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dMatrix.cc

// Matrix manipulations.
/*

Copyright (C) 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002,
              2003, 2004, 2005, 2006, 2007, 2008 John W. Eaton

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 <cfloat>

#include <iostream>
#include <vector>

#include "Array-util.h"
#include "byte-swap.h"
#include "dMatrix.h"
#include "dbleAEPBAL.h"
#include "dbleDET.h"
#include "dbleSCHUR.h"
#include "dbleSVD.h"
#include "dbleCHOL.h"
#include "f77-fcn.h"
#include "lo-error.h"
#include "lo-ieee.h"
#include "lo-mappers.h"
#include "lo-utils.h"
#include "mx-base.h"
#include "mx-m-dm.h"
#include "mx-dm-m.h"
#include "mx-inlines.cc"
#include "oct-cmplx.h"
#include "quit.h"

#if defined (HAVE_FFTW3)
#include "oct-fftw.h"
#endif

// Fortran functions we call.

extern "C"
{
  F77_RET_T
  F77_FUNC (xilaenv, XILAENV) (const octave_idx_type&, F77_CONST_CHAR_ARG_DECL,
                         F77_CONST_CHAR_ARG_DECL,
                         const octave_idx_type&, const octave_idx_type&,
                         const octave_idx_type&, const octave_idx_type&,
                         octave_idx_type&
                         F77_CHAR_ARG_LEN_DECL F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (dgebal, DGEBAL) (F77_CONST_CHAR_ARG_DECL,
                       const octave_idx_type&, double*, const octave_idx_type&, octave_idx_type&,
                       octave_idx_type&, double*, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (dgebak, DGEBAK) (F77_CONST_CHAR_ARG_DECL,
                       F77_CONST_CHAR_ARG_DECL,
                       const octave_idx_type&, const octave_idx_type&, const octave_idx_type&, double*,
                       const octave_idx_type&, double*, const octave_idx_type&, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL
                       F77_CHAR_ARG_LEN_DECL);


  F77_RET_T
  F77_FUNC (dgemm, DGEMM) (F77_CONST_CHAR_ARG_DECL,
                     F77_CONST_CHAR_ARG_DECL,
                     const octave_idx_type&, const octave_idx_type&, const octave_idx_type&,
                     const double&, const double*, const octave_idx_type&,
                     const double*, const octave_idx_type&, const double&,
                     double*, const octave_idx_type&
                     F77_CHAR_ARG_LEN_DECL
                     F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (dgemv, DGEMV) (F77_CONST_CHAR_ARG_DECL,
                     const octave_idx_type&, const octave_idx_type&, const double&,
                     const double*, const octave_idx_type&, const double*,
                     const octave_idx_type&, const double&, double*,
                     const octave_idx_type&
                     F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (xddot, XDDOT) (const octave_idx_type&, const double*, const octave_idx_type&,
                     const double*, const octave_idx_type&, double&);

  F77_RET_T
  F77_FUNC (dgetrf, DGETRF) (const octave_idx_type&, const octave_idx_type&, double*, const octave_idx_type&,
                  octave_idx_type*, octave_idx_type&);

  F77_RET_T
  F77_FUNC (dgetrs, DGETRS) (F77_CONST_CHAR_ARG_DECL, const octave_idx_type&, const octave_idx_type&, 
                       const double*, const octave_idx_type&,
                       const octave_idx_type*, double*, const octave_idx_type&, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (dgetri, DGETRI) (const octave_idx_type&, double*, const octave_idx_type&, const octave_idx_type*,
                       double*, const octave_idx_type&, octave_idx_type&);

  F77_RET_T
  F77_FUNC (dgecon, DGECON) (F77_CONST_CHAR_ARG_DECL, const octave_idx_type&, double*, 
                       const octave_idx_type&, const double&, double&, 
                       double*, octave_idx_type*, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (dgelsy, DGELSY) (const octave_idx_type&, const octave_idx_type&, const octave_idx_type&,
                       double*, const octave_idx_type&, double*,
                       const octave_idx_type&, octave_idx_type*, double&, octave_idx_type&,
                       double*, const octave_idx_type&, octave_idx_type&);

  F77_RET_T
  F77_FUNC (dgelsd, DGELSD) (const octave_idx_type&, const octave_idx_type&, const octave_idx_type&,
                       double*, const octave_idx_type&, double*,
                       const octave_idx_type&, double*, double&, octave_idx_type&,
                       double*, const octave_idx_type&, octave_idx_type*,
                       octave_idx_type&);

  F77_RET_T
  F77_FUNC (dpotrf, DPOTRF) (F77_CONST_CHAR_ARG_DECL, const octave_idx_type&, 
                       double *, const octave_idx_type&, 
                       octave_idx_type& F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (dpocon, DPOCON) (F77_CONST_CHAR_ARG_DECL, const octave_idx_type&, 
                       double*, const octave_idx_type&, const double&,
                       double&, double*, octave_idx_type*,
                       octave_idx_type& F77_CHAR_ARG_LEN_DECL);
  F77_RET_T
  F77_FUNC (dpotrs, DPOTRS) (F77_CONST_CHAR_ARG_DECL, const octave_idx_type&, 
                       const octave_idx_type&, const double*, 
                       const octave_idx_type&, double*, 
                       const octave_idx_type&, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (dtrtri, DTRTRI) (F77_CONST_CHAR_ARG_DECL, F77_CONST_CHAR_ARG_DECL, 
                       const octave_idx_type&, const double*, 
                       const octave_idx_type&, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL
                       F77_CHAR_ARG_LEN_DECL);
  F77_RET_T
  F77_FUNC (dtrcon, DTRCON) (F77_CONST_CHAR_ARG_DECL, F77_CONST_CHAR_ARG_DECL, 
                       F77_CONST_CHAR_ARG_DECL, const octave_idx_type&, 
                       const double*, const octave_idx_type&, double&,
                       double*, octave_idx_type*, octave_idx_type& 
                       F77_CHAR_ARG_LEN_DECL
                       F77_CHAR_ARG_LEN_DECL
                       F77_CHAR_ARG_LEN_DECL);
  F77_RET_T
  F77_FUNC (dtrtrs, DTRTRS) (F77_CONST_CHAR_ARG_DECL, F77_CONST_CHAR_ARG_DECL, 
                       F77_CONST_CHAR_ARG_DECL, const octave_idx_type&, 
                       const octave_idx_type&, const double*, 
                       const octave_idx_type&, double*, 
                       const octave_idx_type&, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL
                       F77_CHAR_ARG_LEN_DECL
                       F77_CHAR_ARG_LEN_DECL);

  // Note that the original complex fft routines were not written for
  // double complex arguments.  They have been modified by adding an
  // implicit double precision (a-h,o-z) statement at the beginning of
  // each subroutine.

  F77_RET_T
  F77_FUNC (cffti, CFFTI) (const octave_idx_type&, Complex*);

  F77_RET_T
  F77_FUNC (cfftf, CFFTF) (const octave_idx_type&, Complex*, Complex*);

  F77_RET_T
  F77_FUNC (cfftb, CFFTB) (const octave_idx_type&, Complex*, Complex*);

  F77_RET_T
  F77_FUNC (dlartg, DLARTG) (const double&, const double&, double&,
                       double&, double&);

  F77_RET_T
  F77_FUNC (dtrsyl, DTRSYL) (F77_CONST_CHAR_ARG_DECL,
                       F77_CONST_CHAR_ARG_DECL,
                       const octave_idx_type&, const octave_idx_type&, const octave_idx_type&,
                       const double*, const octave_idx_type&, const double*,
                       const octave_idx_type&, const double*, const octave_idx_type&,
                       double&, octave_idx_type&
                       F77_CHAR_ARG_LEN_DECL
                       F77_CHAR_ARG_LEN_DECL);

  F77_RET_T
  F77_FUNC (xdlange, XDLANGE) (F77_CONST_CHAR_ARG_DECL, const octave_idx_type&,
                         const octave_idx_type&, const double*,
                         const octave_idx_type&, double*, double&
                         F77_CHAR_ARG_LEN_DECL); 
}

// Matrix class.

Matrix::Matrix (const RowVector& rv)
  : MArray2<double> (1, rv.length (), 0.0)
{
  for (octave_idx_type i = 0; i < rv.length (); i++)
    elem (0, i) = rv.elem (i);
}

Matrix::Matrix (const ColumnVector& cv)
  : MArray2<double> (cv.length (), 1, 0.0)
{
  for (octave_idx_type i = 0; i < cv.length (); i++)
    elem (i, 0) = cv.elem (i);
}

Matrix::Matrix (const DiagMatrix& a)
  : MArray2<double> (a.rows (), a.cols (), 0.0)
{
  for (octave_idx_type i = 0; i < a.length (); i++)
    elem (i, i) = a.elem (i, i);
}

// FIXME -- could we use a templated mixed-type copy function
// here?

Matrix::Matrix (const boolMatrix& a)
  : MArray2<double> (a.rows (), a.cols ())
{
  for (octave_idx_type i = 0; i < a.rows (); i++)
    for (octave_idx_type j = 0; j < a.cols (); j++)
      elem (i, j) = a.elem (i, j);
}

Matrix::Matrix (const charMatrix& a)
  : MArray2<double> (a.rows (), a.cols ())
{
  for (octave_idx_type i = 0; i < a.rows (); i++)
    for (octave_idx_type j = 0; j < a.cols (); j++)
      elem (i, j) = a.elem (i, j);
}

bool
Matrix::operator == (const Matrix& a) const
{
  if (rows () != a.rows () || cols () != a.cols ())
    return false;

  return mx_inline_equal (data (), a.data (), length ());
}

bool
Matrix::operator != (const Matrix& a) const
{
  return !(*this == a);
}

bool
Matrix::is_symmetric (void) const
{
  if (is_square () && rows () > 0)
    {
      for (octave_idx_type i = 0; i < rows (); i++)
      for (octave_idx_type j = i+1; j < cols (); j++)
        if (elem (i, j) != elem (j, i))
          return false;

      return true;
    }

  return false;
}

Matrix&
Matrix::insert (const Matrix& a, octave_idx_type r, octave_idx_type c)
{
  Array2<double>::insert (a, r, c);
  return *this;
}

Matrix&
Matrix::insert (const RowVector& a, octave_idx_type r, octave_idx_type c)
{
  octave_idx_type a_len = a.length ();

  if (r < 0 || r >= rows () || c < 0 || c + a_len > cols ())
    {
      (*current_liboctave_error_handler) ("range error for insert");
      return *this;
    }

  if (a_len > 0)
    {
      make_unique ();

      for (octave_idx_type i = 0; i < a_len; i++)
      xelem (r, c+i) = a.elem (i);
    }

  return *this;
}

Matrix&
Matrix::insert (const ColumnVector& a, octave_idx_type r, octave_idx_type c)
{
  octave_idx_type a_len = a.length ();

  if (r < 0 || r + a_len > rows () || c < 0 || c >= cols ())
    {
      (*current_liboctave_error_handler) ("range error for insert");
      return *this;
    }

  if (a_len > 0)
    {
      make_unique ();

      for (octave_idx_type i = 0; i < a_len; i++)
      xelem (r+i, c) = a.elem (i);
    }

  return *this;
}

Matrix&
Matrix::insert (const DiagMatrix& a, octave_idx_type r, octave_idx_type c)
{
  octave_idx_type a_nr = a.rows ();
  octave_idx_type a_nc = a.cols ();

  if (r < 0 || r + a_nr > rows () || c < 0 || c + a_nc > cols ())
    {
      (*current_liboctave_error_handler) ("range error for insert");
      return *this;
    }

  fill (0.0, r, c, r + a_nr - 1, c + a_nc - 1);

  octave_idx_type a_len = a.length ();

  if (a_len > 0)
    {
      make_unique ();

      for (octave_idx_type i = 0; i < a_len; i++)
      xelem (r+i, c+i) = a.elem (i, i);
    }

  return *this;
}

Matrix&
Matrix::fill (double val)
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr > 0 && nc > 0)
    {
      make_unique ();

      for (octave_idx_type j = 0; j < nc; j++)
      for (octave_idx_type i = 0; i < nr; i++)
        xelem (i, j) = val;
    }

  return *this;
}

Matrix&
Matrix::fill (double val, octave_idx_type r1, octave_idx_type c1, octave_idx_type r2, octave_idx_type c2)
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (r1 < 0 || r2 < 0 || c1 < 0 || c2 < 0
      || r1 >= nr || r2 >= nr || c1 >= nc || c2 >= nc)
    {
      (*current_liboctave_error_handler) ("range error for fill");
      return *this;
    }

  if (r1 > r2) { octave_idx_type tmp = r1; r1 = r2; r2 = tmp; }
  if (c1 > c2) { octave_idx_type tmp = c1; c1 = c2; c2 = tmp; }

  if (r2 >= r1 && c2 >= c1)
    {
      make_unique ();

      for (octave_idx_type j = c1; j <= c2; j++)
      for (octave_idx_type i = r1; i <= r2; i++)
        xelem (i, j) = val;
    }

  return *this;
}

Matrix
Matrix::append (const Matrix& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nr != a.rows ())
    {
      (*current_liboctave_error_handler) ("row dimension mismatch for append");
      return Matrix ();
    }

  octave_idx_type nc_insert = nc;
  Matrix retval (nr, nc + a.cols ());
  retval.insert (*this, 0, 0);
  retval.insert (a, 0, nc_insert);
  return retval;
}

Matrix
Matrix::append (const RowVector& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nr != 1)
    {
      (*current_liboctave_error_handler) ("row dimension mismatch for append");
      return Matrix ();
    }

  octave_idx_type nc_insert = nc;
  Matrix retval (nr, nc + a.length ());
  retval.insert (*this, 0, 0);
  retval.insert (a, 0, nc_insert);
  return retval;
}

Matrix
Matrix::append (const ColumnVector& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nr != a.length ())
    {
      (*current_liboctave_error_handler) ("row dimension mismatch for append");
      return Matrix ();
    }

  octave_idx_type nc_insert = nc;
  Matrix retval (nr, nc + 1);
  retval.insert (*this, 0, 0);
  retval.insert (a, 0, nc_insert);
  return retval;
}

Matrix
Matrix::append (const DiagMatrix& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nr != a.rows ())
    {
      (*current_liboctave_error_handler) ("row dimension mismatch for append");
      return *this;
    }

  octave_idx_type nc_insert = nc;
  Matrix retval (nr, nc + a.cols ());
  retval.insert (*this, 0, 0);
  retval.insert (a, 0, nc_insert);
  return retval;
}

Matrix
Matrix::stack (const Matrix& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nc != a.cols ())
    {
      (*current_liboctave_error_handler)
      ("column dimension mismatch for stack");
      return Matrix ();
    }

  octave_idx_type nr_insert = nr;
  Matrix retval (nr + a.rows (), nc);
  retval.insert (*this, 0, 0);
  retval.insert (a, nr_insert, 0);
  return retval;
}

Matrix
Matrix::stack (const RowVector& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nc != a.length ())
    {
      (*current_liboctave_error_handler)
      ("column dimension mismatch for stack");
      return Matrix ();
    }

  octave_idx_type nr_insert = nr;
  Matrix retval (nr + 1, nc);
  retval.insert (*this, 0, 0);
  retval.insert (a, nr_insert, 0);
  return retval;
}

Matrix
Matrix::stack (const ColumnVector& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nc != 1)
    {
      (*current_liboctave_error_handler)
      ("column dimension mismatch for stack");
      return Matrix ();
    }

  octave_idx_type nr_insert = nr;
  Matrix retval (nr + a.length (), nc);
  retval.insert (*this, 0, 0);
  retval.insert (a, nr_insert, 0);
  return retval;
}

Matrix
Matrix::stack (const DiagMatrix& a) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();
  if (nc != a.cols ())
    {
      (*current_liboctave_error_handler)
      ("column dimension mismatch for stack");
      return Matrix ();
    }

  octave_idx_type nr_insert = nr;
  Matrix retval (nr + a.rows (), nc);
  retval.insert (*this, 0, 0);
  retval.insert (a, nr_insert, 0);
  return retval;
}

Matrix
real (const ComplexMatrix& a)
{
  octave_idx_type a_len = a.length ();
  Matrix retval;
  if (a_len > 0)
    retval = Matrix (mx_inline_real_dup (a.data (), a_len),
                 a.rows (), a.cols ());
  return retval;
}

Matrix
imag (const ComplexMatrix& a)
{
  octave_idx_type a_len = a.length ();
  Matrix retval;
  if (a_len > 0)
    retval = Matrix (mx_inline_imag_dup (a.data (), a_len),
                 a.rows (), a.cols ());
  return retval;
}

Matrix
Matrix::extract (octave_idx_type r1, octave_idx_type c1, octave_idx_type r2, octave_idx_type c2) const
{
  if (r1 > r2) { octave_idx_type tmp = r1; r1 = r2; r2 = tmp; }
  if (c1 > c2) { octave_idx_type tmp = c1; c1 = c2; c2 = tmp; }

  octave_idx_type new_r = r2 - r1 + 1;
  octave_idx_type new_c = c2 - c1 + 1;

  Matrix result (new_r, new_c);

  for (octave_idx_type j = 0; j < new_c; j++)
    for (octave_idx_type i = 0; i < new_r; i++)
      result.xelem (i, j) = elem (r1+i, c1+j);

  return result;
}

Matrix
Matrix::extract_n (octave_idx_type r1, octave_idx_type c1, octave_idx_type nr, octave_idx_type nc) const
{
  Matrix result (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      result.xelem (i, j) = elem (r1+i, c1+j);

  return result;
}

// extract row or column i.

RowVector
Matrix::row (octave_idx_type i) const
{
  octave_idx_type nc = cols ();
  if (i < 0 || i >= rows ())
    {
      (*current_liboctave_error_handler) ("invalid row selection");
      return RowVector ();
    }

  RowVector retval (nc);
  for (octave_idx_type j = 0; j < nc; j++)
    retval.xelem (j) = elem (i, j);

  return retval;
}

ColumnVector
Matrix::column (octave_idx_type i) const
{
  octave_idx_type nr = rows ();
  if (i < 0 || i >= cols ())
    {
      (*current_liboctave_error_handler) ("invalid column selection");
      return ColumnVector ();
    }

  ColumnVector retval (nr);
  for (octave_idx_type j = 0; j < nr; j++)
    retval.xelem (j) = elem (j, i);

  return retval;
}

Matrix
Matrix::inverse (void) const
{
  octave_idx_type info;
  double rcond;
  MatrixType mattype (*this);
  return inverse (mattype, info, rcond, 0, 0);
}

Matrix
Matrix::inverse (octave_idx_type& info) const
{
  double rcond;
  MatrixType mattype (*this);
  return inverse (mattype, info, rcond, 0, 0);
}

Matrix
Matrix::inverse (octave_idx_type& info, double& rcond, int force,
             int calc_cond) const
{
  MatrixType mattype (*this);
  return inverse (mattype, info, rcond, force, calc_cond);
}

Matrix
Matrix::inverse (MatrixType& mattype) const
{
  octave_idx_type info;
  double rcond;
  return inverse (mattype, info, rcond, 0, 0);
}

Matrix
Matrix::inverse (MatrixType &mattype, octave_idx_type& info) const
{
  double rcond;
  return inverse (mattype, info, rcond, 0, 0);
}

Matrix
Matrix::tinverse (MatrixType &mattype, octave_idx_type& info, double& rcond, 
              int force, int calc_cond) const
{
  Matrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr != nc || nr == 0 || nc == 0)
    (*current_liboctave_error_handler) ("inverse requires square matrix");
  else
    {
      int typ = mattype.type ();
      char uplo = (typ == MatrixType::Lower ? 'L' : 'U');
      char udiag = 'N';
      retval = *this;
      double *tmp_data = retval.fortran_vec ();

      F77_XFCN (dtrtri, DTRTRI, (F77_CONST_CHAR_ARG2 (&uplo, 1),
                         F77_CONST_CHAR_ARG2 (&udiag, 1),
                         nr, tmp_data, nr, info 
                         F77_CHAR_ARG_LEN (1)
                         F77_CHAR_ARG_LEN (1)));

      if (f77_exception_encountered)
      (*current_liboctave_error_handler) ("unrecoverable error in dtrtri");
      else
      {
        // Throw-away extra info LAPACK gives so as to not change output.
        rcond = 0.0;
        if (info != 0) 
          info = -1;
        else if (calc_cond) 
          {
            octave_idx_type dtrcon_info = 0;
            char job = '1';

            OCTAVE_LOCAL_BUFFER (double, work, 3 * nr);
            OCTAVE_LOCAL_BUFFER (octave_idx_type, iwork, nr);

            F77_XFCN (dtrcon, DTRCON, (F77_CONST_CHAR_ARG2 (&job, 1),
                               F77_CONST_CHAR_ARG2 (&uplo, 1),
                               F77_CONST_CHAR_ARG2 (&udiag, 1),
                               nr, tmp_data, nr, rcond, 
                               work, iwork, dtrcon_info 
                               F77_CHAR_ARG_LEN (1)
                               F77_CHAR_ARG_LEN (1)
                               F77_CHAR_ARG_LEN (1)));

            if (f77_exception_encountered)
            (*current_liboctave_error_handler) 
              ("unrecoverable error in dtrcon");

            if (dtrcon_info != 0) 
            info = -1;
          }
      }

      if (info == -1 && ! force)
      retval = *this; // Restore matrix contents.
    }

  return retval;
}


Matrix
Matrix::finverse (MatrixType &mattype, octave_idx_type& info, double& rcond, 
              int force, int calc_cond) const
{
  Matrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr != nc || nr == 0 || nc == 0)
    (*current_liboctave_error_handler) ("inverse requires square matrix");
  else
    {
      Array<octave_idx_type> ipvt (nr);
      octave_idx_type *pipvt = ipvt.fortran_vec ();

      retval = *this;
      double *tmp_data = retval.fortran_vec ();

      Array<double> z(1);
      octave_idx_type lwork = -1;

      // Query the optimum work array size.
      F77_XFCN (dgetri, DGETRI, (nc, tmp_data, nr, pipvt, 
                         z.fortran_vec (), lwork, info));

      if (f77_exception_encountered) 
      {
        (*current_liboctave_error_handler)
          ("unrecoverable error in dgetri");
        return retval;
      }

      lwork = static_cast<octave_idx_type> (z(0));
      lwork = (lwork < 2 *nc ? 2*nc : lwork);
      z.resize (lwork);
      double *pz = z.fortran_vec ();

      info = 0;

      // Calculate the norm of the matrix, for later use.
      double anorm = 0;
      if (calc_cond) 
      anorm = retval.abs().sum().row(static_cast<octave_idx_type>(0)).max();

      F77_XFCN (dgetrf, DGETRF, (nc, nc, tmp_data, nr, pipvt, info));

      if (f77_exception_encountered)
      (*current_liboctave_error_handler) ("unrecoverable error in dgetrf");
      else
      {
        // Throw-away extra info LAPACK gives so as to not change output.
        rcond = 0.0;
        if (info != 0) 
          info = -1;
        else if (calc_cond) 
          {
            octave_idx_type dgecon_info = 0;

            // Now calculate the condition number for non-singular matrix.
            char job = '1';
            Array<octave_idx_type> iz (nc);
            octave_idx_type *piz = iz.fortran_vec ();
            F77_XFCN (dgecon, DGECON, (F77_CONST_CHAR_ARG2 (&job, 1),
                               nc, tmp_data, nr, anorm, 
                               rcond, pz, piz, dgecon_info
                               F77_CHAR_ARG_LEN (1)));

            if (f77_exception_encountered)
            (*current_liboctave_error_handler) 
              ("unrecoverable error in dgecon");

            if (dgecon_info != 0) 
            info = -1;
          }

        if (info == -1 && ! force)
          retval = *this; // Restore matrix contents.
        else
          {
            octave_idx_type dgetri_info = 0;

            F77_XFCN (dgetri, DGETRI, (nc, tmp_data, nr, pipvt,
                               pz, lwork, dgetri_info));

            if (f77_exception_encountered)
            (*current_liboctave_error_handler)
              ("unrecoverable error in dgetri");

            if (dgetri_info != 0) 
            info = -1;
          }
      }

      if (info != 0)
      mattype.mark_as_rectangular();
    }

  return retval;
}

Matrix
Matrix::inverse (MatrixType &mattype, octave_idx_type& info, double& rcond, 
             int force, int calc_cond) const
{
  int typ = mattype.type (false);
  Matrix ret;

  if (typ == MatrixType::Unknown)
    typ = mattype.type (*this);

  if (typ == MatrixType::Upper || typ == MatrixType::Lower)
    ret = tinverse (mattype, info, rcond, force, calc_cond);
  else
    {
      if (mattype.is_hermitian ())
      {
        CHOL chol (*this, info, calc_cond);
        if (info == 0)
          {
            if (calc_cond)
            rcond = chol.rcond ();
            else
            rcond = 1.0;
            ret = chol.inverse ();
          }
        else
          mattype.mark_as_unsymmetric ();
      }

      if (!mattype.is_hermitian ())
      ret = finverse(mattype, info, rcond, force, calc_cond);

      if ((mattype.is_hermitian () || calc_cond) && rcond == 0.)
      ret = Matrix (rows (), columns (), octave_Inf);
    }

  return ret;
}

Matrix
Matrix::pseudo_inverse (double tol) const
{
  SVD result (*this, SVD::economy);

  DiagMatrix S = result.singular_values ();
  Matrix U = result.left_singular_matrix ();
  Matrix V = result.right_singular_matrix ();

  ColumnVector sigma = S.diag ();

  octave_idx_type r = sigma.length () - 1;
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (tol <= 0.0)
    {
      if (nr > nc)
      tol = nr * sigma.elem (0) * DBL_EPSILON;
      else
      tol = nc * sigma.elem (0) * DBL_EPSILON;
    }

  while (r >= 0 && sigma.elem (r) < tol)
    r--;

  if (r < 0)
    return Matrix (nc, nr, 0.0);
  else
    {
      Matrix Ur = U.extract (0, 0, nr-1, r);
      DiagMatrix D = DiagMatrix (sigma.extract (0, r)) . inverse ();
      Matrix Vr = V.extract (0, 0, nc-1, r);
      return Vr * D * Ur.transpose ();
    }
}

#if defined (HAVE_FFTW3)

ComplexMatrix
Matrix::fourier (void) const
{
  size_t nr = rows ();
  size_t nc = cols ();

  ComplexMatrix retval (nr, nc);

  size_t npts, nsamples;

  if (nr == 1 || nc == 1)
    {
      npts = nr > nc ? nr : nc;
      nsamples = 1;
    }
  else
    {
      npts = nr;
      nsamples = nc;
    }

  const double *in (fortran_vec ());
  Complex *out (retval.fortran_vec ());

  octave_fftw::fft (in, out, npts, nsamples); 

  return retval;
}

ComplexMatrix
Matrix::ifourier (void) const
{
  size_t nr = rows ();
  size_t nc = cols ();

  ComplexMatrix retval (nr, nc);

  size_t npts, nsamples;

  if (nr == 1 || nc == 1)
    {
      npts = nr > nc ? nr : nc;
      nsamples = 1;
    }
  else
    {
      npts = nr;
      nsamples = nc;
    }

  ComplexMatrix tmp (*this);
  Complex *in (tmp.fortran_vec ());
  Complex *out (retval.fortran_vec ());

  octave_fftw::ifft (in, out, npts, nsamples); 

  return retval;
}

ComplexMatrix
Matrix::fourier2d (void) const
{
  dim_vector dv(rows (), cols ());

  const double *in = fortran_vec ();
  ComplexMatrix retval (rows (), cols ());
  octave_fftw::fftNd (in, retval.fortran_vec (), 2, dv);

  return retval;
}

ComplexMatrix
Matrix::ifourier2d (void) const
{
  dim_vector dv(rows (), cols ());

  ComplexMatrix retval (*this);
  Complex *out (retval.fortran_vec ());

  octave_fftw::ifftNd (out, out, 2, dv);

  return retval;
}

#else

ComplexMatrix
Matrix::fourier (void) const
{
  ComplexMatrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  octave_idx_type npts, nsamples;

  if (nr == 1 || nc == 1)
    {
      npts = nr > nc ? nr : nc;
      nsamples = 1;
    }
  else
    {
      npts = nr;
      nsamples = nc;
    }

  octave_idx_type nn = 4*npts+15;

  Array<Complex> wsave (nn);
  Complex *pwsave = wsave.fortran_vec ();

  retval = ComplexMatrix (*this);
  Complex *tmp_data = retval.fortran_vec ();

  F77_FUNC (cffti, CFFTI) (npts, pwsave);

  for (octave_idx_type j = 0; j < nsamples; j++)
    {
      OCTAVE_QUIT;

      F77_FUNC (cfftf, CFFTF) (npts, &tmp_data[npts*j], pwsave);
    }

  return retval;
}

ComplexMatrix
Matrix::ifourier (void) const
{
  ComplexMatrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  octave_idx_type npts, nsamples;

  if (nr == 1 || nc == 1)
    {
      npts = nr > nc ? nr : nc;
      nsamples = 1;
    }
  else
    {
      npts = nr;
      nsamples = nc;
    }

  octave_idx_type nn = 4*npts+15;

  Array<Complex> wsave (nn);
  Complex *pwsave = wsave.fortran_vec ();

  retval = ComplexMatrix (*this);
  Complex *tmp_data = retval.fortran_vec ();

  F77_FUNC (cffti, CFFTI) (npts, pwsave);

  for (octave_idx_type j = 0; j < nsamples; j++)
    {
      OCTAVE_QUIT;

      F77_FUNC (cfftb, CFFTB) (npts, &tmp_data[npts*j], pwsave);
    }

  for (octave_idx_type j = 0; j < npts*nsamples; j++)
    tmp_data[j] = tmp_data[j] / static_cast<double> (npts);

  return retval;
}

ComplexMatrix
Matrix::fourier2d (void) const
{
  ComplexMatrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  octave_idx_type npts, nsamples;

  if (nr == 1 || nc == 1)
    {
      npts = nr > nc ? nr : nc;
      nsamples = 1;
    }
  else
    {
      npts = nr;
      nsamples = nc;
    }

  octave_idx_type nn = 4*npts+15;

  Array<Complex> wsave (nn);
  Complex *pwsave = wsave.fortran_vec ();

  retval = ComplexMatrix (*this);
  Complex *tmp_data = retval.fortran_vec ();

  F77_FUNC (cffti, CFFTI) (npts, pwsave);

  for (octave_idx_type j = 0; j < nsamples; j++)
    {
      OCTAVE_QUIT;

      F77_FUNC (cfftf, CFFTF) (npts, &tmp_data[npts*j], pwsave);
    }

  npts = nc;
  nsamples = nr;
  nn = 4*npts+15;

  wsave.resize (nn);
  pwsave = wsave.fortran_vec ();

  Array<Complex> tmp (npts);
  Complex *prow = tmp.fortran_vec ();

  F77_FUNC (cffti, CFFTI) (npts, pwsave);

  for (octave_idx_type j = 0; j < nsamples; j++)
    {
      OCTAVE_QUIT;

      for (octave_idx_type i = 0; i < npts; i++)
      prow[i] = tmp_data[i*nr + j];

      F77_FUNC (cfftf, CFFTF) (npts, prow, pwsave);

      for (octave_idx_type i = 0; i < npts; i++)
      tmp_data[i*nr + j] = prow[i];
    }

  return retval;
}

ComplexMatrix
Matrix::ifourier2d (void) const
{
  ComplexMatrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  octave_idx_type npts, nsamples;

  if (nr == 1 || nc == 1)
    {
      npts = nr > nc ? nr : nc;
      nsamples = 1;
    }
  else
    {
      npts = nr;
      nsamples = nc;
    }

  octave_idx_type nn = 4*npts+15;

  Array<Complex> wsave (nn);
  Complex *pwsave = wsave.fortran_vec ();

  retval = ComplexMatrix (*this);
  Complex *tmp_data = retval.fortran_vec ();

  F77_FUNC (cffti, CFFTI) (npts, pwsave);

  for (octave_idx_type j = 0; j < nsamples; j++)
    {
      OCTAVE_QUIT;

      F77_FUNC (cfftb, CFFTB) (npts, &tmp_data[npts*j], pwsave);
    }

  for (octave_idx_type j = 0; j < npts*nsamples; j++)
    tmp_data[j] = tmp_data[j] / static_cast<double> (npts);

  npts = nc;
  nsamples = nr;
  nn = 4*npts+15;

  wsave.resize (nn);
  pwsave = wsave.fortran_vec ();

  Array<Complex> tmp (npts);
  Complex *prow = tmp.fortran_vec ();

  F77_FUNC (cffti, CFFTI) (npts, pwsave);

  for (octave_idx_type j = 0; j < nsamples; j++)
    {
      OCTAVE_QUIT;

      for (octave_idx_type i = 0; i < npts; i++)
      prow[i] = tmp_data[i*nr + j];

      F77_FUNC (cfftb, CFFTB) (npts, prow, pwsave);

      for (octave_idx_type i = 0; i < npts; i++)
      tmp_data[i*nr + j] = prow[i] / static_cast<double> (npts);
    }

  return retval;
}

#endif

DET
Matrix::determinant (void) const
{
  octave_idx_type info;
  double rcond;
  return determinant (info, rcond, 0);
}

DET
Matrix::determinant (octave_idx_type& info) const
{
  double rcond;
  return determinant (info, rcond, 0);
}

DET
Matrix::determinant (octave_idx_type& info, double& rcond, int calc_cond) const
{
  DET retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr == 0 || nc == 0)
    {
      retval = DET (1.0, 0);
    }
  else
    {
      Array<octave_idx_type> ipvt (nr);
      octave_idx_type *pipvt = ipvt.fortran_vec ();

      Matrix atmp = *this;
      double *tmp_data = atmp.fortran_vec ();

      info = 0;

      // Calculate the norm of the matrix, for later use.
      double anorm = 0;
      if (calc_cond) 
      anorm = atmp.abs().sum().row(static_cast<octave_idx_type>(0)).max();

      F77_XFCN (dgetrf, DGETRF, (nr, nr, tmp_data, nr, pipvt, info));

      if (f77_exception_encountered)
      (*current_liboctave_error_handler) ("unrecoverable error in dgetrf");
      else
      {
        // Throw-away extra info LAPACK gives so as to not change output.
        rcond = 0.0;
        if (info != 0) 
          {
            info = -1;
            retval = DET ();
          } 
        else 
          {
            if (calc_cond) 
            {
              // Now calc the condition number for non-singular matrix.
              char job = '1';
              Array<double> z (4 * nc);
              double *pz = z.fortran_vec ();
              Array<octave_idx_type> iz (nc);
              octave_idx_type *piz = iz.fortran_vec ();

              F77_XFCN (dgecon, DGECON, (F77_CONST_CHAR_ARG2 (&job, 1),
                                   nc, tmp_data, nr, anorm, 
                                   rcond, pz, piz, info
                                   F77_CHAR_ARG_LEN (1)));

              if (f77_exception_encountered)
                (*current_liboctave_error_handler) 
                  ("unrecoverable error in dgecon");
            }

            if (info != 0) 
            {
              info = -1;
              retval = DET ();
            } 
            else 
            {
              double c = 1.0;
              int e = 0;

              for (octave_idx_type i = 0; i < nc; i++) 
                {
                  if (ipvt(i) != (i+1))
                  c = -c;

                  c *= atmp(i,i);

                  if (c == 0.0)
                  break;

                  while (fabs (c) < 0.5)
                  {
                    c *= 2.0;
                    e--;
                  }

                  while (fabs (c) >= 2.0)
                  {
                    c /= 2.0;
                    e++;
                  }
                }

              retval = DET (c, e);
            }
          }
      }
    }

  return retval;
}

Matrix
Matrix::utsolve (MatrixType &mattype, const Matrix& b, octave_idx_type& info,
            double& rcond, solve_singularity_handler sing_handler,
            bool calc_cond) const
{
  Matrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr != b.rows ())
    (*current_liboctave_error_handler)
      ("matrix dimension mismatch solution of linear equations");
  else if (nr == 0 || nc == 0 || b.cols () == 0)
    retval = Matrix (nc, b.cols (), 0.0);
  else
    {
      volatile int typ = mattype.type ();

      if (typ == MatrixType::Permuted_Upper ||
        typ == MatrixType::Upper)
      {
        octave_idx_type b_nc = b.cols ();
        rcond = 1.;
        info = 0;

        if (typ == MatrixType::Permuted_Upper)
          {
            (*current_liboctave_error_handler)
            ("permuted triangular matrix not implemented");
          }
        else
          {
            const double *tmp_data = fortran_vec ();

            if (calc_cond)
            {
              char norm = '1';
              char uplo = 'U';
              char dia = 'N';

              Array<double> z (3 * nc);
              double *pz = z.fortran_vec ();
              Array<octave_idx_type> iz (nc);
              octave_idx_type *piz = iz.fortran_vec ();

              F77_XFCN (dtrcon, DTRCON, (F77_CONST_CHAR_ARG2 (&norm, 1), 
                                   F77_CONST_CHAR_ARG2 (&uplo, 1), 
                                   F77_CONST_CHAR_ARG2 (&dia, 1), 
                                   nr, tmp_data, nr, rcond,
                                   pz, piz, info
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)));

              if (f77_exception_encountered)
                (*current_liboctave_error_handler) 
                  ("unrecoverable error in dtrcon");

              if (info != 0) 
                info = -2;

              volatile double rcond_plus_one = rcond + 1.0;

              if (rcond_plus_one == 1.0 || xisnan (rcond))
                {
                  info = -2;

                  if (sing_handler)
                  sing_handler (rcond);
                  else
                  (*current_liboctave_error_handler)
                    ("matrix singular to machine precision, rcond = %g",
                     rcond);
                }
            }

            if (info == 0)
            {
              retval = b;
              double *result = retval.fortran_vec ();

              char uplo = 'U';
              char trans = 'N';
              char dia = 'N';

              F77_XFCN (dtrtrs, DTRTRS, (F77_CONST_CHAR_ARG2 (&uplo, 1), 
                                   F77_CONST_CHAR_ARG2 (&trans, 1), 
                                   F77_CONST_CHAR_ARG2 (&dia, 1), 
                                   nr, b_nc, tmp_data, nr,
                                   result, nr, info
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)));

              if (f77_exception_encountered)
                (*current_liboctave_error_handler) 
                  ("unrecoverable error in dtrtrs");
            }
          }
      }
      else
      (*current_liboctave_error_handler) ("incorrect matrix type");
    }

  return retval;
}

Matrix
Matrix::ltsolve (MatrixType &mattype, const Matrix& b, octave_idx_type& info,
            double& rcond, solve_singularity_handler sing_handler,
            bool calc_cond) const
{
  Matrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr != b.rows ())
    (*current_liboctave_error_handler)
      ("matrix dimension mismatch solution of linear equations");
  else if (nr == 0 || nc == 0 || b.cols () == 0)
    retval = Matrix (nc, b.cols (), 0.0);
  else
    {
      volatile int typ = mattype.type ();

      if (typ == MatrixType::Permuted_Lower ||
        typ == MatrixType::Lower)
      {
        octave_idx_type b_nc = b.cols ();
        rcond = 1.;
        info = 0;

        if (typ == MatrixType::Permuted_Lower)
          {
            (*current_liboctave_error_handler)
            ("permuted triangular matrix not implemented");
          }
        else
          {
            const double *tmp_data = fortran_vec ();

            if (calc_cond)
            {
              char norm = '1';
              char uplo = 'L';
              char dia = 'N';

              Array<double> z (3 * nc);
              double *pz = z.fortran_vec ();
              Array<octave_idx_type> iz (nc);
              octave_idx_type *piz = iz.fortran_vec ();

              F77_XFCN (dtrcon, DTRCON, (F77_CONST_CHAR_ARG2 (&norm, 1), 
                                   F77_CONST_CHAR_ARG2 (&uplo, 1), 
                                   F77_CONST_CHAR_ARG2 (&dia, 1), 
                                   nr, tmp_data, nr, rcond,
                                   pz, piz, info
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)));

              if (f77_exception_encountered)
                (*current_liboctave_error_handler) 
                  ("unrecoverable error in dtrcon");

              if (info != 0) 
                info = -2;

              volatile double rcond_plus_one = rcond + 1.0;

              if (rcond_plus_one == 1.0 || xisnan (rcond))
                {
                  info = -2;

                  if (sing_handler)
                  sing_handler (rcond);
                  else
                  (*current_liboctave_error_handler)
                    ("matrix singular to machine precision, rcond = %g",
                     rcond);
                }
            }

            if (info == 0)
            {
              retval = b;
              double *result = retval.fortran_vec ();

              char uplo = 'L';
              char trans = 'N';
              char dia = 'N';

              F77_XFCN (dtrtrs, DTRTRS, (F77_CONST_CHAR_ARG2 (&uplo, 1), 
                                   F77_CONST_CHAR_ARG2 (&trans, 1), 
                                   F77_CONST_CHAR_ARG2 (&dia, 1), 
                                   nr, b_nc, tmp_data, nr,
                                   result, nr, info
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)
                                   F77_CHAR_ARG_LEN (1)));

              if (f77_exception_encountered)
                (*current_liboctave_error_handler) 
                  ("unrecoverable error in dtrtrs");
            }
          }
      }
      else
      (*current_liboctave_error_handler) ("incorrect matrix type");
    }

  return retval;
}

Matrix
Matrix::fsolve (MatrixType &mattype, const Matrix& b, octave_idx_type& info,
            double& rcond, solve_singularity_handler sing_handler,
            bool calc_cond) const
{
  Matrix retval;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr != nc || nr != b.rows ())
    (*current_liboctave_error_handler)
      ("matrix dimension mismatch solution of linear equations");
  else if (nr == 0 || b.cols () == 0)
    retval = Matrix (nc, b.cols (), 0.0);
  else
    {
      volatile int typ = mattype.type ();
 
     // Calculate the norm of the matrix, for later use.
      double anorm = -1.;

      if (typ == MatrixType::Hermitian)
      {
        info = 0;
        char job = 'L';
        Matrix atmp = *this;
        double *tmp_data = atmp.fortran_vec ();
        anorm = atmp.abs().sum().row(static_cast<octave_idx_type>(0)).max();

        F77_XFCN (dpotrf, DPOTRF, (F77_CONST_CHAR_ARG2 (&job, 1), nr, 
                             tmp_data, nr, info
                             F77_CHAR_ARG_LEN (1)));

        if (f77_exception_encountered)
          (*current_liboctave_error_handler) 
            ("unrecoverable error in dpotrf");
        else
          {
            // Throw-away extra info LAPACK gives so as to not change output.
            rcond = 0.0;
            if (info != 0) 
            {
              info = -2;

              mattype.mark_as_unsymmetric ();
              typ = MatrixType::Full;
            }
            else 
            {
              if (calc_cond)
                {
                  Array<double> z (3 * nc);
                  double *pz = z.fortran_vec ();
                  Array<octave_idx_type> iz (nc);
                  octave_idx_type *piz = iz.fortran_vec ();

                  F77_XFCN (dpocon, DPOCON, (F77_CONST_CHAR_ARG2 (&job, 1),
                                     nr, tmp_data, nr, anorm,
                                     rcond, pz, piz, info
                                     F77_CHAR_ARG_LEN (1)));

                  if (f77_exception_encountered)
                  (*current_liboctave_error_handler) 
                    ("unrecoverable error in dpocon");
            
                  if (info != 0) 
                  info = -2;

                  volatile double rcond_plus_one = rcond + 1.0;

                  if (rcond_plus_one == 1.0 || xisnan (rcond))
                  {
                    info = -2;

                    if (sing_handler)
                      sing_handler (rcond);
                    else
                      (*current_liboctave_error_handler)
                        ("matrix singular to machine precision, rcond = %g",
                         rcond);
                  }
                }

              if (info == 0)
                {
                  retval = b;
                  double *result = retval.fortran_vec ();

                  octave_idx_type b_nc = b.cols ();

                  F77_XFCN (dpotrs, DPOTRS, (F77_CONST_CHAR_ARG2 (&job, 1),
                                     nr, b_nc, tmp_data, nr,
                                     result, b.rows(), info
                                     F77_CHAR_ARG_LEN (1)));
            
                  if (f77_exception_encountered)
                  (*current_liboctave_error_handler)
                    ("unrecoverable error in dpotrs");
                }
              else
                {
                  mattype.mark_as_unsymmetric ();
                  typ = MatrixType::Full;
                }           
            }
          }
      }

      if (typ == MatrixType::Full)
      {
        info = 0;

        Array<octave_idx_type> ipvt (nr);
        octave_idx_type *pipvt = ipvt.fortran_vec ();

        Matrix atmp = *this;
        double *tmp_data = atmp.fortran_vec ();
        if(anorm < 0.)
          anorm = atmp.abs().sum().row(static_cast<octave_idx_type>(0)).max();

        Array<double> z (4 * nc);
        double *pz = z.fortran_vec ();
        Array<octave_idx_type> iz (nc);
        octave_idx_type *piz = iz.fortran_vec ();

        F77_XFCN (dgetrf, DGETRF, (nr, nr, tmp_data, nr, pipvt, info));

        if (f77_exception_encountered)
          (*current_liboctave_error_handler) 
            ("unrecoverable error in dgetrf");
        else
          {
            // Throw-away extra info LAPACK gives so as to not change output.
            rcond = 0.0;
            if (info != 0) 
            {
              info = -2;

              if (sing_handler)
                sing_handler (rcond);
              else
                (*current_liboctave_error_handler)
                  ("matrix singular to machine precision");

              mattype.mark_as_rectangular ();
            }
            else 
            {
              if (calc_cond)
                {
                  // Now calculate the condition number for 
                  // non-singular matrix.
                  char job = '1';
                  F77_XFCN (dgecon, DGECON, (F77_CONST_CHAR_ARG2 (&job, 1),
                                     nc, tmp_data, nr, anorm, 
                                     rcond, pz, piz, info
                                     F77_CHAR_ARG_LEN (1)));
            
                  if (f77_exception_encountered)
                  (*current_liboctave_error_handler) 
                    ("unrecoverable error in dgecon");
            
                  if (info != 0) 
                  info = -2;

                  volatile double rcond_plus_one = rcond + 1.0;

                  if (rcond_plus_one == 1.0 || xisnan (rcond))
                  {
                    info = -2;

                    if (sing_handler)
                      sing_handler (rcond);
                    else
                      (*current_liboctave_error_handler)
                        ("matrix singular to machine precision, rcond = %g",
                         rcond);
                  }
                }

              if (info == 0)
                {
                  retval = b;
                  double *result = retval.fortran_vec ();

                  octave_idx_type b_nc = b.cols ();

                  char job = 'N';
                  F77_XFCN (dgetrs, DGETRS, (F77_CONST_CHAR_ARG2 (&job, 1),
                                     nr, b_nc, tmp_data, nr,
                                     pipvt, result, b.rows(), info
                                     F77_CHAR_ARG_LEN (1)));
            
                  if (f77_exception_encountered)
                  (*current_liboctave_error_handler)
                    ("unrecoverable error in dgetrs");
                }
              else
                mattype.mark_as_rectangular ();
            }
          }
      }
      else if (typ != MatrixType::Hermitian)
      (*current_liboctave_error_handler) ("incorrect matrix type");
    }

  return retval;
}

Matrix
Matrix::solve (MatrixType &typ, const Matrix& b) const
{
  octave_idx_type info;
  double rcond;
  return solve (typ, b, info, rcond, 0);
}

Matrix
Matrix::solve (MatrixType &typ, const Matrix& b, octave_idx_type& info, 
             double& rcond) const
{
  return solve (typ, b, info, rcond, 0);
}

Matrix
Matrix::solve (MatrixType &mattype, const Matrix& b, octave_idx_type& info,
             double& rcond, solve_singularity_handler sing_handler,
             bool singular_fallback) const
{
  Matrix retval;
  int typ = mattype.type ();

  if (typ == MatrixType::Unknown)
    typ = mattype.type (*this);

  // Only calculate the condition number for LU/Cholesky
  if (typ == MatrixType::Upper || typ == MatrixType::Permuted_Upper)
    retval = utsolve (mattype, b, info, rcond, sing_handler, false);
  else if (typ == MatrixType::Lower || typ == MatrixType::Permuted_Lower)
    retval = ltsolve (mattype, b, info, rcond, sing_handler, false);
  else if (typ == MatrixType::Full || typ == MatrixType::Hermitian)
    retval = fsolve (mattype, b, info, rcond, sing_handler, true);
  else if (typ != MatrixType::Rectangular)
    {
      (*current_liboctave_error_handler) ("unknown matrix type");
      return Matrix ();
    }

  // Rectangular or one of the above solvers flags a singular matrix
  if (singular_fallback && mattype.type () == MatrixType::Rectangular)
    {
      octave_idx_type rank;
      retval = lssolve (b, info, rank, rcond);
    }

  return retval;
}

ComplexMatrix
Matrix::solve (MatrixType &typ, const ComplexMatrix& b) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (typ, b);
}

ComplexMatrix
Matrix::solve (MatrixType &typ, const ComplexMatrix& b, 
  octave_idx_type& info) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (typ, b, info);
}

ComplexMatrix
Matrix::solve (MatrixType &typ, const ComplexMatrix& b, octave_idx_type& info,
             double& rcond) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (typ, b, info, rcond);
}

ComplexMatrix
Matrix::solve (MatrixType &typ, const ComplexMatrix& b, octave_idx_type& info,
             double& rcond, solve_singularity_handler sing_handler,
             bool singular_fallback) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (typ, b, info, rcond, sing_handler, singular_fallback);
}

ColumnVector
Matrix::solve (MatrixType &typ, const ColumnVector& b) const
{
  octave_idx_type info; double rcond;
  return solve (typ, b, info, rcond);
}

ColumnVector
Matrix::solve (MatrixType &typ, const ColumnVector& b, 
             octave_idx_type& info) const
{
  double rcond;
  return solve (typ, b, info, rcond);
}

ColumnVector
Matrix::solve (MatrixType &typ, const ColumnVector& b, octave_idx_type& info,
             double& rcond) const
{
  return solve (typ, b, info, rcond, 0);
}

ColumnVector
Matrix::solve (MatrixType &typ, const ColumnVector& b, octave_idx_type& info,
             double& rcond, solve_singularity_handler sing_handler) const
{
  Matrix tmp (b);
  return solve (typ, tmp, info, rcond, sing_handler).column(static_cast<octave_idx_type> (0));
}

ComplexColumnVector
Matrix::solve (MatrixType &typ, const ComplexColumnVector& b) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (typ, b);
}

ComplexColumnVector
Matrix::solve (MatrixType &typ, const ComplexColumnVector& b, 
             octave_idx_type& info) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (typ, b, info);
}

ComplexColumnVector
Matrix::solve (MatrixType &typ, const ComplexColumnVector& b, 
             octave_idx_type& info, double& rcond) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (typ, b, info, rcond);
}

ComplexColumnVector
Matrix::solve (MatrixType &typ, const ComplexColumnVector& b, 
             octave_idx_type& info, double& rcond,
             solve_singularity_handler sing_handler) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve(typ, b, info, rcond, sing_handler);
}

Matrix
Matrix::solve (const Matrix& b) const
{
  octave_idx_type info;
  double rcond;
  return solve (b, info, rcond, 0);
}

Matrix
Matrix::solve (const Matrix& b, octave_idx_type& info) const
{
  double rcond;
  return solve (b, info, rcond, 0);
}

Matrix
Matrix::solve (const Matrix& b, octave_idx_type& info, double& rcond) const
{
  return solve (b, info, rcond, 0);
}

Matrix
Matrix::solve (const Matrix& b, octave_idx_type& info,
             double& rcond, solve_singularity_handler sing_handler) const
{
  MatrixType mattype (*this);
  return solve (mattype, b, info, rcond, sing_handler);
}

ComplexMatrix
Matrix::solve (const ComplexMatrix& b) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b);
}

ComplexMatrix
Matrix::solve (const ComplexMatrix& b, octave_idx_type& info) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b, info);
}

ComplexMatrix
Matrix::solve (const ComplexMatrix& b, octave_idx_type& info, double& rcond) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b, info, rcond);
}

ComplexMatrix
Matrix::solve (const ComplexMatrix& b, octave_idx_type& info, double& rcond,
             solve_singularity_handler sing_handler) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b, info, rcond, sing_handler);
}

ColumnVector
Matrix::solve (const ColumnVector& b) const
{
  octave_idx_type info; double rcond;
  return solve (b, info, rcond);
}

ColumnVector
Matrix::solve (const ColumnVector& b, octave_idx_type& info) const
{
  double rcond;
  return solve (b, info, rcond);
}

ColumnVector
Matrix::solve (const ColumnVector& b, octave_idx_type& info, double& rcond) const
{
  return solve (b, info, rcond, 0);
}

ColumnVector
Matrix::solve (const ColumnVector& b, octave_idx_type& info, double& rcond,
             solve_singularity_handler sing_handler) const
{
  MatrixType mattype (*this);
  return solve (mattype, b, info, rcond, sing_handler);
}

ComplexColumnVector
Matrix::solve (const ComplexColumnVector& b) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b);
}

ComplexColumnVector
Matrix::solve (const ComplexColumnVector& b, octave_idx_type& info) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b, info);
}

ComplexColumnVector
Matrix::solve (const ComplexColumnVector& b, octave_idx_type& info, double& rcond) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b, info, rcond);
}

ComplexColumnVector
Matrix::solve (const ComplexColumnVector& b, octave_idx_type& info, double& rcond,
             solve_singularity_handler sing_handler) const
{
  ComplexMatrix tmp (*this);
  return tmp.solve (b, info, rcond, sing_handler);
}

Matrix
Matrix::lssolve (const Matrix& b) const
{
  octave_idx_type info;
  octave_idx_type rank;
  double rcond;
  return lssolve (b, info, rank, rcond);
}

Matrix
Matrix::lssolve (const Matrix& b, octave_idx_type& info) const
{
  octave_idx_type rank;
  double rcond;
  return lssolve (b, info, rank, rcond);
}

Matrix
Matrix::lssolve (const Matrix& b, octave_idx_type& info,
             octave_idx_type& rank) const
{
  double rcond;
  return lssolve (b, info, rank, rcond);
}

Matrix
Matrix::lssolve (const Matrix& b, octave_idx_type& info,
             octave_idx_type& rank, double &rcond) const
{
  Matrix retval;

  octave_idx_type nrhs = b.cols ();

  octave_idx_type m = rows ();
  octave_idx_type n = cols ();

  if (m != b.rows ())
    (*current_liboctave_error_handler)
      ("matrix dimension mismatch solution of linear equations");
  else if (m == 0 || n == 0 || b.cols () == 0)
    retval = Matrix (n, b.cols (), 0.0);
  else
    {
      volatile octave_idx_type minmn = (m < n ? m : n);
      octave_idx_type maxmn = m > n ? m : n;
      rcond = -1.0;
      if (m != n)
      {
        retval = Matrix (maxmn, nrhs, 0.0);

        for (octave_idx_type j = 0; j < nrhs; j++)
          for (octave_idx_type i = 0; i < m; i++)
            retval.elem (i, j) = b.elem (i, j);
      }
      else
      retval = b;

      Matrix atmp = *this;
      double *tmp_data = atmp.fortran_vec ();

      double *pretval = retval.fortran_vec ();
      Array<double> s (minmn);
      double *ps = s.fortran_vec ();

      // Ask DGELSD what the dimension of WORK should be.
      octave_idx_type lwork = -1;

      Array<double> work (1);

      octave_idx_type smlsiz;
      F77_FUNC (xilaenv, XILAENV) (9, F77_CONST_CHAR_ARG2 ("DGELSD", 6),
                           F77_CONST_CHAR_ARG2 (" ", 1),
                           0, 0, 0, 0, smlsiz
                           F77_CHAR_ARG_LEN (6)
                           F77_CHAR_ARG_LEN (1));

      octave_idx_type mnthr;
      F77_FUNC (xilaenv, XILAENV) (6, F77_CONST_CHAR_ARG2 ("DGELSD", 6),
                           F77_CONST_CHAR_ARG2 (" ", 1),
                           m, n, nrhs, -1, mnthr
                           F77_CHAR_ARG_LEN (6)
                           F77_CHAR_ARG_LEN (1));

      // We compute the size of iwork because DGELSD in older versions
      // of LAPACK does not return it on a query call.
      double dminmn = static_cast<double> (minmn);
      double dsmlsizp1 = static_cast<double> (smlsiz+1);
#if defined (HAVE_LOG2)
      double tmp = log2 (dminmn / dsmlsizp1);
#else
      double tmp = log (dminmn / dsmlsizp1) / log (2.0);
#endif
      octave_idx_type nlvl = static_cast<octave_idx_type> (tmp) + 1;
      if (nlvl < 0)
      nlvl = 0;

      octave_idx_type liwork = 3 * minmn * nlvl + 11 * minmn;
      if (liwork < 1)
      liwork = 1;
      Array<octave_idx_type> iwork (liwork);
      octave_idx_type* piwork = iwork.fortran_vec ();

      F77_XFCN (dgelsd, DGELSD, (m, n, nrhs, tmp_data, m, pretval, maxmn,
                         ps, rcond, rank, work.fortran_vec (),
                         lwork, piwork, info));

      // The workspace query is broken in at least LAPACK 3.0.0
      // through 3.1.1 when n >= mnthr.  The obtuse formula below
      // should provide sufficient workspace for DGELSD to operate
      // efficiently.
      if (n >= mnthr)
      {
        const octave_idx_type wlalsd
          = 9*m + 2*m*smlsiz + 8*m*nlvl + m*nrhs + (smlsiz+1)*(smlsiz+1);

        octave_idx_type addend = m;

        if (2*m-4 > addend)
          addend = 2*m-4;

        if (nrhs > addend)
          addend = nrhs;

        if (n-3*m > addend)
          addend = n-3*m;

        if (wlalsd > addend)
          addend = wlalsd;

        const octave_idx_type lworkaround = 4*m + m*m + addend;

        if (work(0) < lworkaround)
          work(0) = lworkaround;
      }
      else if (m >= n)
      {
        octave_idx_type lworkaround
          = 12*n + 2*n*smlsiz + 8*n*nlvl + n*nrhs + (smlsiz+1)*(smlsiz+1);

        if (work(0) < lworkaround)
          work(0) = lworkaround;
      }

      if (f77_exception_encountered)
      (*current_liboctave_error_handler) 
        ("unrecoverable error in dgelsd");
      else
      {
        lwork = static_cast<octave_idx_type> (work(0));
        work.resize (lwork);

        F77_XFCN (dgelsd, DGELSD, (m, n, nrhs, tmp_data, m, pretval,
                             maxmn, ps, rcond, rank,
                             work.fortran_vec (), lwork, 
                             piwork, info));

        if (f77_exception_encountered)
          (*current_liboctave_error_handler) 
            ("unrecoverable error in dgelsd");
        else 
          {
            if (rank < minmn)
            (*current_liboctave_warning_handler) 
              ("dgelsd: rank deficient %dx%d matrix, rank = %d", m, n, rank);
            if (s.elem (0) == 0.0)
            rcond = 0.0;
            else
            rcond = s.elem (minmn - 1) / s.elem (0);

            retval.resize (n, nrhs);
          }
      }
    }

  return retval;
}

ComplexMatrix
Matrix::lssolve (const ComplexMatrix& b) const
{
  ComplexMatrix tmp (*this);
  octave_idx_type info;
  octave_idx_type rank;
  double rcond;
  return tmp.lssolve (b, info, rank, rcond);
}

ComplexMatrix
Matrix::lssolve (const ComplexMatrix& b, octave_idx_type& info) const
{
  ComplexMatrix tmp (*this);
  octave_idx_type rank;
  double rcond;
  return tmp.lssolve (b, info, rank, rcond);
}

ComplexMatrix
Matrix::lssolve (const ComplexMatrix& b, octave_idx_type& info, 
             octave_idx_type& rank) const
{
  ComplexMatrix tmp (*this);
  double rcond;
  return tmp.lssolve (b, info, rank, rcond);
}

ComplexMatrix
Matrix::lssolve (const ComplexMatrix& b, octave_idx_type& info, 
             octave_idx_type& rank, double& rcond) const
{
  ComplexMatrix tmp (*this);
  return tmp.lssolve (b, info, rank, rcond);
}

ColumnVector
Matrix::lssolve (const ColumnVector& b) const
{
  octave_idx_type info;
  octave_idx_type rank;
  double rcond;
  return lssolve (b, info, rank, rcond);
}

ColumnVector
Matrix::lssolve (const ColumnVector& b, octave_idx_type& info) const
{
  octave_idx_type rank;
  double rcond;
  return lssolve (b, info, rank, rcond);
}

ColumnVector
Matrix::lssolve (const ColumnVector& b, octave_idx_type& info,
             octave_idx_type& rank) const
{
  double rcond;
  return lssolve (b, info, rank, rcond);
}

ColumnVector
Matrix::lssolve (const ColumnVector& b, octave_idx_type& info,
             octave_idx_type& rank, double &rcond) const
{
  ColumnVector retval;

  octave_idx_type nrhs = 1;

  octave_idx_type m = rows ();
  octave_idx_type n = cols ();

  if (m != b.length ())
    (*current_liboctave_error_handler)
      ("matrix dimension mismatch solution of linear equations");
  else if (m == 0 || n == 0)
    retval = ColumnVector (n, 0.0);
  else
    {
      volatile octave_idx_type minmn = (m < n ? m : n);
      octave_idx_type maxmn = m > n ? m : n;
      rcond = -1.0;
 
      if (m != n)
      {
        retval = ColumnVector (maxmn, 0.0);

        for (octave_idx_type i = 0; i < m; i++)
          retval.elem (i) = b.elem (i);
      }
      else
      retval = b;

      Matrix atmp = *this;
      double *tmp_data = atmp.fortran_vec ();

      double *pretval = retval.fortran_vec ();
      Array<double> s (minmn);
      double *ps = s.fortran_vec ();

      // Ask DGELSD what the dimension of WORK should be.
      octave_idx_type lwork = -1;

      Array<double> work (1);

      octave_idx_type smlsiz;
      F77_FUNC (xilaenv, XILAENV) (9, F77_CONST_CHAR_ARG2 ("DGELSD", 6),
                           F77_CONST_CHAR_ARG2 (" ", 1),
                           0, 0, 0, 0, smlsiz
                           F77_CHAR_ARG_LEN (6)
                           F77_CHAR_ARG_LEN (1));

      // We compute the size of iwork because DGELSD in older versions
      // of LAPACK does not return it on a query call.
      double dminmn = static_cast<double> (minmn);
      double dsmlsizp1 = static_cast<double> (smlsiz+1);
#if defined (HAVE_LOG2)
      double tmp = log2 (dminmn / dsmlsizp1);
#else
      double tmp = log (dminmn / dsmlsizp1) / log (2.0);
#endif
      octave_idx_type nlvl = static_cast<octave_idx_type> (tmp) + 1;
      if (nlvl < 0)
      nlvl = 0;

      octave_idx_type liwork = 3 * minmn * nlvl + 11 * minmn;
      if (liwork < 1)
      liwork = 1;
      Array<octave_idx_type> iwork (liwork);
      octave_idx_type* piwork = iwork.fortran_vec ();

      F77_XFCN (dgelsd, DGELSD, (m, n, nrhs, tmp_data, m, pretval, maxmn,
                         ps, rcond, rank, work.fortran_vec (),
                         lwork, piwork, info));

      if (f77_exception_encountered)
      (*current_liboctave_error_handler) 
        ("unrecoverable error in dgelsd");
      else
      {
        lwork = static_cast<octave_idx_type> (work(0));
        work.resize (lwork);

        F77_XFCN (dgelsd, DGELSD, (m, n, nrhs, tmp_data, m, pretval,
                             maxmn, ps, rcond, rank,
                             work.fortran_vec (), lwork, 
                             piwork, info));

        if (f77_exception_encountered)
          (*current_liboctave_error_handler) 
            ("unrecoverable error in dgelsd");
        else if (rank < minmn)
          {
            if (rank < minmn)
            (*current_liboctave_warning_handler) 
              ("dgelsd: rank deficient %dx%d matrix, rank = %d", m, n, rank);
            if (s.elem (0) == 0.0)
            rcond = 0.0;
            else
            rcond = s.elem (minmn - 1) / s.elem (0);
          }

        retval.resize (n, nrhs);
      }
    }

  return retval;
}

ComplexColumnVector
Matrix::lssolve (const ComplexColumnVector& b) const
{
  ComplexMatrix tmp (*this);
  octave_idx_type info;
  octave_idx_type rank;
  double rcond;
  return tmp.lssolve (b, info, rank, rcond);
}

ComplexColumnVector
Matrix::lssolve (const ComplexColumnVector& b, octave_idx_type& info) const
{
  ComplexMatrix tmp (*this);
  octave_idx_type rank;
  double rcond;
  return tmp.lssolve (b, info, rank, rcond);
}

ComplexColumnVector
Matrix::lssolve (const ComplexColumnVector& b, octave_idx_type& info, 
             octave_idx_type& rank) const
{
  ComplexMatrix tmp (*this);
  double rcond;
  return tmp.lssolve (b, info, rank, rcond);
}

ComplexColumnVector
Matrix::lssolve (const ComplexColumnVector& b, octave_idx_type& info, 
             octave_idx_type& rank, double &rcond) const
{
  ComplexMatrix tmp (*this);
  return tmp.lssolve (b, info, rank, rcond);
}

// Constants for matrix exponential calculation.

static double padec [] =
{
  5.0000000000000000e-1,
  1.1666666666666667e-1,
  1.6666666666666667e-2,
  1.6025641025641026e-3,
  1.0683760683760684e-4,
  4.8562548562548563e-6,
  1.3875013875013875e-7,
  1.9270852604185938e-9,
};

Matrix
Matrix::expm (void) const
{
  Matrix retval;

  Matrix m = *this;

  if (numel () == 1)
    return Matrix (1, 1, exp (m(0)));

  octave_idx_type nc = columns ();

  // Preconditioning step 1: trace normalization to reduce dynamic
  // range of poles, but avoid making stable eigenvalues unstable.

  // trace shift value
  volatile double trshift = 0.0;

  for (octave_idx_type i = 0; i < nc; i++)
    trshift += m.elem (i, i);

  trshift /= nc;

  if (trshift > 0.0)
    {
      for (octave_idx_type i = 0; i < nc; i++)
      m.elem (i, i) -= trshift;
    }

  // Preconditioning step 2: balancing; code follows development
  // in AEPBAL

  double *p_m = m.fortran_vec ();

  octave_idx_type info, ilo, ihi, ilos, ihis;
  Array<double> dpermute (nc);
  Array<double> dscale (nc);

  // permutation first
  char job = 'P';
  F77_XFCN (dgebal, DGEBAL, (F77_CONST_CHAR_ARG2 (&job, 1),
                       nc, p_m, nc, ilo, ihi,
                       dpermute.fortran_vec (), info
                       F77_CHAR_ARG_LEN (1)));

  // then scaling
  job = 'S';
  F77_XFCN (dgebal, DGEBAL, (F77_CONST_CHAR_ARG2 (&job, 1),
                       nc, p_m, nc, ilos, ihis,
                       dscale.fortran_vec (), info
                       F77_CHAR_ARG_LEN (1)));

  if (f77_exception_encountered)
    {
      (*current_liboctave_error_handler) ("unrecoverable error in dgebal");
      return retval;
    }

  // Preconditioning step 3: scaling.
  
  ColumnVector work(nc);
  double inf_norm;
  
  F77_XFCN (xdlange, XDLANGE, (F77_CONST_CHAR_ARG2 ("I", 1),
                         nc, nc, m.fortran_vec (), nc,
                         work.fortran_vec (), inf_norm
                         F77_CHAR_ARG_LEN (1)));
  
  if (f77_exception_encountered)
    {
      (*current_liboctave_error_handler) ("unrecoverable error in dlange");
      return retval;
    }

  octave_idx_type sqpow = static_cast<octave_idx_type> (inf_norm > 0.0
                 ? (1.0 + log (inf_norm) / log (2.0))
                 : 0.0);
  
  // Check whether we need to square at all.
  
  if (sqpow < 0)
    sqpow = 0;
  
  if (sqpow > 0)
    {
      if (sqpow > 1023)
      sqpow = 1023;

      double scale_factor = 1.0;
      for (octave_idx_type i = 0; i < sqpow; i++)
      scale_factor *= 2.0;

      m = m / scale_factor;
    }
  
  // npp, dpp: pade' approx polynomial matrices.
  
  Matrix npp (nc, nc, 0.0);
  double *pnpp = npp.fortran_vec ();
  Matrix dpp = npp;
  double *pdpp = dpp.fortran_vec ();
  
  // Now powers a^8 ... a^1.
  
  octave_idx_type minus_one_j = -1;
  for (octave_idx_type j = 7; j >= 0; j--)
    {
      for (octave_idx_type i = 0; i < nc; i++)
      {
        octave_idx_type k = i * nc + i;
        pnpp[k] += padec[j];
        pdpp[k] += minus_one_j * padec[j];
      }      

      npp = m * npp;
      pnpp = npp.fortran_vec ();

      dpp = m * dpp;
      pdpp = dpp.fortran_vec ();

      minus_one_j *= -1;
    }
  
  // Zero power.
  
  dpp = -dpp;
  for (octave_idx_type j = 0; j < nc; j++)
    {
      npp.elem (j, j) += 1.0;
      dpp.elem (j, j) += 1.0;
    }
  
  // Compute pade approximation = inverse (dpp) * npp.

  retval = dpp.solve (npp, info);

  if (info < 0)
    return retval;

  // Reverse preconditioning step 3: repeated squaring.
  
  while (sqpow)
    {
      retval = retval * retval;
      sqpow--;
    }
  
  // Reverse preconditioning step 2: inverse balancing.
  // apply inverse scaling to computed exponential
  for (octave_idx_type i = 0; i < nc; i++)
    for (octave_idx_type j = 0; j < nc; j++)
      retval(i,j) *= dscale(i) / dscale(j);

  OCTAVE_QUIT;

  // construct balancing permutation vector
  Array<octave_idx_type> iperm (nc);
  for (octave_idx_type i = 0; i < nc; i++)
    iperm(i) = i;  // identity permutation

  // trailing permutations must be done in reverse order
  for (octave_idx_type i = nc - 1; i >= ihi; i--)
    {
      octave_idx_type swapidx = static_cast<octave_idx_type> (dpermute(i)) - 1;
      octave_idx_type tmp = iperm(i);
      iperm(i) = iperm(swapidx);
      iperm(swapidx) = tmp;
    }

  // leading permutations in forward order
  for (octave_idx_type i = 0; i < (ilo-1); i++)
    {
      octave_idx_type swapidx = static_cast<octave_idx_type> (dpermute(i)) - 1;
      octave_idx_type tmp = iperm(i);
      iperm(i) = iperm (swapidx);
      iperm(swapidx) = tmp;
    }

  // construct inverse balancing permutation vector
  Array<octave_idx_type> invpvec (nc);
  for (octave_idx_type i = 0; i < nc; i++)
    invpvec(iperm(i)) = i;     // Thanks to R. A. Lippert for this method

  OCTAVE_QUIT;

  Matrix tmpMat = retval;
  for (octave_idx_type i = 0; i < nc; i++)
    for (octave_idx_type j = 0; j < nc; j++)
      retval(i,j) = tmpMat(invpvec(i),invpvec(j));

  // Reverse preconditioning step 1: fix trace normalization. 
  if (trshift > 0.0)
    retval = exp (trshift) * retval;

  return retval;
}

Matrix&
Matrix::operator += (const DiagMatrix& a)
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  octave_idx_type a_nr = a.rows ();
  octave_idx_type a_nc = a.cols ();

  if (nr != a_nr || nc != a_nc)
    {
      gripe_nonconformant ("operator +=", nr, nc, a_nr, a_nc);
      return *this;
    }

  for (octave_idx_type i = 0; i < a.length (); i++)
    elem (i, i) += a.elem (i, i);

  return *this;
}

Matrix&
Matrix::operator -= (const DiagMatrix& a)
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  octave_idx_type a_nr = a.rows ();
  octave_idx_type a_nc = a.cols ();

  if (nr != a_nr || nc != a_nc)
    {
      gripe_nonconformant ("operator -=", nr, nc, a_nr, a_nc);
      return *this;
    }

  for (octave_idx_type i = 0; i < a.length (); i++)
    elem (i, i) -= a.elem (i, i);

  return *this;
}

// unary operations

boolMatrix
Matrix::operator ! (void) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  boolMatrix b (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      b.elem (i, j) = ! elem (i, j);

  return b;
}

// column vector by row vector -> matrix operations

Matrix
operator * (const ColumnVector& v, const RowVector& a)
{
  Matrix retval;

  octave_idx_type len = v.length ();

  if (len != 0)
    {
      octave_idx_type a_len = a.length ();

      retval.resize (len, a_len);
      double *c = retval.fortran_vec ();
        
      F77_XFCN (dgemm, DGEMM, (F77_CONST_CHAR_ARG2 ("N", 1),
                         F77_CONST_CHAR_ARG2 ("N", 1),
                         len, a_len, 1, 1.0, v.data (), len,
                         a.data (), 1, 0.0, c, len
                         F77_CHAR_ARG_LEN (1)
                         F77_CHAR_ARG_LEN (1)));

      if (f77_exception_encountered)
      (*current_liboctave_error_handler)
        ("unrecoverable error in dgemm");
    }

  return retval;
}

// other operations.

Matrix
Matrix::map (d_d_Mapper f) const
{
  Matrix b (*this);
  return b.apply (f);
}

boolMatrix
Matrix::map (b_d_Mapper f) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  boolMatrix retval (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      retval(i,j) = f (elem(i,j));

  return retval;
}

Matrix&
Matrix::apply (d_d_Mapper f)
{
  double *d = fortran_vec (); // Ensures only one reference to my privates!

  for (octave_idx_type i = 0; i < length (); i++)
    d[i] = f (d[i]);

  return *this;
}

bool
Matrix::any_element_is_negative (bool neg_zero) const
{
  octave_idx_type nel = nelem ();

  if (neg_zero)
    {
      for (octave_idx_type i = 0; i < nel; i++)
      if (lo_ieee_signbit (elem (i)))
        return true;
    }
  else
    {
      for (octave_idx_type i = 0; i < nel; i++)
      if (elem (i) < 0)
        return true;
    }

  return false;
}


bool
Matrix::any_element_is_inf_or_nan (void) const
{
  octave_idx_type nel = nelem ();

  for (octave_idx_type i = 0; i < nel; i++)
    {
      double val = elem (i);
      if (xisinf (val) || xisnan (val))
      return true;
    }

  return false;
}

bool
Matrix::any_element_not_one_or_zero (void) const
{
  octave_idx_type nel = nelem ();

  for (octave_idx_type i = 0; i < nel; i++)
    {
      double val = elem (i);
      if (val != 0 && val != 1)
      return true;
    }

  return false;
}

bool
Matrix::all_elements_are_int_or_inf_or_nan (void) const
{
  octave_idx_type nel = nelem ();

  for (octave_idx_type i = 0; i < nel; i++)
    {
      double val = elem (i);
      if (xisnan (val) || D_NINT (val) == val)
      continue;
      else
      return false;
    }

  return true;
}

// Return nonzero if any element of M is not an integer.  Also extract
// the largest and smallest values and return them in MAX_VAL and MIN_VAL.

bool
Matrix::all_integers (double& max_val, double& min_val) const
{
  octave_idx_type nel = nelem ();

  if (nel > 0)
    {
      max_val = elem (0);
      min_val = elem (0);
    }
  else
    return false;

  for (octave_idx_type i = 0; i < nel; i++)
    {
      double val = elem (i);

      if (val > max_val)
      max_val = val;

      if (val < min_val)
      min_val = val;

      if (D_NINT (val) != val)
      return false;
    }

  return true;
}

bool
Matrix::too_large_for_float (void) const
{
  octave_idx_type nel = nelem ();

  for (octave_idx_type i = 0; i < nel; i++)
    {
      double val = elem (i);

      if (! (xisnan (val) || xisinf (val))
        && fabs (val) > FLT_MAX)
      return true;
    }

  return false;
}

// FIXME Do these really belong here?  Maybe they should be
// in a base class?

boolMatrix
Matrix::all (int dim) const
{
  MX_ALL_OP (dim);
}

boolMatrix
Matrix::any (int dim) const
{
  MX_ANY_OP (dim);
}

Matrix
Matrix::cumprod (int dim) const
{
  MX_CUMULATIVE_OP (Matrix, double, *=);
}

Matrix
Matrix::cumsum (int dim) const
{
  MX_CUMULATIVE_OP (Matrix, double, +=);
}

Matrix
Matrix::prod (int dim) const
{
  MX_REDUCTION_OP (Matrix, *=, 1.0, 1.0);
}

Matrix
Matrix::sum (int dim) const
{
  MX_REDUCTION_OP (Matrix, +=, 0.0, 0.0);
}

Matrix
Matrix::sumsq (int dim) const
{
#define ROW_EXPR \
  double d = elem (i, j); \
  retval.elem (i, 0) += d * d

#define COL_EXPR \
  double d = elem (i, j); \
  retval.elem (0, j) += d * d

  MX_BASE_REDUCTION_OP (Matrix, ROW_EXPR, COL_EXPR, 0.0, 0.0);

#undef ROW_EXPR
#undef COL_EXPR
}

Matrix
Matrix::abs (void) const
{
  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  Matrix retval (nr, nc);

  for (octave_idx_type j = 0; j < nc; j++)
    for (octave_idx_type i = 0; i < nr; i++)
      retval (i, j) = fabs (elem (i, j));

  return retval;
}

ColumnVector
Matrix::diag (void) const
{
  return diag (0);
}

ColumnVector
Matrix::diag (octave_idx_type k) const
{
  octave_idx_type nnr = rows ();
  octave_idx_type nnc = cols ();
  if (k > 0)
    nnc -= k;
  else if (k < 0)
    nnr += k;

  ColumnVector d;

  if (nnr > 0 && nnc > 0)
    {
      octave_idx_type ndiag = (nnr < nnc) ? nnr : nnc;

      d.resize (ndiag);

      if (k > 0)
      {
        for (octave_idx_type i = 0; i < ndiag; i++)
          d.elem (i) = elem (i, i+k);
      }
      else if (k < 0)
      {
        for (octave_idx_type i = 0; i < ndiag; i++)
          d.elem (i) = elem (i-k, i);
      }
      else
      {
        for (octave_idx_type i = 0; i < ndiag; i++)
          d.elem (i) = elem (i, i);
      }
    }
  else
    (*current_liboctave_error_handler)
      ("diag: requested diagonal out of range");

  return d;
}

ColumnVector
Matrix::row_min (void) const
{
  Array<octave_idx_type> dummy_idx;
  return row_min (dummy_idx);
}

ColumnVector
Matrix::row_min (Array<octave_idx_type>& idx_arg) const
{
  ColumnVector result;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr > 0 && nc > 0)
    {
      result.resize (nr);
      idx_arg.resize (nr);

      for (octave_idx_type i = 0; i < nr; i++)
        {
        octave_idx_type idx_j;

        double tmp_min = octave_NaN;

        for (idx_j = 0; idx_j < nc; idx_j++)
          {
            tmp_min = elem (i, idx_j);

            if (! xisnan (tmp_min))
            break;
          }

        for (octave_idx_type j = idx_j+1; j < nc; j++)
          {
            double tmp = elem (i, j);

            if (xisnan (tmp))
            continue;
            else if (tmp < tmp_min)
            {
              idx_j = j;
              tmp_min = tmp;
            }
          }

        result.elem (i) = tmp_min;
        idx_arg.elem (i) = xisnan (tmp_min) ? 0 : idx_j;
        }
    }

  return result;
}

ColumnVector
Matrix::row_max (void) const
{
  Array<octave_idx_type> dummy_idx;
  return row_max (dummy_idx);
}

ColumnVector
Matrix::row_max (Array<octave_idx_type>& idx_arg) const
{
  ColumnVector result;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr > 0 && nc > 0)
    {
      result.resize (nr);
      idx_arg.resize (nr);

      for (octave_idx_type i = 0; i < nr; i++)
        {
        octave_idx_type idx_j;

        double tmp_max = octave_NaN;

        for (idx_j = 0; idx_j < nc; idx_j++)
          {
            tmp_max = elem (i, idx_j);

            if (! xisnan (tmp_max))
            break;
          }

        for (octave_idx_type j = idx_j+1; j < nc; j++)
          {
            double tmp = elem (i, j);

            if (xisnan (tmp))
            continue;
            else if (tmp > tmp_max)
            {
              idx_j = j;
              tmp_max = tmp;
            }
          }

        result.elem (i) = tmp_max;
        idx_arg.elem (i) = xisnan (tmp_max) ? 0 : idx_j;
        }
    }

  return result;
}

RowVector
Matrix::column_min (void) const
{
  Array<octave_idx_type> dummy_idx;
  return column_min (dummy_idx);
}

RowVector
Matrix::column_min (Array<octave_idx_type>& idx_arg) const
{
  RowVector result;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr > 0 && nc > 0)
    {
      result.resize (nc);
      idx_arg.resize (nc);

      for (octave_idx_type j = 0; j < nc; j++)
        {
        octave_idx_type idx_i;

        double tmp_min = octave_NaN;

        for (idx_i = 0; idx_i < nr; idx_i++)
          {
            tmp_min = elem (idx_i, j);

            if (! xisnan (tmp_min))
            break;
          }

        for (octave_idx_type i = idx_i+1; i < nr; i++)
          {
            double tmp = elem (i, j);

            if (xisnan (tmp))
            continue;
            else if (tmp < tmp_min)
            {
              idx_i = i;
              tmp_min = tmp;
            }
          }

        result.elem (j) = tmp_min;
        idx_arg.elem (j) = xisnan (tmp_min) ? 0 : idx_i;
        }
    }

  return result;
}

RowVector
Matrix::column_max (void) const
{
  Array<octave_idx_type> dummy_idx;
  return column_max (dummy_idx);
}

RowVector
Matrix::column_max (Array<octave_idx_type>& idx_arg) const
{
  RowVector result;

  octave_idx_type nr = rows ();
  octave_idx_type nc = cols ();

  if (nr > 0 && nc > 0)
    {
      result.resize (nc);
      idx_arg.resize (nc);

      for (octave_idx_type j = 0; j < nc; j++)
        {
        octave_idx_type idx_i;

        double tmp_max = octave_NaN;

        for (idx_i = 0; idx_i < nr; idx_i++)
          {
            tmp_max = elem (idx_i, j);

            if (! xisnan (tmp_max))
            break;
          }

        for (octave_idx_type i = idx_i+1; i < nr; i++)
          {
            double tmp = elem (i, j);

            if (xisnan (tmp))
            continue;
            else if (tmp > tmp_max)
            {
              idx_i = i;
              tmp_max = tmp;
            }
          }

        result.elem (j) = tmp_max;
        idx_arg.elem (j) = xisnan (tmp_max) ? 0 : idx_i;
        }
    }

  return result;
}

std::ostream&
operator << (std::ostream& os, const Matrix& a)
{
  for (octave_idx_type i = 0; i < a.rows (); i++)
    {
      for (octave_idx_type j = 0; j < a.cols (); j++)
      {
        os << " ";
        octave_write_double (os, a.elem (i, j));
      }
      os << "\n";
    }
  return os;
}

std::istream&
operator >> (std::istream& is, Matrix& a)
{
  octave_idx_type nr = a.rows ();
  octave_idx_type nc = a.cols ();

  if (nr < 1 || nc < 1)
    is.clear (std::ios::badbit);
  else
    {
      double tmp;
      for (octave_idx_type i = 0; i < nr; i++)
      for (octave_idx_type j = 0; j < nc; j++)
        {
          tmp = octave_read_double (is);
          if (is)
            a.elem (i, j) = tmp;
          else
            goto done;
        }
    }

 done:

  return is;
}

Matrix
Givens (double x, double y)
{
  double cc, s, temp_r;

  F77_FUNC (dlartg, DLARTG) (x, y, cc, s, temp_r);

  Matrix g (2, 2);

  g.elem (0, 0) = cc;
  g.elem (1, 1) = cc;
  g.elem (0, 1) = s;
  g.elem (1, 0) = -s;

  return g;
}

Matrix
Sylvester (const Matrix& a, const Matrix& b, const Matrix& c)
{
  Matrix retval;

  // FIXME -- need to check that a, b, and c are all the same
  // size.

  // Compute Schur decompositions.

  SCHUR as (a, "U");
  SCHUR bs (b, "U");
  
  // Transform c to new coordinates.

  Matrix ua = as.unitary_matrix ();
  Matrix sch_a = as.schur_matrix ();

  Matrix ub = bs.unitary_matrix ();
  Matrix sch_b = bs.schur_matrix ();
  
  Matrix cx = ua.transpose () * c * ub;
  
  // Solve the sylvester equation, back-transform, and return the
  // solution.

  octave_idx_type a_nr = a.rows ();
  octave_idx_type b_nr = b.rows ();

  double scale;
  octave_idx_type info;

  double *pa = sch_a.fortran_vec ();
  double *pb = sch_b.fortran_vec ();
  double *px = cx.fortran_vec ();

  F77_XFCN (dtrsyl, DTRSYL, (F77_CONST_CHAR_ARG2 ("N", 1),
                       F77_CONST_CHAR_ARG2 ("N", 1),
                       1, a_nr, b_nr, pa, a_nr, pb,
                       b_nr, px, a_nr, scale, info
                       F77_CHAR_ARG_LEN (1)
                       F77_CHAR_ARG_LEN (1)));


  if (f77_exception_encountered)
    (*current_liboctave_error_handler) ("unrecoverable error in dtrsyl");
  else
    {
      // FIXME -- check info?
  
      retval = -ua*cx*ub.transpose ();
    }

  return retval;
}

// matrix by matrix -> matrix operations

/* Simple Dot Product, Matrix-Vector and Matrix-Matrix Unit tests
%!assert([1 2 3] * [ 4 ; 5 ; 6], 32, 1e-14)
%!assert([1 2 ; 3 4 ] * [5 ; 6], [17 ; 39 ], 1e-14)
%!assert([1 2 ; 3 4 ] * [5 6 ; 7 8], [19 22; 43 50], 1e-14)
*/

/* Test some simple identities
%!shared M, cv, rv
%! M = randn(10,10);
%! cv = randn(10,1);
%! rv = randn(1,10);
%!assert([M*cv,M*cv],M*[cv,cv],1e-14)
%!assert([rv*M;rv*M],[rv;rv]*M,1e-14)
%!assert(2*rv*cv,[rv,rv]*[cv;cv],1e-14)
*/


Matrix
operator * (const Matrix& m, const Matrix& a)
{
  Matrix retval;

  octave_idx_type nr = m.rows ();
  octave_idx_type nc = m.cols ();

  octave_idx_type a_nr = a.rows ();
  octave_idx_type a_nc = a.cols ();

  if (nc != a_nr)
    gripe_nonconformant ("operator *", nr, nc, a_nr, a_nc);
  else
    {
      if (nr == 0 || nc == 0 || a_nc == 0)
      retval.resize (nr, a_nc, 0.0);
      else
      {
        octave_idx_type ld  = nr;
        octave_idx_type lda = a_nr;

        retval.resize (nr, a_nc);
        double *c = retval.fortran_vec ();

        if (a_nc == 1)
          {
            if (nr == 1)
            F77_FUNC (xddot, XDDOT) (nc, m.data (), 1, a.data (), 1, *c);
            else
            {
              F77_XFCN (dgemv, DGEMV, (F77_CONST_CHAR_ARG2 ("N", 1),
                                 nr, nc, 1.0,  m.data (), ld,
                                 a.data (), 1, 0.0, c, 1
                                 F77_CHAR_ARG_LEN (1)));

              if (f77_exception_encountered)
                (*current_liboctave_error_handler)
                  ("unrecoverable error in dgemv");
            }
            }
        else
          {
            F77_XFCN (dgemm, DGEMM, (F77_CONST_CHAR_ARG2 ("N", 1),
                               F77_CONST_CHAR_ARG2 ("N", 1),
                               nr, a_nc, nc, 1.0, m.data (),
                               ld, a.data (), lda, 0.0, c, nr
                               F77_CHAR_ARG_LEN (1)
                               F77_CHAR_ARG_LEN (1)));

            if (f77_exception_encountered)
            (*current_liboctave_error_handler)
              ("unrecoverable error in dgemm");
          }
      }
    }

  return retval;
}

// FIXME -- it would be nice to share code among the min/max
// functions below.

#define EMPTY_RETURN_CHECK(T) \
  if (nr == 0 || nc == 0) \
    return T (nr, nc);

Matrix
min (double d, const Matrix& m)
{
  octave_idx_type nr = m.rows ();
  octave_idx_type nc = m.columns ();

  EMPTY_RETURN_CHECK (Matrix);

  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) = xmin (d, m (i, j));
      }

  return result;
}

Matrix
min (const Matrix& m, double d)
{
  octave_idx_type nr = m.rows ();
  octave_idx_type nc = m.columns ();

  EMPTY_RETURN_CHECK (Matrix);

  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) = xmin (m (i, j), d);
      }

  return result;
}

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

  if (nr != b.rows () || nc != b.columns ())
    {
      (*current_liboctave_error_handler)
      ("two-arg min expecting args of same size");
      return Matrix ();
    }

  EMPTY_RETURN_CHECK (Matrix);

  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) = xmin (a (i, j), b (i, j));
      }

  return result;
}

Matrix
max (double d, const Matrix& m)
{
  octave_idx_type nr = m.rows ();
  octave_idx_type nc = m.columns ();

  EMPTY_RETURN_CHECK (Matrix);

  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) = xmax (d, m (i, j));
      }

  return result;
}

Matrix
max (const Matrix& m, double d)
{
  octave_idx_type nr = m.rows ();
  octave_idx_type nc = m.columns ();

  EMPTY_RETURN_CHECK (Matrix);

  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) = xmax (m (i, j), d);
      }

  return result;
}

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

  if (nr != b.rows () || nc != b.columns ())
    {
      (*current_liboctave_error_handler)
      ("two-arg max expecting args of same size");
      return Matrix ();
    }

  EMPTY_RETURN_CHECK (Matrix);

  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) = xmax (a (i, j), b (i, j));
      }

  return result;
}

MS_CMP_OPS(Matrix, , double, )
MS_BOOL_OPS(Matrix, double, 0.0)

SM_CMP_OPS(double, , Matrix, )
SM_BOOL_OPS(double, Matrix, 0.0)

MM_CMP_OPS(Matrix, , Matrix, )
MM_BOOL_OPS(Matrix, Matrix, 0.0)

/*
;;; Local Variables: ***
;;; mode: C++ ***
;;; End: ***
*/

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