/*

Copyright (C) 1996-2011 John W. Eaton
Copyright (C) 2008-2009 Jaroslav Hajek
Copyright (C) 2008-2009 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
<http://www.gnu.org/licenses/>.

*/


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

#include "CmplxCHOL.h"
#include "dbleCHOL.h"
#include "fCmplxCHOL.h"
#include "floatCHOL.h"
#include "SparseCmplxCHOL.h"
#include "SparsedbleCHOL.h"
#include "oct-spparms.h"
#include "sparse-util.h"

#include "ov-re-sparse.h"
#include "ov-cx-sparse.h"
#include "defun-dld.h"
#include "error.h"
#include "gripes.h"
#include "oct-obj.h"
#include "utils.h"

template <class CHOLT>
static octave_value
get_chol_r (const CHOLT& fact)
{
  return octave_value (fact.chol_matrix (), 
                       MatrixType (MatrixType::Upper));
}

template <class CHOLT>
static octave_value
get_chol_l (const CHOLT& fact)
{
  return octave_value (fact.chol_matrix ().transpose (), 
                       MatrixType (MatrixType::Lower));
}

DEFUN_DLD (chol, args, nargout,
  "-*- texinfo -*-\n\
@deftypefn  {Loadable Function} {@var{R} =} chol (@var{A})\n\
@deftypefnx {Loadable Function} {[@var{R}, @var{p}] =} chol (@var{A})\n\
@deftypefnx {Loadable Function} {[@var{R}, @var{p}, @var{Q}] =} chol (@var{S})\n\
@deftypefnx {Loadable Function} {[@var{R}, @var{p}, @var{Q}] =} chol (@var{S}, 'vector')\n\
@deftypefnx {Loadable Function} {[@var{L}, @dots{}] =} chol (@dots{}, 'lower')\n\
@cindex Cholesky factorization\n\
Compute the Cholesky factor, @var{R}, of the symmetric positive definite\n\
matrix @var{A}, where\n\
@tex\n\
$ R^T R = A $.\n\
@end tex\n\
@ifnottex\n\
\n\
@example\n\
@var{R}' * @var{R} = @var{A}.\n\
@end example\n\
\n\
@end ifnottex\n\
\n\
Called with one output argument @code{chol} fails if @var{A} or @var{S} is\n\
not positive definite.  With two or more output arguments @var{p} flags\n\
whether the matrix was positive definite and @code{chol} does not fail.  A\n\
zero value indicated that the matrix was positive definite and the @var{R}\n\
gives the factorization, and @var{p} will have a positive value otherwise.\n\
\n\
If called with 3 outputs then a sparsity preserving row/column permutation\n\
is applied to @var{A} prior to the factorization.  That is @var{R}\n\
is the factorization of @code{@var{A}(@var{Q},@var{Q})} such that\n\
@tex\n\
$ R^T R = Q^T A Q$.\n\
@end tex\n\
@ifnottex\n\
\n\
@example\n\
@var{R}' * @var{R} = @var{Q}' * @var{A} * @var{Q}.\n\
@end example\n\
\n\
@end ifnottex\n\
\n\
The sparsity preserving permutation is generally returned as a matrix.\n\
However, given the flag 'vector', @var{Q} will be returned as a vector\n\
such that\n\
@tex\n\
$ R^T R = A (Q, Q)$.\n\
@end tex\n\
@ifnottex\n\
\n\
@example\n\
@var{R}' * @var{R} = @var{A}(@var{Q}, @var{Q}).\n\
@end example\n\
\n\
@end ifnottex\n\
\n\
Called with either a sparse or full matrix and using the 'lower' flag,\n\
@code{chol} returns the lower triangular factorization such that\n\
@tex\n\
$ L L^T = A $.\n\
@end tex\n\
@ifnottex\n\
\n\
@example\n\
@var{L} * @var{L}' = @var{A}.\n\
@end example\n\
\n\
@end ifnottex\n\
\n\
In general the lower triangular factorization is significantly faster for\n\
sparse matrices.\n\
@seealso{cholinv, chol2inv}\n\
@end deftypefn")
{
  octave_value_list retval;
  int nargin = args.length ();
  bool LLt = false;
  bool vecout = false;

  if (nargin < 1 || nargin > 3 || nargout > 3 
      || (! args(0).is_sparse_type () && nargout > 2))
    {
      print_usage ();
      return retval;
    }

  int n = 1;
  while (n < nargin && ! error_state)
    {
      std::string tmp = args(n++).string_value ();

      if (! error_state )
        {
          if (tmp.compare ("vector") == 0)
            vecout = true;
          else if (tmp.compare ("lower") == 0)
            LLt = true;
          else if (tmp.compare ("upper") == 0)
            LLt = false;
          else
            error ("chol: unexpected second or third input");
        }
      else
        error ("chol: expecting trailing string arguments");
    }

  if (! error_state)
    {
      octave_value arg = args(0);
    
      octave_idx_type nr = arg.rows ();
      octave_idx_type nc = arg.columns ();
      bool natural = (nargout != 3);

      int arg_is_empty = empty_arg ("chol", nr, nc);

      if (arg_is_empty < 0)
        return retval;
      if (arg_is_empty > 0)
        return octave_value (Matrix ());

      if (arg.is_sparse_type ())
        {
          if (arg.is_real_type ())
            {
              SparseMatrix m = arg.sparse_matrix_value ();

              if (! error_state)
                {
                  octave_idx_type info;
                  SparseCHOL fact (m, info, natural);
                  if (nargout == 3)
                    {
                      if (vecout)
                        retval(2) = fact.perm ();
                      else
                        retval(2) = fact.Q();
                    }

                  if (nargout > 1 || info == 0)
                    {
                      retval(1) = fact.P();
                      if (LLt)
                        retval(0) = fact.L();
                      else
                        retval(0) = fact.R();
                    }
                  else
                    error ("chol: input matrix must be positive definite");
                }
            }
          else if (arg.is_complex_type ())
            {
              SparseComplexMatrix m = arg.sparse_complex_matrix_value ();

              if (! error_state)
                {
                  octave_idx_type info;
                  SparseComplexCHOL fact (m, info, natural);

                  if (nargout == 3)
                    {
                      if (vecout)
                        retval(2) = fact.perm ();
                      else
                        retval(2) = fact.Q();
                    }
          
                  if (nargout > 1 || info == 0)
                    {
                      retval(1) = fact.P();
                      if (LLt)
                        retval(0) = fact.L();
                      else
                        retval(0) = fact.R();
                    }
                  else
                    error ("chol: input matrix must be positive definite");
                }
            }
          else
            gripe_wrong_type_arg ("chol", arg);
        }
      else if (arg.is_single_type ())
        {
          if (arg.is_real_type ())
            {
              FloatMatrix m = arg.float_matrix_value ();

              if (! error_state)
                {
                  octave_idx_type info;
                  FloatCHOL fact (m, info);
                  if (nargout == 2 || info == 0)
                    {
                      retval(1) = info;
                      if (LLt)
                        retval(0) = get_chol_l (fact);
                      else
                        retval(0) = get_chol_r (fact);
                    }
                  else
                    error ("chol: input matrix must be positive definite");
                }
            }
          else if (arg.is_complex_type ())
            {
              FloatComplexMatrix m = arg.float_complex_matrix_value ();

              if (! error_state)
                {
                  octave_idx_type info;
                  FloatComplexCHOL fact (m, info);
                  if (nargout == 2 || info == 0)
                    {
                      retval(1) = info;
                      if (LLt)
                        retval(0) = get_chol_l (fact);
                      else
                        retval(0) = get_chol_r (fact);
                    }
                  else
                    error ("chol: input matrix must be positive definite");
                }
            }
          else
            gripe_wrong_type_arg ("chol", arg);
        }
      else
        {
          if (arg.is_real_type ())
            {
              Matrix m = arg.matrix_value ();

              if (! error_state)
                {
                  octave_idx_type info;
                  CHOL fact (m, info);
                  if (nargout == 2 || info == 0)
                    {
                      retval(1) = info;
                      if (LLt)
                        retval(0) = get_chol_l (fact);
                      else
                        retval(0) = get_chol_r (fact);
                    }
                  else
                    error ("chol: input matrix must be positive definite");
                }
            }
          else if (arg.is_complex_type ())
            {
              ComplexMatrix m = arg.complex_matrix_value ();

              if (! error_state)
                {
                  octave_idx_type info;
                  ComplexCHOL fact (m, info);
                  if (nargout == 2 || info == 0)
                    {
                      retval(1) = info;
                      if (LLt)
                        retval(0) = get_chol_l (fact);
                      else
                        retval(0) = get_chol_r (fact);
                    }
                  else
                    error ("chol: input matrix must be positive definite");
                }
            }
          else
            gripe_wrong_type_arg ("chol", arg);
        }
    }

  return retval;
}

/*

%!assert(chol ([2, 1; 1, 1]), [sqrt(2), 1/sqrt(2); 0, 1/sqrt(2)], sqrt (eps));
%!assert(chol (single([2, 1; 1, 1])), single([sqrt(2), 1/sqrt(2); 0, 1/sqrt(2)]), sqrt (eps('single')));

%!error chol ([1, 2; 3, 4]);
%!error chol ([1, 2; 3, 4; 5, 6]);
%!error <Invalid call to chol.*> chol ();
%!error <unexpected second or third input.*> chol (1, 2);

 */

DEFUN_DLD (cholinv, args, ,
  "-*- texinfo -*-\n\
@deftypefn {Loadable Function} {} cholinv (@var{A})\n\
Use the Cholesky@tie{}factorization to compute the inverse of the\n\
symmetric positive definite matrix @var{A}.\n\
@seealso{chol, chol2inv, inv}\n\
@end deftypefn")
{
  octave_value retval;

  int nargin = args.length ();

  if (nargin == 1)
    {
      octave_value arg = args(0);
    
      octave_idx_type nr = arg.rows ();
      octave_idx_type nc = arg.columns ();

      if (nr == 0 || nc == 0)
        retval = Matrix ();
      else
        {
          if (arg.is_sparse_type ())
            {
              if (arg.is_real_type ())
                {
                  SparseMatrix m = arg.sparse_matrix_value ();

                  if (! error_state)
                    {
                      octave_idx_type info;
                      SparseCHOL chol (m, info);
                      if (info == 0)
                        retval = chol.inverse ();
                      else
                        error ("cholinv: A must be positive definite");
                    }
                }
              else if (arg.is_complex_type ())
                {
                  SparseComplexMatrix m = arg.sparse_complex_matrix_value ();

                  if (! error_state)
                    {
                      octave_idx_type info;
                      SparseComplexCHOL chol (m, info);
                      if (info == 0)
                        retval = chol.inverse ();
                      else
                        error ("cholinv: A must be positive definite");
                    }
                }
              else
                gripe_wrong_type_arg ("cholinv", arg);
            }
          else if (arg.is_single_type ())
            {
              if (arg.is_real_type ())
                {
                  FloatMatrix m = arg.float_matrix_value ();

                  if (! error_state)
                    {
                      octave_idx_type info;
                      FloatCHOL chol (m, info);
                      if (info == 0)
                        retval = chol.inverse ();
                      else
                        error ("cholinv: A must be positive definite");
                    }
                }
              else if (arg.is_complex_type ())
                {
                  FloatComplexMatrix m = arg.float_complex_matrix_value ();

                  if (! error_state)
                    {
                      octave_idx_type info;
                      FloatComplexCHOL chol (m, info);
                      if (info == 0)
                        retval = chol.inverse ();
                      else
                        error ("cholinv: A must be positive definite");
                    }
                }
              else
                gripe_wrong_type_arg ("chol", arg);
            }
          else
            {
              if (arg.is_real_type ())
                {
                  Matrix m = arg.matrix_value ();

                  if (! error_state)
                    {
                      octave_idx_type info;
                      CHOL chol (m, info);
                      if (info == 0)
                        retval = chol.inverse ();
                      else
                        error ("cholinv: A must be positive definite");
                    }
                }
              else if (arg.is_complex_type ())
                {
                  ComplexMatrix m = arg.complex_matrix_value ();

                  if (! error_state)
                    {
                      octave_idx_type info;
                      ComplexCHOL chol (m, info);
                      if (info == 0)
                        retval = chol.inverse ();
                      else
                        error ("cholinv: A must be positive definite");
                    }
                }
              else
                gripe_wrong_type_arg ("chol", arg);
            }
        }
    }
  else
    print_usage ();

  return retval;
}

/*

%!shared A, Ainv
%! A = [2,0.2;0.2,1];
%! Ainv = inv(A);
%!test
%! Ainv1 = cholinv(A);
%! assert (norm(Ainv-Ainv1),0,1e-10)
%!testif HAVE_CHOLMOD
%! Ainv2 = inv(sparse(A));
%! assert (norm(Ainv-Ainv2),0,1e-10)
%!testif HAVE_CHOLMOD
%! Ainv3 = cholinv(sparse(A));
%! assert (norm(Ainv-Ainv3),0,1e-10)
%!testif HAVE_CHOLMOD
%! Ainv4 = spcholinv(sparse(A));
%! assert (norm(Ainv-Ainv4),0,1e-10)

*/

DEFUN_DLD (chol2inv, args, ,
  "-*- texinfo -*-\n\
@deftypefn {Loadable Function} {} chol2inv (@var{U})\n\
Invert a symmetric, positive definite square matrix from its Cholesky\n\
decomposition, @var{U}.  Note that @var{U} should be an upper-triangular\n\
matrix with positive diagonal elements.  @code{chol2inv (@var{U})}\n\
provides @code{inv (@var{U}'*@var{U})} but it is much faster than\n\
using @code{inv}.\n\
@seealso{chol, cholinv, inv}\n\
@end deftypefn")
{
  octave_value retval;

  int nargin = args.length ();

  if (nargin == 1)
    {
      octave_value arg = args(0);
    
      octave_idx_type nr = arg.rows ();
      octave_idx_type nc = arg.columns ();

      if (nr == 0 || nc == 0)
        retval = Matrix ();
      else
        {
          if (arg.is_sparse_type ())
            {
              if (arg.is_real_type ())
                {
                  SparseMatrix r = arg.sparse_matrix_value ();

                  if (! error_state)
                    retval = chol2inv (r);
                }
              else if (arg.is_complex_type ())
                {
                  SparseComplexMatrix r = arg.sparse_complex_matrix_value ();

                  if (! error_state)
                    retval = chol2inv (r);
                }
              else
                gripe_wrong_type_arg ("chol2inv", arg);
            }
          else if (arg.is_single_type ())
            {
              if (arg.is_real_type ())
                {
                  FloatMatrix r = arg.float_matrix_value ();

                  if (! error_state)
                    retval = chol2inv (r);
                }
              else if (arg.is_complex_type ())
                {
                  FloatComplexMatrix r = arg.float_complex_matrix_value ();

                  if (! error_state)
                    retval = chol2inv (r);
                }
              else
                gripe_wrong_type_arg ("chol2inv", arg);

            }
          else
            {
              if (arg.is_real_type ())
                {
                  Matrix r = arg.matrix_value ();

                  if (! error_state)
                    retval = chol2inv (r);
                }
              else if (arg.is_complex_type ())
                {
                  ComplexMatrix r = arg.complex_matrix_value ();

                  if (! error_state)
                    retval = chol2inv (r);
                }
              else
                gripe_wrong_type_arg ("chol2inv", arg);
            }
        }
    }
  else
    print_usage ();

  return retval;
}

DEFUN_DLD (cholupdate, args, nargout,
  "-*- texinfo -*-\n\
@deftypefn {Loadable Function} {[@var{R1}, @var{info}] =} cholupdate (@var{R}, @var{u}, @var{op})\n\
Update or downdate a Cholesky@tie{}factorization.  Given an upper triangular\n\
matrix @var{R} and a column vector @var{u}, attempt to determine another\n\
upper triangular matrix @var{R1} such that\n\
@itemize @bullet\n\
@item\n\
@var{R1}'*@var{R1} = @var{R}'*@var{R} + @var{u}*@var{u}'\n\
if @var{op} is \"+\"\n\
\n\
@item\n\
@var{R1}'*@var{R1} = @var{R}'*@var{R} - @var{u}*@var{u}'\n\
if @var{op} is \"-\"\n\
@end itemize\n\
\n\
If @var{op} is \"-\", @var{info} is set to\n\
@itemize\n\
@item 0 if the downdate was successful,\n\
\n\
@item 1 if @var{R}'*@var{R} - @var{u}*@var{u}' is not positive definite,\n\
\n\
@item 2 if @var{R} is singular.\n\
@end itemize\n\
\n\
If @var{info} is not present, an error message is printed in cases 1 and 2.\n\
@seealso{chol, qrupdate}\n\
@end deftypefn")
{
  octave_idx_type nargin = args.length ();

  octave_value_list retval;

  if (nargin > 3 || nargin < 2)
    {
      print_usage ();
      return retval;
    }

  octave_value argr = args(0);
  octave_value argu = args(1);

  if (argr.is_numeric_type () && argu.is_numeric_type ()
      && (nargin < 3 || args(2).is_string ()))
    {
      octave_idx_type n = argr.rows ();

      std::string op = (nargin < 3) ? "+" : args(2).string_value ();

      bool down = op == "-";

      if (down || op == "+")
        if (argr.columns () == n && argu.rows () == n && argu.columns () == 1)
          {
            int err = 0;
            if (argr.is_single_type () || argu.is_single_type ())
              {
                if (argr.is_real_type () && argu.is_real_type ())
                  {
                    // real case
                    FloatMatrix R = argr.float_matrix_value ();
                    FloatColumnVector u = argu.float_column_vector_value ();

                    FloatCHOL fact;
                    fact.set (R);

                    if (down)
                      err = fact.downdate (u);
                    else
                      fact.update (u);

                    retval(0) = get_chol_r (fact);
                  }
                else
                  {
                    // complex case
                    FloatComplexMatrix R = argr.float_complex_matrix_value ();
                    FloatComplexColumnVector u = argu.float_complex_column_vector_value ();

                    FloatComplexCHOL fact;
                    fact.set (R);

                    if (down)
                      err = fact.downdate (u);
                    else
                      fact.update (u);

                    retval(0) = get_chol_r (fact);
                  }
              }
            else
              {
                if (argr.is_real_type () && argu.is_real_type ())
                  {
                    // real case
                    Matrix R = argr.matrix_value ();
                    ColumnVector u = argu.column_vector_value ();

                    CHOL fact;
                    fact.set (R);

                    if (down)
                      err = fact.downdate (u);
                    else
                      fact.update (u);

                    retval(0) = get_chol_r (fact);
                  }
                else
                  {
                    // complex case
                    ComplexMatrix R = argr.complex_matrix_value ();
                    ComplexColumnVector u = argu.complex_column_vector_value ();

                    ComplexCHOL fact;
                    fact.set (R);

                    if (down)
                      err = fact.downdate (u);
                    else
                      fact.update (u);

                    retval(0) = get_chol_r (fact);
                  }
              }

            if (nargout > 1)
              retval(1) = err;
            else if (err == 1)
              error ("cholupdate: downdate violates positiveness");
            else if (err == 2)
              error ("cholupdate: singular matrix");
          }
        else
          error ("cholupdate: dimension mismatch between R and U");
      else
        error ("cholupdate: OP must be \"+\" or \"-\"");
    }
  else
    print_usage ();

  return retval;
}

/*
%!shared A, u, Ac, uc
%! A = [  0.436997  -0.131721   0.124120  -0.061673 ;
%!       -0.131721   0.738529   0.019851  -0.140295 ;
%!        0.124120   0.019851   0.354879  -0.059472 ;
%!       -0.061673  -0.140295  -0.059472   0.600939 ];
%! 
%! u = [  0.98950 ;
%!        0.39844 ;
%!        0.63484 ;
%!        0.13351 ];
%! Ac = [  0.5585528 + 0.0000000i  -0.1662088 - 0.0315341i   0.0107873 + 0.0236411i  -0.0276775 - 0.0186073i ;
%!        -0.1662088 + 0.0315341i   0.6760061 + 0.0000000i   0.0011452 - 0.0475528i   0.0145967 + 0.0247641i ;
%!         0.0107873 - 0.0236411i   0.0011452 + 0.0475528i   0.6263149 - 0.0000000i  -0.1585837 - 0.0719763i ;
%!        -0.0276775 + 0.0186073i   0.0145967 - 0.0247641i  -0.1585837 + 0.0719763i   0.6034234 - 0.0000000i ];
%! 
%! uc = [ 0.54267 + 0.91519i ;
%!        0.99647 + 0.43141i ;
%!        0.83760 + 0.68977i ;
%!        0.39160 + 0.90378i ];



%!test
%! R = chol(A);
%! 
%! R1 = cholupdate(R,u);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - R'*R - u*u',Inf) < 1e1*eps)
%! 
%! R1 = cholupdate(R1,u,"-");
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1 - R,Inf) < 1e1*eps)
%! 
%!test
%! R = chol(Ac);
%! 
%! R1 = cholupdate(R,uc);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - R'*R - uc*uc',Inf) < 1e1*eps)
%! 
%! R1 = cholupdate(R1,uc,"-");
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1 - R,Inf) < 1e1*eps)

%!test
%! R = chol(single(A));
%! 
%! R1 = cholupdate(R,single(u));
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - R'*R - single(u*u'),Inf) < 1e1*eps('single'))
%! 
%! R1 = cholupdate(R1,single(u),"-");
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1 - R,Inf) < 2e1*eps('single'))
%! 
%!test
%! R = chol(single(Ac));
%! 
%! R1 = cholupdate(R,single(uc));
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - R'*R - single(uc*uc'),Inf) < 1e1*eps('single'))
%! 
%! R1 = cholupdate(R1,single(uc),"-");
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1 - R,Inf) < 2e1*eps('single'))
*/

DEFUN_DLD (cholinsert, args, nargout,
  "-*- texinfo -*-\n\
@deftypefn  {Loadable Function} {@var{R1} =} cholinsert (@var{R}, @var{j}, @var{u})\n\
@deftypefnx {Loadable Function} {[@var{R1}, @var{info}] =} cholinsert (@var{R}, @var{j}, @var{u})\n\
Given a Cholesky@tie{}factorization of a real symmetric or complex Hermitian\n\
positive definite matrix @w{@var{A} = @var{R}'*@var{R}}, @var{R}@tie{}upper\n\
triangular, return the Cholesky@tie{}factorization of\n\
@var{A1}, where @w{A1(p,p) = A}, @w{A1(:,j) = A1(j,:)' = u} and\n\
@w{p = [1:j-1,j+1:n+1]}.  @w{u(j)} should be positive.\n\
On return, @var{info} is set to\n\
@itemize\n\
@item 0 if the insertion was successful,\n\
\n\
@item 1 if @var{A1} is not positive definite,\n\
\n\
@item 2 if @var{R} is singular.\n\
@end itemize\n\
\n\
If @var{info} is not present, an error message is printed in cases 1 and 2.\n\
@seealso{chol, cholupdate, choldelete}\n\
@end deftypefn")
{
  octave_idx_type nargin = args.length ();

  octave_value_list retval;

  if (nargin != 3)
    {
      print_usage ();
      return retval;
    }

  octave_value argr = args(0);
  octave_value argj = args(1);
  octave_value argu = args(2);

  if (argr.is_numeric_type () && argu.is_numeric_type ()
      && argj.is_real_scalar ())
    {
      octave_idx_type n = argr.rows ();
      octave_idx_type j = argj.scalar_value ();

      if (argr.columns () == n && argu.rows () == n+1 && argu.columns () == 1)
        {
          if (j > 0 && j <= n+1)
            {
              int err = 0;
              if (argr.is_single_type () || argu.is_single_type ())
                {
                  if (argr.is_real_type () && argu.is_real_type ())
                    {
                      // real case
                      FloatMatrix R = argr.float_matrix_value ();
                      FloatColumnVector u = argu.float_column_vector_value ();

                      FloatCHOL fact;
                      fact.set (R);
                      err = fact.insert_sym (u, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                  else
                    {
                      // complex case
                      FloatComplexMatrix R = argr.float_complex_matrix_value ();
                      FloatComplexColumnVector u = argu.float_complex_column_vector_value ();

                      FloatComplexCHOL fact;
                      fact.set (R);
                      err = fact.insert_sym (u, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                }
              else
                {
                  if (argr.is_real_type () && argu.is_real_type ())
                    {
                      // real case
                      Matrix R = argr.matrix_value ();
                      ColumnVector u = argu.column_vector_value ();

                      CHOL fact;
                      fact.set (R);
                      err = fact.insert_sym (u, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                  else
                    {
                      // complex case
                      ComplexMatrix R = argr.complex_matrix_value ();
                      ComplexColumnVector u = argu.complex_column_vector_value ();

                      ComplexCHOL fact;
                      fact.set (R);
                      err = fact.insert_sym (u, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                }

              if (nargout > 1)
                retval(1) = err;
              else if (err == 1)
                error ("cholinsert: insertion violates positiveness");
              else if (err == 2)
                error ("cholinsert: singular matrix");
              else if (err == 3)
                error ("cholinsert: diagonal element must be real");
            }
          else
            error ("cholinsert: index J out of range");
        }
      else
        error ("cholinsert: dimension mismatch between R and U");
    }
  else
    print_usage ();

  return retval;
}

/*
%!test
%! u2 = [  0.35080 ;
%!         0.63930 ;
%!         3.31057 ;
%!        -0.13825 ;
%!         0.45266 ];
%!
%! R = chol(A);
%! 
%! j = 3; p = [1:j-1, j+1:5];
%! R1 = cholinsert(R,j,u2); A1 = R1'*R1;
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(A1(p,p) - A,Inf) < 1e1*eps)
%! 
%!test
%! u2 = [  0.35080  + 0.04298i;
%!         0.63930  + 0.23778i;
%!         3.31057  + 0.00000i;
%!        -0.13825  + 0.19879i;
%!         0.45266  + 0.50020i];
%!
%! R = chol(Ac);
%! 
%! j = 3; p = [1:j-1, j+1:5];
%! R1 = cholinsert(R,j,u2); A1 = R1'*R1;
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(A1(p,p) - Ac,Inf) < 1e1*eps)
%! 

%!test
%! u2 = single ([  0.35080 ;
%!                 0.63930 ;
%!                 3.31057 ;
%!                -0.13825 ;
%!                 0.45266 ]);
%!
%! R = chol(single(A));
%! 
%! j = 3; p = [1:j-1, j+1:5];
%! R1 = cholinsert(R,j,u2); A1 = R1'*R1;
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(A1(p,p) - A,Inf) < 1e1*eps('single'))
%! 
%!test
%! u2 = single ([  0.35080  + 0.04298i;
%!                 0.63930  + 0.23778i;
%!                 3.31057  + 0.00000i;
%!                -0.13825  + 0.19879i;
%!                 0.45266  + 0.50020i]);
%!
%! R = chol(single(Ac));
%! 
%! j = 3; p = [1:j-1, j+1:5];
%! R1 = cholinsert(R,j,u2); A1 = R1'*R1;
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(A1(p,p) - single(Ac),Inf) < 2e1*eps('single'))
%! 
*/

DEFUN_DLD (choldelete, args, ,
  "-*- texinfo -*-\n\
@deftypefn {Loadable Function} {@var{R1} =} choldelete (@var{R}, @var{j})\n\
Given a Cholesky@tie{}factorization of a real symmetric or complex Hermitian\n\
positive definite matrix @w{@var{A} = @var{R}'*@var{R}}, @var{R}@tie{}upper\n\
triangular, return the Cholesky@tie{}factorization of @w{A(p,p)}, where\n\
@w{p = [1:j-1,j+1:n+1]}.\n\
@seealso{chol, cholupdate, cholinsert}\n\
@end deftypefn")
{
  octave_idx_type nargin = args.length ();

  octave_value_list retval;

  if (nargin != 2)
    {
      print_usage ();
      return retval;
    }

  octave_value argr = args(0);
  octave_value argj = args(1);

  if (argr.is_numeric_type () && argj.is_real_scalar ())
    {
      octave_idx_type n = argr.rows ();
      octave_idx_type j = argj.scalar_value ();

      if (argr.columns () == n)
        {
          if (j > 0 && j <= n)
            {
              if (argr.is_single_type ())
                {
                  if (argr.is_real_type ())
                    {
                      // real case
                      FloatMatrix R = argr.float_matrix_value ();

                      FloatCHOL fact;
                      fact.set (R);
                      fact.delete_sym (j-1);

                      retval(0) = get_chol_r (fact);
                    }
                  else
                    {
                      // complex case
                      FloatComplexMatrix R = argr.float_complex_matrix_value ();

                      FloatComplexCHOL fact;
                      fact.set (R);
                      fact.delete_sym (j-1);

                      retval(0) = get_chol_r (fact);
                    }
                }
              else
                {
                  if (argr.is_real_type ())
                    {
                      // real case
                      Matrix R = argr.matrix_value ();

                      CHOL fact;
                      fact.set (R);
                      fact.delete_sym (j-1);

                      retval(0) = get_chol_r (fact);
                    }
                  else
                    {
                      // complex case
                      ComplexMatrix R = argr.complex_matrix_value ();

                      ComplexCHOL fact;
                      fact.set (R);
                      fact.delete_sym (j-1);

                      retval(0) = get_chol_r (fact);
                    }
                }
            }
          else
            error ("choldelete: index J out of range");
        }
      else
        error ("choldelete: matrix R must be square");
    }
  else
    print_usage ();

  return retval;
}

/*
%!test
%! R = chol(A);
%! 
%! j = 3; p = [1:j-1,j+1:4];
%! R1 = choldelete(R,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - A(p,p),Inf) < 1e1*eps)
%! 
%!test
%! R = chol(Ac);
%! 
%! j = 3; p = [1:j-1,j+1:4];
%! R1 = choldelete(R,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - Ac(p,p),Inf) < 1e1*eps)

%!test
%! R = chol(single(A));
%! 
%! j = 3; p = [1:j-1,j+1:4];
%! R1 = choldelete(R,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - single(A(p,p)),Inf) < 1e1*eps('single'))
%! 
%!test
%! R = chol(single(Ac));
%! 
%! j = 3; p = [1:j-1,j+1:4];
%! R1 = choldelete(R,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - single(Ac(p,p)),Inf) < 1e1*eps('single'))
*/

DEFUN_DLD (cholshift, args, ,
  "-*- texinfo -*-\n\
@deftypefn {Loadable Function} {@var{R1} =} cholshift (@var{R}, @var{i}, @var{j})\n\
Given a Cholesky@tie{}factorization of a real symmetric or complex Hermitian\n\
positive definite matrix @w{@var{A} = @var{R}'*@var{R}}, @var{R}@tie{}upper\n\
triangular, return the Cholesky@tie{}factorization of\n\
@w{@var{A}(p,p)}, where @w{p} is the permutation @*\n\
@code{p = [1:i-1, shift(i:j, 1), j+1:n]} if @w{@var{i} < @var{j}} @*\n\
 or @*\n\
@code{p = [1:j-1, shift(j:i,-1), i+1:n]} if @w{@var{j} < @var{i}}.  @*\n\
\n\
@seealso{chol, cholinsert, choldelete}\n\
@end deftypefn")
{
  octave_idx_type nargin = args.length ();

  octave_value_list retval;

  if (nargin != 3)
    {
      print_usage ();
      return retval;
    }

  octave_value argr = args(0);
  octave_value argi = args(1);
  octave_value argj = args(2);

  if (argr.is_numeric_type () && argi.is_real_scalar () && argj.is_real_scalar ())
    {
      octave_idx_type n = argr.rows ();
      octave_idx_type i = argi.scalar_value ();
      octave_idx_type j = argj.scalar_value ();

      if (argr.columns () == n)
        {
          if (j > 0 && j <= n+1 && i > 0 && i <= n+1)
            {

              if (argr.is_single_type () && argi.is_single_type () && 
                  argj.is_single_type ())
                {
                  if (argr.is_real_type ())
                    {
                      // real case
                      FloatMatrix R = argr.float_matrix_value ();

                      FloatCHOL fact;
                      fact.set (R);
                      fact.shift_sym (i-1, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                  else
                    {
                      // complex case
                      FloatComplexMatrix R = argr.float_complex_matrix_value ();

                      FloatComplexCHOL fact;
                      fact.set (R);
                      fact.shift_sym (i-1, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                }
              else
                {
                  if (argr.is_real_type ())
                    {
                      // real case
                      Matrix R = argr.matrix_value ();

                      CHOL fact;
                      fact.set (R);
                      fact.shift_sym (i-1, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                  else
                    {
                      // complex case
                      ComplexMatrix R = argr.complex_matrix_value ();

                      ComplexCHOL fact;
                      fact.set (R);
                      fact.shift_sym (i-1, j-1);

                      retval(0) = get_chol_r (fact);
                    }
                }
            }
          else
            error ("cholshift: index I or J is out of range");
        }
      else
        error ("cholshift: R must be a square matrix");
    }
  else
    print_usage ();

  return retval;
}

/*
%!test
%! R = chol(A);
%! 
%! i = 1; j = 3; p = [1:i-1, shift(i:j,-1), j+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - A(p,p),Inf) < 1e1*eps)
%! 
%! j = 1; i = 3; p = [1:j-1, shift(j:i,+1), i+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - A(p,p),Inf) < 1e1*eps)
%! 
%!test
%! R = chol(Ac);
%! 
%! i = 1; j = 3; p = [1:i-1, shift(i:j,-1), j+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - Ac(p,p),Inf) < 1e1*eps)
%! 
%! j = 1; i = 3; p = [1:j-1, shift(j:i,+1), i+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - Ac(p,p),Inf) < 1e1*eps)

%!test
%! R = chol(single(A));
%! 
%! i = 1; j = 3; p = [1:i-1, shift(i:j,-1), j+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - single(A(p,p)),Inf) < 1e1*eps('single'))
%! 
%! j = 1; i = 3; p = [1:j-1, shift(j:i,+1), i+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - single(A(p,p)),Inf) < 1e1*eps('single'))
%! 
%!test
%! R = chol(single(Ac));
%! 
%! i = 1; j = 3; p = [1:i-1, shift(i:j,-1), j+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - single(Ac(p,p)),Inf) < 1e1*eps('single'))
%! 
%! j = 1; i = 3; p = [1:j-1, shift(j:i,+1), i+1:4];
%! R1 = cholshift(R,i,j);
%! 
%! assert(norm(triu(R1)-R1,Inf) == 0)
%! assert(norm(R1'*R1 - single(Ac(p,p)),Inf) < 1e1*eps('single'))
*/