#
# GENERATED WITH PDL::PP from lib/PDL/MatrixOps.pd! Don't modify!
#
package PDL::MatrixOps;

our @EXPORT_OK = qw(identity gurney stretcher inv det determinant eigens_sym eigens svd lu_decomp lu_decomp2 lu_backsub simq squaretotri tritosquare tricpy tritosym mstack augment );
our %EXPORT_TAGS = (Func=>\@EXPORT_OK);

use PDL::Core;
use PDL::Exporter;
use DynaLoader;


   
   our @ISA = ( 'PDL::Exporter','DynaLoader' );
   push @PDL::Core::PP, __PACKAGE__;
   bootstrap PDL::MatrixOps ;








#line 14 "lib/PDL/MatrixOps.pd"

=encoding utf8

use strict;
use warnings;

=head1 NAME

PDL::MatrixOps -- Some Useful Matrix Operations

=head1 SYNOPSIS

    $inv = $x->inv;

    $det = $x->det;

    ($lu,$perm,$par) = $x->lu_decomp;
    $y = lu_backsub($lu,$perm,$z); # solve $x x $y = $z

=head1 DESCRIPTION

PDL::MatrixOps is PDL's built-in matrix manipulation code.  It
contains utilities for many common matrix operations: inversion,
determinant finding, eigenvalue/vector finding, singular value
decomposition, etc.  PDL::MatrixOps routines are written in a mixture
of Perl and C, so that they are reliably present even when there is no
external library available (e.g.
L<PDL::LinearAlgebra> or any of the PDL::GSL family of modules).

Matrix manipulation, particularly with large matrices, is a
challenging field and no one algorithm is suitable in all cases.  The
utilities here use general-purpose algorithms that work acceptably for
many cases but might not scale well to very large or pathological
(near-singular) matrices.

Except as noted, the matrices are PDLs whose 0th dimension ranges over
column and whose 1st dimension ranges over row.  The matrices appear
correctly when printed.

These routines should work OK with L<PDL::Matrix> objects
as well as with normal PDLs.

=head1 TIPS ON MATRIX OPERATIONS

Like most computer languages, PDL addresses matrices in (column,row)
order in most cases; this corresponds to (X,Y) coordinates in the
matrix itself, counting rightwards and downwards from the upper left
corner.  This means that if you print a PDL that contains a matrix,
the matrix appears correctly on the screen, but if you index a matrix
element, you use the indices in the reverse order that you would in a
math textbook.  If you prefer your matrices indexed in (row, column)
order, you can try using the L<PDL::Matrix> object, which
includes an implicit exchange of the first two dimensions but should
be compatible with most of these matrix operations.  TIMTOWDTI.

Matrices, row vectors, and column vectors can be multiplied with the 'x'
operator (which is, of course, broadcastable):

    $m3 = $m1 x $m2;
    $col_vec2 = $m1 x $col_vec1;
    $row_vec2 = $row_vec1 x $m1;
    $scalar = $row_vec x $col_vec;

Because of the (column,row) addressing order, 1-D PDLs are treated as
_row_ vectors; if you want a _column_ vector you must add a dummy dimension:

    $rowvec  = pdl(1,2);             # row vector
    $colvec  = $rowvec->slice('*1'); # 1x2 column vector
    $matrix  = pdl([[3,4],[6,2]]);   # 2x2 matrix
    $rowvec2 = $rowvec x $matrix;    # right-multiplication by matrix
    $colvec  = $matrix x $colvec;    # left-multiplication by matrix
    $m2      = $matrix x $rowvec;    # Throws an error

Implicit broadcasting works correctly with most matrix operations, but
you must be extra careful that you understand the dimensionality.  In
particular, matrix multiplication and other matrix ops need nx1 PDLs
as row vectors and 1xn PDLs as column vectors.  In most cases you must
explicitly include the trailing 'x1' dimension in order to get the expected
results when you broadcast over multiple row vectors.

When broadcasting over matrices, it's very easy to get confused about
which dimension goes where. It is useful to include comments with
every expression, explaining what you think each dimension means:

	$x = xvals(360)*3.14159/180;        # (angle)
	$rot = cat(cat(cos($x),sin($x)),    # rotmat: (col,row,angle)
	           cat(-sin($x),cos($x)));

=head1 ACKNOWLEDGEMENTS

MatrixOps includes algorithms and pre-existing code from several
origins.  In particular, C<eigens_sym> is the work of Stephen Moshier,
C<svd> uses an SVD subroutine written by Bryant Marks, and C<eigens>
uses a subset of the Small Scientific Library by Kenneth Geisshirt.
They are free software, distributable under same terms as PDL itself.

=head1 NOTES

This is intended as a general-purpose linear algebra package for
small-to-mid sized matrices.  The algorithms may not scale well to
large matrices (hundreds by hundreds) or to near singular matrices.

If there is something you want that is not here, please add and
document it!

=cut

use Carp;
use strict;
#line 137 "lib/PDL/MatrixOps.pm"


=head1 FUNCTIONS

=cut





#line 131 "lib/PDL/MatrixOps.pd"

=head2 identity

=for sig

  Signature: (n; [o]a(n,n))

=for ref

Return an identity matrix of the specified size.  If you hand in a
scalar, its value is the size of the identity matrix; if you hand in a
dimensioned PDL, the 0th dimension is the first two dimensions of the
matrix, with higher dimensions preserved.

=cut

sub identity {
  my $n = shift;
  my $out =
    !(my $was_pdl = UNIVERSAL::isa($n,'PDL')) ? zeroes($n,$n) :
    $n->getndims == 0 ? zeroes($n->type, $n->at(0),$n->at(0)) :
    undef;
  if (!defined $out) {
    my @dims = $n->dims;
    $out = zeroes($n->type, @dims[0, 0, 2..$#dims]);
  }
  $out->diagonal(0,1)++;
  $was_pdl ? bless $out, ref($n) : $out;
}

#line 163 "lib/PDL/MatrixOps.pd"

=head2 gurney

=for ref

Turns a vector into non-square matrix. Takes two dims, one must
match vector dim 0.

This is useful for dealing with a full SVD:

=for example

  ($u,$s,$v) = svd($in, 1);
  $reconstructed = $u x gurney($s, $in->dims) x $v->t; # same as $in

=cut

*gurney = \&PDL::gurney;
sub PDL::gurney {
  my ($diag, $d0, $d1) = @_;
  Carp::confess "gurney: given undef \$d0 or \$d1" if grep !defined $d0, $d1;
  my $diag_len = $diag->dim(0);
  Carp::confess "diagonal length $diag_len does not match either given dim $d0 or $d1" if !grep $diag_len == $_, $d0, $d1;
  my $diff = abs($d0 - $d1);
  my $mat = stretcher($diag);
  return $mat if !$diff;
  return $mat->append(zeroes($diag->type, $diff)) if $d0 > $d1;
  $mat->t->append(zeroes($diag->type, $diff))->t;
}

#line 197 "lib/PDL/MatrixOps.pd"

=head2 stretcher

=for sig

  Signature: (a(n); [o]b(n,n))

=for usage

  $mat = stretcher($eigenvalues);

=for ref

Return a diagonal matrix with the specified diagonal elements.
Preserves higher dimensions.
As of 2.096, it will also have the same datatype.

=cut

sub stretcher {
  my $in = shift;
  my $out = zeroes($in->type, $in->dim(0), $in->dims);
  $out->diagonal(0,1) += $in;
  $out;
}

#line 229 "lib/PDL/MatrixOps.pd"

=head2 inv

=for sig

  Signature: (a(m,m); sv opt )

=for usage

  $a1 = inv($a, {$opt});

=for ref

Invert a square matrix.

You feed in an NxN matrix in $a, and get back its inverse (if it
exists).  The code is inplace-aware, so you can get back the inverse
in $a itself if you want -- though temporary storage is used either
way.  You can cache the LU decomposition in an output option variable.

C<inv> uses C<lu_decomp> by default; that is a numerically stable
(pivoting) LU decomposition method.

OPTIONS:

=over 3

=item * s

Boolean value indicating whether to complain if the matrix is singular.  If
this is false, singular matrices cause inverse to barf.  If it is true, then
singular matrices cause inverse to return undef.

=item * lu (I/O)

This value contains a list ref with the LU decomposition, permutation,
and parity values for C<$a>.  If you do not mention the key, or if the
value is undef, then inverse calls C<lu_decomp>.  If the key exists with
an undef value, then the output of C<lu_decomp> is stashed here (unless
the matrix is singular).  If the value exists, then it is assumed to
hold the LU decomposition.

=item * det (Output)

If this key exists, then the determinant of C<$a> get stored here,
whether or not the matrix is singular.

=back

=cut

*PDL::inv = \&inv;
sub inv {
  my $x = shift;

  barf "inverse needs a square PDL as a matrix\n"
    unless UNIVERSAL::isa($x,'PDL') &&
	   $x->ndims >= 2 &&
	   $x->dim(0) == $x->dim(1)
	   ;

  my $opt = shift;
  $opt = {} unless defined($opt);
  my $is_inplace = $x->is_inplace;
  $x->set_inplace(0);

  my ($lu,$perm,$par);
  if(exists($opt->{lu}) &&
     ref $opt->{lu} eq 'ARRAY' &&
     ref $opt->{lu}->[0] eq 'PDL') {
	    ($lu,$perm,$par) = @{$opt->{lu}};
  } else {
    ($lu,$perm,$par) = lu_decomp($x);
    @{$opt->{lu}} = ($lu,$perm,$par)
     if(ref $opt->{lu} eq 'ARRAY');
  }

  my $det = (defined $lu) ? $lu->diagonal(0,1)->prodover * $par : pdl(0);
  $opt->{det} = $det
    if exists($opt->{det});

  unless($det->nelem > 1 || $det) {
    return undef
      if $opt->{s};
    barf("PDL::inv: got a singular matrix or LU decomposition\n");
  }

  my $out = lu_backsub($lu,$perm,$par,identity($x))->transpose->sever;
  return $out if !$is_inplace;
  $x .= $out;
  $x;
}

#line 329 "lib/PDL/MatrixOps.pd"

=head2 det

=for sig

  Signature: (a(m,m); sv opt)

=for usage

  $det = det($a,{opt});

=for ref

Determinant of a square matrix using LU decomposition (for large matrices)

You feed in a square matrix, you get back the determinant.  Some
options exist that allow you to cache the LU decomposition of the
matrix (note that the LU decomposition is invalid if the determinant
is zero!).  The LU decomposition is cacheable, in case you want to
re-use it.  This method of determinant finding is more rapid than
recursive-descent on large matrices, and if you reuse the LU
decomposition it's essentially free.

OPTIONS:

=over 3

=item * lu (I/O)

Provides a cache for the LU decomposition of the matrix.  If you
provide the key but leave the value undefined, then the LU decomposition
goes in here; if you put an LU decomposition here, it will be used and
the matrix will not be decomposed again.

=back

=cut

*PDL::det = \&det;
sub det {
  my ($x, $opt) = @_;
  $opt = {} unless defined($opt);
  my($lu,$perm,$par);
  if(exists ($opt->{lu}) and (ref $opt->{lu} eq 'ARRAY')) {
    ($lu,$perm,$par) = @{$opt->{lu}};
  } else {
    ($lu,$perm,$par) = lu_decomp($x);
    $opt->{lu} = [$lu,$perm,$par]
      if(exists($opt->{lu}));
  }
  defined $lu ? $lu->diagonal(0,1)->prodover * $par : PDL->zeroes(sbyte,1);
}

#line 388 "lib/PDL/MatrixOps.pd"

=head2 determinant

=for sig

  Signature: (a(m,m))

=for usage

  $det = determinant($x);

=for ref

Determinant of a square matrix, using recursive descent (broadcastable).

This is the traditional, robust recursive determinant method taught in
most linear algebra courses.  It scales like C<O(n!)> (and hence is
pitifully slow for large matrices) but is very robust because no
division is involved (hence no division-by-zero errors for singular
matrices).  It's also broadcastable, so you can find the determinants of
a large collection of matrices all at once if you want.

Matrices up to 3x3 are handled by direct multiplication; larger matrices
are handled by recursive descent to the 3x3 case.

The LU-decomposition method L</det> is faster in isolation for
single matrices larger than about 4x4, and is much faster if you end up
reusing the LU decomposition of C<$a> (NOTE: check performance and
broadcasting benchmarks with new code).

=cut

*PDL::determinant = \&determinant;
sub determinant {
  my($x) = shift;
  my($n);
  return undef unless(
		      UNIVERSAL::isa($x,'PDL') &&
		      $x->getndims >= 2 &&
		      ($n = $x->dim(0)) == $x->dim(1)
		      );

  return $x->clump(2) if($n==1);
  if($n==2) {
    my($y) = $x->clump(2);
    return $y->index(0)*$y->index(3) - $y->index(1)*$y->index(2);
  }
  if($n==3) {
    my($y) = $x->clump(2);
    my $y3 = $y->index(3);
    my $y4 = $y->index(4);
    my $y5 = $y->index(5);
    my $y6 = $y->index(6);
    my $y7 = $y->index(7);
    my $y8 = $y->index(8);

    return (
	 $y->index(0) * ( $y4 * $y8 - $y5 * $y7 )
      +  $y->index(1) * ( $y5 * $y6 - $y3 * $y8 )
      +  $y->index(2) * ( $y3 * $y7 - $y4 * $y6 )
	     );
  }

  my($i);
  my($sum) = zeroes($x->slice('(0),(0)'));

  # Do middle submatrices
  for $i(1..$n-2) {
    my $el = $x->slice("($i),(0)");
    next if( ($el==0)->all );  # Optimize away unnecessary recursion
    $sum += $el * (1-2*($i%2)) *
      determinant($x->slice("0:".($i-1).",1:-1")->
		  append($x->slice(($i+1).":-1,1:-1")));
  }

  # Do beginning and end submatrices
  $sum += $x->slice("(0),(0)")  * determinant($x->slice('1:-1,1:-1'));
  $sum -= $x->slice("(-1),(0)") * determinant($x->slice('0:-2,1:-1')) * (1 - 2*($n % 2));

  return $sum;
}
#line 467 "lib/PDL/MatrixOps.pm"


=head2 eigens_sym

=for sig

 Signature: ([phys]a(m); [o,phys]ev(n,n); [o,phys]e(n))
 Types: (double)

=for ref

Eigenvalues and -vectors of a symmetric square matrix.  If passed
an asymmetric matrix, the routine will warn and symmetrize it, by taking
the average value.  That is, it will solve for 0.5*($a+$a->transpose).

It's broadcastable, so if C<$a> is 3x3x100, it's treated as 100 separate 3x3
matrices, and both C<$ev> and C<$e> get extra dimensions accordingly.

If called in scalar context it hands back only the eigenvalues.  Ultimately,
it should switch to a faster algorithm in this case (as discarding the
eigenvectors is wasteful).

The algorithm used is due to J. von Neumann, which was a rediscovery of
L<Jacobi's Method|http://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm> .

The eigenvectors are returned in COLUMNS of the returned PDL.  That
makes it slightly easier to access individual eigenvectors, since the
0th dim of the output PDL runs across the eigenvectors and the 1st dim
runs across their components.

=for example

  ($ev,$e) = eigens_sym $x;  # Make eigenvector matrix
  $vector = $ev->slice($n);       # Select nth eigenvector as a column-vector
  $vector = $ev->slice("($n)");     # Select nth eigenvector as a row-vector
  # or to sort in descending order:
  $indices = qsorti($ev)->slice("-1:0");
  $sorted_ev = $ev->slice($indices);    # sort values
  $sorted_e = $e->slice(':', $indices); # sort eigenvectors

As of 2.096, the eigenvalues are returned in ascending order.

To compare with L<PDL::LinearAlgebra>:

  use PDL::LinearAlgebra;
  ($val, $rvec) = msymeigen($A = pdl([3,4], [4,-3]),1,1);
  print $val->slice(1) * $rvec->slice(1);
  #[
  # [        -4.472136]
  # [        -2.236068]
  #]
  print $A x $rvec->slice(1);
  #[
  # [        -4.472136]
  # [        -2.236068]
  #]
  ($rvec, $val) = eigens_sym($A); # note return values other way round
  # otherwise the same

=for usage

    ($ev, $e) = eigens_sym($x); # e-vects & e-values
    $e = eigens_sym($x);        # just eigenvalues

=pod

Broadcasts over its inputs.

=for bad

C<eigens_sym> ignores the bad-value flag of the input ndarrays.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




sub PDL::eigens_sym {
  my ($x) = @_;
  my @d = $x->dims;
  barf "Need real square matrix for eigens_sym"
        if @d < 2 or $d[0] != $d[1];
  my ($sym) = 0.5*($x + $x->transpose);
  my ($err) = PDL::max(abs($sym));
  barf "Need symmetric component non-zero for eigens_sym" if $err == 0;
  $err = PDL::max(abs($x-$sym))/$err;
  warn "Using symmetrized version of the matrix in eigens_sym"
    if $err > 1e-5 && $PDL::debug;
  PDL::_eigens_sym_int($sym->squaretotri, my $ev=PDL->null, my $e=PDL->null);
  return ($ev->transpose, $e) if wantarray;
  $e;                #just eigenvalues
}



*eigens_sym = \&PDL::eigens_sym;






=head2 eigens

=for sig

 Signature: ([phys]a(n,n); complex [o,phys]ev(n,n); complex [o,phys]e(n))
 Types: (double)

=for ref

Complex eigenvalues and -vectors of a real square matrix.

(See also L<"eigens_sym"|/eigens_sym>, for eigenvalues and -vectors
of a real, symmetric, square matrix).

The eigens function will attempt to compute the eigenvalues and
eigenvectors of a square matrix with real components.  If the matrix
is symmetric, the same underlying code as L<"eigens_sym"|/eigens_sym>
is used.  If asymmetric, the eigenvalues and eigenvectors are computed
with algorithms from the sslib library. These are a slightly modified
version of EISPACK's C<rg> code.

Not all square matrices are diagonalizable.  If you feed in a
non-diagonalizable matrix, then the algorithm may fail, in which case
an exception will be thrown.

C<eigens> is broadcastable, so you can solve 100 eigenproblems by
feeding in a 3x3x100 array. Both C<$ev> and C<$e> get extra dimensions accordingly.

If called in scalar context C<eigens> hands back only the eigenvalues.  This
is somewhat wasteful, as it calculates the eigenvectors anyway.

The eigenvectors are returned in COLUMNS of the returned PDL (ie the
the 0 dimension).  That makes it slightly easier to access individual
eigenvectors, since the 0th dim of the output PDL runs across the
eigenvectors and the 1st dim runs across their components.

=for example

  ($ev,$e) = eigens $x;  # Make eigenvector matrix
  $vector = $ev->slice($n);   # Select nth eigenvector as a column-vector
  $vector = $ev->slice("($n)"); # Select nth eigenvector as a row-vector
  # or to sort in descending order:
  $indices = qsorti($ev)->slice("-1:0");
  $sorted_ev = $ev->slice($indices);    # sort values
  $sorted_e = $e->slice(':', $indices); # sort eigenvectors

To compare with L<PDL::LinearAlgebra>:

  use PDL::LinearAlgebra;
  ($val, $lvec, $rvec) = meigen($A = pdl([4,-1], [2,1]),1,1);
  print $val->slice(1) * $rvec->slice(1);
  #[
  # [0.894427190999916]
  # [ 1.78885438199983]
  #]
  print $A x $rvec->slice(1);
  #[
  # [0.894427190999916]
  # [ 1.78885438199983]
  #]
  ($rvec, $val) = eigens($A); # note return values other way round
  # otherwise the same

=for usage

  ($ev, $e) = eigens($x); # e'vects & e'vals
  $e = eigens($x);        # just eigenvalues

=pod

Broadcasts over its inputs.

=for bad

C<eigens> ignores the bad-value flag of the input ndarrays.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




sub PDL::eigens {
  my ($x) = @_;
  my @d = $x->dims;
  barf "Need real square matrix for eigens"
        if @d < 2 or $d[0] != $d[1];
  my $deviation = PDL::max(abs($x - $x->transpose))/PDL::max(abs($x));
  if ( $deviation <= 1e-5 ) {
    PDL::_eigens_sym_int($x->squaretotri, my $ev=PDL->null, my $e=PDL->null);
    return $ev->transpose, $e   if wantarray;
    return $e;  #just eigenvalues
  } else {
    PDL::_eigens_int($x, my $ev=PDL->null, my $e=PDL->null);
    return $ev, $e if wantarray;
    return $e;  #just eigenvalues
  }
}



*eigens = \&PDL::eigens;






=head2 svd

=for sig

 Signature: (a(n,m);
    [o]u(nUCol=CALC($COMP(full) ? $SIZE(m) : PDLMIN($SIZE(m),$SIZE(n))),m);
    [o]s(nSV=CALC(PDLMIN($SIZE(m),$SIZE(n))));
    [o]v(nVCol=CALC($COMP(full) ? $SIZE(n) : PDLMIN($SIZE(m),$SIZE(n))),n);
    [t]sbig(nSVbig=CALC($COMP(full) ? PDLMAX($SIZE(m),$SIZE(n)) : 0));
    [t]vbig(vbigcols=CALC(
        $COMP(full) ? PDLMAX($SIZE(n),$SIZE(m)) : 0
      ),vbigrows=CALC(
        PDLMAX($SIZE(n),$SIZE(m))
      ));
    [t]w(wcols=CALC(
        $COMP(full) ? PDLMAX($SIZE(m),$SIZE(n)) :
        $SIZE(m) >= $SIZE(n) ? 0 :
        $SIZE(m)
      ),wrows=CALC(
        ($COMP(full) && $SIZE(m) >= $SIZE(n)) ? $SIZE(m) : $SIZE(n)
      ));; int full)
 Types: (double)

=for usage

 ($u, $s, $v) = svd($a);        # using default value of full=0
 ($u, $s, $v) = svd($a, $full); # overriding default
 svd($a, $u, $s, $v, $full);    # all arguments given
 ($u, $s, $v) = $a->svd;        # method call
 ($u, $s, $v) = $a->svd($full);
 $a->svd($u, $s, $v, $full);

=for ref

Singular value decomposition of a matrix.

Given an m x n matrix C<$a> that has m rows and n columns,
C<svd> computes matrices C<$u> and C<$v>, and a vector of the singular
values C<$s>.
C<svd> computes the "thin" (or "economy") SVD of C<$a>, such that C<$u> is m
x min(m,n), C<$v> is min(m,n) x n, and there are min(m,n) singular values.
As of 2.104, you do not need to transpose this yourself to ensure
the input is "tall" (or square).
The original matrix can be reconstructed as follows:

    ($u,$s,$v) = svd($x);
    $x_copy = $u x stretcher($s) x $v->transpose;

Also as of 2.104, you can now request the full SVD by passing
a true value as the C<$full> parameter. In that case, C<$u> will
be m x m, and C<$v> will be n x n. To reconstruct the original
input, use:

  ($u,$s,$v) = svd($in, 1);
  $in_copy = $u x gurney($s, $in->dims) x $v->t; # same as $in

If m==n, C<$u> and C<$v> can be thought of as rotation matrices that
convert from the original matrix's singular coordinates to final
coordinates, and from original coordinates to singular coordinates,
respectively, and C<stretcher($s)> is a diagonal scaling matrix.

This uses J.C. Nash's implementation of a one-sided Jacobi rotation.
Use LAPACK (i.e. L<PDL::LinearAlgebra>) for better performance if
you need it.

EXAMPLE

The computing literature has loads of examples of how to use SVD.
Here's a trivial example (used in L<PDL::Transform::map|PDL::Transform/map>)
of how to make a matrix less, er, singular, without changing the
orientation of the ellipsoid of transformation:

    { my($r1,$s,$r2) = svd $x;
      $s++;             # fatten all singular values
      $r2 *= $s;        # implicit broadcasting for cheap mult.
      $x .= $r2 x $r1;  # a gets r2 x ess x r1
    }

=pod

Broadcasts over its inputs.

=for bad

C<svd> ignores the bad-value flag of the input ndarrays.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




*svd = \&PDL::svd;





#line 775 "lib/PDL/MatrixOps.pd"

=head2 lu_decomp

=for sig

  Signature: (a(m,m); [o]lu(m,m); [o]perm(m); [o]parity)

=for ref

LU decompose a matrix, with row permutation

=for usage

  ($lu, $perm, $parity) = lu_decomp($x);
  $lu = lu_decomp($x, $perm, $par);  # $perm and $par are outputs!
  lu_decomp($x->inplace,$perm,$par); # Everything in place.

=for description

C<lu_decomp> returns an LU decomposition of a square matrix,
using Crout's method with partial pivoting. It's ported
from I<Numerical Recipes>. The partial pivoting keeps it
numerically stable but means a little more overhead from
broadcasting.

C<lu_decomp> decomposes the input matrix into matrices L and
U such that LU = A, L is a subdiagonal matrix, and U is a
superdiagonal matrix. By convention, the diagonal of L is
all 1's.

The single output matrix contains all the variable elements
of both the L and U matrices, stacked together. Because the
method uses pivoting (rearranging the lower part of the
matrix for better numerical stability), you have to permute
input vectors before applying the L and U matrices. The
permutation is returned either in the second argument or, in
list context, as the second element of the list. You need
the permutation for the output to make any sense, so be sure
to get it one way or the other.

LU decomposition is the answer to a lot of matrix questions,
including inversion and determinant-finding, and C<lu_decomp>
is used by L</inv>.

If you pass in C<$perm> and C<$parity>, they either must be
predeclared PDLs of the correct size ($perm is an n-vector,
C<$parity> is a scalar) or scalars.

If the matrix is singular, then the LU decomposition might
not be defined; in those cases, C<lu_decomp> silently returns
undef. Some singular matrices LU-decompose just fine, and
those are handled OK but give a zero determinant (and hence
can't be inverted).

C<lu_decomp> uses pivoting, which rearranges the values in the
matrix for more numerical stability. This makes it really
good for large and even near-singular matrices. There is
a non-pivoting version C<lu_decomp2> available which is
from 5 to 60 percent faster for typical problems at
the expense of failing to compute a result in some cases.

Now that the C<lu_decomp> is broadcasted, it is the recommended
LU decomposition routine.  It no longer falls back to C<lu_decomp2>.

C<lu_decomp> is ported from I<Numerical Recipes> to PDL. It
should probably be implemented in C.

=cut

*PDL::lu_decomp = \&lu_decomp;

sub lu_decomp {
   my($in) = shift;
   my($permute) = shift;
   my($parity) = shift;
   my($sing_ok) = shift;

   my $TINY = 1e-30;

   barf("lu_decomp requires a square (2D) PDL\n")
   if(!UNIVERSAL::isa($in,'PDL') ||
      $in->ndims < 2 ||
      $in->dim(0) != $in->dim(1));

   my($n) = $in->dim(0);
   my($n1) = $n; $n1--;

   my($inplace) = $in->is_inplace;
   my($out) = ($inplace) ? $in : $in->copy;

   if(defined $permute) {
      barf('lu_decomp: permutation vector must match the matrix')
      if(!UNIVERSAL::isa($permute,'PDL') ||
         $permute->ndims != 1 ||
         $permute->dim(0) != $out->dim(0));
      $permute .= PDL->xvals($in->dim(0));
   } else {
      $permute = $in->slice("(0)")->xvals;
   }

   if(defined $parity) {
      barf('lu_decomp: parity must be a scalar PDL')
      if(!UNIVERSAL::isa($parity,'PDL') ||
         $parity->dim(0) != 1);
      $parity .= 1.0;
   } else {
      $parity = $in->slice('(0),(0)')->ones;
   }

   my($scales) = $in->copy->abs->maximum; # elementwise by rows

   if(($scales==0)->sum) {
      return undef;
   }

   # Some holding tanks
   my($tmprow) = $out->slice('(0)')->zeroes;
   $tmprow = $tmprow->double if $tmprow->type < double;
   my($tmpval) = $tmprow->slice('(0)')->sever;

   my($col,$row);
   for $col(0..$n1) {
      for $row(1..$n1) {
         my($klim) = $row<$col ? $row : $col;
         if($klim > 0) {
            $klim--;
            my($el) = $out->index2d($col,$row);
            $el -= ( $out->slice("($col),0:$klim") *
               $out->slice("0:$klim,($row)") )->sumover;
         }

      }

      # Figure a_ij, with pivoting

      if($col < $n1) {
         # Find the maximum value in the rest of the row
         my $sl = $out->slice("($col),$col:$n1");
         my $wh = $sl->abs->maximum_ind;
         my $big = $sl->index($wh)->sever;

         # Permute if necessary to make the diagonal the maximum
         # if($wh != 0)
         { # Permute rows to place maximum element on diagonal.
            my $whc = $wh+$col;

            my $sl1 = $out->mv(1,0)->index($whc->slice("*$n"));
            my $sl2 = $out->slice(":,($col)");
            $tmprow .= $sl1;  $sl1 .= $sl2;  $sl2 .= $tmprow;

            $sl1 = $permute->index($whc);
            $sl2 = $permute->index($col);
            $tmpval .= $sl1; $sl1 .= $sl2; $sl2 .= $tmpval;

            $parity->where($wh>0) *= -1.0;
         }

         # LAPACK cgetrf does not try fix singularity so nor do we, even though NR does
         my $notbig = $big->where(abs($big) < $TINY);
         return if !$notbig->isempty;

         # Divide by the diagonal element (which is now the largest element)
         my $tout;
         ($tout = $out->slice("($col),".($col+1).":$n1")) /= $big->slice('*1');
      } # end of pivoting part
   } # end of column loop

   wantarray ? ($out,$permute,$parity) : $out;
}

#line 950 "lib/PDL/MatrixOps.pd"

=head2 lu_decomp2

=for sig

  Signature: (a(m,m); [o]lu(m,m))

=for ref

LU decompose a matrix, with no row permutation

=for usage

  ($lu, $perm, $parity) = lu_decomp2($x);

  $lu = lu_decomp2($x,$perm,$parity);   # or
  $lu = lu_decomp2($x);                 # $perm and $parity are optional

  lu_decomp($x->inplace,$perm,$parity); # or
  lu_decomp($x->inplace);               # $perm and $parity are optional

=for description

C<lu_decomp2> works just like L</lu_decomp>, but it does B<no>
pivoting at all.  For compatibility with L</lu_decomp>, it
will give you a permutation list and a parity scalar if you ask
for them -- but they are always trivial.

Because C<lu_decomp2> does not pivot, it is numerically B<unstable> --
that means it is less precise than L</lu_decomp>, particularly for
large or near-singular matrices.  There are also specific types of
non-singular matrices that confuse it (e.g. ([0,-1,0],[1,0,0],[0,0,1]),
which is a 90 degree rotation matrix but which confuses C<lu_decomp2>).

On the other hand, if you want to invert rapidly a few hundred thousand
small matrices and don't mind missing one or two, it could be the ticket.
It can be up to 60% faster at the expense of possible failure of the
decomposition for some of the input matrices.

The output is a single matrix that contains the LU decomposition of C<$a>;
you can even do it in-place, thereby destroying C<$a>, if you want.  See
L</lu_decomp> for more information about LU decomposition.

C<lu_decomp2> is ported from I<Numerical Recipes> into PDL.

=cut

*PDL::lu_decomp2 = \&lu_decomp2;

sub lu_decomp2 {
  my($in) = shift;
  my($perm) = shift;
  my($par) = shift;

  my($sing_ok) = shift;

  my $TINY = 1e-30;

  barf("lu_decomp2 requires a square (2D) PDL\n")
    if(!UNIVERSAL::isa($in,'PDL') ||
       $in->ndims < 2 ||
       $in->dim(0) != $in->dim(1));

  my($n) = $in->dim(0);
  my($n1) = $n; $n1--;

  my($inplace) = $in->is_inplace;
  my($out) = ($inplace) ? $in : $in->copy;

  if(defined $perm) {
    barf('lu_decomp2: permutation vector must match the matrix')
      if(!UNIVERSAL::isa($perm,'PDL') ||
	 $perm->ndims != 1 ||
	 $perm->dim(0) != $out->dim(0));
    $perm .= PDL->xvals($in->dim(0));
  } else {
    $perm = PDL->xvals($in->dim(0));
  }

  if(defined $par) {
    barf('lu_decomp: parity must be a scalar PDL')
      if(!UNIVERSAL::isa($par,'PDL') ||
	 $par->nelem != 1);
    $par .= 1.0;
  } else {
    $par = pdl(1.0);
  }

  my $diagonal = $out->diagonal(0,1);

  my($col,$row);
  for $col(0..$n1) {
    for $row(1..$n1) {
      my($klim) = $row<$col ? $row : $col;
      if($klim > 0) {
	$klim--;
	my($el) = $out->index2d($col,$row);

	$el -= ( $out->slice("($col),0:$klim") *
		 $out->slice("0:$klim,($row)") )->sumover;
      }

    }

    # Figure a_ij, with no pivoting
    if($col < $n1) {
      # Divide the rest of the column by the diagonal element
      $out->slice("($col),".($col+1).":$n1") /= $diagonal->index($col)->dummy(0,$n1-$col);
    }

  } # end of column loop

  wantarray ? ($out,$perm,$par) : $out;
}

#line 1071 "lib/PDL/MatrixOps.pd"

=head2 lu_backsub

=for sig

  Signature: (lu(m,m); perm(m); b(m))

=for ref

Solve A x = B for matrix A, by back substitution into A's LU decomposition.

=for usage

  ($lu,$perm,$par) = lu_decomp($A);

  $x = lu_backsub($lu,$perm,$par,$A);  # or
  $x = lu_backsub($lu,$perm,$B);       # $par is not required for lu_backsub

  lu_backsub($lu,$perm,$B->inplace); # modify $B in-place

  $x = lu_backsub(lu_decomp($A),$B); # (ignores parity value from lu_decomp)

  # starting from square matrix A and columns matrix B, with mathematically
  # correct dimensions
  $A = identity(4) + ones(4, 4);
  $A->slice('2,0') .= 0; # break symmetry to see if need transpose
  $B = sequence(2, 4);
  # all these functions take B as rows, interpret as though notional columns
  # mathematically confusing but can't change as back-compat and also
  # familiar to Fortran users, so just transpose inputs and outputs

  # using lu_backsub
  ($lu,$perm,$par) = lu_decomp($A);
  $x = lu_backsub($lu,$perm,$par, $B->transpose)->transpose;

  # using simq
  # remove all active dims
  @A_dims = $A->dims; @B_dims = $B->transpose->dims;
  splice @A_dims, 0, 2; splice @B_dims, 0, 1;
  @broadcast = PDL::Core::dims_filled(\@A_dims, \@B_dims);
  # simq modifies A, so need 1 copy per broadcast else non-first run has wrong A
  ($x) = simq($A->dupN(1,1,map +($A_dims[$_]//1)==1?$broadcast[$_]:1, 0..$#broadcast)->copy, $B->transpose, 0);
  $x = $x->transpose;

  # or with PDL::LinearAlgebra wrappers of LAPACK
  $x = msolve($A, $B);

  # or with LAPACK
  use PDL::LinearAlgebra::Real;
  getrf($lu=$A->copy, $ipiv=null, $info=null);
  getrs($lu, 1, $x=$B->transpose->copy, $ipiv, $info=null); # again, need transpose
  $x=$x->transpose;

  # or with GSL
  use PDL::GSL::LINALG;
  LU_decomp(my $lu=$A->copy, my $p=null, my $signum=null);
  # $B and $x, first dim is because GSL treats as vector, higher dims broadcast
  # so we transpose in and back out
  LU_solve($lu, $p, $B->transpose, my $x=null);
  $x=$x->transpose;

  # proof of the pudding is in the eating:
  print $A x $x;

=for description

Given the LU decomposition of a square matrix (from L</lu_decomp>),
C<lu_backsub> does back substitution into the matrix to solve
C<a x = b> for given vector C<b>.  It is separated from the
C<lu_decomp> method so that you can call the cheap C<lu_backsub>
multiple times and not have to do the expensive LU decomposition
more than once.

C<lu_backsub> acts on single vectors and broadcasts in the usual
way, which means that it treats C<$y> as the I<transpose>
of the input.  If you want to process a matrix, you must
hand in the I<transpose> of the matrix, and then transpose
the output when you get it back. that is because pdls are
indexed by (col,row), and matrices are (row,column) by
convention, so a 1-D pdl corresponds to a row vector, not a
column vector.

If C<$lu> is dense and you have more than a few points to
solve for, it is probably cheaper to find C<a^-1> with
L</inv>, and just multiply C<x = a^-1 b>.) in fact,
L</inv> works by calling C<lu_backsub> with the identity
matrix.

C<lu_backsub> is ported from section 2.3 of I<Numerical Recipes>.
It is written in PDL but should probably be implemented in C.

=cut

*PDL::lu_backsub = \&lu_backsub;

sub lu_backsub {
   my ($lu, $perm, $y, $par);
   print STDERR "lu_backsub: entering debug version...\n" if $PDL::debug;
   if(@_==3) {
      ($lu, $perm, $y) = @_;
   } elsif(@_==4) {
      ($lu, $perm, $par, $y) = @_;
   }

   barf("lu_backsub: LU decomposition is undef -- probably from a singular matrix.\n")
   unless defined($lu);

   barf("Usage: \$x = lu_backsub(\$lu,\$perm,\$y); all must be PDLs\n")
   unless(UNIVERSAL::isa($lu,'PDL') &&
      UNIVERSAL::isa($perm,'PDL') &&
      UNIVERSAL::isa($y,'PDL'));

   my $n = $y->dim(0);
   my $n1 = $n; $n1--;

   # Make sure broadcasting dimensions are compatible.
   # There are two possible sources of broadcast dims:
   #
   # (1) over multiple LU (i.e., $lu,$perm) instances
   # (2) over multiple  B (i.e., $y) column instances
   #
   # The full dimensions of the function call looks like
   #
   #   lu_backsub( lu(m,m,X), perm(m,X), b(m,Y) )
   #
   # where X is the list of extra LU dims and Y is
   # the list of extra B dims.  We have several possible
   # cases:
   #
   # (1) Check that m dims are compatible
   my $ludims = pdl($lu->dims);
   my $permdims = pdl($perm->dims);
   my $bdims = pdl($y->dims);

   print STDERR "lu_backsub: called with args:  \$lu$ludims, \$perm$permdims, \$y$bdims\n" if $PDL::debug;

   my $m = $ludims->slice("(0)");  # this is the sig dimension
   unless ( ($ludims->slice(0) == $m) and ($ludims->slice(1) == $m) and
      ($permdims->slice(0) == $m) and ($bdims->slice(0) == $m)) {
      barf "lu_backsub: mismatched sig dimensions";
   }

   my $lunumthr = $ludims->dim(0)-2;
   my $permnumthr = $permdims->dim(0)-1;
   my $bnumthr = $bdims->dim(0)-1;
   unless ( ($lunumthr == $permnumthr) and ($ludims->slice("1:-1") == $permdims)->all )  {
      barf "lu_backsub: \$lu and \$perm broadcast dims not equal! \n";
   }

   # (2) If X == Y then default broadcasting is ok
   if ( ($bnumthr==$permnumthr) and ($bdims==$permdims)->all) {
      print STDERR "lu_backsub: have explicit broadcast dims, goto BROADCAST_OK\n" if $PDL::debug;
      goto BROADCAST_OK;
   }

   # (3) If X == (x,Y) then add x dummy to lu,perm

   # (4) If ndims(X) > ndims(Y) then must have #3

   # (5) If ndims(X) < ndims(Y) then foreach
   #     non-trivial leading dim in X (x0,x1,..)
   #     insert dummy (x0,x1) into lu and perm

   # This means that broadcasting occurs over all
   # leading non-trivial (not length 1) dims of
   # B unless all the broadcast dims are explicitly
   # matched to the LU dims.

BROADCAST_OK:

   # Permute the vector and make a copy if necessary.
   my $out = $y->dummy(1,$y->dim(0))->index($perm->dummy(1));
   $out = $out->sever if !$y->is_inplace;
   print STDERR "lu_backsub: starting with \$out" . pdl($out->dims) . "\n" if $PDL::debug;

   # Make sure broadcasting over lu happens OK...

   if($out->ndims < $lu->ndims-1) {
      print STDERR "lu_backsub: adjusting dims for \$out" . pdl($out->dims) . "\n" if $PDL::debug;
      do {
         $out = $out->dummy(-1,$lu->dim($out->ndims+1));
      } while($out->ndims < $lu->ndims-1);
   }

   ## Do forward substitution into L
   my $row; my $r1;

   for $row(1..$n1) {
      $r1 = $row-1;
      $out->index($row) -= ($lu->slice("0:$r1,$row") *
         $out->slice("0:$r1")
      )->sumover;
   }

   ## Do backward substitution into U, and normalize by the diagonal
   my $ludiag = $lu->diagonal(0,1);
   $out->index($n1) /= $ludiag->index($n1)->dummy(0); # TODO: check broadcasting

   for ($row=$n1; $row>0; $row--) {
      $r1 = $row-1;
      $out->index($r1) -= ($lu->slice("$row:$n1,$r1") * # TODO: check broadcast dims
         $out->slice("$row:$n1")
      )->sumover;
      $out->index($r1) /= $ludiag->index($r1)->dummy(0); # TODO: check broadcast dims
   }

   if ($y->is_inplace) {
     $y->setdims([$out->dims]) if !PDL::all($y->shape == $out->shape); # assgn needs same shape
     $y .= $out;
   }
   $out;
}
#line 1277 "lib/PDL/MatrixOps.pm"


=head2 simq

=for sig

 Signature: ([io,phys]a(n,n); [phys]b(n); [o,phys]x(n); int [o,phys]ips(n); int flag)
 Types: (double)

=for ref

Solution of simultaneous linear equations, C<a x = b>.
B<NB does not broadcast well>.

=for usage

  # remove all active dims
  @A_dims = $A->dims; @B_dims = $B->transpose->dims;
  splice @A_dims, 0, 2; splice @B_dims, 0, 1;
  @broadcast = PDL::Core::dims_filled(\@A_dims, \@B_dims);
  # simq modifies A, so need 1 copy per broadcast else non-first run has wrong A
  ($x) = simq($newA = $A->dupN(1,1,map +($A_dims[$_]//1)==1?$broadcast[$_]:1, 0..$#broadcast)->copy, $B->transpose, 0);
  # $newA now contains modified $A for uses in further calls with flag=-1
  $x = $x->transpose;

C<$a> is an C<n x n> matrix (i.e., a vector of length C<n*n>), stored row-wise:
that is, C<a(i,j) = a[ij]>, where C<ij = i*n + j>.

This corresponds to normal PDL usage.  The contents of matrix a may be
altered (but may be required for subsequent calls with flag = -1).

C<$y>, C<$x>, C<$ips> are vectors of length C<n>.

Set C<flag=0> to solve.
Set C<flag=-1> to do a new back substitution for
different C<$y> vector using the same a matrix previously reduced when
C<flag=0> (the C<$ips> vector generated in the previous solution is also
required).

For this function to work well with broadcasting, it will need the LU
decomposition part split out, so that for solving C<A x = B> only C<x>
would be written to.

See also L</lu_backsub>, which does the same thing with a slightly
less opaque interface.

=pod

Broadcasts over its inputs.

=for bad

C<simq> ignores the bad-value flag of the input ndarrays.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




*simq = \&PDL::simq;






=head2 squaretotri

=for sig

 Signature: (a(n,n); [o]b(m=CALC(($SIZE(n) * ($SIZE(n)+1))/2)))
 Types: (sbyte byte short ushort long ulong indx ulonglong longlong
   float double ldouble cfloat cdouble cldouble)

=for usage

 $b = squaretotri($a);
 squaretotri($a, $b);  # all arguments given
 $b = $a->squaretotri; # method call
 $a->squaretotri($b);

=for ref

Convert a lower-triangular square matrix to triangular vector storage.
Ignores upper half of input.

=pod

Broadcasts over its inputs.

=for bad

C<squaretotri> does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




*squaretotri = \&PDL::squaretotri;






=head2 tritosquare

=for sig

 Signature: (a(m); [o]b(n,n))
 Types: (sbyte byte short ushort long ulong indx ulonglong longlong
   float double ldouble cfloat cdouble cldouble)

=for usage

 $b = tritosquare($a);
 tritosquare($a, $b);  # all arguments given
 $b = $a->tritosquare; # method call
 $a->tritosquare($b);

=for ref

Convert a triangular vector to lower-triangular square matrix storage.
Does not touch upper half of output.

See also L</tritosym>, L</tricpy>.

=pod

Broadcasts over its inputs.

=for bad

C<tritosquare> does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




*tritosquare = \&PDL::tritosquare;






=head2 tricpy

=for sig

 Signature: (A(m,n);[o] C(m,n); int uplo)
 Types: (sbyte byte short ushort long ulong indx ulonglong longlong
   float double ldouble cfloat cdouble cldouble)

=for usage

 $C = tricpy($A);        # using default of uplo=0
 $C = tricpy($A, $uplo);
 tricpy($A, $uplo, $C);  # all arguments given
 $C = $A->tricpy;        # method call
 $C = $A->tricpy($uplo);
 $A->tricpy($uplo, $C);

=for ref

Copy triangular part to another matrix.

C<uplo> meanings to copy: -1 strict upper triangular, 0 = upper, 1
lower, 2 strict lower. The "strict" modes are as of 2.104.

Originally by Grégory Vanuxem.
See also L</tritosym>, L</tritosquare>.

=pod

Broadcasts over its inputs.

=for bad

C<tricpy> does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




*tricpy = \&PDL::tricpy;





#line 1423 "lib/PDL/MatrixOps.pd"

=head2 tritosym

=for ref

Returns symmetric or Hermitian matrix from lower or upper triangular matrix.
Supports inplace and broadcasting.

=for usage

 PDL = tritosym(PDL, SCALAR(uplo), SCALAR(conj))
 uplo : UPPER = 0 | LOWER = 1, default = 0
 conj : Hermitian = 1 | Symmetric = 0, default = 0;

Originally by Grégory Vanuxem. Added to PDL in 2.104.
See also L</tricpy>, L</tritosquare>.

=cut

*tritosym = \&PDL::tritosym;
sub PDL::tritosym {
  my @dims = $_[0]->dims;
  barf "Require square array but got (@dims)" unless @dims >= 2 && $dims[0] == $dims[1];
  my ($m, $upper, $conj) = @_;
  $conj = 0 if $m->type->real;
  my $b = $m->new_or_inplace;
  ($conj ? $m->conj : $m)->tricpy($upper ? 2 : -1, $b->t); # strict
  $b->diagonal(0,1)->im .= 0 if $conj;
  $b;
}
#line 1507 "lib/PDL/MatrixOps.pm"


=head2 mstack

=for sig

 Signature: (x(n,m);y(n,p);[o]out(n,q=CALC($SIZE(m)+$SIZE(p))))
 Types: (sbyte byte short ushort long ulong indx ulonglong longlong
   float double ldouble cfloat cdouble cldouble)

=for usage

 $out = mstack($x, $y);
 mstack($x, $y, $out);    # all arguments given
 $out = $x->mstack($y);   # method call
 $x->mstack($y, $out);
 $x->mstack($y) .= $data; # usable as lvalue

=for ref

Combine two 2D ndarrays into a single ndarray, along the second
("vertical") dim.

Originally by Grégory Vanuxem.

=pod

Broadcasts over its inputs.
Creates data-flow back and forth by default.

=for bad

C<mstack> does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




*mstack = \&PDL::mstack;






=head2 augment

=for sig

 Signature: (x(n); y(p);[o]out(q=CALC($SIZE(n)+$SIZE(p))))
 Types: (sbyte byte short ushort long ulong indx ulonglong longlong
   float double ldouble cfloat cdouble cldouble)

=for usage

 $out = augment($x, $y);
 augment($x, $y, $out);    # all arguments given
 $out = $x->augment($y);   # method call
 $x->augment($y, $out);
 $x->augment($y) .= $data; # usable as lvalue

=for ref

Combine two ndarrays into a single ndarray along the 0-th ("horizontal") dim.

Originally by Grégory Vanuxem.

=pod

Broadcasts over its inputs.
Creates data-flow back and forth by default.

=for bad

C<augment> does not process bad values.
It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

=cut




*augment = \&PDL::augment;







#line 1496 "lib/PDL/MatrixOps.pd"

=head1 AUTHOR

Copyright (C) 2002 Craig DeForest (deforest@boulder.swri.edu),
R.J.R. Williams (rjrw@ast.leeds.ac.uk), Karl Glazebrook
(kgb@aaoepp.aao.gov.au).  There is no warranty.  You are allowed to
redistribute and/or modify this work under the same conditions as PDL
itself.  If this file is separated from the PDL distribution, then the
PDL copyright notice should be included in this file.

=cut
#line 1612 "lib/PDL/MatrixOps.pm"

# Exit with OK status

1;
