/*

Copyright (C) 1994-2018 John W. Eaton
Copyright (C) 2008-2009 Jaroslav Hajek
Copyright (C) 2009 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
<https://www.gnu.org/licenses/>.

*/

#if defined (HAVE_CONFIG_H)
#  include "config.h"
#endif

#include <algorithm>

#include "Array.h"
#include "CColVector.h"
#include "CMatrix.h"
#include "CRowVector.h"
#include "dColVector.h"
#include "dMatrix.h"
#include "dRowVector.h"
#include "fCColVector.h"
#include "fCMatrix.h"
#include "fCRowVector.h"
#include "fColVector.h"
#include "fMatrix.h"
#include "fRowVector.h"
#include "lo-error.h"
#include "lo-lapack-proto.h"
#include "lo-qrupdate-proto.h"
#include "oct-cmplx.h"
#include "oct-locbuf.h"
#include "oct-sort.h"
#include "qr.h"

namespace octave
{
  namespace math
  {
    template <typename T>
    qr<T>::qr (const T& q_arg, const T& r_arg)
      : q (q_arg), r (r_arg)
    {
      octave_idx_type q_nr = q.rows ();
      octave_idx_type q_nc = q.cols ();

      octave_idx_type r_nr = r.rows ();
      octave_idx_type r_nc = r.cols ();

      if (! (q_nc == r_nr && (q_nr == q_nc || (q_nr > q_nc && r_nr == r_nc))))
        (*current_liboctave_error_handler) ("QR dimensions mismatch");
    }

    template <typename T>
    typename qr<T>::type
    qr<T>::get_type (void) const
    {
      type retval;

      if (! q.isempty () && q.issquare ())
        retval = qr<T>::std;
      else if (q.rows () > q.cols () && r.issquare ())
        retval = qr<T>::economy;
      else
        retval = qr<T>::raw;

      return retval;
    }

    template <typename T>
    bool
    qr<T>::regular (void) const
    {
      bool retval = true;

      octave_idx_type k = std::min (r.rows (), r.cols ());

      for (octave_idx_type i = 0; i < k; i++)
        {
          if (r(i, i) == ELT_T ())
            {
              retval = false;
              break;
            }
        }

      return retval;
    }

#if ! defined (HAVE_QRUPDATE)

    // Replacement update methods.

    void
    warn_qrupdate_once (void)
    {
      static bool warned = false;

      if (! warned)
        {
          (*current_liboctave_warning_with_id_handler)
            ("Octave:missing-dependency",
             "In this version of Octave, QR & Cholesky updating routines "
             "simply update the matrix and recalculate factorizations. "
             "To use fast algorithms, link Octave with the qrupdate library. "
             "See <http://sourceforge.net/projects/qrupdate>.");

          warned = true;
        }
    }

    template <typename T>
    void
    qr<T>::update (const CV_T& u, const CV_T& v)
    {
      warn_qrupdate_once ();

      octave_idx_type m = q.rows ();
      octave_idx_type n = r.cols ();

      if (u.numel () != m || v.numel () != n)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      init (q*r + T (u) * T (v).hermitian (), get_type ());
    }

    template <typename T>
    void
    qr<T>::update (const T& u, const T& v)
    {
      warn_qrupdate_once ();

      octave_idx_type m = q.rows ();
      octave_idx_type n = r.cols ();

      if (u.rows () != m || v.rows () != n || u.cols () != v.cols ())
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      init (q*r + u * v.hermitian (), get_type ());
    }

    template <typename T, typename CV_T>
    static
    T
    insert_col (const T& a, octave_idx_type i, const CV_T& x)
    {
      T retval (a.rows (), a.cols () + 1);
      retval.assign (idx_vector::colon, idx_vector (0, i),
                     a.index (idx_vector::colon, idx_vector (0, i)));
      retval.assign (idx_vector::colon, idx_vector (i), x);
      retval.assign (idx_vector::colon, idx_vector (i+1, retval.cols ()),
                     a.index (idx_vector::colon, idx_vector (i, a.cols ())));
      return retval;
    }

    template <typename T, typename RV_T>
    static
    T
    insert_row (const T& a, octave_idx_type i, const RV_T& x)
    {
      T retval (a.rows () + 1, a.cols ());
      retval.assign (idx_vector (0, i), idx_vector::colon,
                     a.index (idx_vector (0, i), idx_vector::colon));
      retval.assign (idx_vector (i), idx_vector::colon, x);
      retval.assign (idx_vector (i+1, retval.rows ()), idx_vector::colon,
                     a.index (idx_vector (i, a.rows ()), idx_vector::colon));
      return retval;
    }

    template <typename T>
    static
    T
    delete_col (const T& a, octave_idx_type i)
    {
      T retval = a;
      retval.delete_elements (1, idx_vector (i));
      return retval;
    }

    template <typename T>
    static
    T
    delete_row (const T& a, octave_idx_type i)
    {
      T retval = a;
      retval.delete_elements (0, idx_vector (i));
      return retval;
    }

    template <typename T>
    static
    T
    shift_cols (const T& a, octave_idx_type i, octave_idx_type j)
    {
      octave_idx_type n = a.cols ();
      Array<octave_idx_type> p (dim_vector (n, 1));
      for (octave_idx_type k = 0; k < n; k++) p(k) = k;
      if (i < j)
        {
          for (octave_idx_type k = i; k < j; k++) p(k) = k+1;
          p(j) = i;
        }
      else if (j < i)
        {
          p(j) = i;
          for (octave_idx_type k = j+1; k < i+1; k++) p(k) = k-1;
        }

      return a.index (idx_vector::colon, idx_vector (p));
    }

    template <typename T>
    void
    qr<T>::insert_col (const CV_T& u, octave_idx_type j)
    {
      warn_qrupdate_once ();

      octave_idx_type m = q.rows ();
      octave_idx_type n = r.cols ();

      if (u.numel () != m)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > n)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      init (math::insert_col (q*r, j, u), get_type ());
    }

    template <typename T>
    void
    qr<T>::insert_col (const T& u, const Array<octave_idx_type>& j)
    {
      warn_qrupdate_once ();

      octave_idx_type m = q.rows ();
      octave_idx_type n = r.cols ();

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, ASCENDING);
      octave_idx_type nj = js.numel ();
      bool dups = false;
      for (octave_idx_type i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      if (u.numel () != m || u.cols () != nj)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (nj > 0 && (js(0) < 0 || js(nj-1) > n))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          T a = q*r;
          for (octave_idx_type i = 0; i < nj; i++)
            a = math::insert_col (a, js(i), u.column (i));

          init (a, get_type ());
        }
    }

    template <typename T>
    void
    qr<T>::delete_col (octave_idx_type j)
    {
      warn_qrupdate_once ();

      octave_idx_type n = r.cols ();

      if (j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      init (math::delete_col (q*r, j), get_type ());
    }

    template <typename T>
    void
    qr<T>::delete_col (const Array<octave_idx_type>& j)
    {
      warn_qrupdate_once ();

      octave_idx_type n = r.cols ();

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, DESCENDING);
      octave_idx_type nj = js.numel ();
      bool dups = false;
      for (octave_idx_type i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      if (nj > 0 && (js(0) > n-1 || js(nj-1) < 0))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          T a = q*r;
          for (octave_idx_type i = 0; i < nj; i++)
            a = math::delete_col (a, js(i));

          init (a, get_type ());
        }
    }

    template <typename T>
    void
    qr<T>::insert_row (const RV_T& u, octave_idx_type j)
    {
      warn_qrupdate_once ();

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

      if (! q.issquare () || u.numel () != n)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > m)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      init (math::insert_row (q*r, j, u), get_type ());
    }

    template <typename T>
    void
    qr<T>::delete_row (octave_idx_type j)
    {
      warn_qrupdate_once ();

      octave_idx_type m = r.rows ();

      if (! q.issquare ())
        (*current_liboctave_error_handler) ("qrdelete: dimensions mismatch");

      if (j < 0 || j > m-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      init (math::delete_row (q*r, j), get_type ());
    }

    template <typename T>
    void
    qr<T>::shift_cols (octave_idx_type i, octave_idx_type j)
    {
      warn_qrupdate_once ();

      octave_idx_type n = r.cols ();

      if (i < 0 || i > n-1 || j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrshift: index out of range");

      init (math::shift_cols (q*r, i, j), get_type ());
    }

#endif

    // Specializations.

    template <>
    void
    qr<Matrix>::form (octave_idx_type n_arg, Matrix& afact, double *tau,
                      type qr_type)
    {
      F77_INT n = to_f77_int (n_arg);
      F77_INT m = to_f77_int (afact.rows ());
      F77_INT min_mn = std::min (m, n);
      F77_INT info;

      if (qr_type == qr<Matrix>::raw)
        {
          for (F77_INT j = 0; j < min_mn; j++)
            {
              F77_INT limit = (j < min_mn - 1 ? j : min_mn - 1);
              for (F77_INT i = limit + 1; i < m; i++)
                afact.elem (i, j) *= tau[j];
            }

          r = afact;
        }
      else
        {
          // Attempt to minimize copying.
          if (m >= n)
            {
              // afact will become q.
              q = afact;
              F77_INT k = (qr_type == qr<Matrix>::economy ? n : m);
              r = Matrix (k, n);
              for (F77_INT j = 0; j < n; j++)
                {
                  F77_INT i = 0;
                  for (; i <= j; i++)
                    r.xelem (i, j) = afact.xelem (i, j);
                  for (; i < k; i++)
                    r.xelem (i, j) = 0;
                }
              afact = Matrix (); // optimize memory
            }
          else
            {
              // afact will become r.
              q = Matrix (m, m);
              for (F77_INT j = 0; j < m; j++)
                for (F77_INT i = j + 1; i < m; i++)
                  {
                    q.xelem (i, j) = afact.xelem (i, j);
                    afact.xelem (i, j) = 0;
                  }
              r = afact;
            }

          if (m > 0)
            {
              F77_INT k = to_f77_int (q.cols ());
              // workspace query.
              double rlwork;
              F77_XFCN (dorgqr, DORGQR, (m, k, min_mn, q.fortran_vec (), m,
                                         tau, &rlwork, -1, info));

              // allocate buffer and do the job.
              F77_INT lwork = static_cast<F77_INT> (rlwork);
              lwork = std::max (lwork, static_cast<F77_INT> (1));
              OCTAVE_LOCAL_BUFFER (double, work, lwork);
              F77_XFCN (dorgqr, DORGQR, (m, k, min_mn, q.fortran_vec (), m,
                                         tau, work, lwork, info));
            }
        }
    }

    template <>
    void
    qr<Matrix>::init (const Matrix& a, type qr_type)
    {
      F77_INT m = to_f77_int (a.rows ());
      F77_INT n = to_f77_int (a.cols ());

      F77_INT min_mn = (m < n ? m : n);
      OCTAVE_LOCAL_BUFFER (double, tau, min_mn);

      F77_INT info = 0;

      Matrix afact = a;
      if (m > n && qr_type == qr<Matrix>::std)
        afact.resize (m, m);

      if (m > 0)
        {
          // workspace query.
          double rlwork;
          F77_XFCN (dgeqrf, DGEQRF, (m, n, afact.fortran_vec (), m, tau,
                                     &rlwork, -1, info));

          // allocate buffer and do the job.
          F77_INT lwork = static_cast<F77_INT> (rlwork);
          lwork = std::max (lwork, static_cast<F77_INT> (1));
          OCTAVE_LOCAL_BUFFER (double, work, lwork);
          F77_XFCN (dgeqrf, DGEQRF, (m, n, afact.fortran_vec (), m, tau,
                                     work, lwork, info));
        }

      form (n, afact, tau, qr_type);
    }

#if defined (HAVE_QRUPDATE)

    template <>
    void
    qr<Matrix>::update (const ColumnVector& u, const ColumnVector& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT v_nel = to_f77_int (v.numel ());

      if (u_nel != m || v_nel != n)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      ColumnVector utmp = u;
      ColumnVector vtmp = v;
      OCTAVE_LOCAL_BUFFER (double, w, 2*k);
      F77_XFCN (dqr1up, DQR1UP, (m, n, k, q.fortran_vec (), m,
                                 r.fortran_vec (), k, utmp.fortran_vec (),
                                 vtmp.fortran_vec (), w));
    }

    template <>
    void
    qr<Matrix>::update (const Matrix& u, const Matrix& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_rows = to_f77_int (u.rows ());
      F77_INT u_cols = to_f77_int (u.cols ());

      F77_INT v_rows = to_f77_int (v.rows ());
      F77_INT v_cols = to_f77_int (v.cols ());

      if (u_rows != m || v_rows != n || u_cols != v_cols)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      OCTAVE_LOCAL_BUFFER (double, w, 2*k);
      for (volatile F77_INT i = 0; i < u_cols; i++)
        {
          ColumnVector utmp = u.column (i);
          ColumnVector vtmp = v.column (i);
          F77_XFCN (dqr1up, DQR1UP, (m, n, k, q.fortran_vec (), m,
                                     r.fortran_vec (), k, utmp.fortran_vec (),
                                     vtmp.fortran_vec (), w));
        }
    }

    template <>
    void
    qr<Matrix>::insert_col (const ColumnVector& u, octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());

      if (u_nel != m)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > n)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (k < m)
        {
          q.resize (m, k+1);
          r.resize (k+1, n+1);
        }
      else
        r.resize (k, n+1);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      ColumnVector utmp = u;
      OCTAVE_LOCAL_BUFFER (double, w, k);
      F77_XFCN (dqrinc, DQRINC, (m, n, k, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr, j + 1,
                                 utmp.data (), w));
    }

    template <>
    void
    qr<Matrix>::insert_col (const Matrix& u, const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, ASCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT u_cols = to_f77_int (u.cols ());

      if (u_nel != m || u_cols != nj)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg < 0 || js_end > n))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT kmax = std::min (k + nj, m);
          if (k < m)
            {
              q.resize (m, kmax);
              r.resize (kmax, n + nj);
            }
          else
            r.resize (k, n + nj);

          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (double, w, kmax);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              ColumnVector utmp = u.column (jsi(i));
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (dqrinc, DQRINC, (m, n + ii, std::min (kmax, k + ii),
                                         q.fortran_vec (), ldq,
                                         r.fortran_vec (), ldr, js_elt + 1,
                                         utmp.data (), w));
            }
        }
    }

    template <>
    void
    qr<Matrix>::delete_col (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (double, w, k);
      F77_XFCN (dqrdec, DQRDEC, (m, n, k, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr, j + 1, w));

      if (k < m)
        {
          q.resize (m, k-1);
          r.resize (k-1, n-1);
        }
      else
        r.resize (k, n-1);
    }

    template <>
    void
    qr<Matrix>::delete_col (const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, DESCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg > n-1 || js_end < 0))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (double, w, k);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (dqrdec, DQRDEC, (m, n - ii, (k == m ? k : k - ii),
                                         q.fortran_vec (), ldq,
                                         r.fortran_vec (), ldr,
                                         js_elt + 1, w));
            }

          if (k < m)
            {
              q.resize (m, k - nj);
              r.resize (k - nj, n - nj);
            }
          else
            r.resize (k, n - nj);
        }
    }

    template <>
    void
    qr<Matrix>::insert_row (const RowVector& u, octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = std::min (m, n);

      F77_INT u_nel = to_f77_int (u.numel ());

      if (! q.issquare () || u_nel != n)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > m)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      q.resize (m + 1, m + 1);
      r.resize (m + 1, n);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      RowVector utmp = u;
      OCTAVE_LOCAL_BUFFER (double, w, k);
      F77_XFCN (dqrinr, DQRINR, (m, n, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr,
                                 j + 1, utmp.fortran_vec (), w));

    }

    template <>
    void
    qr<Matrix>::delete_row (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (! q.issquare ())
        (*current_liboctave_error_handler) ("qrdelete: dimensions mismatch");

      if (j < 0 || j > m-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (double, w, 2*m);
      F77_XFCN (dqrder, DQRDER, (m, n, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr, j + 1, w));

      q.resize (m - 1, m - 1);
      r.resize (m - 1, n);
    }

    template <>
    void
    qr<Matrix>::shift_cols (octave_idx_type i_arg, octave_idx_type j_arg)
    {
      F77_INT i = to_f77_int (i_arg);
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (i < 0 || i > n-1 || j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrshift: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (double, w, 2*k);
      F77_XFCN (dqrshc, DQRSHC, (m, n, k,
                                 q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr,
                                 i + 1, j + 1, w));
    }

#endif

    template <>
    void
    qr<FloatMatrix>::form (octave_idx_type n_arg, FloatMatrix& afact,
                           float *tau, type qr_type)
    {
      F77_INT n = to_f77_int (n_arg);
      F77_INT m = to_f77_int (afact.rows ());
      F77_INT min_mn = std::min (m, n);
      F77_INT info;

      if (qr_type == qr<FloatMatrix>::raw)
        {
          for (F77_INT j = 0; j < min_mn; j++)
            {
              F77_INT limit = (j < min_mn - 1 ? j : min_mn - 1);
              for (F77_INT i = limit + 1; i < m; i++)
                afact.elem (i, j) *= tau[j];
            }

          r = afact;
        }
      else
        {
          // Attempt to minimize copying.
          if (m >= n)
            {
              // afact will become q.
              q = afact;
              F77_INT k = (qr_type == qr<FloatMatrix>::economy ? n : m);
              r = FloatMatrix (k, n);
              for (F77_INT j = 0; j < n; j++)
                {
                  F77_INT i = 0;
                  for (; i <= j; i++)
                    r.xelem (i, j) = afact.xelem (i, j);
                  for (; i < k; i++)
                    r.xelem (i, j) = 0;
                }
              afact = FloatMatrix (); // optimize memory
            }
          else
            {
              // afact will become r.
              q = FloatMatrix (m, m);
              for (F77_INT j = 0; j < m; j++)
                for (F77_INT i = j + 1; i < m; i++)
                  {
                    q.xelem (i, j) = afact.xelem (i, j);
                    afact.xelem (i, j) = 0;
                  }
              r = afact;
            }

          if (m > 0)
            {
              F77_INT k = to_f77_int (q.cols ());
              // workspace query.
              float rlwork;
              F77_XFCN (sorgqr, SORGQR, (m, k, min_mn, q.fortran_vec (), m,
                                         tau, &rlwork, -1, info));

              // allocate buffer and do the job.
              F77_INT lwork = static_cast<F77_INT> (rlwork);
              lwork = std::max (lwork, static_cast<F77_INT> (1));
              OCTAVE_LOCAL_BUFFER (float, work, lwork);
              F77_XFCN (sorgqr, SORGQR, (m, k, min_mn, q.fortran_vec (), m,
                                         tau, work, lwork, info));
            }
        }
    }

    template <>
    void
    qr<FloatMatrix>::init (const FloatMatrix& a, type qr_type)
    {
      F77_INT m = to_f77_int (a.rows ());
      F77_INT n = to_f77_int (a.cols ());

      F77_INT min_mn = (m < n ? m : n);
      OCTAVE_LOCAL_BUFFER (float, tau, min_mn);

      F77_INT info = 0;

      FloatMatrix afact = a;
      if (m > n && qr_type == qr<FloatMatrix>::std)
        afact.resize (m, m);

      if (m > 0)
        {
          // workspace query.
          float rlwork;
          F77_XFCN (sgeqrf, SGEQRF, (m, n, afact.fortran_vec (), m, tau,
                                     &rlwork, -1, info));

          // allocate buffer and do the job.
          F77_INT lwork = static_cast<F77_INT> (rlwork);
          lwork = std::max (lwork, static_cast<F77_INT> (1));
          OCTAVE_LOCAL_BUFFER (float, work, lwork);
          F77_XFCN (sgeqrf, SGEQRF, (m, n, afact.fortran_vec (), m, tau,
                                     work, lwork, info));
        }

      form (n, afact, tau, qr_type);
    }

#if defined (HAVE_QRUPDATE)

    template <>
    void
    qr<FloatMatrix>::update (const FloatColumnVector& u, const FloatColumnVector& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT v_nel = to_f77_int (v.numel ());

      if (u_nel != m || v_nel != n)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      FloatColumnVector utmp = u;
      FloatColumnVector vtmp = v;
      OCTAVE_LOCAL_BUFFER (float, w, 2*k);
      F77_XFCN (sqr1up, SQR1UP, (m, n, k, q.fortran_vec (), m,
                                 r.fortran_vec (), k, utmp.fortran_vec (),
                                 vtmp.fortran_vec (), w));
    }

    template <>
    void
    qr<FloatMatrix>::update (const FloatMatrix& u, const FloatMatrix& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_rows = to_f77_int (u.rows ());
      F77_INT u_cols = to_f77_int (u.cols ());

      F77_INT v_rows = to_f77_int (v.rows ());
      F77_INT v_cols = to_f77_int (v.cols ());

      if (u_rows != m || v_rows != n || u_cols != v_cols)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      OCTAVE_LOCAL_BUFFER (float, w, 2*k);
      for (volatile F77_INT i = 0; i < u_cols; i++)
        {
          FloatColumnVector utmp = u.column (i);
          FloatColumnVector vtmp = v.column (i);
          F77_XFCN (sqr1up, SQR1UP, (m, n, k, q.fortran_vec (), m,
                                     r.fortran_vec (), k, utmp.fortran_vec (),
                                     vtmp.fortran_vec (), w));
        }
    }

    template <>
    void
    qr<FloatMatrix>::insert_col (const FloatColumnVector& u,
                                 octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());

      if (u_nel != m)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > n)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (k < m)
        {
          q.resize (m, k+1);
          r.resize (k+1, n+1);
        }
      else
        r.resize (k, n+1);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      FloatColumnVector utmp = u;
      OCTAVE_LOCAL_BUFFER (float, w, k);
      F77_XFCN (sqrinc, SQRINC, (m, n, k, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr, j + 1,
                                 utmp.data (), w));
    }

    template <>
    void
    qr<FloatMatrix>::insert_col (const FloatMatrix& u,
                                 const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, ASCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT u_cols = to_f77_int (u.cols ());

      if (u_nel != m || u_cols != nj)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg < 0 || js_end > n))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT kmax = std::min (k + nj, m);
          if (k < m)
            {
              q.resize (m, kmax);
              r.resize (kmax, n + nj);
            }
          else
            r.resize (k, n + nj);

          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (float, w, kmax);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              FloatColumnVector utmp = u.column (jsi(i));
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (sqrinc, SQRINC, (m, n + ii, std::min (kmax, k + ii),
                                         q.fortran_vec (), ldq,
                                         r.fortran_vec (), ldr, js_elt + 1,
                                         utmp.data (), w));
            }
        }
    }

    template <>
    void
    qr<FloatMatrix>::delete_col (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (float, w, k);
      F77_XFCN (sqrdec, SQRDEC, (m, n, k, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr, j + 1, w));

      if (k < m)
        {
          q.resize (m, k-1);
          r.resize (k-1, n-1);
        }
      else
        r.resize (k, n-1);
    }

    template <>
    void
    qr<FloatMatrix>::delete_col (const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, DESCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg > n-1 || js_end < 0))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (float, w, k);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (sqrdec, SQRDEC, (m, n - ii, (k == m ? k : k - ii),
                                         q.fortran_vec (), ldq,
                                         r.fortran_vec (), ldr,
                                         js_elt + 1, w));
            }

          if (k < m)
            {
              q.resize (m, k - nj);
              r.resize (k - nj, n - nj);
            }
          else
            r.resize (k, n - nj);
        }
    }

    template <>
    void
    qr<FloatMatrix>::insert_row (const FloatRowVector& u,
                                 octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = std::min (m, n);

      F77_INT u_nel = to_f77_int (u.numel ());

      if (! q.issquare () || u_nel != n)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > m)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      q.resize (m + 1, m + 1);
      r.resize (m + 1, n);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      FloatRowVector utmp = u;
      OCTAVE_LOCAL_BUFFER (float, w, k);
      F77_XFCN (sqrinr, SQRINR, (m, n, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr,
                                 j + 1, utmp.fortran_vec (), w));

    }

    template <>
    void
    qr<FloatMatrix>::delete_row (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (! q.issquare ())
        (*current_liboctave_error_handler) ("qrdelete: dimensions mismatch");

      if (j < 0 || j > m-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (float, w, 2*m);
      F77_XFCN (sqrder, SQRDER, (m, n, q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr, j + 1,
                                 w));

      q.resize (m - 1, m - 1);
      r.resize (m - 1, n);
    }

    template <>
    void
    qr<FloatMatrix>::shift_cols (octave_idx_type i_arg, octave_idx_type j_arg)
    {
      F77_INT i = to_f77_int (i_arg);
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (i < 0 || i > n-1 || j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrshift: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (float, w, 2*k);
      F77_XFCN (sqrshc, SQRSHC, (m, n, k,
                                 q.fortran_vec (), ldq,
                                 r.fortran_vec (), ldr,
                                 i + 1, j + 1, w));
    }

#endif

    template <>
    void
    qr<ComplexMatrix>::form (octave_idx_type n_arg, ComplexMatrix& afact,
                             Complex *tau, type qr_type)
    {
      F77_INT n = to_f77_int (n_arg);
      F77_INT m = to_f77_int (afact.rows ());
      F77_INT min_mn = std::min (m, n);
      F77_INT info;

      if (qr_type == qr<ComplexMatrix>::raw)
        {
          for (F77_INT j = 0; j < min_mn; j++)
            {
              F77_INT limit = (j < min_mn - 1 ? j : min_mn - 1);
              for (F77_INT i = limit + 1; i < m; i++)
                afact.elem (i, j) *= tau[j];
            }

          r = afact;
        }
      else
        {
          // Attempt to minimize copying.
          if (m >= n)
            {
              // afact will become q.
              q = afact;
              F77_INT k = (qr_type == qr<ComplexMatrix>::economy ? n : m);
              r = ComplexMatrix (k, n);
              for (F77_INT j = 0; j < n; j++)
                {
                  F77_INT i = 0;
                  for (; i <= j; i++)
                    r.xelem (i, j) = afact.xelem (i, j);
                  for (; i < k; i++)
                    r.xelem (i, j) = 0;
                }
              afact = ComplexMatrix (); // optimize memory
            }
          else
            {
              // afact will become r.
              q = ComplexMatrix (m, m);
              for (F77_INT j = 0; j < m; j++)
                for (F77_INT i = j + 1; i < m; i++)
                  {
                    q.xelem (i, j) = afact.xelem (i, j);
                    afact.xelem (i, j) = 0;
                  }
              r = afact;
            }

          if (m > 0)
            {
              F77_INT k = to_f77_int (q.cols ());
              // workspace query.
              Complex clwork;
              F77_XFCN (zungqr, ZUNGQR, (m, k, min_mn,
                                         F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                         m, F77_DBLE_CMPLX_ARG (tau),
                                         F77_DBLE_CMPLX_ARG (&clwork), -1,
                                         info));

              // allocate buffer and do the job.
              F77_INT lwork = static_cast<F77_INT> (clwork.real ());
              lwork = std::max (lwork, static_cast<F77_INT> (1));
              OCTAVE_LOCAL_BUFFER (Complex, work, lwork);
              F77_XFCN (zungqr, ZUNGQR, (m, k, min_mn,
                                         F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                         m, F77_DBLE_CMPLX_ARG (tau),
                                         F77_DBLE_CMPLX_ARG (work), lwork,
                                         info));
            }
        }
    }

    template <>
    void
    qr<ComplexMatrix>::init (const ComplexMatrix& a, type qr_type)
    {
      F77_INT m = to_f77_int (a.rows ());
      F77_INT n = to_f77_int (a.cols ());

      F77_INT min_mn = (m < n ? m : n);
      OCTAVE_LOCAL_BUFFER (Complex, tau, min_mn);

      F77_INT info = 0;

      ComplexMatrix afact = a;
      if (m > n && qr_type == qr<ComplexMatrix>::std)
        afact.resize (m, m);

      if (m > 0)
        {
          // workspace query.
          Complex clwork;
          F77_XFCN (zgeqrf, ZGEQRF, (m, n,
                                     F77_DBLE_CMPLX_ARG (afact.fortran_vec ()),
                                     m, F77_DBLE_CMPLX_ARG (tau),
                                     F77_DBLE_CMPLX_ARG (&clwork), -1, info));

          // allocate buffer and do the job.
          F77_INT lwork = static_cast<F77_INT> (clwork.real ());
          lwork = std::max (lwork, static_cast<F77_INT> (1));
          OCTAVE_LOCAL_BUFFER (Complex, work, lwork);
          F77_XFCN (zgeqrf, ZGEQRF, (m, n,
                                     F77_DBLE_CMPLX_ARG (afact.fortran_vec ()),
                                     m, F77_DBLE_CMPLX_ARG (tau),
                                     F77_DBLE_CMPLX_ARG (work), lwork, info));
        }

      form (n, afact, tau, qr_type);
    }

#if defined (HAVE_QRUPDATE)

    template <>
    void
    qr<ComplexMatrix>::update (const ComplexColumnVector& u,
                               const ComplexColumnVector& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT v_nel = to_f77_int (v.numel ());

      if (u_nel != m || v_nel != n)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      ComplexColumnVector utmp = u;
      ComplexColumnVector vtmp = v;
      OCTAVE_LOCAL_BUFFER (Complex, w, k);
      OCTAVE_LOCAL_BUFFER (double, rw, k);
      F77_XFCN (zqr1up, ZQR1UP, (m, n, k, F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                 m, F77_DBLE_CMPLX_ARG (r.fortran_vec ()), k,
                                 F77_DBLE_CMPLX_ARG (utmp.fortran_vec ()),
                                 F77_DBLE_CMPLX_ARG (vtmp.fortran_vec ()),
                                 F77_DBLE_CMPLX_ARG (w), rw));
    }

    template <>
    void
    qr<ComplexMatrix>::update (const ComplexMatrix& u, const ComplexMatrix& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_rows = to_f77_int (u.rows ());
      F77_INT u_cols = to_f77_int (u.cols ());

      F77_INT v_rows = to_f77_int (v.rows ());
      F77_INT v_cols = to_f77_int (v.cols ());

      if (u_rows != m || v_rows != n || u_cols != v_cols)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      OCTAVE_LOCAL_BUFFER (Complex, w, k);
      OCTAVE_LOCAL_BUFFER (double, rw, k);
      for (volatile F77_INT i = 0; i < u_cols; i++)
        {
          ComplexColumnVector utmp = u.column (i);
          ComplexColumnVector vtmp = v.column (i);
          F77_XFCN (zqr1up, ZQR1UP, (m, n, k,
                                     F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                     m, F77_DBLE_CMPLX_ARG (r.fortran_vec ()), k,
                                     F77_DBLE_CMPLX_ARG (utmp.fortran_vec ()),
                                     F77_DBLE_CMPLX_ARG (vtmp.fortran_vec ()),
                                     F77_DBLE_CMPLX_ARG (w), rw));
        }
    }

    template <>
    void
    qr<ComplexMatrix>::insert_col (const ComplexColumnVector& u,
                                   octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());

      if (u_nel != m)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > n)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (k < m)
        {
          q.resize (m, k+1);
          r.resize (k+1, n+1);
        }
      else
        r.resize (k, n+1);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      ComplexColumnVector utmp = u;
      OCTAVE_LOCAL_BUFFER (double, rw, k);
      F77_XFCN (zqrinc, ZQRINC, (m, n, k, F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                 ldq, F77_DBLE_CMPLX_ARG (r.fortran_vec ()),
                                 ldr, j + 1,
                                 F77_CONST_DBLE_CMPLX_ARG (utmp.data ()), rw));
    }

    template <>
    void
    qr<ComplexMatrix>::insert_col (const ComplexMatrix& u,
                                   const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, ASCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT u_cols = to_f77_int (u.cols ());

      if (u_nel != m || u_cols != nj)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg < 0 || js_end > n))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT kmax = std::min (k + nj, m);
          if (k < m)
            {
              q.resize (m, kmax);
              r.resize (kmax, n + nj);
            }
          else
            r.resize (k, n + nj);

          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (double, rw, kmax);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              ComplexColumnVector utmp = u.column (jsi(i));
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (zqrinc, ZQRINC, (m, n + ii, std::min (kmax, k + ii),
                                         F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                         ldq,
                                         F77_DBLE_CMPLX_ARG (r.fortran_vec ()),
                                         ldr, js_elt + 1,
                                         F77_CONST_DBLE_CMPLX_ARG (utmp.data ()),
                                         rw));
            }
        }
    }

    template <>
    void
    qr<ComplexMatrix>::delete_col (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (double, rw, k);
      F77_XFCN (zqrdec, ZQRDEC, (m, n, k, F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                 ldq, F77_DBLE_CMPLX_ARG (r.fortran_vec ()),
                                 ldr, j + 1, rw));

      if (k < m)
        {
          q.resize (m, k-1);
          r.resize (k-1, n-1);
        }
      else
        r.resize (k, n-1);
    }

    template <>
    void
    qr<ComplexMatrix>::delete_col (const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, DESCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg > n-1 || js_end < 0))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (double, rw, k);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (zqrdec, ZQRDEC, (m, n - ii, (k == m ? k : k - ii),
                                         F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                         ldq,
                                         F77_DBLE_CMPLX_ARG (r.fortran_vec ()),
                                         ldr, js_elt + 1, rw));
            }

          if (k < m)
            {
              q.resize (m, k - nj);
              r.resize (k - nj, n - nj);
            }
          else
            r.resize (k, n - nj);
        }
    }

    template <>
    void
    qr<ComplexMatrix>::insert_row (const ComplexRowVector& u,
                                   octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = std::min (m, n);

      F77_INT u_nel = to_f77_int (u.numel ());

      if (! q.issquare () || u_nel != n)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > m)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      q.resize (m + 1, m + 1);
      r.resize (m + 1, n);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      ComplexRowVector utmp = u;
      OCTAVE_LOCAL_BUFFER (double, rw, k);
      F77_XFCN (zqrinr, ZQRINR, (m, n, F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                 ldq, F77_DBLE_CMPLX_ARG (r.fortran_vec ()),
                                 ldr, j + 1,
                                 F77_DBLE_CMPLX_ARG (utmp.fortran_vec ()), rw));

    }

    template <>
    void
    qr<ComplexMatrix>::delete_row (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (! q.issquare ())
        (*current_liboctave_error_handler) ("qrdelete: dimensions mismatch");

      if (j < 0 || j > m-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (Complex, w, m);
      OCTAVE_LOCAL_BUFFER (double, rw, m);
      F77_XFCN (zqrder, ZQRDER, (m, n, F77_DBLE_CMPLX_ARG (q.fortran_vec ()),
                                 ldq, F77_DBLE_CMPLX_ARG (r.fortran_vec ()),
                                 ldr, j + 1, F77_DBLE_CMPLX_ARG (w), rw));

      q.resize (m - 1, m - 1);
      r.resize (m - 1, n);
    }

    template <>
    void
    qr<ComplexMatrix>::shift_cols (octave_idx_type i_arg,
                                   octave_idx_type j_arg)
    {
      F77_INT i = to_f77_int (i_arg);
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (i < 0 || i > n-1 || j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrshift: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (Complex, w, k);
      OCTAVE_LOCAL_BUFFER (double, rw, k);
      F77_XFCN (zqrshc, ZQRSHC, (m, n, k,
                                 F77_DBLE_CMPLX_ARG (q.fortran_vec ()), ldq,
                                 F77_DBLE_CMPLX_ARG (r.fortran_vec ()), ldr,
                                 i + 1, j + 1, F77_DBLE_CMPLX_ARG (w), rw));
    }

#endif

    template <>
    void
    qr<FloatComplexMatrix>::form (octave_idx_type n_arg, FloatComplexMatrix& afact,
                                  FloatComplex *tau, type qr_type)
    {
      F77_INT n = to_f77_int (n_arg);
      F77_INT m = to_f77_int (afact.rows ());
      F77_INT min_mn = std::min (m, n);
      F77_INT info;

      if (qr_type == qr<FloatComplexMatrix>::raw)
        {
          for (F77_INT j = 0; j < min_mn; j++)
            {
              F77_INT limit = (j < min_mn - 1 ? j : min_mn - 1);
              for (F77_INT i = limit + 1; i < m; i++)
                afact.elem (i, j) *= tau[j];
            }

          r = afact;
        }
      else
        {
          // Attempt to minimize copying.
          if (m >= n)
            {
              // afact will become q.
              q = afact;
              F77_INT k = (qr_type == qr<FloatComplexMatrix>::economy ? n : m);
              r = FloatComplexMatrix (k, n);
              for (F77_INT j = 0; j < n; j++)
                {
                  F77_INT i = 0;
                  for (; i <= j; i++)
                    r.xelem (i, j) = afact.xelem (i, j);
                  for (; i < k; i++)
                    r.xelem (i, j) = 0;
                }
              afact = FloatComplexMatrix (); // optimize memory
            }
          else
            {
              // afact will become r.
              q = FloatComplexMatrix (m, m);
              for (F77_INT j = 0; j < m; j++)
                for (F77_INT i = j + 1; i < m; i++)
                  {
                    q.xelem (i, j) = afact.xelem (i, j);
                    afact.xelem (i, j) = 0;
                  }
              r = afact;
            }

          if (m > 0)
            {
              F77_INT k = to_f77_int (q.cols ());
              // workspace query.
              FloatComplex clwork;
              F77_XFCN (cungqr, CUNGQR, (m, k, min_mn,
                                         F77_CMPLX_ARG (q.fortran_vec ()), m,
                                         F77_CMPLX_ARG (tau),
                                         F77_CMPLX_ARG (&clwork), -1, info));

              // allocate buffer and do the job.
              F77_INT lwork = static_cast<F77_INT> (clwork.real ());
              lwork = std::max (lwork, static_cast<F77_INT> (1));
              OCTAVE_LOCAL_BUFFER (FloatComplex, work, lwork);
              F77_XFCN (cungqr, CUNGQR, (m, k, min_mn,
                                         F77_CMPLX_ARG (q.fortran_vec ()), m,
                                         F77_CMPLX_ARG (tau),
                                         F77_CMPLX_ARG (work), lwork, info));
            }
        }
    }

    template <>
    void
    qr<FloatComplexMatrix>::init (const FloatComplexMatrix& a, type qr_type)
    {
      F77_INT m = to_f77_int (a.rows ());
      F77_INT n = to_f77_int (a.cols ());

      F77_INT min_mn = (m < n ? m : n);
      OCTAVE_LOCAL_BUFFER (FloatComplex, tau, min_mn);

      F77_INT info = 0;

      FloatComplexMatrix afact = a;
      if (m > n && qr_type == qr<FloatComplexMatrix>::std)
        afact.resize (m, m);

      if (m > 0)
        {
          // workspace query.
          FloatComplex clwork;
          F77_XFCN (cgeqrf, CGEQRF, (m, n, F77_CMPLX_ARG (afact.fortran_vec ()),
                                     m, F77_CMPLX_ARG (tau),
                                     F77_CMPLX_ARG (&clwork), -1, info));

          // allocate buffer and do the job.
          F77_INT lwork = static_cast<F77_INT> (clwork.real ());
          lwork = std::max (lwork, static_cast<F77_INT> (1));
          OCTAVE_LOCAL_BUFFER (FloatComplex, work, lwork);
          F77_XFCN (cgeqrf, CGEQRF, (m, n, F77_CMPLX_ARG (afact.fortran_vec ()),
                                     m, F77_CMPLX_ARG (tau),
                                     F77_CMPLX_ARG (work), lwork, info));
        }

      form (n, afact, tau, qr_type);
    }

#if defined (HAVE_QRUPDATE)

    template <>
    void
    qr<FloatComplexMatrix>::update (const FloatComplexColumnVector& u,
                                    const FloatComplexColumnVector& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT v_nel = to_f77_int (v.numel ());

      if (u_nel != m || v_nel != n)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      FloatComplexColumnVector utmp = u;
      FloatComplexColumnVector vtmp = v;
      OCTAVE_LOCAL_BUFFER (FloatComplex, w, k);
      OCTAVE_LOCAL_BUFFER (float, rw, k);
      F77_XFCN (cqr1up, CQR1UP, (m, n, k, F77_CMPLX_ARG (q.fortran_vec ()),
                                 m, F77_CMPLX_ARG (r.fortran_vec ()), k,
                                 F77_CMPLX_ARG (utmp.fortran_vec ()),
                                 F77_CMPLX_ARG (vtmp.fortran_vec ()),
                                 F77_CMPLX_ARG (w), rw));
    }

    template <>
    void
    qr<FloatComplexMatrix>::update (const FloatComplexMatrix& u,
                                    const FloatComplexMatrix& v)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_rows = to_f77_int (u.rows ());
      F77_INT u_cols = to_f77_int (u.cols ());

      F77_INT v_rows = to_f77_int (v.rows ());
      F77_INT v_cols = to_f77_int (v.cols ());

      if (u_rows != m || v_rows != n || u_cols != v_cols)
        (*current_liboctave_error_handler) ("qrupdate: dimensions mismatch");

      OCTAVE_LOCAL_BUFFER (FloatComplex, w, k);
      OCTAVE_LOCAL_BUFFER (float, rw, k);
      for (volatile F77_INT i = 0; i < u_cols; i++)
        {
          FloatComplexColumnVector utmp = u.column (i);
          FloatComplexColumnVector vtmp = v.column (i);
          F77_XFCN (cqr1up, CQR1UP, (m, n, k, F77_CMPLX_ARG (q.fortran_vec ()),
                                     m, F77_CMPLX_ARG (r.fortran_vec ()), k,
                                     F77_CMPLX_ARG (utmp.fortran_vec ()),
                                     F77_CMPLX_ARG (vtmp.fortran_vec ()),
                                     F77_CMPLX_ARG (w), rw));
        }
    }

    template <>
    void
    qr<FloatComplexMatrix>::insert_col (const FloatComplexColumnVector& u,
                                        octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      F77_INT u_nel = to_f77_int (u.numel ());

      if (u_nel != m)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > n)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (k < m)
        {
          q.resize (m, k+1);
          r.resize (k+1, n+1);
        }
      else
        r.resize (k, n+1);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      FloatComplexColumnVector utmp = u;
      OCTAVE_LOCAL_BUFFER (float, rw, k);
      F77_XFCN (cqrinc, CQRINC, (m, n, k, F77_CMPLX_ARG (q.fortran_vec ()), ldq,
                                 F77_CMPLX_ARG (r.fortran_vec ()), ldr, j + 1,
                                 F77_CONST_CMPLX_ARG (utmp.data ()), rw));
    }

    template <>
    void
    qr<FloatComplexMatrix>::insert_col (const FloatComplexMatrix& u,
                                        const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, ASCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT u_nel = to_f77_int (u.numel ());
      F77_INT u_cols = to_f77_int (u.cols ());

      if (u_nel != m || u_cols != nj)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg < 0 || js_end > n))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT kmax = std::min (k + nj, m);
          if (k < m)
            {
              q.resize (m, kmax);
              r.resize (kmax, n + nj);
            }
          else
            r.resize (k, n + nj);

          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (float, rw, kmax);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (cqrinc, CQRINC, (m, n + ii, std::min (kmax, k + ii),
                                         F77_CMPLX_ARG (q.fortran_vec ()), ldq,
                                         F77_CMPLX_ARG (r.fortran_vec ()), ldr,
                                         js_elt + 1,
                                         F77_CONST_CMPLX_ARG (u.column (jsi(i)).data ()),
                                         rw));
            }
        }
    }

    template <>
    void
    qr<FloatComplexMatrix>::delete_col (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (float, rw, k);
      F77_XFCN (cqrdec, CQRDEC, (m, n, k, F77_CMPLX_ARG (q.fortran_vec ()), ldq,
                                 F77_CMPLX_ARG (r.fortran_vec ()), ldr, j + 1,
                                 rw));

      if (k < m)
        {
          q.resize (m, k-1);
          r.resize (k-1, n-1);
        }
      else
        r.resize (k, n-1);
    }

    template <>
    void
    qr<FloatComplexMatrix>::delete_col (const Array<octave_idx_type>& j)
    {
      F77_INT m = to_f77_int (q.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = to_f77_int (q.cols ());

      Array<octave_idx_type> jsi;
      Array<octave_idx_type> js = j.sort (jsi, 0, DESCENDING);
      F77_INT nj = to_f77_int (js.numel ());
      bool dups = false;
      for (F77_INT i = 0; i < nj - 1; i++)
        dups = dups && js(i) == js(i+1);

      if (dups)
        (*current_liboctave_error_handler) ("qrinsert: duplicate index detected");

      F77_INT js_beg = to_f77_int (js(0));
      F77_INT js_end = to_f77_int (js(nj-1));

      if (nj > 0 && (js_beg > n-1 || js_end < 0))
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      if (nj > 0)
        {
          F77_INT ldq = to_f77_int (q.rows ());
          F77_INT ldr = to_f77_int (r.rows ());

          OCTAVE_LOCAL_BUFFER (float, rw, k);
          for (volatile F77_INT i = 0; i < nj; i++)
            {
              F77_INT ii = i;
              F77_INT js_elt = to_f77_int (js(ii));
              F77_XFCN (cqrdec, CQRDEC, (m, n - ii, (k == m ? k : k - ii),
                                         F77_CMPLX_ARG (q.fortran_vec ()), ldq,
                                         F77_CMPLX_ARG (r.fortran_vec ()), ldr,
                                         js_elt + 1, rw));
            }

          if (k < m)
            {
              q.resize (m, k - nj);
              r.resize (k - nj, n - nj);
            }
          else
            r.resize (k, n - nj);
        }
    }

    template <>
    void
    qr<FloatComplexMatrix>::insert_row (const FloatComplexRowVector& u,
                                        octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());
      F77_INT k = std::min (m, n);

      F77_INT u_nel = to_f77_int (u.numel ());

      if (! q.issquare () || u_nel != n)
        (*current_liboctave_error_handler) ("qrinsert: dimensions mismatch");

      if (j < 0 || j > m)
        (*current_liboctave_error_handler) ("qrinsert: index out of range");

      q.resize (m + 1, m + 1);
      r.resize (m + 1, n);

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      FloatComplexRowVector utmp = u;
      OCTAVE_LOCAL_BUFFER (float, rw, k);
      F77_XFCN (cqrinr, CQRINR, (m, n, F77_CMPLX_ARG (q.fortran_vec ()), ldq,
                                 F77_CMPLX_ARG (r.fortran_vec ()), ldr,
                                 j + 1, F77_CMPLX_ARG (utmp.fortran_vec ()),
                                 rw));

    }

    template <>
    void
    qr<FloatComplexMatrix>::delete_row (octave_idx_type j_arg)
    {
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (! q.issquare ())
        (*current_liboctave_error_handler) ("qrdelete: dimensions mismatch");

      if (j < 0 || j > m-1)
        (*current_liboctave_error_handler) ("qrdelete: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (FloatComplex, w, m);
      OCTAVE_LOCAL_BUFFER (float, rw, m);
      F77_XFCN (cqrder, CQRDER, (m, n, F77_CMPLX_ARG (q.fortran_vec ()), ldq,
                                 F77_CMPLX_ARG (r.fortran_vec ()), ldr, j + 1,
                                 F77_CMPLX_ARG (w), rw));

      q.resize (m - 1, m - 1);
      r.resize (m - 1, n);
    }

    template <>
    void
    qr<FloatComplexMatrix>::shift_cols (octave_idx_type i_arg,
                                        octave_idx_type j_arg)
    {
      F77_INT i = to_f77_int (i_arg);
      F77_INT j = to_f77_int (j_arg);

      F77_INT m = to_f77_int (q.rows ());
      F77_INT k = to_f77_int (r.rows ());
      F77_INT n = to_f77_int (r.cols ());

      if (i < 0 || i > n-1 || j < 0 || j > n-1)
        (*current_liboctave_error_handler) ("qrshift: index out of range");

      F77_INT ldq = to_f77_int (q.rows ());
      F77_INT ldr = to_f77_int (r.rows ());

      OCTAVE_LOCAL_BUFFER (FloatComplex, w, k);
      OCTAVE_LOCAL_BUFFER (float, rw, k);
      F77_XFCN (cqrshc, CQRSHC, (m, n, k,
                                 F77_CMPLX_ARG (q.fortran_vec ()), ldq,
                                 F77_CMPLX_ARG (r.fortran_vec ()), ldr,
                                 i + 1, j + 1, F77_CMPLX_ARG (w), rw));
    }

#endif

    // Instantiations we need.

    template class qr<Matrix>;

    template class qr<FloatMatrix>;

    template class qr<ComplexMatrix>;

    template class qr<FloatComplexMatrix>;
  }
}
