//                  +---+----+
//   complex_matrix | 2 |  3 |
//                  +---+----+

// -*- 1 -*-
FloatMatrix
xdiv (const FloatMatrix& a, const FloatDiagMatrix& b)
{ return mdm_div_impl (a, b); }

// -*- 2 -*-
FloatComplexMatrix
xdiv (const FloatComplexMatrix& a, const FloatDiagMatrix& b)
{ return mdm_div_impl (a, b); }

// -*- 3 -*-
FloatComplexMatrix
xdiv (const FloatComplexMatrix& a, const FloatComplexDiagMatrix& b)
{ return mdm_div_impl (a, b); }

template <class MT, class DMT>
MT
dmm_leftdiv_impl (const DMT& d, const MT& a)
{
  if (! mx_leftdiv_conform (d, a, blas_no_trans))
    return MT ();

  octave_idx_type m = d.cols ();
  octave_idx_type n = a.cols ();
  octave_idx_type k = a.rows ();
  octave_idx_type l = d.length ();
  MT x (m, n);
  typedef typename DMT::element_type S;
  typedef typename MT::element_type T;
  const T *aa = a.data ();
  const S *dd = d.data ();
  T *xx = x.fortran_vec ();

  for (octave_idx_type j = 0; j < n; j++)
    {
      for (octave_idx_type i = 0; i < l; i++)
        xx[i] = dd[i] != S () ? aa[i] / dd[i] : T ();
      for (octave_idx_type i = l; i < m; i++)
        xx[i] = T ();
      aa += k; xx += m;
    }

  return x;
}

// Left division functions.
//
//       op2 \ op1:         m   cm
//                        +---+----+
//   diag_matrix          | 1 |  2 |
//                        +---+----+
//   complex_diag_matrix  |   |  3 |
//                        +---+----+

// -*- 1 -*-
Matrix
xleftdiv (const DiagMatrix& a, const Matrix& b)
{ return dmm_leftdiv_impl (a, b); }

// -*- 2 -*-
ComplexMatrix
xleftdiv (const DiagMatrix& a, const ComplexMatrix& b)
{ return dmm_leftdiv_impl (a, b); }

// -*- 3 -*-
ComplexMatrix
xleftdiv (const ComplexDiagMatrix& a, const ComplexMatrix& b)
{ return dmm_leftdiv_impl (a, b); }

// Left division functions, float type.
//
//       op2 \ op1:         m   cm
//                        +---+----+
//   diag_matrix          | 1 |  2 |
//                        +---+----+
//   complex_diag_matrix  |   |  3 |
//                        +---+----+

// -*- 1 -*-
FloatMatrix
xleftdiv (const FloatDiagMatrix& a, const FloatMatrix& b)
{ return dmm_leftdiv_impl (a, b); }

// -*- 2 -*-
FloatComplexMatrix
xleftdiv (const FloatDiagMatrix& a, const FloatComplexMatrix& b)
{ return dmm_leftdiv_impl (a, b); }

// -*- 3 -*-
FloatComplexMatrix
xleftdiv (const FloatComplexDiagMatrix& a, const FloatComplexMatrix& b)
{ return dmm_leftdiv_impl (a, b); }

// Diagonal by diagonal matrix division.

template <class MT, class DMT>
MT
dmdm_div_impl (const MT& a, const DMT& d)
{
  if (! mx_div_conform (a, d))
    return MT ();

  octave_idx_type m = a.rows ();
  octave_idx_type n = d.rows ();
  octave_idx_type k = d.cols ();
  octave_idx_type l = std::min (m, n);
  octave_idx_type lk = std::min (l, k);
  MT x (m, n);
  typedef typename DMT::element_type S;
  typedef typename MT::element_type T;
  const T *aa = a.data ();
  const S *dd = d.data ();
  T *xx = x.fortran_vec ();

  for (octave_idx_type i = 0; i < lk; i++)
    xx[i] = dd[i] != S () ? aa[i] / dd[i] : T ();
  for (octave_idx_type i = lk; i < l; i++)
    xx[i] = T ();

  return x;
}

// Right division functions.
//
//       op2 / op1:        dm  cdm
//            +--        +---+----+
//   diag_matrix         | 1 |    |
//                       +---+----+
//   complex_diag_matrix | 2 |  3 |
//                       +---+----+

// -*- 1 -*-
DiagMatrix
xdiv (const DiagMatrix& a, const DiagMatrix& b)
{ return dmdm_div_impl (a, b); }

// -*- 2 -*-
ComplexDiagMatrix
xdiv (const ComplexDiagMatrix& a, const DiagMatrix& b)
{ return dmdm_div_impl (a, b); }

// -*- 3 -*-
ComplexDiagMatrix
xdiv (const ComplexDiagMatrix& a, const ComplexDiagMatrix& b)
{ return dmdm_div_impl (a, b); }

// Right division functions, float type.
//
//       op2 / op1:        dm  cdm
//            +--        +---+----+
//   diag_matrix         | 1 |    |
//                       +---+----+
//   complex_diag_matrix | 2 |  3 |
//                       +---+----+

// -*- 1 -*-
FloatDiagMatrix
xdiv (const FloatDiagMatrix& a, const FloatDiagMatrix& b)
{ return dmdm_div_impl (a, b); }

// -*- 2 -*-
FloatComplexDiagMatrix
xdiv (const FloatComplexDiagMatrix& a, const FloatDiagMatrix& b)
{ return dmdm_div_impl (a, b); }

// -*- 3 -*-
FloatComplexDiagMatrix
xdiv (const FloatComplexDiagMatrix& a, const FloatComplexDiagMatrix& b)
{ return dmdm_div_impl (a, b); }

template <class MT, class DMT>
MT
dmdm_leftdiv_impl (const DMT& d, const MT& a)
{
  if (! mx_leftdiv_conform (d, a, blas_no_trans))
    return MT ();

  octave_idx_type m = d.cols ();
  octave_idx_type n = a.cols ();
  octave_idx_type k = d.rows ();
  octave_idx_type l = std::min (m, n);
  octave_idx_type lk = std::min (l, k);
  MT x (m, n);
  typedef typename DMT::element_type S;
  typedef typename MT::element_type T;
  const T *aa = a.data ();
  const S *dd = d.data ();
  T *xx = x.fortran_vec ();

  for (octave_idx_type i = 0; i < lk; i++)
    xx[i] = dd[i] != S () ? aa[i] / dd[i] : T ();
  for (octave_idx_type i = lk; i < l; i++)
    xx[i] = T ();

  return x;
}

// Left division functions.
//
//       op2 \ op1:         dm  cdm
//                        +---+----+
//   diag_matrix          | 1 |  2 |
//                        +---+----+
//   complex_diag_matrix  |   |  3 |
//                        +---+----+

// -*- 1 -*-
DiagMatrix
xleftdiv (const DiagMatrix& a, const DiagMatrix& b)
{ return dmdm_leftdiv_impl (a, b); }

// -*- 2 -*-
ComplexDiagMatrix
xleftdiv (const DiagMatrix& a, const ComplexDiagMatrix& b)
{ return dmdm_leftdiv_impl (a, b); }

// -*- 3 -*-
ComplexDiagMatrix
xleftdiv (const ComplexDiagMatrix& a, const ComplexDiagMatrix& b)
{ return dmdm_leftdiv_impl (a, b); }

// Left division functions, float type.
//
//       op2 \ op1:         dm  cdm
//                        +---+----+
//   diag_matrix          | 1 |  2 |
//                        +---+----+
//   complex_diag_matrix  |   |  3 |
//                        +---+----+

// -*- 1 -*-
FloatDiagMatrix
xleftdiv (const FloatDiagMatrix& a, const FloatDiagMatrix& b)
{ return dmdm_leftdiv_impl (a, b); }

// -*- 2 -*-
FloatComplexDiagMatrix
xleftdiv (const FloatDiagMatrix& a, const FloatComplexDiagMatrix& b)
{ return dmdm_leftdiv_impl (a, b); }

// -*- 3 -*-
FloatComplexDiagMatrix
xleftdiv (const FloatComplexDiagMatrix& a, const FloatComplexDiagMatrix& b)
{ return dmdm_leftdiv_impl (a, b); }