Provided by: libpdl-linearalgebra-perl_0.12-3build1_amd64 bug

NAME

       PDL::LinearAlgebra::Real - PDL interface to the real lapack linear algebra programming
       library

SYNOPSIS

        use PDL::LinearAlgebra::Real;

        $a = random (100,100);
        $s = zeroes(100);
        $u = zeroes(100,100);
        $v = zeroes(100,100);
        $info = 0;
        $job = 0;
        gesdd($a, $job, $info, $s , $u, $v);

       Blas vector routine use increment.

DESCRIPTION

       This module provides an interface to parts of the real lapack library.  These routines
       accept either float or double piddles.

FUNCTIONS

   gesvd
         Signature: ([io,phys]A(m,n); int jobu(); int jobvt(); [o,phys]s(r); [o,phys]U(p,q); [o,phys]VT(s,t); int [o,phys]info())

       Computes the singular value decomposition (SVD) of a real M-by-N matrix A.

       The SVD is written

        A = U * SIGMA * V'

       where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U
       is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix.  The diagonal
       elements of SIGMA are the singular values of A; they are real and non-negative, and are
       returned in descending order.  The first min(m,n) columns of U and V are the left and
       right singular vectors of A.

       Note that the routine returns VT = V', not V.

           jobu:   Specifies options for computing all or part of the matrix U:
                   = 0:  no columns of U (no left singular vectors) are
                           computed.
                   = 1:  all M columns of U are returned in array U:
                   = 2:  the first min(m,n) columns of U (the left singular
                           vectors) are returned in the array U;
                   = 3:  the first min(m,n) columns of U (the left singular
                           vectors) are overwritten on the array A;

           jobvt:  Specifies options for computing all or part of the matrix
                   V':
                   = 0:  no rows of V' (no right singular vectors) are
                           computed.
                   = 1:  all N rows of V' are returned in the array VT;
                   = 2:  the first min(m,n) rows of V' (the right singular
                           vectors) are returned in the array VT;
                   = 3:  the first min(m,n) rows of V' (the right singular
                           vectors) are overwritten on the array A;

                   jobvt and jobu cannot both be 3.

           A:      On entry, the M-by-N matrix A.
                   On exit,
                   if jobu = 3,  A is overwritten with the first min(m,n)
                                   columns of U (the left singular vectors,
                                   stored columnwise);
                   if jobvt = 3, A is overwritten with the first min(m,n)
                                   rows of V' (the right singular vectors,
                                   stored rowwise);
                   if jobu != 3 and jobvt != 3, the contents of A
                                   are destroyed.

           s:      The singular values of A, sorted so that s(i) >= s(i+1).

           U:      If jobu = 1, U contains the M-by-M orthogonal matrix U;
                   if jobu = 3, U contains the first min(m,n) columns of U
                   (the left singular vectors, stored columnwise);
                   if jobu = 0 or 3, U is not referenced.
                   Min size  = [1,1].

           VT:     If jobvt = 1, VT contains the N-by-N orthogonal matrix
                   V';
                   if jobvt = 2, VT contains the first min(m,n) rows of
                   V' (the right singular vectors, stored rowwise);
                   if jobvt = 0 or 3, VT is not referenced.
                   Min size  = [1,1].

           info:   = 0:  successful exit.
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  if bdsqr did not converge, info specifies how many
                         superdiagonals of an intermediate bidiagonal form B
                         did not converge to zero.

        $a = random (float, 100,100);
        $s = zeroes(float, 100);
        $u = zeroes(float, 100,100);
        $vt = zeroes(float, 100,100);
        $info = pdl(long, 0);
        gesvd($a, 2, 2, $s , $u, $vt, $info);

       gesvd ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gesdd
         Signature: ([io,phys]A(m,n); int job(); [o,phys]s(r); [o,phys]U(p,q); [o,phys]VT(s,t); int [o,phys]info())

       Computes the singular value decomposition (SVD) of a real M-by-N matrix A.

       This routine use the Coppen's divide and conquer algorithm.  It is much faster than the
       simple driver for large matrices, but uses more workspace.

           job:    Specifies options for computing all or part of matrix:

                   = 0:  no columns of U or rows of V' are computed;
                   = 1:  all M columns of U and all N rows of V' are
                           returned in the arrays U and VT;
                   = 2:  the first min(M,N) columns of U and the first
                           min(M,N) rows of V' are returned in the arrays U
                           and VT;
                   = 3:  If M >= N, the first N columns of U are overwritten
                           on the array A and all rows of V' are returned in
                           the array VT;
                           otherwise, all columns of U are returned in the
                           array U and the first M rows of V' are overwritten
                           on the array A.

           A:      On entry, the M-by-N matrix A.
                   On exit,
                   if job = 3,  A is overwritten with the first N columns
                                   of U (the left singular vectors, stored
                                   columnwise) if M >= N;
                                   A is overwritten with the first M rows
                                   of V' (the right singular vectors, stored
                                   rowwise) otherwise.
                   if job != 3, the contents of A are destroyed.

           s:      The singular values of A, sorted so that s(i) >= s(i+1).

           U:      If job = 1 or job = 3 and M < N, U contains the M-by-M
                   orthogonal matrix U;
                   if job = 2, U contains the first min(M,N) columns of U
                   (the left singular vectors, stored columnwise);
                   if job = 3 and M >= N, or job = 0, U is not referenced.
                   Min size  = [1,1].

           VT:     If job = 1 or job = 3 and M >= N, VT contains the
                   N-by-N orthogonal matrix V';
                   if job = 2, VT contains the first min(M,N) rows of
                   V' (the right singular vectors, stored rowwise);
                   if job = 3 and M < N, or job = 0, VT is not referenced.
                   Min size  = [1,1].

           info:   = 0:  successful exit.
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  bdsdc did not converge, updating process failed.

        $lines = 50;
        $columns = 100;
        $a = random (float, $lines, $columns);
        $min = $lines < $columns ? $lines : $columns;
        $s = zeroes(float, $min);
        $u = zeroes(float, $lines, $lines);
        $vt = zeroes(float, $columns, $columns);
        $info = long (0);
        gesdd($a, 1, $s , $u, $vt, $info);

       gesdd ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ggsvd
         Signature: ([io,phys]A(m,n); int jobu(); int jobv(); int jobq(); [io,phys]B(p,n); int [o,phys]k(); int [o,phys]l();[o,phys]alpha(n);[o,phys]beta(n); [o,phys]U(q,r); [o,phys]V(s,t); [o,phys]Q(u,v); int [o,phys]iwork(n); int [o,phys]info())

       Computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A
       and P-by-N real matrix B:

               U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )

               where U, V and Q are orthogonal matrices, and Z' is the transpose
               of Z.

       Let K+L = the effective numerical rank of the matrix (A',B')', then R is a K+L-by-K+L
       nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
       matrices and of the following structures, respectively:

               If M-K-L >= 0,

                               K  L
                  D1 =     K ( I  0 )
                           L ( 0  C )
                       M-K-L ( 0  0 )

                             K  L
                  D2 =   L ( 0  S )
                       P-L ( 0  0 )

                           N-K-L  K    L
             ( 0 R ) = K (  0   R11  R12 )
                       L (  0    0   R22 )

           where

             C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
             S = diag( BETA(K+1),  ... , BETA(K+L) ),
             C**2 + S**2 = I.

             R is stored in A(1:K+L,N-K-L+1:N) on exit.

           If M-K-L < 0,

                             K M-K K+L-M
                  D1 =   K ( I  0    0   )
                       M-K ( 0  C    0   )

                               K M-K K+L-M
                  D2 =   M-K ( 0  S    0  )
                       K+L-M ( 0  0    I  )
                         P-L ( 0  0    0  )

                              N-K-L  K   M-K  K+L-M
             ( 0 R ) =     K ( 0    R11  R12  R13  )
                         M-K ( 0     0   R22  R23  )
                       K+L-M ( 0     0    0   R33  )

           where

             C = diag( ALPHA(K+1), ... , ALPHA(M) ),
             S = diag( BETA(K+1),  ... , BETA(M) ),
             C**2 + S**2 = I.

             (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
             ( 0  R22 R23 )
             in B(M-K+1:L,N+M-K-L+1:N) on exit.

       The routine computes C, S, R, and optionally the orthogonal transformation matrices U, V
       and Q.

       In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and B implicitly
       gives the SVD of A*inv(B):

                                A*inv(B) = U*(D1*inv(D2))*V'.

       If ( A',B')' has orthonormal columns, then the GSVD of A and B is also equal to the CS
       decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the
       eigenvalue problem:

                                A'*A x = lambda* B'*B x.

       In some literature, the GSVD of A and B is presented in the form

                            U'*A*X = ( 0 D1 ),   V'*B*X = ( 0 D2 )
                            where U and V are orthogonal and X is nonsingular, D1 and D2 are "diagonal".

       The former GSVD form can be converted to the latter form by taking the nonsingular matrix
       X as

                                X = Q*( I   0    )
                                      ( 0 inv(R) ).

           Arguments
           =========

           jobu:   = 0:  U is not computed.
                   = 1:  Orthogonal matrix U is computed;

           jobv:   = 0:  V is not computed.
                   = 1:  Orthogonal matrix V is computed;

           jobq:   = 0:  Q is not computed.
                   = 1:  Orthogonal matrix Q is computed;

           k:
           l:      On exit, k and l specify the dimension of the subblocks
                   described in the Purpose section.
                   k + l = effective numerical rank of (A',B')'.

           A:      On entry, the M-by-N matrix A.
                   On exit, A contains the triangular matrix R, or part of R.

           B:      On entry, the P-by-N matrix B.
                   On exit, B contains the triangular matrix R if M-k-l < 0.

           alpha:
           beta:   On exit, alpha and beta contain the generalized singular
                   value pairs of A and B;
                     alpha(1:k) = 1,
                     beta(1:k)  = 0,
                   and if M-k-l >= 0,
                     alpha(k+1:k+l) = C,
                     beta(k+1:k+l)  = S,
                   or if M-k-l < 0,
                     alpha(k+1:M)=C, alpha(M+1:k+l)=0
                     beta(k+1:M) =S, beta(M+1:k+l) =1
                   and
                     alpha(k+l+1:N) = 0
                     beta(k+l+1:N)  = 0

           U:      If jobu = 1, U contains the M-by-M orthogonal matrix U.
                   If jobu = 0, U is not referenced.
                   Need a minimum array of (1,1) if jobu = 0;

           V:      If jobv = 1, V contains the P-by-P orthogonal matrix V.
                   If jobv = 0, V is not referenced.
                   Need a minimum array of (1,1) if jobv = 0;

           Q:      If jobq = 1, Q contains the N-by-N orthogonal matrix Q.
                   If jobq = 0, Q is not referenced.
                   Need a minimum array of (1,1) if jobq = 0;

           iwork:  On exit, iwork stores the sorting information. More
                   precisely, the following loop will sort alpha
                      for I = k+1, min(M,k+l)
                          swap alpha(I) and alpha(iwork(I))
                      endfor
                   such that alpha(1) >= alpha(2) >= ... >= alpha(N).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  if info = 1, the Jacobi-type procedure failed to
                         converge.  For further details, see subroutine tgsja.

        $k = null;
        $l = null;
        $A = random(5,6);
        $B = random(7,6);
        $alpha = zeroes(6);
        $beta = zeroes(6);
        $U = zeroes(5,5);
        $V = zeroes(7,7);
        $Q = zeroes(6,6);
        $iwork = zeroes(long, 6);
        $info = null;
        ggsvd($A,1,1,1,$B,$k,$l,$alpha, $beta,$U, $V, $Q, $iwork,$info);

       ggsvd ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   geev
         Signature: ([phys]A(n,n); int jobvl(); int jobvr(); [o,phys]wr(n); [o,phys]wi(n); [o,phys]vl(m,m); [o,phys]vr(p,p); int [o,phys]info())

       Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the
       left and/or right eigenvectors.

       The right eigenvector v(j) of A satisfies:
        A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue.

       The left eigenvector u(j) of A satisfies:
        u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugate transpose of u(j).

       The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest
       component real.

           Arguments
           =========

           jobvl:  = 0: left eigenvectors of A are not computed;
                   = 1: left eigenvectors of A are computed.

           jobvr:  = 0: right eigenvectors of A are not computed;
                   = 1: right eigenvectors of A are computed.

           A:      A is overwritten.

           wr:
           wi:     wr and wi contain the real and imaginary parts,
                   respectively, of the computed eigenvalues.  Complex
                   conjugate pairs of eigenvalues appear consecutively
                   with the eigenvalue having the positive imaginary part
                   first.

           vl:     If jobvl = 1, the left eigenvectors u(j) are stored one
                   after another in the columns of vl, in the same order
                   as their eigenvalues else  vl is not referenced.
                   If the j-th eigenvalue is real, then u(j) = vl(:,j),
                   the j-th column of vl.
                   If the j-th and (j+1)-st eigenvalues form a complex
                   conjugate pair, then u(j) = vl(:,j) + i*vl(:,j+1) and
                   u(j+1) = vl(:,j) - i*vl(:,j+1).
                   Min size  = [1].

           vr:     If jobvr = 1, the right eigenvectors v(j) are stored one
                   after another in the columns of vr, in the same order
                   as their eigenvalues else vr is not referenced.
                   If the j-th eigenvalue is real, then v(j) = vr(:,j),
                   the j-th column of vr.
                   If the j-th and (j+1)-st eigenvalues form a complex
                   conjugate pair, then v(j) = vr(:,j) + i*vr(:,j+1) and
                   v(j+1) = vr(:,j) - i*vr(:,j+1).
                   Min size  = [1].

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  if info = i, the QR algorithm failed to compute all the
                         eigenvalues, and no eigenvectors have been computed;
                         elements i+1:N of wr and wi contain eigenvalues which
                         have converged.

        $a = random (5, 5);
        $wr = zeroes(5);
        $wi = zeroes($wr);
        $vl = zeroes($a);
        $vr = zeroes($a);
        $info = null;
        geev($a, 1, 1, $wr, $wi, $vl, $vr, $info);

       geev ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   geevx
         Signature: ([io,phys]A(n,n);  int jobvl(); int jobvr(); int balance(); int sense(); [o,phys]wr(n); [o,phys]wi(n); [o,phys]vl(m,m); [o,phys]vr(p,p); int [o,phys]ilo(); int [o,phys]ihi(); [o,phys]scale(n); [o,phys]abnrm(); [o,phys]rconde(q); [o,phys]rcondv(r); int [o,phys]info())

       Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the
       left and/or right eigenvectors.

       Optionally also, it computes a balancing transformation to improve the conditioning of the
       eigenvalues and eigenvectors (ilo, ihi, scale, and abnrm), reciprocal condition numbers
       for the eigenvalues (rconde), and reciprocal condition numbers for the right eigenvectors
       (rcondv).

       The right eigenvector v(j) of A satisfies:

        A * v(j) = lambda(j) * v(j)
        where lambda(j) is its eigenvalue.

       The left eigenvector u(j) of A satisfies:

        u(j)**H * A = lambda(j) * u(j)**H
        where u(j)**H denotes the conjugate transpose of u(j).

       The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest
       component real.

       Balancing a matrix means permuting the rows and columns to make it more nearly upper
       triangular, and applying a diagonal similarity transformation D * A * D**(-1), where D is
       a diagonal matrix, to make its rows and columns closer in norm and the condition numbers
       of its eigenvalues and eigenvectors smaller.  The computed reciprocal condition numbers
       correspond to the balanced matrix.  Permuting rows and columns will not change the
       condition numbers (in exact arithmetic) but diagonal scaling will.  For further
       explanation of balancing, see section 4.10.2 of the LAPACK Users' Guide.

           Arguments
           =========

           balance:
                   Indicates how the input matrix should be diagonally scaled
                   and/or permuted to improve the conditioning of its
                   eigenvalues.
                   = 0: Do not diagonally scale or permute;
                   = 1: Perform permutations to make the matrix more nearly
                          upper triangular. Do not diagonally scale;
                   = 2: Diagonally scale the matrix, i.e. replace A by
                          D*A*D**(-1), where D is a diagonal matrix chosen
                          to make the rows and columns of A more equal in
                          norm. Do not permute;
                   = 3: Both diagonally scale and permute A.

                   Computed reciprocal condition numbers will be for the matrix
                   after balancing and/or permuting. Permuting does not change
                   condition numbers (in exact arithmetic), but balancing does.

           jobvl:   = 0: left eigenvectors of A are not computed;
                   = 1: left eigenvectors of A are computed.
                   If sense = 1 or 3, jobvl must = 1.

           jobvr;  = 0: right eigenvectors of A are not computed;
                   = 1: right eigenvectors of A are computed.
                   If sense = 1 or 3, jobvr must = 1.

           sense:  Determines which reciprocal condition numbers are computed.
                   = 0: None are computed;
                   = 1: Computed for eigenvalues only;
                   = 2: Computed for right eigenvectors only;
                   = 3: Computed for eigenvalues and right eigenvectors.

                   If sense = 1 or 3, both left and right eigenvectors
                   must also be computed (jobvl = 1 and jobvr = 1).

           A:      The N-by-N matrix.
                   It is overwritten.  If jobvl = 1 or
                   jobvr = 1, A contains the real Schur form of the balanced
                   version of the input matrix A.

           wr
           wi:     wr and wi contain the real and imaginary parts,
                   respectively, of the computed eigenvalues.  Complex
                   conjugate pairs of eigenvalues will appear consecutively
                   with the eigenvalue having the positive imaginary part
                   first.

           vl:     If jobvl = 1, the left eigenvectors u(j) are stored one
                   after another in the columns of vl, in the same order
                   as their eigenvalues else vl is not referenced.
                   If the j-th eigenvalue is real, then u(j) = vl(:,j),
                   the j-th column of vl.
                   If the j-th and (j+1)-st eigenvalues form a complex
                   conjugate pair, then u(j) = vl(:,j) + i*vl(:,j+1) and
                   u(j+1) = vl(:,j) - i*vl(:,j+1).
                   Min size  = [1].

           vr:     If jobvr = 1, the right eigenvectors v(j) are stored one
                   after another in the columns of vr, in the same order
                   as their eigenvalues else vr is not referenced.
                   If the j-th eigenvalue is real, then v(j) = vr(:,j),
                   the j-th column of vr.
                   If the j-th and (j+1)-st eigenvalues form a complex
                   conjugate pair, then v(j) = vr(:,j) + i*vr(:,j+1) and
                   v(j+1) = vr(:,j) - i*vr(:,j+1).
                   Min size  = [1].

           ilo,ihi:Integer values determined when A was
                   balanced.  The balanced A(i,j) = 0 if I > J and
                   J = 1,...,ilo-1 or I = ihi+1,...,N.

           scale:  Details of the permutations and scaling factors applied
                   when balancing A.  If P(j) is the index of the row and column
                   interchanged with row and column j, and D(j) is the scaling
                   factor applied to row and column j, then
                   scale(J) = P(J),    for J = 1,...,ilo-1
                            = D(J),    for J = ilo,...,ihi
                            = P(J)     for J = ihi+1,...,N.
                   The order in which the interchanges are made is N to ihi+1,
                   then 1 to ilo-1.

           abnrm:  The one-norm of the balanced matrix (the maximum
                   of the sum of absolute values of elements of any column).

           rconde: rconde(j) is the reciprocal condition number of the j-th
                   eigenvalue.

           rcondv: rcondv(j) is the reciprocal condition number of the j-th
                   right eigenvector.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  if info = i, the QR algorithm failed to compute all the
                         eigenvalues, and no eigenvectors or condition numbers
                         have been computed; elements 1:ilo-1 and i+1:N of wr
                         and wi contain eigenvalues which have converged.

        $a = random (5,5);
        $wr = zeroes(5);
        $wi = zeroes(5);
        $vl = zeroes(5,5);
        $vr = zeroes(5,5);
        $ilo = null;
        $ihi = null;
        $scale  = zeroes(5);
        $abnrm = null;
        $rconde = zeroes(5);
        $rcondv = zeroes(5);
        $info = null;
        geevx($a, 1,1,3,3,$wr, $wi, $vl, $vr, $ilo, $ihi, $scale, $abnrm,$rconde, $rcondv, $info);

       geevx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ggev
         Signature: ([phys]A(n,n); int jobvl();int jobvr();[phys]B(n,n);[o,phys]alphar(n);[o,phys]alphai(n);[o,phys]beta(n);[o,phys]VL(m,m);[o,phys]VR(p,p);int [o,phys]info())

       Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized
       eigenvalues, and optionally, the left and/or right generalized eigenvectors.

       A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio
       alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the
       pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both
       being zero.

       The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

               A * v(j) = lambda(j) * B * v(j).

       The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

               u(j)**H * A  = lambda(j) * u(j)**H * B .

               where u(j)**H is the conjugate-transpose of u(j).

           Arguments
           =========

           jobvl:  = 0:  do not compute the left generalized eigenvectors;
                   = 1:  compute the left generalized eigenvectors.

           jobvr:  = 0:  do not compute the right generalized eigenvectors;
                   = 1:  compute the right generalized eigenvectors.

           A:      On entry, the matrix A in the pair (A,B).
                   On exit, A has been overwritten.

           B:      On entry, the matrix B in the pair (A,B).
                   On exit, B has been overwritten.

           alphar:
           alphai:
           beta:   On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
                   be the generalized eigenvalues.  If alphai(j) is zero, then
                   the j-th eigenvalue is real; if positive, then the j-th and
                   (j+1)-st eigenvalues are a complex conjugate pair, with
                   alphai(j+1) negative.

                   Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
                   may easily over- or underflow, and beta(j) may even be zero.
                   Thus, the user should avoid naively computing the ratio
                   alpha/beta.  However, alphar and alphai will be always less
                   than and usually comparable with norm(A) in magnitude, and
                   beta always less than and usually comparable with norm(B).

           VL:     If jobvl = 1, the left eigenvectors u(j) are stored one
                   after another in the columns of VL, in the same order as
                   their eigenvalues. If the j-th eigenvalue is real, then
                   u(j) = VL(:,j), the j-th column of VL. If the j-th and
                   (j+1)-th eigenvalues form a complex conjugate pair, then
                   u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
                   Each eigenvector will be scaled so the largest component have
                   abs(real part)+abs(imag. part)=1.
                   Not referenced if jobvl = 0.

           VR:     If jobvr = 1, the right eigenvectors v(j) are stored one
                   after another in the columns of VR, in the same order as
                   their eigenvalues. If the j-th eigenvalue is real, then
                   v(j) = VR(:,j), the j-th column of VR. If the j-th and
                   (j+1)-th eigenvalues form a complex conjugate pair, then
                   v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
                   Each eigenvector will be scaled so the largest component have
                   abs(real part)+abs(imag. part)=1.
                   Not referenced if jobvr = 0.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   = 1,...,N:
                         The QZ iteration failed.  No eigenvectors have been
                         calculated, but alphar(j), alphai(j), and beta(j)
                         should be correct for j=info+1,...,N.
                   > N:  =N+1: other than QZ iteration failed in hgeqz.
                         =N+2: error return from tgevc.

        $a = random(5,5);
        $b = random(5,5);
        $alphar = zeroes(5);
        $alphai = zeroes(5);
        $beta = zeroes(5);
        $vl = zeroes(5,5);
        $vr = zeroes(5,5);
        ggev($a, 1, 1, $b, $alphar, $alphai, $beta, $vl, $vr, ($info=null));

       ggev ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ggevx
         Signature: ([io,phys]A(n,n);int balanc();int jobvl();int jobvr();int sense();[io,phys]B(n,n);[o,phys]alphar(n);[o,phys]alphai(n);[o,phys]beta(n);[o,phys]VL(m,m);[o,phys]VR(p,p);int [o,phys]ilo();int [o,phys]ihi();[o,phys]lscale(n);[o,phys]rscale(n);[o,phys]abnrm();[o,phys]bbnrm();[o,phys]rconde(r);[o,phys]rcondv(s);int [o,phys]info())

       Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized
       eigenvalues, and optionally, the left and/or right generalized eigenvectors.

       Optionally also, it computes a balancing transformation to improve the conditioning of the
       eigenvalues and eigenvectors (ilo, ihi, lscale, rscale, abnrm, and bbnrm), reciprocal
       condition numbers for the eigenvalues (rconde), and reciprocal condition numbers for the
       right eigenvectors (rcondv).

       A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio
       alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the
       pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both
       being zero.

       The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

               A * v(j) = lambda(j) * B * v(j) .

       The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

               u(j)**H * A  = lambda(j) * u(j)**H * B.

               where u(j)**H is the conjugate-transpose of u(j).

       Further Details ===============

       Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate
       eigenvalues, second, applying diagonal similarity transformation to the rows and columns
       to make the rows and columns as close in norm as possible. The computed reciprocal
       condition numbers correspond to the balanced matrix. Permuting rows and columns will not
       change the condition numbers (in exact arithmetic) but diagonal scaling will.  For further
       explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide.

       An approximate error bound on the chordal distance between the i-th computed generalized
       eigenvalue w and the corresponding exact eigenvalue lambda is

               chord(w, lambda) <= EPS * norm(abnrm, bbnrm) / rconde(I)

       An approximate error bound for the angle between the i-th computed eigenvector vl(i) or
       vr(i) is given by

               EPS * norm(abnrm, bbnrm) / DIF(i).

       For further explanation of the reciprocal condition numbers rconde and rcondv, see section
       4.11 of LAPACK User's Guide.

           Arguments
           =========

           balanc: Specifies the balance option to be performed.
                   = 0:  do not diagonally scale or permute;
                   = 1:  permute only;
                   = 2:  scale only;
                   = 3:  both permute and scale.
                   Computed reciprocal condition numbers will be for the
                   matrices after permuting and/or balancing. Permuting does
                   not change condition numbers (in exact arithmetic), but
                   balancing does.

           jobvl:  = 0:  do not compute the left generalized eigenvectors;
                   = 1:  compute the left generalized eigenvectors.

           jobvr:  = 0:  do not compute the right generalized eigenvectors;
                   = 1:  compute the right generalized eigenvectors.

           sense:  Determines which reciprocal condition numbers are computed.
                   = 0: none are computed;
                   = 1: computed for eigenvalues only;
                   = 2: computed for eigenvectors only;
                   = 3: computed for eigenvalues and eigenvectors.

           A:      On entry, the matrix A in the pair (A,B).
                   On exit, A has been overwritten. If jobvl=1 or jobvr=1
                   or both, then A contains the first part of the real Schur
                   form of the "balanced" versions of the input A and B.

           B:      On entry, the matrix B in the pair (A,B).
                   On exit, B has been overwritten. If jobvl=1 or jobvr=1
                   or both, then B contains the second part of the real Schur
                   form of the "balanced" versions of the input A and B.

           alphar:
           alphai:
           beta:   On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
                   be the generalized eigenvalues.  If alphai(j) is zero, then
                   the j-th eigenvalue is real; if positive, then the j-th and
                   (j+1)-st eigenvalues are a complex conjugate pair, with
                   alphai(j+1) negative.

                   Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
                   may easily over- or underflow, and beta(j) may even be zero.
                   Thus, the user should avoid naively computing the ratio
                   ALPHA/beta. However, alphar and alphai will be always less
                   than and usually comparable with norm(A) in magnitude, and
                   beta always less than and usually comparable with norm(B).

           vl:     If jobvl = 1, the left eigenvectors u(j) are stored one
                   after another in the columns of vl, in the same order as
                   their eigenvalues. If the j-th eigenvalue is real, then
                   u(j) = vl(:,j), the j-th column of vl. If the j-th and
                   (j+1)-th eigenvalues form a complex conjugate pair, then
                   u(j) = vl(:,j)+i*vl(:,j+1) and u(j+1) = vl(:,j)-i*vl(:,j+1).
                   Each eigenvector will be scaled so the largest component have
                   abs(real part) + abs(imag. part) = 1.
                   Not referenced if jobvl = 0.

           vr:     If jobvr = 1, the right eigenvectors v(j) are stored one
                   after another in the columns of vr, in the same order as
                   their eigenvalues. If the j-th eigenvalue is real, then
                   v(j) = vr(:,j), the j-th column of vr. If the j-th and
                   (j+1)-th eigenvalues form a complex conjugate pair, then
                   v(j) = vr(:,j)+i*vr(:,j+1) and v(j+1) = vr(:,j)-i*vr(:,j+1).
                   Each eigenvector will be scaled so the largest component have
                   abs(real part) + abs(imag. part) = 1.
                   Not referenced if jobvr = 0.

           ilo,ihi:ilo and ihi are integer values such that on exit
                   A(i,j) = 0 and B(i,j) = 0 if i > j and
                   j = 1,...,ilo-1 or i = ihi+1,...,N.
                   If balanc = 0 or 2, ilo = 1 and ihi = N.

           lscale: Details of the permutations and scaling factors applied
                   to the left side of A and B.  If PL(j) is the index of the
                   row interchanged with row j, and DL(j) is the scaling
                   factor applied to row j, then
                     lscale(j) = PL(j)  for j = 1,...,ilo-1
                               = DL(j)  for j = ilo,...,ihi
                               = PL(j)  for j = ihi+1,...,N.
                   The order in which the interchanges are made is N to ihi+1,
                   then 1 to ilo-1.

           rscale: Details of the permutations and scaling factors applied
                   to the right side of A and B.  If PR(j) is the index of the
                   column interchanged with column j, and DR(j) is the scaling
                   factor applied to column j, then
                     rscale(j) = PR(j)  for j = 1,...,ilo-1
                               = DR(j)  for j = ilo,...,ihi
                               = PR(j)  for j = ihi+1,...,N
                   The order in which the interchanges are made is N to ihi+1,
                   then 1 to ilo-1.

           abnrm:  The one-norm of the balanced matrix A.

           bbnrm:  The one-norm of the balanced matrix B.

           rconde: If sense = 1 or 3, the reciprocal condition numbers of
                   the selected eigenvalues, stored in consecutive elements of
                   the array. For a complex conjugate pair of eigenvalues two
                   consecutive elements of rconde are set to the same value.
                   Thus rconde(j), rcondv(j), and the j-th columns of vl and vr
                   all correspond to the same eigenpair (but not in general the
                   j-th eigenpair, unless all eigenpairs are selected).
                   If sense = 2, rconde is not referenced.

           rcondv: If sense = 2 or 3, the estimated reciprocal condition
                   numbers of the selected eigenvectors, stored in consecutive
                   elements of the array. For a complex eigenvector two
                   consecutive elements of rcondv are set to the same value. If
                   the eigenvalues cannot be reordered to compute rcondv(j),
                   rcondv(j) is set to 0; this can only occur when the true
                   value would be very small anyway.
                   If sense = 1, rcondv is not referenced.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   = 1,...,N:
                         The QZ iteration failed.  No eigenvectors have been
                         calculated, but alphar(j), alphai(j), and beta(j)
                         should be correct for j=info+1,...,N.
                   > N:  =N+1: other than QZ iteration failed in hgeqz.
                         =N+2: error return from tgevc.

        $a = random(5,5);
        $b = random(5,5);
        $alphar = zeroes(5);
        $alphai = zeroes(5);
        $beta = zeroes(5);
        $vl = zeroes(5,5);
        $vr = zeroes(5,5);
        $lscale = zeroes(5);
        $rscale = zeroes(5);
        $ilo = null;
        $ihi = null;
        $abnrm = null;
        $bbnrm = null;
        $rconde = zeroes(5);
        $rcondv = zeroes(5);
        ggevx($a, 3, 1, 1, 3, $b, $alphar, $alphai, $beta, $vl, $vr,
        $ilo, $ihi, $lscale, $rscale, $abnrm, $bbnrm, $rconde,$rcondv,($info=null));

       ggevx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gees
         Signature: ([io,phys]A(n,n);  int jobvs(); int sort(); [o,phys]wr(n); [o,phys]wi(n); [o,phys]vs(p,p); int [o,phys]sdim(); int [o,phys]info(); SV* select_func)

       Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T,
       and, optionally, the matrix of Schur vectors Z.  This gives the Schur factorization A =
       Z*T*Z'.

       Optionally, it also orders the eigenvalues on the diagonal of the real Schur form so that
       selected eigenvalues are at the top left.  The leading columns of Z then form an
       orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.

       A matrix is in real Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2
       blocks. 2-by-2 blocks will be standardized in the form

               [  a  b  ]
               [  c  a  ]
               where b*c < 0.

       The eigenvalues of such a block are a +- sqrt(bc).

           Arguments
           =========

           jobvs:  = 0: Schur vectors are not computed;
                   = 1: Schur vectors are computed.

           sort:   Specifies whether or not to order the eigenvalues on the
                   diagonal of the Schur form.
                   = 0: Eigenvalues are not ordered;
                   = 1: Eigenvalues are ordered (see select_func).

           select_func:
                   If sort = 1, select_func is used to select eigenvalues to sort
                   to the top left of the Schur form.
                   If sort = 0, select_func is not referenced.
                   An eigenvalue wr(j)+sqrt(-1)*wi(j) is selected if
                   select_func(SCALAR(wr(j)), SCALAR(wi(j))) is true; i.e.,
                   if either one of a complex conjugate pair of eigenvalues
                   is selected, then both complex eigenvalues are selected.
                   Note that a selected complex eigenvalue may no longer
                   satisfy select_func(wr(j),wi(j)) = 1 after ordering, since
                   ordering may change the value of complex eigenvalues
                   (especially if the eigenvalue is ill-conditioned); in this
                   case info is set to N+2 (see info below).

           A:      The N-by-N matrix A.
                   On exit, A has been overwritten by its real Schur form T.

           sdim:   If sort = 0, sdim = 0.
                   If sort = 1, sdim = number of eigenvalues (after sorting)
                                  for which select_func is true. (Complex conjugate
                                  pairs for which select_func is true for either
                                  eigenvalue count as 2.)

           wr:
           wi:     wr and wi contain the real and imaginary parts,
                   respectively, of the computed eigenvalues in the same order
                   that they appear on the diagonal of the output Schur form T.
                   Complex conjugate pairs of eigenvalues will appear
                   consecutively with the eigenvalue having the positive
                   imaginary part first.

           vs:     If jobvs = 1, vs contains the orthogonal matrix Z of Schur
                   vectors else vs is not referenced.

           info    = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value.
                   > 0: if info = i, and i is
                      <= N: the QR algorithm failed to compute all the
                            eigenvalues; elements 1:ILO-1 and i+1:N of wr and wi
                            contain those eigenvalues which have converged; if
                            jobvs = 1, vs contains the matrix which reduces A
                            to its partially converged Schur form.
                      = N+1: the eigenvalues could not be reordered because some
                            eigenvalues were too close to separate (the problem
                            is very ill-conditioned);
                      = N+2: after reordering, roundoff changed values of some
                            complex eigenvalues so that leading eigenvalues in
                            the Schur form no longer satisfy select_func = 1  This
                            could also be caused by underflow due to scaling.

        sub select_function{
               my ($a, $b ) = @_;
               # Stable "continuous time" eigenspace
               return $a < 0 ? 1 : 0;
        }
        $A = random (5,5);
        $wr= zeroes(5);
        $wi = zeroes(5);
        $vs = zeroes(5,5);
        $sdim  = null;
        $info = null;
        gees($A, 1,1, $wr, $wi, $vs, $sdim, $info,\&select_function);

       gees ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   geesx
         Signature: ([io,phys]A(n,n);  int jobvs(); int sort(); int sense(); [o,phys]wr(n); [o,phys]wi(n); [o,phys]vs(p,p); int [o,phys]sdim(); [o,phys]rconde();[o,phys]rcondv(); int [o,phys]info(); SV* select_func)

       Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T,
       and, optionally, the matrix of Schur vectors Z.  This gives the Schur factorization A =
       Z*T*Z'.

       Optionally, it also orders the eigenvalues on the diagonal of the real Schur form so that
       selected eigenvalues are at the top left; computes a reciprocal condition number for the
       average of the selected eigenvalues (rconde); and computes a reciprocal condition number
       for the right invariant subspace corresponding to the selected eigenvalues (rcondv).  The
       leading columns of Z form an orthonormal basis for this invariant subspace.

       For further explanation of the reciprocal condition numbers rconde and rcondv, see Section
       4.10 of the LAPACK Users' Guide (where these quantities are called s and sep
       respectively).

       A real matrix is in real Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2
       blocks. 2-by-2 blocks will be standardized in the form

               [  a  b  ]
               [  c  a  ]
               where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

           Arguments
           =========

           jobvs:  = 0: Schur vectors are not computed;
                   = 1: Schur vectors are computed.

           sort:   Specifies whether or not to order the eigenvalues on the
                   diagonal of the Schur form.
                   = 0: Eigenvalues are not ordered;
                   = 1: Eigenvalues are ordered (see select_func).

           select_func:
                   If sort = 1, select_func is used to select eigenvalues to sort
                   to the top left of the Schur form else select_func is not referenced.
                   An eigenvalue wr(j)+sqrt(-1)*wi(j) is selected if
                   select_func(wr(j),wi(j)) is true; i.e., if either one of a
                   complex conjugate pair of eigenvalues is selected, then both
                   are.  Note that a selected complex eigenvalue may no longer
                   satisfy select_func(wr(j),wi(j)) = 1 after ordering, since
                   ordering may change the value of complex eigenvalues
                   (especially if the eigenvalue is ill-conditioned); in this
                   case info may be set to N+3 (see info below).

           sense:  Determines which reciprocal condition numbers are computed.
                   = 0: None are computed;
                   = 1: Computed for average of selected eigenvalues only;
                   = 2: Computed for selected right invariant subspace only;
                   = 3: Computed for both.
                   If sense = 1, 2 or 3, sort must equal 1.

           A:      On entry, the N-by-N matrix A.
                   On exit, A is overwritten by its real Schur form T.

           sdim:   If sort = 0, sdim = 0.
                   If sort = 1, sdim = number of eigenvalues (after sorting)
                                  for which select_func is 1. (Complex conjugate
                                  pairs for which select_func is 1 for either
                                  eigenvalue count as 2.)

           wr:
           wi:     wr and wi contain the real and imaginary parts, respectively,
                   of the computed eigenvalues, in the same order that they
                   appear on the diagonal of the output Schur form T.  Complex
                   conjugate pairs of eigenvalues appear consecutively with the
                   eigenvalue having the positive imaginary part first.

           vs      If jobvs = 1, vs contains the orthogonal matrix Z of Schur
                   vectors else vs is not referenced.

           rconde: If sense = 1 or 3, rconde contains the reciprocal
                   condition number for the average of the selected eigenvalues.
                   Not referenced if sense = 0 or 2.

           rcondv: If sense = 2 or 3, rcondv contains the reciprocal
                   condition number for the selected right invariant subspace.
                   Not referenced if sense = 0 or 1.

           info:   = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value.
                   > 0: if info = i, and i is
                      <= N: the QR algorithm failed to compute all the
                            eigenvalues; elements 1:ilo-1 and i+1:N of wr and wi
                            contain those eigenvalues which have converged; if
                            jobvs = 1, vs contains the transformation which
                            reduces A to its partially converged Schur form.
                      = N+1: the eigenvalues could not be reordered because some
                            eigenvalues were too close to separate (the problem
                            is very ill-conditioned);
                      = N+2: after reordering, roundoff changed values of some
                            complex eigenvalues so that leading eigenvalues in
                            the Schur form no longer satisfy select_func=1  This
                            could also be caused by underflow due to scaling.

        sub select_function{
               my ($a, $b) = @_;
               # Stable "discrete time" eigenspace
               return sqrt($a**2 + $b**2) < 1 ? 1 : 0;
        }
        $A = random (5,5);
        $wr= zeroes(5);
        $wi = zeroes(5);
        $vs = zeroes(5,5);
        $sdim  = null;
        $rconde = null;
        $rcondv = null;
        $info = null;
        geesx($A, 1,1, 3, $wr, $wi, $vs, $sdim, $rconde, $rcondv, $info, \&select_function);

       geesx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gges
         Signature: ([io,phys]A(n,n); int jobvsl();int jobvsr();int sort();[io,phys]B(n,n);[o,phys]alphar(n);[o,phys]alphai(n);[o,phys]beta(n);[o,phys]VSL(m,m);[o,phys]VSR(p,p);int [o,phys]sdim();int [o,phys]info(); SV* select_func)

       Computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized
       eigenvalues, the generalized real Schur form (S,T), optionally, the left and/or right
       matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization

               (A,B) = ( (VSL)*S*(VSR)', (VSL)*T*(VSR)' )

       Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues
       appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the
       upper triangular matrix T.The leading columns of VSL and VSR then form an orthonormal
       basis for the corresponding left and right eigenspaces (deflating subspaces).

       (If only the generalized eigenvalues are needed, use the driver ggev instead, which is
       faster.)

       A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta
       = w, such that  A - w*B is singular.  It is usually represented as the pair (alpha,beta),
       as there is a reasonable interpretation for beta=0 or both being zero.

       A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with
       non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks.
       1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be
       "standardized" by making the corresponding elements of T have the form:

               [  a  0  ]
               [  0  b  ]

       and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair
       of generalized eigenvalues.

           Arguments
           =========

           jobvsl: = 0:  do not compute the left Schur vectors;
                   = 1:  compute the left Schur vectors.

           jobvsr: = 0:  do not compute the right Schur vectors;
                   = 1:  compute the right Schur vectors.

           sort:   Specifies whether or not to order the eigenvalues on the
                   diagonal of the generalized Schur form.
                   = 0:  Eigenvalues are not ordered;
                   = 1:  Eigenvalues are ordered (see delztg);

           delztg: If sort = 0, delztg is not referenced.
                   If sort = 1, delztg is used to select eigenvalues to sort
                   to the top left of the Schur form.
                   An eigenvalue (alphar(j)+alphai(j))/beta(j) is selected if
                   delztg(alphar(j),alphai(j),beta(j)) is true; i.e. if either
                   one of a complex conjugate pair of eigenvalues is selected,
                   then both complex eigenvalues are selected.

                   Note that in the ill-conditioned case, a selected complex
                   eigenvalue may no longer satisfy delztg(alphar(j),alphai(j),
                   beta(j)) = 1 after ordering. info is to be set to N+2
                   in this case.

           A:      On entry, the first of the pair of matrices.
                   On exit, A has been overwritten by its generalized Schur
                   form S.

           B:      On entry, the second of the pair of matrices.
                   On exit, B has been overwritten by its generalized Schur
                   form T.

           sdim:   If sort = 0, sdim = 0.
                   If sort = 1, sdim = number of eigenvalues (after sorting)
                   for which delztg is true.  (Complex conjugate pairs for which
                   delztg is true for either eigenvalue count as 2.)

           alphar:
           alphai:
           beta:   On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
                   be the generalized eigenvalues.  alphar(j) + alphai(j)*i,
                   and  beta(j),j=1,...,N are the diagonals of the complex Schur
                   form (S,T) that would result if the 2-by-2 diagonal blocks of
                   the real Schur form of (A,B) were further reduced to
                   triangular form using 2-by-2 complex unitary transformations.
                   If alphai(j) is zero, then the j-th eigenvalue is real; if
                   positive, then the j-th and (j+1)-st eigenvalues are a
                   complex conjugate pair, with alphai(j+1) negative.

                   Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
                   may easily over- or underflow, and beta(j) may even be zero.
                   Thus, the user should avoid naively computing the ratio.
                   However, alphar and alphai will be always less than and
                   usually comparable with norm(A) in magnitude, and beta always
                   less than and usually comparable with norm(B).

           VSL:    If jobvsl = 1, VSL will contain the left Schur vectors.
                   Not referenced if jobvsl = 0.
                   The leading dimension must always be >=1.

           VSR:    If jobvsr = 1, VSR will contain the right Schur vectors.
                   Not referenced if jobvsr = 0.
                   The leading dimension must always be >=1.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   = 1,...,N:
                         The QZ iteration failed.  (A,B) are not in Schur
                         form, but alphar(j), alphai(j), and beta(j) should
                         be correct for j=info+1,...,N.
                   > N:  =N+1: other than QZ iteration failed in hgeqz.
                         =N+2: after reordering, roundoff changed values of
                               some complex eigenvalues so that leading
                               eigenvalues in the Generalized Schur form no
                               longer satisfy delztg=1  This could also
                               be caused due to scaling.
                         =N+3: reordering failed in tgsen.

        sub my_select{
               my ($zr, $zi, $d) = @_;
               # stable generalized eigenvalues for continuous time
               return ( ($zr < 0 && $d > 0 ) || ($zr > 0 && $d < 0) ) ?  1 : 0;
        }
        $a = random(5,5);
        $b = random(5,5);
        $sdim = null;
        $alphar = zeroes(5);
        $alphai = zeroes(5);
        $beta = zeroes(5);
        $vsl = zeroes(5,5);
        $vsr = zeroes(5,5);
        gges($a, 1, 1, 1, $b, $alphar, $alphai, $beta, $vsl, $vsr, $sdim,($info=null), \&my_select);

       gges ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ggesx
         Signature: ([io,phys]A(n,n); int jobvsl();int jobvsr();int sort();int sense();[io,phys]B(n,n);[o,phys]alphar(n);[o,phys]alphai(n);[o,phys]beta(n);[o,phys]VSL(m,m);[o,phys]VSR(p,p);int [o,phys]sdim();[o,phys]rconde(q);[o,phys]rcondv(r);int [o,phys]info(); SV* select_func)

       Computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized
       eigenvalues, the real Schur form (S,T), and, optionally, the left and/or right matrices of
       Schur vectors (VSL and VSR).  This gives the generalized Schur factorization

               (A,B) = ( (VSL) S (VSR)', (VSL) T (VSR)' )

       Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues
       appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the
       upper triangular matrix T; computes a reciprocal condition number for the average of the
       selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right
       and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The
       leading columns of VSL and VSR then form an orthonormal basis for the corresponding left
       and right eigenspaces (deflating subspaces).

       A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta
       = w, such that  A - w*B is singular.  It is usually represented as the pair (alpha,beta),
       as there is a reasonable interpretation for beta=0 or for both being zero.

       A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with
       non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks.
       1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be
       "standardized" by making the corresponding elements of T have the form:

               [  a  0  ]
               [  0  b  ]

       and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair
       of generalized eigenvalues.

       Further details ===============

       An approximate (asymptotic) bound on the average absolute error of the selected
       eigenvalues is

               EPS * norm((A, B)) / RCONDE( 1 ).

       An approximate (asymptotic) bound on the maximum angular error in the computed deflating
       subspaces is

               EPS * norm((A, B)) / RCONDV( 2 ).

       See LAPACK User's Guide, section 4.11 for more information.

           Arguments
           =========

           jobvsl: = 0:  do not compute the left Schur vectors;
                   = 1:  compute the left Schur vectors.

           jobvsr: = 0:  do not compute the right Schur vectors;
                   = 1:  compute the right Schur vectors.

           sort:   Specifies whether or not to order the eigenvalues on the
                   diagonal of the generalized Schur form.
                   = 0:  Eigenvalues are not ordered;
                   = 1:  Eigenvalues are ordered (see delztg);

           delztg: If sort = 0, delztg is not referenced.
                   If sort = 1, delztg is used to select eigenvalues to sort
                   to the top left of the Schur form.
                   An eigenvalue (alphar(j)+alphai(j))/beta(j) is selected if
                   delztg(alphar(j),alphai(j),beta(j)) is true; i.e. if either
                   one of a complex conjugate pair of eigenvalues is selected,
                   then both complex eigenvalues are selected.

                   Note that in the ill-conditioned case, a selected complex
                   eigenvalue may no longer satisfy delztg(alphar(j),alphai(j),
                   beta(j)) = 1 after ordering. info is to be set to N+2
                   in this case.

           sense:  Determines which reciprocal condition numbers are computed.
                   = 0 : None are computed;
                   = 1 : Computed for average of selected eigenvalues only;
                   = 2 : Computed for selected deflating subspaces only;
                   = 3 : Computed for both.
                   If sense = 1, 2, or 3, sort must equal 1.

           A:      On entry, the first of the pair of matrices.
                   On exit, A has been overwritten by its generalized Schur
                   form S.

           B:      On entry, the second of the pair of matrices.
                   On exit, B has been overwritten by its generalized Schur
                   form T.

           sdim:   If sort = 0, sdim = 0.
                   If sort = 1, sdim = number of eigenvalues (after sorting)
                   for which delztg is true.  (Complex conjugate pairs for which
                   delztg is true for either eigenvalue count as 2.)

           alphar:
           alphai:
           beta:   On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
                   be the generalized eigenvalues.  alphar(j) + alphai(j)*i,
                   and  beta(j),j=1,...,N are the diagonals of the complex Schur
                   form (S,T) that would result if the 2-by-2 diagonal blocks of
                   the real Schur form of (A,B) were further reduced to
                   triangular form using 2-by-2 complex unitary transformations.
                   If alphai(j) is zero, then the j-th eigenvalue is real; if
                   positive, then the j-th and (j+1)-st eigenvalues are a
                   complex conjugate pair, with alphai(j+1) negative.

                   Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
                   may easily over- or underflow, and beta(j) may even be zero.
                   Thus, the user should avoid naively computing the ratio.
                   However, alphar and alphai will be always less than and
                   usually comparable with norm(A) in magnitude, and beta always
                   less than and usually comparable with norm(B).

           VSL:    If jobvsl = 1, VSL will contain the left Schur vectors.
                   Not referenced if jobvsl = 0.
                   The leading dimension must always be >=1.

           VSR:    If jobvsr = 1, VSR will contain the right Schur vectors.
                   Not referenced if jobvsr = 0.
                   The leading dimension must always be >=1.

           rconde: If sense = 1 or 3, rconde(1) and rconde(2) contain the
                   reciprocal condition numbers for the average of the selected
                   eigenvalues.
                   Not referenced if sense = 0 or 2.

           rcondv: If sense = 2 or 3, rcondv(1) and rcondv(2) contain the
                   reciprocal condition numbers for the selected deflating
                   subspaces.
                   Not referenced if sense = 0 or 1.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   = 1,...,N:
                         The QZ iteration failed.  (A,B) are not in Schur
                         form, but alphar(j), alphai(j), and beta(j) should
                         be correct for j=info+1,...,N.
                   > N:  =N+1: other than QZ iteration failed in hgeqz.
                         =N+2: after reordering, roundoff changed values of
                               some complex eigenvalues so that leading
                               eigenvalues in the Generalized Schur form no
                               longer satisfy delztg=1  This could also
                               be caused due to scaling.
                         =N+3: reordering failed in tgsen.

        sub my_select{
               my ($zr, $zi, $d) = @_;
               # Eigenvalue : (ZR/D) + sqrt(-1)*(ZI/D)
               # stable generalized eigenvalues for discrete time
               return (sqrt($zr**2 + $zi**2) < abs($d) ) ?  1 : 0;

        }
        $a = random(5,5);
        $b = random(5,5);
        $sdim = null;
        $alphar = zeroes(5);
        $alphai = zeroes(5);
        $beta = zeroes(5);
        $vsl = zeroes(5,5);
        $vsr = zeroes(5,5);
        $rconde = zeroes(2);
        $rcondv = zeroes(2);
        ggesx($a, 1, 1, 1, 3,$b, $alphar, $alphai, $beta, $vsl, $vsr, $sdim, $rconde, $rcondv, ($info=null), \&my_select);

       ggesx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   syev
         Signature: ([io,phys]A(n,n);  int jobz(); int uplo(); [o,phys]w(n); int [o,phys]info())

       Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.

           Arguments
           =========

           jobz:   = 0:  Compute eigenvalues only;
                   = 1:  Compute eigenvalues and eigenvectors.

           uplo    = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of A contains the
                   upper triangular part of the matrix A.  If uplo = 1,
                   the leading N-by-N lower triangular part of A contains
                   the lower triangular part of the matrix A.
                   On exit, if jobz = 1, then if info = 0, A contains the
                   orthonormal eigenvectors of the matrix A.
                   If jobz = 0, then on exit the lower triangle (if uplo=1)
                   or the upper triangle (if uplo=0) of A, including the
                   diagonal, is destroyed.

           w:      If info = 0, the eigenvalues in ascending order.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, the algorithm failed to converge; i
                         off-diagonal elements of an intermediate tridiagonal
                         form did not converge to zero.

        # Assume $a is symmetric ;)
        $a = random (5,5);
        syev($a, 1,1, (my $w = zeroes(5)), (my $info=null));

       syev ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   syevd
         Signature: ([io,phys]A(n,n);  int jobz(); int uplo(); [o,phys]w(n); int [o,phys]info())

       Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If
       eigenvectors are desired, it uses a divide and conquer algorithm.

       The divide and conquer algorithm makes very mild assumptions about floating point
       arithmetic. It will work on machines with a guard digit in add/subtract, or on those
       binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray
       C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without
       guard digits, but we know of none.

       Because of large use of BLAS of level 3, syevd needs N**2 more workspace than syevx.

           Arguments
           =========

           jobz:   = 0:  Compute eigenvalues only;
                   = 1:  Compute eigenvalues and eigenvectors.

           uplo    = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of A contains the
                   upper triangular part of the matrix A.  If uplo = 1,
                   the leading N-by-N lower triangular part of A contains
                   the lower triangular part of the matrix A.
                   On exit, if jobz = 1, then if info = 0, A contains the
                   orthonormal eigenvectors of the matrix A.
                   If jobz = 0, then on exit the lower triangle (if uplo=1)
                   or the upper triangle (if uplo=0) of A, including the
                   diagonal, is destroyed.

           w:      If info = 0, the eigenvalues in ascending order.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, the algorithm failed to converge; i
                         off-diagonal elements of an intermediate tridiagonal
                         form did not converge to zero.

        # Assume $a is symmetric ;)
        $a = random (5,5);
        syevd($a, 1,1, (my $w = zeroes(5)), (my $info=null));

       syevd ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   syevx
         Signature: ([phys]A(n,n);  int jobz(); int range(); int uplo(); [phys]vl(); [phys]vu(); int [phys]il(); int [phys]iu(); [phys]abstol(); int [o,phys]m(); [o,phys]w(n); [o,phys]z(p,q);int [o,phys]ifail(r); int [o,phys]info())

       Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.
       Eigenvalues and eigenvectors can be selected by specifying either a range of values or a
       range of indices for the desired eigenvalues.

           Arguments
           =========

           jobz:   = 0:  Compute eigenvalues only;
                   = 1:  Compute eigenvalues and eigenvectors.

           range:  = 0: all eigenvalues will be found.
                   = 1: all eigenvalues in the half-open interval (vl,vu]
                          will be found.
                   = 1: the il-th through iu-th eigenvalues will be found.

           uplo    = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of A contains the
                   upper triangular part of the matrix A.  If uplo = 1,
                   the leading N-by-N lower triangular part of A contains
                   the lower triangular part of the matrix A.
                   On exit, the lower triangle (if uplo=1) or the upper
                   triangle (if uplo=0) of A, including the diagonal, is
                   destroyed.

           vl:
           vu:     If range=1, the lower and upper bounds of the interval to
                   be searched for eigenvalues. vl < vu.
                   Not referenced if range = 0 or 2.

           il:
           iu:     If range=2, the indices (in ascending order) of the
                   smallest and largest eigenvalues to be returned.
                   1 <= il <= iu <= N, if N > 0; il = 1 and iu = 0 if N = 0.
                   Not referenced if range = 0 or 1.

           abstol: The absolute error tolerance for the eigenvalues.
                   An approximate eigenvalue is accepted as converged
                   when it is determined to lie in an interval [a,b]
                   of width less than or equal to

                           abstol + EPS *   max( |a|,|b| ) ,

                   where EPS is the machine precision.  If abstol is less than
                   or equal to zero, then  EPS*|T|  will be used in its place,
                   where |T| is the 1-norm of the tridiagonal matrix obtained
                   by reducing A to tridiagonal form.

                   Eigenvalues will be computed most accurately when abstol is
                   set to twice the underflow threshold 2*lamch(1), not zero.
                   If this routine returns with info>0, indicating that some
                   eigenvectors did not converge, try setting abstol to
                   2*lamch(1).

                   See "Computing Small Singular Values of Bidiagonal Matrices
                   with Guaranteed High Relative Accuracy," by Demmel and
                   Kahan, LAPACK Working Note #3.

           m:      The total number of eigenvalues found.  0 <= m <= N.
                   If range = 0, m = N, and if range = 2, m = iu-il+1.

           w:      On normal exit, the first M elements contain the selected
                   eigenvalues in ascending order.

           z:      If jobz = 1, then if info = 0, the first m columns of z
                   contain the orthonormal eigenvectors of the matrix A
                   corresponding to the selected eigenvalues, with the i-th
                   column of z holding the eigenvector associated with w(i).
                   If an eigenvector fails to converge, then that column of z
                   contains the latest approximation to the eigenvector, and the
                   index of the eigenvector is returned in ifail.
                   If jobz = 0, then z is not referenced.
                   Note: the user must ensure that at least max(1,m) columns are
                   supplied in the array z; if range = 1, the exact value of m
                   is not known in advance and an upper bound must be used.

           ifail:   If jobz = 1, then if info = 0, the first m elements of
                   ifail are zero.  If info > 0, then ifail contains the
                   indices of the eigenvectors that failed to converge.
                   If jobz = 0, then ifail is not referenced.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, then i eigenvectors failed to converge.
                         Their indices are stored in array ifail.

        # Assume $a is symmetric ;)
        $a = random (5,5);
        $unfl = lamch(1);
        $ovfl = lamch(9);
        labad($unfl, $ovfl);
        $abstol = $unfl + $unfl;
        $m = null;
        $info = null;
        $ifail = zeroes(5);
        $w = zeroes(5);
        $z = zeroes(5,5);
        syevx($a, 1,0,1,0,0,0,0,$abstol, $m, $w, $z ,$ifail, $info);

       syevx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   syevr
         Signature: ([phys]A(n,n);  int jobz(); int range(); int uplo(); [phys]vl(); [phys]vu(); int [phys]il(); int [phys]iu();[phys]abstol();int [o,phys]m();[o,phys]w(n); [o,phys]z(p,q);int [o,phys]isuppz(r); int [o,phys]info())

       Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix T.
       Eigenvalues and eigenvectors can be selected by specifying either a range of values or a
       range of indices for the desired eigenvalues.

       Whenever possible, syevr calls stegr to compute the eigenspectrum using Relatively Robust
       Representations.  stegr computes eigenvalues by the dqds algorithm, while orthogonal
       eigenvectors are computed from various "good" L D L^T representations (also known as
       Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as
       possible. More specifically, the various steps of the algorithm are as follows. For the
       i-th unreduced block of T,

              (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
                   is a relatively robust representation,
              (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
                  relative accuracy by the dqds algorithm,
              (c) If there is a cluster of close eigenvalues, "choose" sigma_i
                  close to the cluster, and go to step (a),
              (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
                  compute the corresponding eigenvector by forming a
                  rank-revealing twisted factorization.

       The desired accuracy of the output can be specified by the input parameter abstol.

       For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal
       eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical
       Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

       Note 1 : syevr calls stegr when the full spectrum is requested on machines which conform
       to the ieee-754 floating point standard.  syevr calls stebz and stein on non-ieee machines
       and when partial spectrum requests are made.

       Normal execution of stegr may create NaNs and infinities and hence may abort due to a
       floating point exception in environments which do not handle NaNs and infinities in the
       ieee standard default manner.

           Arguments
           =========

           jobz:   = 0:  Compute eigenvalues only;
                   = 1:  Compute eigenvalues and eigenvectors.

           range:  = 0: all eigenvalues will be found.
                   = 1: all eigenvalues in the half-open interval (vl,vu]
                          will be found.
                   = 2: the il-th through iu-th eigenvalues will be found.
          ********* For range = 1 or 2 and iu - il < N - 1, stebz and
          ********* stein are called

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of A contains the
                   upper triangular part of the matrix A.  If uplo = 1,
                   the leading N-by-N lower triangular part of A contains
                   the lower triangular part of the matrix A.
                   On exit, the lower triangle (if uplo=1) or the upper
                   triangle (if uplo=0) of A, including the diagonal, is
                   destroyed.

           vl:
           vu:     If range=1, the lower and upper bounds of the interval to
                   be searched for eigenvalues. vl < vu.
                   Not referenced if range = 0 or 2.

           il:
           iu:     If range=2, the indices (in ascending order) of the
                   smallest and largest eigenvalues to be returned.
                   1 <= il <= iu <= N, if N > 0; il = 1 and iu = 0 if N = 0.
                   Not referenced if range = 0 or 1.

           abstol: The absolute error tolerance for the eigenvalues.
                   An approximate eigenvalue is accepted as converged
                   when it is determined to lie in an interval [a,b]
                   of width less than or equal to

                           abstol + EPS *   max( |a|,|b| ) ,

                   where EPS is the machine precision.  If abstol is less than
                   or equal to zero, then  EPS*|T|  will be used in its place,
                   where |T| is the 1-norm of the tridiagonal matrix obtained
                   by reducing A to tridiagonal form.

                   See "Computing Small Singular Values of Bidiagonal Matrices
                   with Guaranteed High Relative Accuracy," by Demmel and
                   Kahan, LAPACK Working Note #3.

                   If high relative accuracy is important, set abstol to
                   lamch(1).  Doing so will guarantee that
                   eigenvalues are computed to high relative accuracy when
                   possible in future releases.  The current code does not
                   make any guarantees about high relative accuracy, but
                   furure releases will. See J. Barlow and J. Demmel,
                   "Computing Accurate Eigensystems of Scaled Diagonally
                   Dominant Matrices", LAPACK Working Note #7, for a discussion
                   of which matrices define their eigenvalues to high relative
                   accuracy.

           m:      The total number of eigenvalues found.  0 <= m <= N.
                   If range = 0, m = N, and if range = 2, m = iu-il+1.

           w:      The first m elements contain the selected eigenvalues in
                   ascending order.

           z:      If jobz = 1, then if info = 0, the first m columns of z
                   contain the orthonormal eigenvectors of the matrix A
                   corresponding to the selected eigenvalues, with the i-th
                   column of z holding the eigenvector associated with w(i).
                   If jobz = 0, then z is not referenced.
                   Note: the user must ensure that at least max(1,m) columns are
                   supplied in the array z; if range = 1, the exact value of m
                   is not known in advance and an upper bound must be used.

           isuppz: array of int, dimension ( 2*max(1,m) )
                   The support of the eigenvectors in z, i.e., the indices
                   indicating the nonzero elements in z. The i-th eigenvector
                   is nonzero only in elements isuppz( 2*i-1 ) through
                   isuppz( 2*i ).
          ********* Implemented only for range = 0 or 2 and iu - il = N - 1

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  Internal error

        # Assume $a is symmetric ;)
        $a = random (5,5);
        $unfl = lamch(1);
        $ovfl = lamch(9);
        labad($unfl, $ovfl);
        $abstol = $unfl + $unfl;
        $m = null;
        $info = null;
        $isuppz = zeroes(10);
        $w = zeroes(5);
        $z = zeroes(5,5);
        syevr($a, 1,0,1,0,0,0,0,$abstol, $m, $w, $z ,$isuppz, $info);

       syevr ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sygv
         Signature: ([io,phys]A(n,n);int [phys]itype();int jobz(); int uplo();[io,phys]B(n,n);[o,phys]w(n); int [o,phys]info())

       Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized
       symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or
       B*A*x=(lambda)*x.  Here A and B are assumed to be symmetric and B is also positive
       definite.

           Arguments
           =========

           itype:  Specifies the problem type to be solved:
                   = 1:  A*x = (lambda)*B*x
                   = 2:  A*B*x = (lambda)*x
                   = 3:  B*A*x = (lambda)*x

           jobz:   = 0:  Compute eigenvalues only;
                   = 1:  Compute eigenvalues and eigenvectors.

           uplo:   = 0:  Upper triangles of A and B are stored;
                   = 1:  Lower triangles of A and B are stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of A contains the
                   upper triangular part of the matrix A.  If uplo = 1,
                   the leading N-by-N lower triangular part of A contains
                   the lower triangular part of the matrix A.

                   On exit, if jobz = 1, then if info = 0, A contains the
                   matrix Z of eigenvectors.  The eigenvectors are normalized
                   as follows:
                   if itype = 1 or 2, Z'*B*Z = I;
                   if itype = 3, Z'*inv(B)*Z = I.
                   If jobz = 0, then on exit the upper triangle (if uplo=0)
                   or the lower triangle (if uplo=1) of A, including the
                   diagonal, is destroyed.

           B:      On entry, the symmetric positive definite matrix B.
                   If uplo = 0, the leading N-by-N upper triangular part of B
                   contains the upper triangular part of the matrix B.
                   If uplo = 1, the leading N-by-N lower triangular part of B
                   contains the lower triangular part of the matrix B.

                   On exit, if info <= N, the part of B containing the matrix is
                   overwritten by the triangular factor U or L from the Cholesky
                   factorization B = U'*U or B = L*L'.

           W:      If info = 0, the eigenvalues in ascending order.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  potrf or syev returned an error code:
                      <= N:  if info = i, syev failed to converge;
                             i off-diagonal elements of an intermediate
                             tridiagonal form did not converge to zero;
                      > N:   if info = N + i, for 1 <= i <= N, then the leading
                             minor of order i of B is not positive definite.
                             The factorization of B could not be completed and
                             no eigenvalues or eigenvectors were computed.

        # Assume $a is symmetric ;)
        $a = random (5,5);
        # Assume $a is symmetric and positive definite ;)
        $b = random (5,5);
        sygv($a, 1,1, 0, $b, (my $w = zeroes(5)), (my $info=null));

       sygv ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sygvd
         Signature: ([io,phys]A(n,n);int [phys]itype();int jobz(); int uplo();[io,phys]B(n,n);[o,phys]w(n); int [o,phys]info())

       Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized
       symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or
       B*A*x=(lambda)*x.  Here A and B are assumed to be symmetric and B is also positive
       definite.

       The divide and conquer algorithm makes very mild assumptions about floating point
       arithmetic. It will work on machines with a guard digit in add/subtract, or on those
       binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray
       C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without
       guard digits, but we know of none.

           Arguments
           =========

           itype:  Specifies the problem type to be solved:
                   = 1:  A*x = (lambda)*B*x
                   = 2:  A*B*x = (lambda)*x
                   = 3:  B*A*x = (lambda)*x

           jobz:   = 0:  Compute eigenvalues only;
                   = 1:  Compute eigenvalues and eigenvectors.

           uplo:   = 0:  Upper triangles of A and B are stored;
                   = 1:  Lower triangles of A and B are stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of A contains the
                   upper triangular part of the matrix A.  If uplo = 1,
                   the leading N-by-N lower triangular part of A contains
                   the lower triangular part of the matrix A.

                   On exit, if jobz = 1, then if info = 0, A contains the
                   matrix Z of eigenvectors.  The eigenvectors are normalized
                   as follows:
                   if itype = 1 or 2, Z'*B*Z = I;
                   if itype = 3, Z'*inv(B)*Z = I.
                   If jobz = 0, then on exit the upper triangle (if uplo=0)
                   or the lower triangle (if uplo=1) of A, including the
                   diagonal, is destroyed.

           B:      On entry, the symmetric positive definite matrix B.
                   If uplo = 0, the leading N-by-N upper triangular part of B
                   contains the upper triangular part of the matrix B.
                   If uplo = 1, the leading N-by-N lower triangular part of B
                   contains the lower triangular part of the matrix B.

                   On exit, if info <= N, the part of B containing the matrix is
                   overwritten by the triangular factor U or L from the Cholesky
                   factorization B = U'*U or B = L*L'.

           W:      If info = 0, the eigenvalues in ascending order.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  potrf or syev returned an error code:
                      <= N:  if info = i, syevd failed to converge;
                             i off-diagonal elements of an intermediate
                             tridiagonal form did not converge to zero;
                      > N:   if info = N + i, for 1 <= i <= N, then the leading
                             minor of order i of B is not positive definite.
                             The factorization of B could not be completed and
                             no eigenvalues or eigenvectors were computed.

        # Assume $a is symmetric ;)
        $a = random (5,5);
        # Assume $b is symmetric positive definite ;)
        $b = random (5,5);
        sygvd($a, 1,1, 0, $b, (my $w = zeroes(5)), (my $info=null));

       sygvd ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sygvx
         Signature: ([io,phys]A(n,n);int [phys]itype();int jobz();int range(); int uplo();[io,phys]B(n,n);[phys]vl();[phys]vu();int [phys]il();int [phys]iu();[phys]abstol();int [o,phys]m();[o,phys]w(n); [o,phys]Z(p,q);int [o,phys]ifail(r);int [o,phys]info())

       Computes selected eigenvalues, and optionally, eigenvectors of a real generalized
       symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or
       B*A*x=(lambda)*x.  Here A and B are assumed to be symmetric and B is also positive
       definite.  Eigenvalues and eigenvectors can be selected by specifying either a range of
       values or a range of indices for the desired eigenvalues.

           Arguments
           =========

           itype:  Specifies the problem type to be solved:
                   = 1:  A*x = (lambda)*B*x
                   = 2:  A*B*x = (lambda)*x
                   = 3:  B*A*x = (lambda)*x

           jobz:   = 0:  Compute eigenvalues only;
                   = 1:  Compute eigenvalues and eigenvectors.

           range:  = 0: all eigenvalues will be found.
                   = 1: all eigenvalues in the half-open interval (vl,vu]
                          will be found.
                   = 2: the il-th through iu-th eigenvalues will be found.

           uplo:   = 0:  Upper triangle of A and B are stored;
                   = 1:  Lower triangle of A and B are stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of A contains the
                   upper triangular part of the matrix A.  If uplo = 1,
                   the leading N-by-N lower triangular part of A contains
                   the lower triangular part of the matrix A.

                   On exit, the lower triangle (if uplo=1) or the upper
                   triangle (if uplo=0) of A, including the diagonal, is
                   destroyed.

           B:      On entry, the symmetric matrix B.  If uplo = 0, the
                   leading N-by-N upper triangular part of B contains the
                   upper triangular part of the matrix B.  If uplo = 1,
                   the leading N-by-N lower triangular part of B contains
                   the lower triangular part of the matrix B.

                   On exit, if info <= N, the part of B containing the matrix is
                   overwritten by the triangular factor U or L from the Cholesky
                   factorization B = U'*U or B = L*L'.

           vl:
           vu:     If range=1, the lower and upper bounds of the interval to
                   be searched for eigenvalues. vl < vu.
                   Not referenced if range = 0 or 2.

           il:
           iu:     If range=2, the indices (in ascending order) of the
                   smallest and largest eigenvalues to be returned.
                   1 <= il <= iu <= N, if N > 0; il = 1 and iu = 0 if N = 0.
                   Not referenced if range = 0 or 1.

           abstol: The absolute error tolerance for the eigenvalues.
                   An approximate eigenvalue is accepted as converged
                   when it is determined to lie in an interval [a,b]
                   of width less than or equal to

                           abstol + EPS *   max( |a|,|b| ) ,

                   where EPS is the machine precision.  If abstol is less than
                   or equal to zero, then  EPS*|T|  will be used in its place,
                   where |T| is the 1-norm of the tridiagonal matrix obtained
                   by reducing A to tridiagonal form.

                   Eigenvalues will be computed most accurately when abstol is
                   set to twice the underflow threshold 2*lamch(1), not zero.
                   If this routine returns with info>0, indicating that some
                   eigenvectors did not converge, try setting abstol to
                   2* lamch(1).

           m:      The total number of eigenvalues found.  0 <= m <= N.
                   If range = 0, m = N, and if range = 2, m = iu-il+1.

           w:      On normal exit, the first m elements contain the selected
                   eigenvalues in ascending order.

           Z:      If jobz = 0, then Z is not referenced.
                   If jobz = 1, then if info = 0, the first m columns of Z
                   contain the orthonormal eigenvectors of the matrix A
                   corresponding to the selected eigenvalues, with the i-th
                   column of Z holding the eigenvector associated with w(i).
                   The eigenvectors are normalized as follows:
                   if itype = 1 or 2, Z'*B*Z = I;
                   if itype = 3, Z'*inv(B)*Z = I.

                   If an eigenvector fails to converge, then that column of Z
                   contains the latest approximation to the eigenvector, and the
                   index of the eigenvector is returned in ifail.
                   Note: the user must ensure that at least max(1,m) columns are
                   supplied in the array Z; if range = 1, the exact value of m
                   is not known in advance and an upper bound must be used.

           ifail:  If jobz = 1, then if info = 0, the first M elements of
                   ifail are zero.  If info > 0, then ifail contains the
                   indices of the eigenvectors that failed to converge.
                   If jobz = 0, then ifail is not referenced.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  potrf or syevx returned an error code:
                      <= N:  if info = i, syevx failed to converge;
                             i eigenvectors failed to converge.  Their indices
                             are stored in array ifail.
                      > N:   if info = N + i, for 1 <= i <= N, then the leading
                             minor of order i of B is not positive definite.
                             The factorization of B could not be completed and
                             no eigenvalues or eigenvectors were computed.

        # Assume $a is symmetric ;)
        $a = random (5,5);
        # Assume $b is symmetric positive definite ;)
        $b = random (5,5);
        $unfl = lamch(1);
        $ovfl = lamch(9);
        labad($unfl, $ovfl);
        $abstol = $unfl + $unfl;
        $m = null;
        $w=zeroes(5);
        $z = zeroes(5,5);
        $ifail = zeroes(5);
        sygvx($a, 1,1, 0,0, $b, 0, 0, 0, 0, $abstol, $m, $w, $z,$ifail,(my $info=null));

       sygvx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gesv
         Signature: ([io,phys]A(n,n);  [io,phys]B(n,m); int [o,phys]ipiv(n); int [o,phys]info())

       Computes the solution to a real system of linear equations

               A * X = B,
               where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

       The LU decomposition with partial pivoting and row interchanges is used to factor A as

               A = P * L * U,
               where P is a permutation matrix, L is unit lower triangular, and U is
               upper triangular.

       The factored form of A is then used to solve the system of equations A * X = B.

           Arguments
           =========

           A:      On entry, the N-by-N coefficient matrix A.
                   On exit, the factors L and U from the factorization
                   A = P*L*U; the unit diagonal elements of L are not stored.

           ipiv:   The pivot indices that define the permutation matrix P;
                   row i of the matrix was interchanged with row ipiv(i).

           B:      On entry, the N-by-NRHS matrix of right hand side matrix B.
                   On exit, if info = 0, the N-by-NRHS solution matrix X.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, U(i,i) is exactly zero.  The factorization
                         has been completed, but the factor U is exactly
                         singular, so the solution could not be computed.

        $a = random (5,5);
        $a = transpose($a);
        $b = random (5,5);
        $b = transpose($b);
        gesv($a,$b, (my $ipiv=zeroes(5)),(my $info=null));
        print "The solution matrix X is :". transpose($b)."\n" unless $info;

       gesv ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gesvx
         Signature: ([io,phys]A(n,n); int trans(); int fact(); [io,phys]B(n,m); [io,phys]af(n,n); int [io,phys]ipiv(n); int [io]equed(); [io,phys]r(n); [io,phys]c(n); [o,phys]X(n,m); [o,phys]rcond(); [o,phys]ferr(m); [o,phys]berr(m);[o,phys]rpvgrw();int [o,phys]info())

       Uses the LU factorization to compute the solution to a real system of linear equations

               A * X = B,
               where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

       Error bounds on the solution and a condition estimate are also provided.

       The following steps are performed:

       1. If fact = 2, real scaling factors are computed to equilibrate the system:

                  trans = 0:  diag(r)*A*diag(c)     *inv(diag(c))*X = diag(c)*B
                  trans = 1: (diag(r)*A*diag(c))' *inv(diag(r))*X = diag(c)*B
                  trans = 2: (diag(r)*A*diag(c))**H *inv(diag(r))*X = diag(c)*B

          Whether or not the system will be equilibrated depends on the scaling of the matrix A,
          but if equilibration is used, A is overwritten by diag(r)*A*diag(c) and B by diag(r)*B
          (if trans=0) or diag(c)*B (if trans = 1 or 2).

       2. If fact = 1 or 2, the LU decomposition is used to factor the matrix A (after
          equilibration if fact = 2) as

                  A = P * L * U,
                  where P is a permutation matrix, L is a unit lower triangular
                  matrix, and U is upper triangular.

       3. If some U(i,i)=0, so that U is exactly singular, then the routine returns with info =
          i. Otherwise, the factored form of A is used to estimate the condition number of the
          matrix A.  If the reciprocal of the condition number is less than machine precision,
          info = N+1 is returned as a warning, but the routine still goes on to solve for X and
          compute error bounds as described below.

       4. The system of equations is solved for X using the factored form of A.

       5. Iterative refinement is applied to improve the computed solution matrix and calculate
          error bounds and backward error estimates for it.

       6. If equilibration was used, the matrix X is premultiplied by diag(c) (if trans = 0) or
          diag(r) (if trans = 1 or 2) so that it solves the original system before equilibration.

           Arguments
           =========

           fact:   Specifies whether or not the factored form of the matrix A is
                   supplied on entry, and if not, whether the matrix A should be
                   equilibrated before it is factored.
                   = 0:  On entry, af and ipiv contain the factored form of A.
                           If equed is not 0, the matrix A has been
                           equilibrated with scaling factors given by r and c.
                           A, af, and ipiv are not modified.
                   = 1:  The matrix A will be copied to af and factored.
                   = 2:  The matrix A will be equilibrated if necessary, then
                           copied to af and factored.

           trans:  Specifies the form of the system of equations:
                   = 0:  A * X = B     (No transpose)
                   = 1:  A' * X = B  (Transpose)
                   = 2:  A**H * X = B  (Transpose)

           A:      On entry, the N-by-N matrix A.  If fact = 0 and equed is
                   not 0, then A must have been equilibrated by the scaling
                   factors in r and/or c.  A is not modified if fact = 0 or
                   1, or if fact = 2 and equed = 0 on exit.

                   On exit, if equed != 0, A is scaled as follows:
                   equed = 1:  A := diag(r) * A
                   equed = 2:  A := A * diag(c)
                   equed = 3:  A := diag(r) * A * diag(c).

           af:     If fact = 0, then af is an input argument and on entry
                   contains the factors L and U from the factorization
                   A = P*L*U as computed by getrf.  If equed != 0, then
                   af is the factored form of the equilibrated matrix A.

                   If fact = 1, then af is an output argument and on exit
                   returns the factors L and U from the factorization A = P*L*U
                   of the original matrix A.

                   If fact = 2, then af is an output argument and on exit
                   returns the factors L and U from the factorization A = P*L*U
                   of the equilibrated matrix A (see the description of A for
                   the form of the equilibrated matrix).

           ipiv:   If fact = 0, then ipiv is an input argument and on entry
                   contains the pivot indices from the factorization A = P*L*U
                   as computed by getrf; row i of the matrix was interchanged
                   with row ipiv(i).

                   If fact = 1, then ipiv is an output argument and on exit
                   contains the pivot indices from the factorization A = P*L*U
                   of the original matrix A.

                   If fact = 2, then ipiv is an output argument and on exit
                   contains the pivot indices from the factorization A = P*L*U
                   of the equilibrated matrix A.

           equed:  Specifies the form of equilibration that was done.
                   = 0:  No equilibration (always true if fact = 1).
                   = 1:  Row equilibration, i.e., A has been premultiplied by
                           diag(r).
                   = 2:  Column equilibration, i.e., A has been postmultiplied
                           by diag(c).
                   = 3:  Both row and column equilibration, i.e., A has been
                           replaced by diag(r) * A * diag(c).
                   equed is an input argument if fact = 0; otherwise, it is an
                   output argument.

           r:      The row scale factors for A.  If equed = 1 or 3, A is
                   multiplied on the left by diag(r); if equed = 0 or 2, r
                   is not accessed.  r is an input argument if fact = 0;
                   otherwise, r is an output argument.  If fact = 0 and
                   equed = 1 or 3, each element of r must be positive.

           c:      The column scale factors for A.  If equed = 2 or 3, A is
                   multiplied on the right by diag(c); if equed = 0 or 1, c
                   is not accessed.  c is an input argument if fact = 0;
                   otherwise, c is an output argument.  If fact = 0 and
                   equed = 2 or 3, each element of c must be positive.

           B:      On entry, the N-by-NRHS right hand side matrix B.
                   On exit,
                   if equed = 0, B is not modified;
                   if trans = 0 and equed = 1 or 3, B is overwritten by
                   diag(r)*B;
                   if trans = 1 or 2 and equed = 2 or 3, B is
                   overwritten by diag(c)*B.

           X:      If info = 0 or info = N+1, the N-by-NRHS solution matrix X
                   to the original system of equations.  Note that A and B are
                   modified on exit if equed != 0, and the solution to the
                   equilibrated system is inv(diag(c))*X if trans = 0 and
                   equed = 2 or 3, or inv(diag(r))*X if trans = 1 or 2
                   and equed = 1 or 3.

           rcond:  The estimate of the reciprocal condition number of the matrix
                   A after equilibration (if done).  If rcond is less than the
                   machine precision (in particular, if rcond = 0), the matrix
                   is singular to working precision.  This condition is
                   indicated by a return code of info > 0.

           ferr:   The estimated forward error bound for each solution vector
                   X(j) (the j-th column of the solution matrix X).
                   If XTRUE is the true solution corresponding to X(j), ferr(j)
                   is an estimated upper bound for the magnitude of the largest
                   element in (X(j) - XTRUE) divided by the magnitude of the
                   largest element in X(j).  The estimate is as reliable as
                   the estimate for rcond, and is almost always a slight
                   overestimate of the true error.

           berr:   The componentwise relative backward error of each solution
                   vector X(j) (i.e., the smallest relative change in
                   any element of A or B that makes X(j) an exact solution).

           rpvgrw: Contains the reciprocal pivot growth factor norm(A)/norm(U).
                   The "max absolute element" norm is used. If it is much less
                   than 1, then the stability of the LU factorization of the
                   (equilibrated) matrix A could be poor. This also means that
                   the solution X, condition estimator rcond, and forward error
                   bound ferr could be unreliable. If factorization fails with
                   0<info<=N, then it contains the reciprocal pivot growth factor
                   for the leading info columns of A.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, and i is
                         <= N:  U(i,i) is exactly zero.  The factorization has
                                been completed, but the factor U is exactly
                                singular, so the solution and error bounds
                                could not be computed. rcond = 0 is returned.
                         = N+1: U is nonsingular, but rcond is less than machine
                                precision, meaning that the matrix is singular
                                to working precision.  Nevertheless, the
                                solution and error bounds are computed because
                                there are a number of situations where the
                                computed solution can be more accurate than the
                                value of rcond would suggest.

        $a= random(5,5);
        $b = random(5,5);
        $a = transpose($a);
        $b = transpose($b);
        $rcond = pdl(0);
        $rpvgrw = pdl(0);
        $equed = pdl(long,0);
        $info = pdl(long,0);
        $berr = zeroes(5);
        $ipiv = zeroes(5);
        $ferr = zeroes(5);
        $r = zeroes(5);
        $c = zeroes(5);
        $X = zeroes(5,5);
        $af = zeroes(5,5);
        gesvx($a,0, 2, $b, $af, $ipiv, $equed, $r, $c, $X, $rcond, $ferr, $berr, $rpvgrw, $info);
        print "The solution matrix X is :". transpose($X)."\n" unless $info;

       gesvx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sysv
         Signature: ([io,phys]A(n,n);  int uplo(); [io,phys]B(n,m); int [o,phys]ipiv(n); int [o,phys]info())

       Computes the solution to a real system of linear equations

               A * X = B,
               where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
               matrices.

       The diagonal pivoting method is used to factor A as

               A = U * D * U',  if uplo = 0, or
               A = L * D * L',  if uplo = 1,
               where U (or L) is a product of permutation and unit upper (lower)
               triangular matrices, and D is symmetric and block diagonal with
               1-by-1 and 2-by-2 diagonal blocks.

       The factored form of A is then used to solve the system of equations A * X = B.

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the leading
                   N-by-N upper triangular part of A contains the upper
                   triangular part of the matrix A, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of A contains the lower
                   triangular part of the matrix A, and the strictly upper
                   triangular part of A is not referenced.

                   On exit, if info = 0, the block diagonal matrix D and the
                   multipliers used to obtain the factor U or L from the
                   factorization A = U*D*U' or A = L*D*L' as computed by
                   sytrf.

           ipiv:   Details of the interchanges and the block structure of D, as
                   determined by sytrf.  If ipiv(k) > 0, then rows and columns
                   k and ipiv(k) were interchanged, and D(k,k) is a 1-by-1
                   diagonal block.  If uplo = 0 and ipiv(k) = ipiv(k-1) < 0,
                   then rows and columns k-1 and -ipiv(k) were interchanged and
                   D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If uplo = 1 and
                   ipiv(k) = ipiv(k+1) < 0, then rows and columns k+1 and
                   -ipiv(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
                   diagonal block.

           B:      On entry, the N-by-NRHS right hand side matrix B.
                   On exit, if info = 0, the N-by-NRHS solution matrix X.

           info:   = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value
                   > 0: if info = i, D(i,i) is exactly zero.  The factorization
                        has been completed, but the block diagonal matrix D is
                        exactly singular, so the solution could not be computed.

        # Assume $a is symmetric ;)
        $a = random (5,5);
        $a = transpose($a);
        $b = random(4,5);
        $b = transpose($b);
        sysv($a, 1, $b, (my $ipiv=zeroes(5)),(my $info=null));
        print "The solution matrix X is :". transpose($b)."\n" unless $info;

       sysv ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sysvx
         Signature: ([phys]A(n,n); int uplo(); int fact(); [phys]B(n,m); [io,phys]af(n,n); int [io,phys]ipiv(n); [o,phys]X(n,m); [o,phys]rcond(); [o,phys]ferr(m); [o,phys]berr(m); int [o,phys]info())

       Uses the diagonal pivoting factorization to compute the solution to a real system of
       linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-
       NRHS matrices.

       Error bounds on the solution and a condition estimate are also provided.

       The following steps are performed:

       1. If fact = 0, the diagonal pivoting method is used to factor A.  The form of the
          factorization is

                  A = U * D * U',  if uplo = 0, or
                  A = L * D * L',  if uplo = 1,
                  where U (or L) is a product of permutation and unit upper (lower)
                  triangular matrices, and D is symmetric and block diagonal with
                  1-by-1 and 2-by-2 diagonal blocks.

       2. If some D(i,i)=0, so that D is exactly singular, then the routine returns with info =
          i. Otherwise, the factored form of A is used
                 to estimate the condition number of the matrix A.  If the
                 reciprocal of the condition number is less than machine precision,
                 info = N+1 is returned as a warning, but the routine still goes on
                 to solve for X and compute error bounds as described below.

       3. The system of equations is solved for X using the factored form of A.

       4. Iterative refinement is applied to improve the computed solution matrix and calculate
          error bounds and backward error estimates for it.

           Arguments
           =========

           fact:   Specifies whether or not the factored form of A has been
                   supplied on entry.
                   = 0:  The matrix A will be copied to af and factored.
                   = 1:  On entry, af and ipiv contain the factored form of
                           A.  af and ipiv will not be modified.

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      The symmetric matrix A.  If uplo = 0, the leading N-by-N
                   upper triangular part of A contains the upper triangular part
                   of the matrix A, and the strictly lower triangular part of A
                   is not referenced.  If uplo = 1, the leading N-by-N lower
                   triangular part of A contains the lower triangular part of
                   the matrix A, and the strictly upper triangular part of A is
                   not referenced.

           af:     If fact = 1, then af is an input argument and on entry
                   contains the block diagonal matrix D and the multipliers used
                   to obtain the factor U or L from the factorization
                   A = U*D*U' or A = L*D*L' as computed by sytrf.

                   If fact = 0, then af is an output argument and on exit
                   returns the block diagonal matrix D and the multipliers used
                   to obtain the factor U or L from the factorization
                   A = U*D*U' or A = L*D*L'.

           ipiv:   If fact = 1, then ipiv is an input argument and on entry
                   contains details of the interchanges and the block structure
                   of D, as determined by sytrf.
                   If ipiv(k) > 0, then rows and columns k and ipiv(k) were
                   interchanged and D(k,k) is a 1-by-1 diagonal block.
                   If uplo = 0 and ipiv(k) = ipiv(k-1) < 0, then rows and
                   columns k-1 and -ipiv(k) were interchanged and D(k-1:k,k-1:k)
                   is a 2-by-2 diagonal block.  If uplo = 1 and ipiv(k) =
                   ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were
                   interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

                   If fact = 0, then ipiv is an output argument and on exit
                   contains details of the interchanges and the block structure
                   of D, as determined by sytrf.

           B:      The N-by-NRHS right hand side matrix B.

           X:      If info = 0 or info = N+1, the N-by-NRHS solution matrix X.

           rcond:  The estimate of the reciprocal condition number of the matrix
                   A.  If rcond is less than the machine precision (in
                   particular, if rcond = 0), the matrix is singular to working
                   precision.  This condition is indicated by a return code of
                   info > 0.

           ferr:   The estimated forward error bound for each solution vector
                   X(j) (the j-th column of the solution matrix X).
                   If XTRUE is the true solution corresponding to X(j), ferr(j)
                   is an estimated upper bound for the magnitude of the largest
                   element in (X(j) - XTRUE) divided by the magnitude of the
                   largest element in X(j).  The estimate is as reliable as
                   the estimate for rcond, and is almost always a slight
                   overestimate of the true error.

           berr:   The componentwise relative backward error of each solution
                   vector X(j) (i.e., the smallest relative change in
                   any element of A or B that makes X(j) an exact solution).

           info:   = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value
                   > 0: if info = i, and i is
                         <= N:  D(i,i) is exactly zero.  The factorization
                                has been completed but the factor D is exactly
                                singular, so the solution and error bounds could
                                not be computed. rcond = 0 is returned.
                         = N+1: D is nonsingular, but rcond is less than machine
                                precision, meaning that the matrix is singular
                                to working precision.  Nevertheless, the
                                solution and error bounds are computed because
                                there are a number of situations where the
                                computed solution can be more accurate than the
                                value of rcond would suggest.

        $a= random(5,5);
        $b = random(10,5);
        $a = transpose($a);
        $b = transpose($b);
        $X = zeroes($b);
        $af = zeroes($a);
        $ipiv = zeroes(long, 5);
        $rcond = pdl(0);
        $ferr = zeroes(10);
        $berr = zeroes(10);
        $info = pdl(long, 0);
        # Assume $a is  symmetric
        sysvx($a, 0, 0, $b,$af, $ipiv, $X, $rcond, $ferr, $berr,$info);
        print "The solution matrix X is :". transpose($X)."\n";

       sysvx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   posv
         Signature: ([io,phys]A(n,n);  int uplo(); [io,phys]B(n,m); int [o,phys]info())

       Computes the solution to a real system of linear equations

               A * X = B,
               where A is an N-by-N symmetric positive definite matrix and X and B
               are N-by-NRHS matrices.

       The Cholesky decomposition is used to factor A as

               A = U'* U,  if uplo = 0, or
               A = L * L',  if uplo = 1,
               where U is an upper triangular matrix and L is a lower triangular
               matrix.

       The factored form of A is then used to solve the system of equations A * X = B.

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the leading
                   N-by-N upper triangular part of A contains the upper
                   triangular part of the matrix A, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of A contains the lower
                   triangular part of the matrix A, and the strictly upper
                   triangular part of A is not referenced.

                   On exit, if info = 0, the factor U or L from the Cholesky
                   factorization A = U'*U or A = L*L'.

           B:      On entry, the N-by-NRHS right hand side matrix B.
                   On exit, if info = 0, the N-by-NRHS solution matrix X.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, the leading minor of order i of A is not
                         positive definite, so the factorization could not be
                         completed, and the solution has not been computed.

        # Assume $a is symmetric positive definite ;)
        $a = random (5,5);
        $a = transpose($a);
        $b = random(4,5);
        $b = transpose($b);
        posv($a, 1, $b, (my $info=null));
        print "The solution matrix X is :". transpose($b)."\n" unless $info;

       posv ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   posvx
         Signature: ([io,phys]A(n,n); int uplo(); int fact(); [io,phys]B(n,m); [io,phys]af(n,n); int [io]equed(); [io,phys]s(n); [o,phys]X(n,m); [o,phys]rcond(); [o,phys]ferr(m); [o,phys]berr(m); int [o,phys]info())

       Uses the Cholesky factorization A = U'*U or A = L*L' to compute the solution to a real
       system of linear equations

               A * X = B,
               where A is an N-by-N symmetric positive definite matrix and X and B
               are N-by-NRHS matrices.

       Error bounds on the solution and a condition estimate are also provided.

       The following steps are performed:

       1. If fact = 2, real scaling factors are computed to equilibrate the system:

                  diag(s) * A * diag(s) * inv(diag(s)) * X = diag(s) * B

          Whether or not the system will be equilibrated depends on the scaling of the matrix A,
          but if equilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.

       2. If fact = 1 or 2, the Cholesky decomposition is used to factor the matrix A (after
          equilibration if fact = 2) as

                  A = U'* U,  if uplo = 0, or
                  A = L * L',  if uplo = 1,
                  where U is an upper triangular matrix and L is a lower triangular
                  matrix.

       3. If the leading i-by-i principal minor is not positive definite, then the routine
          returns with info = i. Otherwise, the factored form of A is used to estimate the
          condition number of the matrix A.  If the reciprocal of the condition number is less
          than machine precision, info = N+1 is returned as a warning, but the routine still goes
          on to solve for X and compute error bounds as described below.

       4. The system of equations is solved for X using the factored form of A.

       5. Iterative refinement is applied to improve the computed solution matrix and calculate
          error bounds and backward error estimates for it.

       6. If equilibration was used, the matrix X is premultiplied by diag(s) so that it solves
          the original system before equilibration.

           Arguments
           =========

           fact:   Specifies whether or not the factored form of the matrix A is
                   supplied on entry, and if not, whether the matrix A should be
                   equilibrated before it is factored.
                   = 0:  On entry, af contains the factored form of A.
                           If equed = 1, the matrix A has been equilibrated
                           with scaling factors given by s.  A and af will not
                           be modified.
                   = 1:  The matrix A will be copied to af and factored.
                   = 2:  The matrix A will be equilibrated if necessary, then
                           copied to af and factored.

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A, except if fact = 0 and
                   equed = 1, then A must contain the equilibrated matrix
                   diag(s)*A*diag(s).  If uplo = 0, the leading
                   N-by-N upper triangular part of A contains the upper
                   triangular part of the matrix A, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of A contains the lower
                   triangular part of the matrix A, and the strictly upper
                   triangular part of A is not referenced.  A is not modified if
                   fact = 0 or 1, or if fact = 2 and equed = 0 on exit.

                   On exit, if fact = 2 and equed = 1, A is overwritten by
                   diag(s)*A*diag(s).

           af:     If fact = 0, then af is an input argument and on entry
                   contains the triangular factor U or L from the Cholesky
                   factorization A = U'*U or A = L*L', in the same storage
                   format as A.  If equed != 0, then af is the factored form
                   of the equilibrated matrix diag(s)*A*diag(s).

                   If fact = 1, then af is an output argument and on exit
                   returns the triangular factor U or L from the Cholesky
                   factorization A = U'*U or A = L*L' of the original
                   matrix A.

                   If fact = 2, then af is an output argument and on exit
                   returns the triangular factor U or L from the Cholesky
                   factorization A = U'*U or A = L*L' of the equilibrated
                   matrix A (see the description of A for the form of the
                   equilibrated matrix).

           equed:  Specifies the form of equilibration that was done.
                   = 0:  No equilibration (always true if fact = 1).
                   = 1:  Equilibration was done, i.e., A has been replaced by
                           diag(s) * A * diag(s).
                   equed is an input argument if fact = 0; otherwise, it is an
                   output argument.

           s:      The scale factors for A; not accessed if equed = 0.  s is
                   an input argument if fact = 0; otherwise, s is an output
                   argument.  If fact = 0 and equed = 1, each element of s
                   must be positive.

           B:      On entry, the N-by-NRHS right hand side matrix B.
                   On exit, if equed = 0, B is not modified; if equed = 1,
                   B is overwritten by diag(s) * B.

           X:      If info = 0 or info = N+1, the N-by-NRHS solution matrix X to
                   the original system of equations.  Note that if equed = 1,
                   A and B are modified on exit, and the solution to the
                   equilibrated system is inv(diag(s))*X.

           rcond:  The estimate of the reciprocal condition number of the matrix
                   A after equilibration (if done).  If rcond is less than the
                   machine precision (in particular, if rcond = 0), the matrix
                   is singular to working precision.  This condition is
                   indicated by a return code of info > 0.

           ferr:   The estimated forward error bound for each solution vector
                   X(j) (the j-th column of the solution matrix X).
                   If XTRUE is the true solution corresponding to X(j), FERR(j)
                   is an estimated upper bound for the magnitude of the largest
                   element in (X(j) - XTRUE) divided by the magnitude of the
                   largest element in X(j).  The estimate is as reliable as
                   the estimate for rcond, and is almost always a slight
                   overestimate of the true error.

           berr:   The componentwise relative backward error of each solution
                   vector X(j) (i.e., the smallest relative change in
                   any element of A or B that makes X(j) an exact solution).

           info:   = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value
                   > 0: if info = i, and i is
                         <= N:  the leading minor of order i of A is
                                not positive definite, so the factorization
                                could not be completed, and the solution has not
                                been computed. rcond = 0 is returned.
                         = N+1: U is nonsingular, but rcond is less than machine
                                precision, meaning that the matrix is singular
                                to working precision.  Nevertheless, the
                                solution and error bounds are computed because
                                there are a number of situations where the
                                computed solution can be more accurate than the
                                value of rcond would suggest.

        $a= random(5,5);
        $b = random(5,5);
        $a = transpose($a);
        $b = transpose($b);
        # Assume $a is symmetric positive definite
        $rcond = pdl(0);
        $equed = pdl(long,0);
        $info = pdl(long,0);
        $berr = zeroes(5);
        $ferr = zeroes(5);
        $s = zeroes(5);
        $X = zeroes(5,5);
        $af = zeroes(5,5);
        posvx($a,0,2,$b,$af, $equed, $s, $X, $rcond, $ferr, $berr,$info);
        print "The solution matrix X is :". transpose($X)."\n" unless $info;

       posvx ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gels
         Signature: ([io,phys]A(m,n); int trans(); [io,phys]B(p,q);int [o,phys]info())

       Solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A,
       or its transpose, using a QR or LQ factorization of A.  It is assumed that A has full
       rank.

       The following options are provided:

       1. If trans = 0 and m >= n:  find the least squares solution of an overdetermined system,
          i.e., solve the least squares problem minimize || B - A*X ||.

       2. If trans = 0 and m < n:  find the minimum norm solution of an underdetermined system A
          * X = B.

       3. If trans = 1 and m >= n:  find the minimum norm solution of an undetermined system A' *
          X = B.

       4. If trans = 1 and m < n:  find the least squares solution of an overdetermined system,
          i.e., solve the least squares problem minimize || B - A' * X ||.

       Several right hand side vectors b and solution vectors x can be handled in a single call;
       they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS
       solution matrix X.

           Arguments
           =========

           trans:  = 0: the linear system involves A;
                   = 1: the linear system involves A'.

           A:      On entry, the M-by-N matrix A.
                   On exit,
                     if M >= N, A is overwritten by details of its QR
                                factorization as returned by geqrf;
                     if M <  N, A is overwritten by details of its LQ
                                factorization as returned by gelqf.

           B:      On entry, the matrix B of right hand side vectors, stored
                   columnwise; B is M-by-NRHS if trans = 0, or N-by-NRHS
                   if trans = 1.
                   On exit, B is overwritten by the solution vectors, stored
                   columnwise:
                   if trans = 0 and m >= n, rows 1 to n of B contain the least
                   squares solution vectors; the residual sum of squares for the
                   solution in each column is given by the sum of squares of
                   elements N+1 to M in that column;
                   if trans = 0 and m < n, rows 1 to N of B contain the
                   minimum norm solution vectors;
                   if trans = 1 and m >= n, rows 1 to M of B contain the
                   minimum norm solution vectors;
                   if trans = 1 and m < n, rows 1 to M of B contain the
                   least squares solution vectors; the residual sum of squares
                   for the solution in each column is given by the sum of
                   squares of elements M+1 to N in that column.
                   The leading dimension of the array B >= max(1,M,N).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a= random(7,5);
        # $b will contain X
        # TODO better example with slice
        $b = random(7,6);
        gels($a, 1, $b, ($info = null));

       gels ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gelsy
         Signature: ([io,phys]A(m,n); [io,phys]B(p,q); [phys]rcond(); int [io,phys]jpvt(n); int [o,phys]rank();int [o,phys]info())

       Computes the minimum-norm solution to a real linear least squares problem:

               minimize || A * X - B ||

       using a complete orthogonal factorization of A.

       A is an M-by-N matrix which may be rank-deficient.

       Several right hand side vectors b and solution vectors x can be handled in a single call;
       they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS
       solution matrix X.

       The routine first computes a QR factorization with column pivoting:

               A * P = Q * [ R11 R12 ]
                           [  0  R22 ]

               with R11 defined as the largest leading submatrix whose estimated
               condition number is less than 1/rcond.  The order of R11, rank,
               is the effective rank of A.

       Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal
       transformations from the right, arriving at the complete orthogonal factorization:

               A * P = Q * [ T11 0 ] * Z
                           [  0  0 ]

       The minimum-norm solution is then

               X = P * Z' [ inv(T11)*Q1'*B ]
                          [         0      ]
               where Q1 consists of the first rank columns of Q.

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit, A has been overwritten by details of its
                   complete orthogonal factorization.

           B:      On entry, the M-by-NRHS right hand side matrix B.
                   On exit, the N-by-NRHS solution matrix X.
                   The leading dimension of the array B >= max(1,M,N).

           jpvt:   On entry, if jpvt(i) != 0, the i-th column of A is permuted
                   to the front of AP, otherwise column i is a free column.
                   On exit, if jpvt(i) = k, then the i-th column of AP
                   was the k-th column of A.

           rcond:  rcond is used to determine the effective rank of A, which
                   is defined as the order of the largest leading triangular
                   submatrix R11 in the QR factorization with pivoting of A,
                   whose estimated condition number < 1/rcond.

           rank:   The effective rank of A, i.e., the order of the submatrix
                   R11.  This is the same as the order of the submatrix T11
                   in the complete orthogonal factorization of A.

           info:   = 0: successful exit
                   < 0: If info = -i, the i-th argument had an illegal value.

        $a= random(7,5);
        # $b will contain X
        # TODO better example with slice
        $b = random(7,6);
        $jpvt = zeroes(long, 5);
        $eps = lamch(0);
        #Threshold for rank estimation
        $rcond = sqrt($eps) - (sqrt($eps) - $eps) / 2;
        gelsy($a, $b, $rcond, $jpvt,($rank=null),($info = null));

       gelsy ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gelss
         Signature: ([io,phys]A(m,n); [io,phys]B(p,q); [phys]rcond(); [o,phys]s(r); int [o,phys]rank();int [o,phys]info())

       Computes the minimum norm solution to a real linear least squares problem:

               Minimize 2-norm(| b - A*x |).

       using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be
       rank-deficient.

       Several right hand side vectors b and solution vectors x can be handled in a single call;
       they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS
       solution matrix X.

       The effective rank of A is determined by treating as zero those singular values which are
       less than rcond times the largest singular value.

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit, the first min(m,n) rows of A are overwritten with
                   its right singular vectors, stored rowwise.

           B:      On entry, the M-by-NRHS right hand side matrix B.
                   On exit, B is overwritten by the N-by-NRHS solution
                   matrix X.  If m >= n and rank = n, the residual
                   sum-of-squares for the solution in the i-th column is given
                   by the sum of squares of elements n+1:m in that column.
                   The leading dimension of the array B >= max(1,M,N).

           s:      The singular values of A in decreasing order.
                   The condition number of A in the 2-norm = s(1)/s(min(m,n)).

           rcond:  rcond is used to determine the effective rank of A.
                   Singular values s(i) <= rcond*s(1) are treated as zero.
                   If rcond < 0, machine precision is used instead.

           rank:   The effective rank of A, i.e., the number of singular values
                   which are greater than rcond*s(1).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  the algorithm for computing the SVD failed to converge;
                         if info = i, i off-diagonal elements of an intermediate
                         bidiagonal form did not converge to zero.

        $a= random(7,5);
        # $b will contain X
        # TODO better example with slice
        $b = random(7,6);
        $eps = lamch(0);
        $s =zeroes(5);
        #Threshold for rank estimation
        $rcond = sqrt($eps) - (sqrt($eps) - $eps) / 2;
        gelss($a, $b, $rcond, $s, ($rank=null),($info = null));

       gelss ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gelsd
         Signature: ([io,phys]A(m,n); [io,phys]B(p,q); [phys]rcond(); [o,phys]s(r); int [o,phys]rank();int [o,phys]info())

       Computes the minimum-norm solution to a real linear least squares problem:

               minimize 2-norm(| b - A*x |)

       using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be
       rank-deficient.

       Several right hand side vectors b and solution vectors x can be handled in a single call;
       they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS
       solution matrix X.

       The problem is solved in three steps:

       1. Reduce the coefficient matrix A to bidiagonal form with Householder transformations,
          reducing the original problem into a "bidiagonal least squares problem" (BLS)

       2. Solve the BLS using a divide and conquer approach.

       3. Apply back all the Householder transformationss to solve the original least squares
          problem.

       The effective rank of A is determined by treating as zero those singular values which are
       less than rcond times the largest singular value.

       The divide and conquer algorithm makes very mild assumptions about floating point
       arithmetic. It will work on machines with a guard digit in add/subtract, or on those
       binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray
       C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without
       guard digits, but we know of none.

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit, A has been destroyed.

           B:      On entry, the M-by-NRHS right hand side matrix B.
                   On exit, B is overwritten by the N-by-NRHS solution
                   matrix X.  If m >= n and rank = n, the residual
                   sum-of-squares for the solution in the i-th column is given
                   by the sum of squares of elements n+1:m in that column.
                   The leading dimension of the array B >= max(1,M,N).

           s:      The singular values of A in decreasing order.
                   The condition number of A in the 2-norm = s(1)/s(min(m,n)).

           rcond:  rcond is used to determine the effective rank of A.
                   Singular values s(i) <= rcond*s(1) are treated as zero.
                   If rcond < 0, machine precision is used instead.

           rank:   The effective rank of A, i.e., the number of singular values
                   which are greater than rcond*s(1).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  the algorithm for computing the SVD failed to converge;
                         if info = i, i off-diagonal elements of an intermediate
                         bidiagonal form did not converge to zero.

        $a= random(7,5);
        # $b will contain X
        # TODO better example with slice
        $b = random(7,6);
        $eps = lamch(0);
        $s =zeroes(5);
        #Threshold for rank estimation
        $rcond = sqrt($eps) - (sqrt($eps) - $eps) / 2;
        gelsd($a, $b, $rcond, $s, ($rank=null),($info = null));

       gelsd ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gglse
         Signature: ([phys]A(m,n); [phys]B(p,n);[io,phys]c(m);[phys]d(p);[o,phys]x(n);int [o,phys]info())

       Solves the linear equality-constrained least squares (LSE) problem:

               minimize || c - A*x ||_2   subject to   B*x = d

               where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
               M-vector, and d is a given P-vector. It is assumed that
               P <= N <= M+P, and

                    rank(B) = P and  rank( ( A ) ) = N.
                                         ( ( B ) )

       These conditions ensure that the LSE problem has a unique solution, which is obtained
       using a GRQ factorization of the matrices B and A.

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit, A is destroyed.

           B:      On entry, the P-by-N matrix B.
                   On exit, B is destroyed.

           c:      On entry, c contains the right hand side vector for the
                   least squares part of the LSE problem.
                   On exit, the residual sum of squares for the solution
                   is given by the sum of squares of elements N-P+1 to M of
                   vector c.

           d:      On entry, d contains the right hand side vector for the
                   constrained equation.
                   On exit, d is destroyed.

           x:      On exit, x is the solution of the LSE problem.

           info:   = 0:  successful exit.
                   < 0:  if info = -i, the i-th argument had an illegal value.

        $a = random(7,5);
        $b = random(4,5);
        $c = random(7);
        $d = random(4);
        $x = zeroes(5);
        gglse($a, $b, $c, $d, $x, ($info=null));

       gglse ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ggglm
         Signature: ([phys]A(n,m); [phys]B(n,p);[phys]d(n);[o,phys]x(m);[o,phys]y(p);int [o,phys]info())

       Solves a general Gauss-Markov linear model (GLM) problem:

               minimize || y ||_2   subject to   d = A*x + B*y
                  x

               where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
               given N-vector. It is assumed that M <= N <= M+P, and

                      rank(A) = M    and    rank( A B ) = N.

       Under these assumptions, the constrained equation is always consistent, and there is a
       unique solution x and a minimal 2-norm solution y, which is obtained using a generalized
       QR factorization of A and B.

       In particular, if matrix B is square nonsingular, then the problem GLM is equivalent to
       the following weighted linear least squares problem

               minimize || inv(B)*(d-A*x) ||_2
                  x

               where inv(B) denotes the inverse of B.

           Arguments
           =========

           A:      On entry, the N-by-M matrix A.
                   On exit, A is destroyed.

           B:      On entry, the N-by-P matrix B.
                   On exit, B is destroyed.

           d:      On entry, d is the left hand side of the GLM equation.
                   On exit, d is destroyed.

           x:
           y:      On exit, x and y are the solutions of the GLM problem.

           info:   = 0:  successful exit.
                   < 0:  if info = -i, the i-th argument had an illegal value.

        $a = random(7,5);
        $b = random(7,4);
        $d = random(7);
        $x = zeroes(5);
        $y = zeroes(4);
        ggglm($a, $b, $d, $x, $y,($info=null));

       ggglm ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   getrf
         Signature: ([io,phys]A(m,n); int [o,phys]ipiv(p); int [o,phys]info())

       Computes an LU factorization of a general M-by-N matrix A using partial pivoting with row
       interchanges.

       The factorization has the form

               A = P * L * U

               where P is a permutation matrix, L is lower triangular with unit
               diagonal elements (lower trapezoidal if m > n), and U is upper
               triangular (upper trapezoidal if m < n).

       This is the right-looking Level 3 BLAS version of the algorithm.

           Arguments
           =========

           A:      On entry, the M-by-N matrix to be factored.
                   On exit, the factors L and U from the factorization
                   A = P*L*U; the unit diagonal elements of L are not stored.

           ipiv:  The pivot indices; for 1 <= i <= min(M,N), row i of the
                   matrix was interchanged with row ipiv(i).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, U(i,i) is exactly zero. The factorization
                         has been completed, but the factor U is exactly
                         singular, and division by zero will occur if it is used
                         to solve a system of equations.

        $a = random (float, 100,50);
        $ipiv = zeroes(long, 50);
        $info = null;
        getrf($a, $ipiv, $info);

       getrf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   getf2
         Signature: ([io,phys]A(m,n); int [o,phys]ipiv(p); int [o,phys]info())

       Computes an LU factorization of a general M-by-N matrix A using partial pivoting with row
       interchanges.

       The factorization has the form

               A = P * L * U

               where P is a permutation matrix, L is lower triangular with unit
               diagonal elements (lower trapezoidal if m > n), and U is upper
               triangular (upper trapezoidal if m < n).

       This is the right-looking Level 2 BLAS version of the algorithm.

           Arguments
           =========

           A:      On entry, the M-by-N matrix to be factored.
                   On exit, the factors L and U from the factorization
                   A = P*L*U; the unit diagonal elements of L are not stored.

           ipiv:  The pivot indices; for 1 <= i <= min(M,N), row i of the
                   matrix was interchanged with row ipiv(i).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, U(i,i) is exactly zero. The factorization
                         has been completed, but the factor U is exactly
                         singular, and division by zero will occur if it is used
                         to solve a system of equations.

        $a = random (float, 100,50);
        $ipiv = zeroes(long, 50);
        $info = null;
        getf2($a, $ipiv, $info);

       getf2 ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sytrf
         Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]ipiv(n); int [o,phys]info())

       Computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal
       pivoting method.  The form of the factorization is

               A = U*D*U'  or  A = L*D*L'
               where U (or L) is a product of permutation and unit upper (lower)
               triangular matrices, and D is symmetric and block diagonal with
               1-by-1 and 2-by-2 diagonal blocks.

       This is the blocked version of the algorithm, calling Level 3 BLAS.

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the leading
                   N-by-N upper triangular part of A contains the upper
                   triangular part of the matrix A, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of A contains the lower
                   triangular part of the matrix A, and the strictly upper
                   triangular part of A is not referenced.

                   On exit, the block diagonal matrix D and the multipliers used
                   to obtain the factor U or L (see below for further details).

           ipiv:   Details of the interchanges and the block structure of D.
                   If ipiv(k) > 0, then rows and columns k and ipiv(k) were
                   interchanged and D(k,k) is a 1-by-1 diagonal block.
                   If uplo = 0 and ipiv(k) = ipiv(k-1) < 0, then rows and
                   columns k-1 and -ipiv(k) were interchanged and D(k-1:k,k-1:k)
                   is a 2-by-2 diagonal block.  If uplo = 1 and ipiv(k) =
                   ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were
                   interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, D(i,i) is exactly zero.  The factorization
                         has been completed, but the block diagonal matrix D is
                         exactly singular, and division by zero will occur if it
                         is used to solve a system of equations.

           Further Details
           ===============

           If uplo = 0, then A = U*D*U', where
              U = P(n)*U(n)* ... *P(k)U(k)* ...,
           i.e., U is a product of terms P(k)*U(k), where k decreases from n to
           1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
           and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
           defined by ipiv(k), and U(k) is a unit upper triangular matrix, such
           that if the diagonal block D(k) is of order s (s = 1 or 2), then

                      (   I    v    0   )   k-s
              U(k) =  (   0    I    0   )   s
                      (   0    0    I   )   n-k
                         k-s   s   n-k

           If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
           If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
           and A(k,k), and v overwrites A(1:k-2,k-1:k).

           If uplo = 1, then A = L*D*L', where
              L = P(1)*L(1)* ... *P(k)*L(k)* ...,
           i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
           n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
           and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
           defined by ipiv(k), and L(k) is a unit lower triangular matrix, such
           that if the diagonal block D(k) is of order s (s = 1 or 2), then

                      (   I    0     0   )  k-1
              L(k) =  (   0    I     0   )  s
                      (   0    v     I   )  n-k-s+1
                         k-1   s  n-k-s+1

           If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
           If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
           and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).

        $a = random(100,100);
        $ipiv = zeroes(100);
        $info = null;
        # Assume $a is symmetric
        sytrf($a, 0, $ipiv, $info);

       sytrf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sytf2
         Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]ipiv(n); int [o,phys]info())

       Computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal
       pivoting method.  The form of the factorization is

               A = U*D*U'  or  A = L*D*L'
               where U (or L) is a product of permutation and unit upper (lower)
               triangular matrices, and D is symmetric and block diagonal with
               1-by-1 and 2-by-2 diagonal blocks.

       This is the unblocked version of the algorithm, calling Level 2 BLAS.

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the leading
                   N-by-N upper triangular part of A contains the upper
                   triangular part of the matrix A, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of A contains the lower
                   triangular part of the matrix A, and the strictly upper
                   triangular part of A is not referenced.

                   On exit, the block diagonal matrix D and the multipliers used
                   to obtain the factor U or L (see below for further details).

           ipiv:   Details of the interchanges and the block structure of D.
                   If ipiv(k) > 0, then rows and columns k and ipiv(k) were
                   interchanged and D(k,k) is a 1-by-1 diagonal block.
                   If uplo = 0 and ipiv(k) = ipiv(k-1) < 0, then rows and
                   columns k-1 and -ipiv(k) were interchanged and D(k-1:k,k-1:k)
                   is a 2-by-2 diagonal block.  If uplo = 1 and ipiv(k) =
                   ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were
                   interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, D(i,i) is exactly zero.  The factorization
                         has been completed, but the block diagonal matrix D is
                         exactly singular, and division by zero will occur if it
                         is used to solve a system of equations.

           For further details see sytrf

        $a = random(100,100);
        $ipiv = zeroes(100);
        $info = null;
        # Assume $a is symmetric
        sytf2($a, 0, $ipiv, $info);

       sytf2 ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   potrf
         Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]info())

       Computes the Cholesky factorization of a real symmetric positive definite matrix A.

       The factorization has the form

               A = U' * U,  if uplo = 0, or
               A = L  * L',  if uplo = 1,
               where U is an upper triangular matrix and L is lower triangular.

       This is the block version of the algorithm, calling Level 3 BLAS.

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the leading
                   N-by-N upper triangular part of A contains the upper
                   triangular part of the matrix A, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of A contains the lower
                   triangular part of the matrix A, and the strictly upper
                   triangular part of A is not referenced.

                   On exit, if info = 0, the factor U or L from the Cholesky
                   factorization A = U'*U or A = L*L'.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, the leading minor of order i is not
                         positive definite, and the factorization could not be
                         completed.

        $a = random(100,100);
        # Assume $a is symmetric positive definite
        potrf($a, 0, ($info = null));

       potrf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   potf2
         Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]info())

       Computes the Cholesky factorization of a real symmetric positive definite matrix A.

       The factorization has the form

               A = U' * U,  if uplo = 0, or
               A = L  * L',  if uplo = 1,
               where U is an upper triangular matrix and L is lower triangular.

       This is the unblocked version of the algorithm, calling Level 2 BLAS.

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the symmetric matrix A.  If uplo = 0, the leading
                   N-by-N upper triangular part of A contains the upper
                   triangular part of the matrix A, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of A contains the lower
                   triangular part of the matrix A, and the strictly upper
                   triangular part of A is not referenced.

                   On exit, if info = 0, the factor U or L from the Cholesky
                   factorization A = U'*U or A = L*L'.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, the leading minor of order i is not
                         positive definite, and the factorization could not be
                         completed.

        $a = random(100,100);
        # Assume $a is symmetric positive definite
        potf2($a, 0, ($info = null));

       potf2 ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   getri
         Signature: ([io,phys]A(n,n); int [phys]ipiv(n); int [o,phys]info())

       Computes the inverse of a matrix using the LU factorization computed by "getrf".

       This method inverts U and then computes inv(A) by solving the system

           inv(A)*L = inv(U) for inv(A).

           Arguments
           =========

           A:      On entry, the factors L and U from the factorization
                   A = P*L*U as computed by getrf.
                   On exit, if info = 0, the inverse of the original matrix A.

           ipiv:   The pivot indices from getrf; for 1<=i<=N, row i of the
                   matrix was interchanged with row ipiv(i).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, U(i,i) is exactly zero; the matrix is
                         singular and its inverse could not be computed.

        $a = random (float, 100, 100);
        $ipiv = zeroes(long, 100);
        $info = null;
        getrf($a, $ipiv, $info);
        if ($info == 0){
               getri($a, $ipiv, $info);
        }
        print "Inverse of \$a is :\n $a" unless $info;

       getri ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sytri
         Signature: ([io,phys]A(n,n); int uplo(); int [phys]ipiv(n); int [o,phys]info())

       Computes the inverse of a real symmetric indefinite matrix A using the factorization A =
       U*D*U' or A = L*D*L' computed by "sytrf".

           Arguments
           =========

           uplo:   Specifies whether the details of the factorization are stored
                   as an upper or lower triangular matrix.
                   = 0:  Upper triangular, form is A = U*D*U';
                   = 1:  Lower triangular, form is A = L*D*L'.

           A:      On entry, the block diagonal matrix D and the multipliers
                   used to obtain the factor U or L as computed by sytrf.

                   On exit, if info = 0, the (symmetric) inverse of the original
                   matrix.  If uplo = 0, the upper triangular part of the
                   inverse is formed and the part of A below the diagonal is not
                   referenced; if uplo = 1 the lower triangular part of the
                   inverse is formed and the part of A above the diagonal is
                   not referenced.

           ipiv:   Details of the interchanges and the block structure of D
                   as determined by sytrf.

           info:   = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value
                   > 0: if info = i, D(i,i) = 0; the matrix is singular and its
                        inverse could not be computed.

        $a = random (float, 100, 100);
        # assume $a is symmetric
        $ipiv = zeroes(long, 100);
        sytrf($a, 0, $ipiv, ($info=null));
        if ($info == 0){
               sytri($a, 0, $ipiv, $info);
        }
        print "Inverse of \$a is :\n $a" unless $info;

       sytri ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   potri
         Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]info())

       Computes the inverse of a real symmetric positive definite matrix A using the Cholesky
       factorization A = U'*U or A = L*L' computed by "potrf".

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      On entry, the triangular factor U or L from the Cholesky
                   factorization A = U'*U or A = L*L', as computed by
                   potrf.
                   On exit, the upper or lower triangle of the (symmetric)
                   inverse of A, overwriting the input factor U or L.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, the (i,i) element of the factor U or L is
                         zero, and the inverse could not be computed.

        $a = random (float, 100, 100);
        # Assume $a is symmetric positive definite
        potrf($a, 0, ($info = null));
        if ($info == 0){ # Hum... is it positive definite????
               potri($a, 0,$info);
        }
        print "Inverse of \$a is :\n $a" unless $info;

       potri ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   trtri
         Signature: ([io,phys]A(n,n); int uplo(); int diag(); int [o,phys]info())

       Computes the inverse of a real upper or lower triangular matrix A.

       This is the Level 3 BLAS version of the algorithm.

           Arguments
           =========

           uplo:   = 0:  A is upper triangular;
                   = 1:  A is lower triangular.

           diag:   = 0:  A is non-unit triangular;
                   = 1:  A is unit triangular.

           A:      On entry, the triangular matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of the array A contains
                   the upper triangular matrix, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of the array A contains
                   the lower triangular matrix, and the strictly upper
                   triangular part of A is not referenced.  If diag = 1, the
                   diagonal elements of A are also not referenced and are
                   assumed to be 1.
                   On exit, the (triangular) inverse of the original matrix, in
                   the same storage format.

           info:   = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value
                   > 0: if info = i, A(i,i) is exactly zero.  The triangular
                        matrix is singular and its inverse can not be computed.

        $a = random (float, 100, 100);
        # assume $a is upper triangular
        trtri($a, 1, ($info=null));
        print "Inverse of \$a is :\n transpose($a)" unless $info;

       trtri ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   trti2
         Signature: ([io,phys]A(n,n); int uplo(); int diag(); int [o,phys]info())

       Computes the inverse of a real upper or lower triangular matrix A.

       This is the Level 2 BLAS version of the algorithm.

           Arguments
           =========

           uplo:   = 0:  A is upper triangular;
                   = 1:  A is lower triangular.

           diag:   = 0:  A is non-unit triangular;
                   = 1:  A is unit triangular.

           A:      On entry, the triangular matrix A.  If uplo = 0, the
                   leading N-by-N upper triangular part of the array A contains
                   the upper triangular matrix, and the strictly lower
                   triangular part of A is not referenced.  If uplo = 1, the
                   leading N-by-N lower triangular part of the array A contains
                   the lower triangular matrix, and the strictly upper
                   triangular part of A is not referenced.  If diag = 1, the
                   diagonal elements of A are also not referenced and are
                   assumed to be 1.
                   On exit, the (triangular) inverse of the original matrix, in
                   the same storage format.

           info:   = 0: successful exit
                   < 0: if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 100);
        # assume $a is upper triangular
        trtri2($a, 1, ($info=null));
        print "Inverse of \$a is :\n transpose($a)" unless $info;

       trti2 ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   getrs
         Signature: ([phys]A(n,n); int trans(); [io,phys]B(n,m); int [phys]ipiv(n); int [o,phys]info())

       Solves a system of linear equations

               A * X = B  or  A' * X = B

       with a general N-by-N matrix A using the LU factorization computed by getrf.

           Arguments
           =========

           trans:  Specifies the form of the system of equations:
                   = 0:  A * X = B  (No transpose)
                   = 1:  A'* X = B  (Transpose)

           A:      The factors L and U from the factorization A = P*L*U
                   as computed by getrf.

           ipiv:   The pivot indices from getrf; for 1<=i<=N, row i of the
                   matrix was interchanged with row ipiv(i).

           B:      On entry, the right hand side matrix B.
                   On exit, the solution matrix X.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 100);
        $ipiv = zeroes(long, 100);
        $b = random(100,50);
        getrf($a, $ipiv, ($info=null));
        if ($info == 0){
               getrs($a, 0, $b, $ipiv, $info);
        }
        print "X is :\n $b" unless $info;

       getrs ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sytrs
         Signature: ([phys]A(n,n); int uplo();[io,phys]B(n,m); int [phys]ipiv(n); int [o,phys]info())

       Solves a system of linear equations A*X = B with a real symmetric matrix A using the
       factorization A = U*D*U' or A = L*D*L' computed by "sytrf".

           Arguments
           =========

           uplo:   Specifies whether the details of the factorization are stored
                   as an upper or lower triangular matrix.
                   = 0:  Upper triangular, form is A = U*D*U';
                   = 1:  Lower triangular, form is A = L*D*L'.

           A:      The block diagonal matrix D and the multipliers used to
                   obtain the factor U or L as computed by sytrf.

           ipiv:   Details of the interchanges and the block structure of D
                   as determined by sytrf.

           B:      On entry, the right hand side matrix B.
                   On exit, the solution matrix X.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 100);
        $b = random(50,100);
        $a = transpose($a);
        $b = transpose($b);
        # Assume $a is symmetric
        sytrf($a, 0, ($ipiv=zeroes(100)), ($info=null));
        if ($info == 0){
               sytrs($a, 0, $b, $ipiv, $info);
        }
        print("X is :\n".transpose($b))unless $info;

       sytrs ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   potrs
         Signature: ([phys]A(n,n); int uplo(); [io,phys]B(n,m); int [o,phys]info())

       Solves a system of linear equations A*X = B with a symmetric positive definite matrix A
       using the Cholesky factorization A = U'*U or A = L*L' computed by "potrf".

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      The triangular factor U or L from the Cholesky factorization
                   A = U'*U or A = L*L', as computed by potrf.

           B:      On entry, the right hand side matrix B.
                   On exit, the solution matrix X.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 100);
        $b = random(50,100);
        $a = transpose($a);
        $b = transpose($b);
        # Assume $a is symmetric positive definite
        potrf($a, 0, ($info=null));
        if ($info == 0){
               potrs($a, 0, $b, $info);
        }
        print("X is :\n".transpose($b))unless $info;

       potrs ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   trtrs
         Signature: ([phys]A(n,n); int uplo(); int trans(); int diag();[io,phys]B(n,m); int [o,phys]info())

       Solves a triangular system of the form

               A * X = B  or  A' * X = B,

               where A is a triangular matrix of order N, and B is an N-by-NRHS
               matrix.

       A check is made to verify that A is nonsingular.

           Arguments
           =========

           uplo:   = 0:  A is upper triangular;
                   = 1:  A is lower triangular.

           trans:  Specifies the form of the system of equations:
                   = 0:  A * X = B  (No transpose)
                   = 1:  A**T * X = B  (Transpose)

           diag:   = 0:  A is non-unit triangular;
                   = 1:  A is unit triangular.

           A:      The triangular matrix A.  If uplo = 0, the leading N-by-N
                   upper triangular part of the array A contains the upper
                   triangular matrix, and the strictly lower triangular part of
                   A is not referenced.  If uplo = 1, the leading N-by-N lower
                   triangular part of the array A contains the lower triangular
                   matrix, and the strictly upper triangular part of A is not
                   referenced.  If diag = 1, the diagonal elements of A are
                   also not referenced and are assumed to be 1.

           B:      On entry, the right hand side matrix B.
                   On exit, if info = 0, the solution matrix X.

           info    = 0:  successful exit
                   < 0: if info = -i, the i-th argument had an illegal value
                   > 0: if info = i, the i-th diagonal element of A is zero,
                        indicating that the matrix is singular and the solutions
                        X have not been computed.

        # Assume $a is upper triangular
        $a = random (float, 100, 100);
        $b = random(50,100);
        $a = transpose($a);
        $b = transpose($b);
        $info = null;
        trtrs($a, 0, 0, 0, $b, $info);
        print("X is :\n".transpose($b))unless $info;

       trtrs ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   latrs
         Signature: ([phys]A(n,n); int uplo(); int trans(); int diag(); int normin();[io,phys]x(n); [o,phys]scale();[io,phys]cnorm(n);int [o,phys]info())

       Solves one of the triangular systems

               A *x = s*b  or  A'*x = s*b

       with scaling to prevent overflow.  Here A is an upper or lower triangular matrix, A'
       denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor,
       usually less than or equal to 1, chosen so that the components of x will be less than the
       overflow threshold.  If the unscaled problem will not cause overflow, the Level 2 BLAS
       routine "trsv" is called.  If the matrix A is singular (A(j,j) = 0 for some j), then s is
       set to 0 and a non-trivial solution to A*x = 0 is returned.

       Further Details ======= =======

       A rough bound on x is computed; if that is less than overflow, trsv is called, otherwise,
       specific code is used which checks for possible overflow or divide-by-zero at every
       operation.

       A columnwise scheme is used for solving A*x = b.  The basic algorithm if A is lower
       triangular is

                x[1:n] := b[1:n]
                for j = 1, ..., n
                     x(j) := x(j) / A(j,j)
                     x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
                end

       Define bounds on the components of x after j iterations of the loop:

              M(j) = bound on x[1:j]
              G(j) = bound on x[j+1:n]

       Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

           Then for iteration j+1 we have
              M(j+1) <= G(j) / | A(j+1,j+1) |
              G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
                     <= G(j) ( 1 + cnorm(j+1) / | A(j+1,j+1) | )

           where cnorm(j+1) is greater than or equal to the infinity-norm of
           column j+1 of A, not counting the diagonal.

       Hence

              G(j) <= G(0) product ( 1 + cnorm(i) / | A(i,i) | )
                           1<=i<=j
           and

              |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + cnorm(i) / |A(i,i)| )
                                            1<=i< j

       Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the reciprocal of the
       largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow).

       The bound on x(j) is also used to determine when a step in the columnwise method can be
       performed without fear of overflow.  If the computed bound is greater than a large
       constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j)
       to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

       Similarly, a row-wise scheme is used to solve A'*x = b.  The basic algorithm for A upper
       triangular is

                for j = 1, ..., n
                     x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
                end

       We simultaneously compute two bounds

                G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
                M(j) = bound on x(i), 1<=i<=j

       The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint
       G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.  Then the bound on x(j) is

                M(j) <= M(j-1) * ( 1 + cnorm(j) ) / | A(j,j) |

                     <= M(0) * product ( ( 1 + cnorm(i) ) / |A(i,i)| )
                               1<=i<=j

       and we can safely call trsv if 1/M(n) and 1/G(n) are both greater than max(underflow,
       1/overflow).

           Arguments
           =========

           uplo:   Specifies whether the matrix A is upper or lower triangular.
                   = 0:  Upper triangular
                   = 1:  Lower triangular

           trans:  Specifies the operation applied to A.
                   = 0:  Solve A * x = s*b  (No transpose)
                   = 1:  Solve A'* x = s*b  (Transpose)

           diag:   Specifies whether or not the matrix A is unit triangular.
                   = 0:  Non-unit triangular
                   = 1:  Unit triangular

           normin: Specifies whether cnorm has been set or not.
                   = 1:  cnorm contains the column norms on entry
                   = 0:  cnorm is not set on entry.  On exit, the norms will
                           be computed and stored in cnorm.

           A:      The triangular matrix A.  If uplo = 0, the leading n by n
                   upper triangular part of the array A contains the upper
                   triangular matrix, and the strictly lower triangular part of
                   A is not referenced.  If uplo = 1, the leading n by n lower
                   triangular part of the array A contains the lower triangular
                   matrix, and the strictly upper triangular part of A is not
                   referenced.  If diag = 1, the diagonal elements of A are
                   also not referenced and are assumed to be 1.

           x:      On entry, the right hand side b of the triangular system.
                   On exit, x is overwritten by the solution vector x.

           scale:  The scaling factor s for the triangular system
                      A * x = s*b  or  A'* x = s*b.
                   If scale = 0, the matrix A is singular or badly scaled, and
                   the vector x is an exact or approximate solution to A*x = 0.

           cnorm:  If normin = 0, cnorm is an output argument and cnorm(j)
                   returns the 1-norm of the offdiagonal part of the j-th column
                   of A.
                   If normin = 1, cnorm is an input argument and cnorm(j)
                   contains the norm of the off-diagonal part of the j-th column
                   of A.  If trans = 0, cnorm(j) must be greater than or equal
                   to the infinity-norm, and if trans = 1, cnorm(j)
                   must be greater than or equal to the 1-norm.

           info:   = 0:  successful exit
                   < 0:  if info = -k, the k-th argument had an illegal value

        # Assume $a is upper triangular
        $a = random (float, 100, 100);
        $b = random(100);
        $a = transpose($a);
        $info = null;
        $scale= null;
        $cnorm = zeroes(100);
        latrs($a, 0, 0, 0, 0,$b, $scale, $cnorm,$info);

       latrs ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gecon
         Signature: ([phys]A(n,n); int norm(); [phys]anorm(); [o,phys]rcond();int [o,phys]info())

       Estimates the reciprocal of the condition number of a general real matrix A, in either the
       1-norm or the infinity-norm, using the LU factorization computed by "getrf".

       An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is
       computed as

              rcond = 1 / ( norm(A) * norm(inv(A)) ).

           Arguments
           =========

           norm:   Specifies whether the 1-norm condition number or the
                   infinity-norm condition number is required:
                   = 0:  Infinity-norm.
                   = 1:  1-norm;

           A:      The factors L and U from the factorization A = P*L*U
                   as computed by getrf.

           anorm:  If norm = 0, the infinity-norm of the original matrix A.
                   If norm = 1, the 1-norm of the original matrix A.

           rcond:  The reciprocal of the condition number of the matrix A,
                   computed as rcond = 1/(norm(A) * norm(inv(A))).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 100);
        $anorm  = $a->lange(1);
        $ipiv = zeroes(long, 100);
        $info = null;
        getrf($a, $ipiv, $info);
        ($rcond, $info) = gecon($a, 1, $anorm) unless $info != 0;

       gecon ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   sycon
         Signature: ([phys]A(n,n); int uplo(); int ipiv(n); [phys]anorm(); [o,phys]rcond();int [o,phys]info())

       Estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric
       matrix A using the factorization A = U*D*U' or A = L*D*L' computed by "sytrf".

       An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is
       computed as rcond = 1 / (anorm * norm(inv(A))).

           Arguments
           =========

           uplo:   Specifies whether the details of the factorization are stored
                   as an upper or lower triangular matrix.
                   = 0:  Upper triangular, form is A = U*D*U';
                   = 1:  Lower triangular, form is A = L*D*L'.

           A:      The block diagonal matrix D and the multipliers used to
                   obtain the factor U or L as computed by sytrf.

           ipiv:   Details of the interchanges and the block structure of D
                   as determined by sytrf.

           anorm:  The 1-norm of the original matrix A.

           rcond:  The reciprocal of the condition number of the matrix A,
                   computed as rcond = 1/(anorm * aimvnm), where ainvnm is an
                   estimate of the 1-norm of inv(A) computed in this routine.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.

        # Assume $a is symmetric
        $a = random (float, 100, 100);
        $anorm  = $a->lansy(1,1);
        $ipiv = zeroes(long, 100);
        $info = null;
        sytrf($a, 1,$ipiv, $info);
        ($rcond, $info) = sycon($a, 1, $anorm) unless $info != 0;

       sycon ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   pocon
         Signature: ([phys]A(n,n); int uplo(); [phys]anorm(); [o,phys]rcond();int [o,phys]info())

       Estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric
       positive definite matrix using the Cholesky factorization A = U'*U or A = L*L' computed by
       "potrf".

       An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is
       computed as rcond = 1 / (anorm * norm(inv(A))).

           Arguments
           =========

           uplo:   = 0:  Upper triangle of A is stored;
                   = 1:  Lower triangle of A is stored.

           A:      The triangular factor U or L from the Cholesky factorization
                   A = U'*U or A = L*L', as computed by potrf.

           anorm:  The 1-norm of the matrix A.

           rcond:  The reciprocal of the condition number of the matrix A,
                   computed as rcond = 1/(anorm * ainvnm), where ainvnm is an
                   estimate of the 1-norm of inv(A) computed in this routine.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        # Assume $a is symmetric positive definite
        $a = random (float, 100, 100);
        $anorm  = $a->lansy(1,1);
        $info = null;
        potrf($a,  0, $info);
        ($rcond, $info) = pocon($a, 1, $anorm) unless $info != 0;

       pocon ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   trcon
         Signature: ([phys]A(n,n); int norm();int uplo();int diag(); [o,phys]rcond();int [o,phys]info())

       Estimates the reciprocal of the condition number of a triangular matrix A, in either the
       1-norm or the infinity-norm.

       The norm of A is computed and an estimate is obtained for norm(inv(A)), then the
       reciprocal of the condition number is computed as

               rcond = 1 / ( norm(A) * norm(inv(A)) ).

           Arguments
           =========

           norm:   Specifies whether the 1-norm condition number or the
                   infinity-norm condition number is required:
                   = 0:        Infinity-norm.
                   = 1:        1-norm;

           uplo:   = 0:  A is upper triangular;
                   = 1:  A is lower triangular.

           diag:   = 0:  A is non-unit triangular;
                   = 1:  A is unit triangular.

           A:      The triangular matrix A.  If uplo = 0, the leading N-by-N
                   upper triangular part of the array A contains the upper
                   triangular matrix, and the strictly lower triangular part of
                   A is not referenced.  If uplo = 1, the leading N-by-N lower
                   triangular part of the array A contains the lower triangular
                   matrix, and the strictly upper triangular part of A is not
                   referenced.  If diag = 1, the diagonal elements of A are
                   also not referenced and are assumed to be 1.

           rcond:  The reciprocal of the condition number of the matrix A,
                   computed as rcond = 1/(norm(A) * norm(inv(A))).

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        # Assume $a is upper triangular
        $a = random (float, 100, 100);
        $info = null;
        ($rcond, $info) = trcon($a, 1, 1, 0) unless $info != 0;

       trcon ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   geqp3
         Signature: ([io,phys]A(m,n); int [io,phys]jpvt(n); [o,phys]tau(k); int [o,phys]info())

       geqp3 computes a QR factorization  using Level 3 BLAS with column pivoting of a matrix A:

                       A*P = Q*R

       The matrix Q is represented as a product of elementary reflectors

               Q = H(1) H(2) . . . H(k), where k = min(m,n).

       Each H(i) has the form

               H(i) = I - tau * v * v'

               where tau is a real/complex scalar, and v is a real/complex vector
               with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
               A(i+1:m,i), and tau in tau(i).

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit, the upper triangle of the array contains the
                   min(M,N)-by-N upper trapezoidal matrix R; the elements below
                   the diagonal, together with the array tau, represent the
                   orthogonal matrix Q as a product of min(M,N) elementary
                   reflectors.

           jpvt:   On entry, if jpvt(J)!=0, the J-th column of A is permuted
                   to the front of A*P (a leading column); if jpvt(J)=0,
                   the J-th column of A is a free column.
                   On exit, if jpvt(J)=K, then the J-th column of A*P was the
                   the K-th column of A.

           tau:    The scalar factors of the elementary reflectors.

           info:   = 0: successful exit.
                   < 0: if info = -i, the i-th argument had an illegal value.

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        $jpvt = zeroes(long, 50);
        geqp3($a, $jpvt, $tau, $info);

       geqp3 ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   geqrf
         Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())

       geqrf computes a QR factorization of a matrix A:

               A = Q * R

       The matrix Q is represented as a product of elementary reflectors

               Q = H(1) H(2) . . . H(k), where k = min(m,n).

       Each H(i) has the form

               H(i) = I - tau * v * v'

               where tau is a real/complex scalar, and v is a real/complex vector
               with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
               A(i+1:m,i), and tau in tau(i).

           Arguments
           =========

           A:      On exit, the elements on and above the diagonal of the array
                   contain the min(M,N)-by-N upper trapezoidal matrix R (R is
                   upper triangular if m >= n); the elements below the diagonal,
                   with the array TAU, represent the orthogonal matrix Q as a
                   product of min(m,n) elementary reflectors.

           tau:    The scalar factors of the elementary reflectors.

           info:   = 0: successful exit.
                   < 0: if info = -i, the i-th argument had an illegal value.

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        geqrf($a, $tau, $info);

       geqrf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   orgqr
         Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())

       Generates an M-by-N real matrix Q with orthonormal columns, which is defined as the first
       N columns of a product of K elementary reflectors of order M

               Q  =  H(1) H(2) . . . H(k)

               as returned by geqrf or geqp3.

           Arguments
           =========

           A:      On entry, the i-th column must contain the vector which
                   defines the elementary reflector H(i), for i = 1,2,...,k, as
                   returned by geqrf or geqp3 in the first k columns of its array
                   argument A.
                   On exit, the M-by-N matrix Q.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by geqrf or geqp3.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument has an illegal value

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        geqrf($a, $tau, $info);
        orgqr($a, $tau, $info) unless $info != 0;

       orgqr ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ormqr
         Signature: ([phys]A(p,k); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())

       Overwrites the general real M-by-N matrix C with

                           side = 0     side = 1
           trans = 0:      Q * C          C * Q
           trans = 1:      Q' * C       C * Q'

           where Q is a real orthogonal matrix defined as the product of k
           elementary reflectors

                 Q = H(1) H(2) . . . H(k)

               as returned by geqrf or geqp3.

       Q is of order M if "side" = 0 and of order N if "side" = 1.

           Arguments
           =========

           side:   = 0: apply Q or Q' from the Left;
                   = 1: apply Q or Q' from the Right.

           trans:  = 0:  No transpose, apply Q;
                   = 1:  Transpose, apply Q'.

           A:      The i-th column must contain the vector which defines the
                   elementary reflector H(i), for i = 1,2,...,k, as returned by
                   geqrf or geqp3 in the first k columns of its array argument A.
                   A is modified by the routine but restored on exit.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by geqrf or geqp3.

           C:      On entry, the M-by-N matrix C.
                   On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 50, 100);
        $a = transpose($a);
        $info = null;
        $tau = zeroes(float, 50);
        geqrf($a, $tau, $info);
        $c = random(70,50);
        # $c will contain the result
        $c->reshape(70,100);
        $c = transpose($c);
        ormqr($a, $tau, $c, $info);

       ormqr ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gelqf
         Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())

       Computes an LQ factorization of a real M-by-N matrix A:

               A = L * Q.

       The matrix Q is represented as a product of elementary reflectors

              Q = H(k) . . . H(2) H(1), where k = min(m,n).

       Each H(i) has the form

               H(i) = I - tau * v * v'

               where tau is a real scalar, and v is a real vector with
               v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
               and tau in tau(i).

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit, the elements on and below the diagonal of the array
                   contain the m-by-min(m,n) lower trapezoidal matrix L (L is
                   lower triangular if m <= n); the elements above the diagonal,
                   with the array tau, represent the orthogonal matrix Q as a
                   product of elementary reflectors.

           tau:    The scalar factors of the elementary reflectors.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        gelqf($a, $tau, $info);

       gelqf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   orglq
         Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())

       Generates an M-by-N real matrix Q with orthonormal rows, which is defined as the first M
       rows of a product of K elementary reflectors of order N

               Q  =  H(k) . . . H(2) H(1)

               as returned by gelqf.

           Arguments
           =========

           A:      On entry, the i-th row must contain the vector which defines
                   the elementary reflector H(i), for i = 1,2,...,k, as returned
                   by gelqf in the first k rows of its array argument A.
                   On exit, the M-by-N matrix Q.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by gelqf.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument has an illegal value

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        gelqf($a, $tau, $info);
        orglq($a, $tau, $info) unless $info != 0;

       orglq ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ormlq
         Signature: ([phys]A(k,p); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())

       Overwrites the general real M-by-N matrix C with

                           side = 0     side = 1
           trans = 0:      Q * C          C * Q
           trans = 1:      Q' * C       C * Q'

           where Q is a real orthogonal matrix defined as the product of k
           elementary reflectors

                 Q = H(k) . . . H(2) H(1)

           as returned by gelqf.

       Q is of order M if "side" = 0 and of order N if "side" = 1.

           Arguments
           =========

           side:   = 0: apply Q or Q' from the Left;
                   = 1: apply Q or Q' from the Right.

           trans:  = 0:  No transpose, apply Q;
                   = 1:  Transpose, apply Q'.

           A:      The i-th row must contain the vector which defines the
                   elementary reflector H(i), for i = 1,2,...,k, as returned by
                   gelqf in the first k rows of its array argument A.
                   A is modified by the routine but restored on exit.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by gelqf.

           C:      On entry, the M-by-N matrix C.
                   On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 50, 100);
        $a = transpose($a);
        $info = null;
        $tau = zeroes(float, 50);
        gelqf($a, $tau, $info);
        $c = random(70,50);
        # $c will contain the result
        $c->reshape(70,100);
        $c = transpose($c);
        ormlq($a, $tau, $c, $info);

       ormlq ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   geqlf
         Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())

       Computes a QL factorization of a real M-by-N matrix A:

               A = Q * L

       The matrix Q is represented as a product of elementary reflectors

               Q = H(k) . . . H(2) H(1), where k = min(m,n).

       Each H(i) has the form

               H(i) = I - tau * v * v'

               where tau is a real scalar, and v is a real vector with
               v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
               A(1:m-k+i-1,n-k+i), and tau in TAU(i).

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit,
                   if m >= n, the lower triangle of the subarray
                   A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
                   if m <= n, the elements on and below the (n-m)-th
                   superdiagonal contain the M-by-N lower trapezoidal matrix L;
                   the remaining elements, with the array tau, represent the
                   orthogonal matrix Q as a product of elementary reflectors.

           tau:    The scalar factors of the elementary reflectors.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        geqlf($a, $tau, $info);

       geqlf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   orgql
         Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())

       Generates an M-by-N real matrix Q with orthonormal columns, which is defined as the last N
       columns of a product of K elementary reflectors of order M

               Q  =  H(k) . . . H(2) H(1)

               as returned by geqlf.

           Arguments
           =========

           A:      On entry, the (n-k+i)-th column must contain the vector which
                   defines the elementary reflector H(i), for i = 1,2,...,k, as
                   returned by geqlf in the last k columns of its array
                   argument A.
                   On exit, the M-by-N matrix Q.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by geqlf.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument has an illegal value

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        geqlf($a, $tau, $info);
        orgql($a, $tau, $info) unless $info != 0;

       orgql ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ormql
         Signature: ([phys]A(p,k); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())

       Overwrites the general real M-by-N matrix C with

                           side = 0     side = 1
           trans = 0:      Q * C          C * Q
           trans = 1:      Q' * C       C * Q'

           where Q is a real orthogonal matrix defined as the product of k
           elementary reflectors

                 Q = H(k) . . . H(2) H(1)

           as returned by geqlf.

       Q is of order M if "side" = 0 and of order N if "side" = 1.

           Arguments
           =========

           side:   = 0: apply Q or Q' from the Left;
                   = 1: apply Q or Q' from the Right.

           trans:  = 0:  No transpose, apply Q;
                   = 1:  Transpose, apply Q'.

           A:      The i-th row must contain the vector which defines the
                   elementary reflector H(i), for i = 1,2,...,k, as returned by
                   geqlf in the last k rows of its array argument A.
                   A is modified by the routine but restored on exit.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by geqlf.

           C:      On entry, the M-by-N matrix C.
                   On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 50, 100);
        $a = transpose($a);
        $info = null;
        $tau = zeroes(float, 50);
        geqlf($a, $tau, $info);
        $c = random(70,50);
        # $c will contain the result
        $c->reshape(70,100);
        $c = transpose($c);
        ormql($a, $tau, $c, $info);

       ormql ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gerqf
         Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())

       Computes an RQ factorization of a real M-by-N matrix A:

               A = R * Q.

       The matrix Q is represented as a product of elementary reflectors

               Q = H(1) H(2) . . . H(k), where k = min(m,n).

       Each H(i) has the form

               H(i) = I - tau * v * v'

               where tau is a real scalar, and v is a real vector with
               v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
               A(m-k+i,1:n-k+i-1), and tau in TAU(i).

           Arguments
           =========

           A:      On entry, the M-by-N matrix A.
                   On exit,
                   if m <= n, the upper triangle of the subarray
                   A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
                   if m >= n, the elements on and above the (m-n)-th subdiagonal
                   contain the M-by-N upper trapezoidal matrix R;
                   the remaining elements, with the array tau, represent the
                   orthogonal matrix Q as a product of min(m,n) elementary
                   reflectors (see Further Details).

           tau:    The scalar factors of the elementary reflectors.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        gerqf($a, $tau, $info);

       gerqf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   orgrq
         Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())

       Generates an M-by-N real matrix Q with orthonormal rows, which is defined as the last M
       rows of a product of K elementary reflectors of order N

               Q  =  H(1) H(2) . . . H(k)

               as returned by gerqf.

           Arguments
           =========

           A:      On entry, the (m-k+i)-th row must contain the vector which
                   defines the elementary reflector H(i), for i = 1,2,...,k, as
                   returned by gerqf in the last k rows of its array argument
                   A.
                   On exit, the M-by-N matrix Q.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by gerqf.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument has an illegal value

        $a = random (float, 100, 50);
        $info = null;
        $tau = zeroes(float, 50);
        gerqf($a, $tau, $info);
        orgrq($a, $tau, $info) unless $info != 0;

       orgrq ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ormrq
         Signature: ([phys]A(k,p); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())

       Overwrites the general real M-by-N matrix C with

                           side = 0     side = 1
           trans = 0:      Q * C          C * Q
           trans = 1:      Q' * C       C * Q'

           where Q is a real orthogonal matrix defined as the product of k
           elementary reflectors

                 Q = H(1) H(2) . . . H(k)

               as returned by gerqf.

       Q is of order M if "side" = 0 and of order N if "side" = 1.

           Arguments
           =========

           side:   = 0: apply Q or Q' from the Left;
                   = 1: apply Q or Q' from the Right.

           trans:  = 0:  No transpose, apply Q;
                   = 1:  Transpose, apply Q'.

           A:      The i-th row must contain the vector which defines the
                   elementary reflector H(i), for i = 1,2,...,k, as returned by
                   gerqf in the last k rows of its array argument A.
                   A is modified by the routine but restored on exit.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by gerqf.

           C:      On entry, the M-by-N matrix C.
                   On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 50, 100);
        $a = transpose($a);
        $info = null;
        $tau = zeroes(float, 50);
        gerqf($a, $tau, $info);
        $c = random(70,50);
        # $c will contain the result
        $c->reshape(70,100);
        $c = transpose($c);
        ormrq($a, $tau, $c, $info);

       ormrq ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   tzrzf
         Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())

       Reduces the M-by-N ( M <= N ) real upper trapezoidal matrix A to upper triangular form by
       means of orthogonal transformations.

       The upper trapezoidal matrix A is factored as

               A = ( R  0 ) * Z,

               where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
               triangular matrix.

       The factorization is obtained by Householder's method.  The kth transformation matrix, Z(
       k ), which is used to introduce zeros into the ( m - k + 1 )th row of A, is given in the
       form

              Z( k ) = ( I     0   ),
                       ( 0  T( k ) )

           where

              T( k ) = I - tau*u( k )*u( k )',   u( k ) = (   1    ),
                                                           (   0    )
                                                           ( z( k ) )

       tau is a scalar and z( k ) is an ( n - m ) element vector.  tau and z( k ) are chosen to
       annihilate the elements of the kth row of X.

       The scalar tau is returned in the kth element of "tau" and the vector u( k ) in the kth
       row of A, such that the elements of z( k ) are in  a( k, m + 1 ), ..., a( k, n ). The
       elements of R are returned in the upper triangular part of A.

       Z is given by

              Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

           Arguments
           =========

           A:      On entry, the leading M-by-N upper trapezoidal part of the
                   array A must contain the matrix to be factorized.
                   On exit, the leading M-by-M upper triangular part of A
                   contains the upper triangular matrix R, and elements M+1 to
                   N of the first M rows of A, with the array tau, represent the
                   orthogonal matrix Z as a product of M elementary reflectors.

           tau:    The scalar factors of the elementary reflectors.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 50, 100);
        $info = null;
        $tau = zeroes(float, 50);
        tzrzf($a, $tau, $info);

       tzrzf ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   ormrz
         Signature: ([phys]A(k,p); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())

       Overwrites the general real M-by-N matrix C with

                           side = 0     side = 1
           trans = 0:      Q * C          C * Q
           trans = 1:      Q' * C       C * Q'

           where Q is a real orthogonal matrix defined as the product of k
           elementary reflectors

                 Q = H(1) H(2) . . . H(k)

           as returned by tzrzf.

       Q is of order M if "side" = 0 and of order N if "side" = 1.

           Arguments
           =========

           side:   = 0: apply Q or Q' from the Left;
                   = 1: apply Q or Q' from the Right.

           trans:  = 0:  No transpose, apply Q;
                   = 1:  Transpose, apply Q'.

           A:      The i-th row must contain the vector which defines the
                   elementary reflector H(i), for i = 1,2,...,k, as returned by
                   tzrzf in the last k rows of its array argument A.
                   A is modified by the routine but restored on exit.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by tzrzf.

           C:      On entry, the M-by-N matrix C.
                   On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (float, 50, 100);
        $a = transpose($a);
        $info = null;
        $tau = zeroes(float, 50);
        tzrzf($a, $tau, $info);
        $c = random(70,50);
        # $c will contain the result
        $c->reshape(70,100);
        $c = transpose($c);
        ormrz($a, $tau, $c, $info);

       ormrz ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gehrd
         Signature: ([io,phys]A(n,n); int [phys]ilo();int [phys]ihi();[o,phys]tau(k); int [o,phys]info())

       Reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity
       transformation:  Q' * A * Q = H .

       Further Details ===============

       The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

               Q = H(ilo) H(ilo+1) . . . H(ihi-1).

       Each H(i) has the form

               H(i) = I - tau * v * v'
               where tau is a real scalar, and v is a real vector with
               v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
               exit in A(i+2:ihi,i), and tau in tau(i).

       The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi =
       6:

               on entry,                        on exit,

               ( a   a   a   a   a   a   a )    (  a   a   h   h   h   h   a )
               (     a   a   a   a   a   a )    (      a   h   h   h   h   a )
               (     a   a   a   a   a   a )    (      h   h   h   h   h   h )
               (     a   a   a   a   a   a )    (      v2  h   h   h   h   h )
               (     a   a   a   a   a   a )    (      v2  v3  h   h   h   h )
               (     a   a   a   a   a   a )    (      v2  v3  v4  h   h   h )
               (                         a )    (                          a )

               where a denotes an element of the original matrix A, h denotes a
               modified element of the upper Hessenberg matrix H, and vi denotes an
               element of the vector defining H(i).

           Arguments
           =========

           ilo:
           ihi:    It is assumed that A is already upper triangular in rows
                   and columns 1:ilo-1 and ihi+1:N. ilo and ihi are normally
                   set by a previous call to gebal; otherwise they should be
                   set to 1 and N respectively. See Further Details.
                   1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.

           A:      On entry, the N-by-N general matrix to be reduced.
                   On exit, the upper triangle and the first subdiagonal of A
                   are overwritten with the upper Hessenberg matrix H, and the
                   elements below the first subdiagonal, with the array tau,
                   represent the orthogonal matrix Q as a product of elementary
                   reflectors. See Further Details.

           tau:    The scalar factors of the elementary reflectors (see Further
                   Details). Elements 1:ilo-1 and ihi:N-1 of tau are set to
                   zero. (dimension (N-1))

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.

        $a = random (50, 50);
        $info = null;
        $tau = zeroes(50);
        gehrd($a, 1, 50, $tau, $info);

       gehrd ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   orghr
         Signature: ([io,phys]A(n,n); int [phys]ilo();int [phys]ihi();[phys]tau(k); int [o,phys]info())

       Generates a real orthogonal matrix Q which is defined as the product of ihi-ilo elementary
       reflectors of order N, as returned by "gehrd":

               Q = H(ilo) H(ilo+1) . . . H(ihi-1).

           Arguments
           =========

           ilo:
           ihi:   ilo and ihi must have the same values as in the previous call
                   of gehrd. Q is equal to the unit matrix except in the
                   submatrix Q(ilo+1:ihi,ilo+1:ihi).
                   1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.

           A:      On entry, the vectors which define the elementary reflectors,
                   as returned by gehrd.
                   On exit, the N-by-N orthogonal matrix Q.

           tau:    tau(i) must contain the scalar factor of the elementary
                   reflector H(i), as returned by gehrd.(dimension (N-1))

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (50, 50);
        $info = null;
        $tau = zeroes(50);
        gehrd($a, 1, 50, $tau, $info);
        orghr($a, 1, 50, $tau, $info);

       orghr ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   hseqr
         Signature: ([io,phys]H(n,n); int job();int compz();int [phys]ilo();int [phys]ihi();[o,phys]wr(n); [o,phys]wi(n);[o,phys]Z(m,m); int [o,phys]info())

       Computes the eigenvalues of a real upper Hessenberg matrix H and, optionally, the matrices
       T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular
       matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors.

       Optionally Z may be postmultiplied into an input orthogonal matrix Q, so that this routine
       can give the Schur factorization of a matrix A which has been reduced to the Hessenberg
       form H by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

           Arguments
           =========

           job:    = 0:  compute eigenvalues only;
                   = 1:  compute eigenvalues and the Schur form T.

           compz:  = 0:  no Schur vectors are computed;
                   = 1:  Z is initialized to the unit matrix and the matrix Z
                           of Schur vectors of H is returned;
                   = 2:  Z must contain an orthogonal matrix Q on entry, and
                           the product Q*Z is returned.

           ilo:
           ihi:    It is assumed that H is already upper triangular in rows
                   and columns 1:ilo-1 and ihi+1:N. ilo and ihi are normally
                   set by a previous call to gebal, and then passed to gehrd
                   when the matrix output by gebal is reduced to Hessenberg
                   form. Otherwise ilo and ihi should be set to 1 and N
                   respectively.
                   1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.

           H:      On entry, the upper Hessenberg matrix H.
                   On exit, if job = 1, H contains the upper quasi-triangular
                   matrix T from the Schur decomposition (the Schur form);
                   2-by-2 diagonal blocks (corresponding to complex conjugate
                   pairs of eigenvalues) are returned in standard form, with
                   H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If job = 0,
                   the contents of H are unspecified on exit.

           wr:
           wi:     The real and imaginary parts, respectively, of the computed
                   eigenvalues. If two eigenvalues are computed as a complex
                   conjugate pair, they are stored in consecutive elements of
                   wr and wi, say the i-th and (i+1)th, with wi(i) > 0 and
                   wi(i+1) < 0. If job = 1, the eigenvalues are stored in the
                   same order as on the diagonal of the Schur form returned in
                   H, with wr(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
                   diagonal block, wi(i) = sqrt(H(i+1,i)*H(i,i+1)) and
                   wi(i+1) = -wi(i).

           Z:      If compz = 0: Z is not referenced.
                   If compz = 1: on entry, Z need not be set, and on exit, Z
                   contains the orthogonal matrix Z of the Schur vectors of H.
                   If compz = 2: on entry Z must contain an N-by-N matrix Q,
                   which is assumed to be equal to the unit matrix except for
                   the submatrix Z(ilo:ihi,ilo:ihi); on exit Z contains Q*Z.
                   Normally Q is the orthogonal matrix generated by orghr after
                   the call to gehrd which formed the Hessenberg matrix H.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value
                   > 0:  if info = i, hseqr failed to compute all of the
                         eigenvalues in a total of 30*(ihi-ilo+1) iterations;
                         elements 1:ilo-1 and i+1:n of wr and wi contain those
                         eigenvalues which have been successfully computed.

        $a = random (50, 50);
        $info = null;
        $tau = zeroes(50);
        $z= zeroes(1,1);
        gehrd($a, 1, 50, $tau, $info);
        hseqr($a,0,0,1,50,($wr=null),($wi=null),$z,$info);

       hseqr ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   trevc
         Signature: ([io,phys]T(n,n); int side();int howmny();int [phys]select(q);[io,phys]VL(m,r); [io,phys]VR(p,s);int [o,phys]m(); int [o,phys]info())

       Computes some or all of the right and/or left eigenvectors of a real upper quasi-
       triangular matrix T.

       The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w
       are defined by:

               T*x = w*x,     y'*T = w*y'
               where y' denotes the conjugate transpose of the vector y.

       If all eigenvectors are requested, the routine may either return the matrices X and/or Y
       of right or left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input
       orthogonal matrix. If T was obtained from the real-Schur factorization of an original
       matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or left eigenvectors of A.

       T must be in Schur canonical form (as returned by hseqr), that is, block upper triangular
       with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal
       elements equal and its off-diagonal elements of opposite sign.  Corresponding to each
       2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only
       one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue
       with positive imaginary part.

       Further Details ===============

       The algorithm used in this program is basically backward (forward) substitution, with
       scaling to make the the code robust against possible overflow.

       Each eigenvector is normalized so that the element of largest magnitude has magnitude 1;
       here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

           Arguments
           =========

           side:   = 0 :  compute both right and left eigenvectors;
                   = 1 :  compute right eigenvectors only;
                   = 2 :  compute left eigenvectors only.

           howmny: = 0:  compute all right and/or left eigenvectors;
                   = 1:  compute all right and/or left eigenvectors,
                           and backtransform them using the input matrices
                           supplied in VR and/or VL;
                   = 2:  compute selected right and/or left eigenvectors,
                           specified by the logical array select.

           select: If howmny = 2, select specifies the eigenvectors to be
                   computed.
                   If howmny = 0 or 1, select is not referenced.
                   To select the real eigenvector corresponding to a real
                   eigenvalue w(j), select(j) must be set to TRUE.  To select
                   the complex eigenvector corresponding to a complex conjugate
                   pair w(j) and w(j+1), either select(j) or select(j+1) must be
                   set to TRUE; then on exit select(j) is TRUE and
                   select(j+1) is FALSE.

           T:      The upper quasi-triangular matrix T in Schur canonical form.

           VL:     On entry, if side = 2 or 0 and howmny = 1, VL must
                   contain an N-by-N matrix Q (usually the orthogonal matrix Q
                   of Schur vectors returned by hseqr).
                   On exit, if side = 2 or 0, VL contains:
                   if howmny = 0, the matrix Y of left eigenvectors of T;
                                    VL has the same quasi-lower triangular form
                                    as T'. If T(i,i) is a real eigenvalue, then
                                    the i-th column VL(i) of VL  is its
                                    corresponding eigenvector. If T(i:i+1,i:i+1)
                                    is a 2-by-2 block whose eigenvalues are
                                    complex-conjugate eigenvalues of T, then
                                    VL(i)+sqrt(-1)*VL(i+1) is the complex
                                    eigenvector corresponding to the eigenvalue
                                    with positive real part.
                   if howmny = 1, the matrix Q*Y;
                   if howmny = 2, the left eigenvectors of T specified by
                                    select, stored consecutively in the columns
                                    of VL, in the same order as their
                                    eigenvalues.
                   A complex eigenvector corresponding to a complex eigenvalue
                   is stored in two consecutive columns, the first holding the
                   real part, and the second the imaginary part.
                   If side = 1, VL is not referenced.

           VR:     On entry, if side = 1 or 0 and howmny = 1, VR must
                   contain an N-by-N matrix Q (usually the orthogonal matrix Q
                   of Schur vectors returned by hseqr).
                   On exit, if side = 1 or 0, VR contains:
                   if howmny = 0, the matrix X of right eigenvectors of T;
                                    VR has the same quasi-upper triangular form
                                    as T. If T(i,i) is a real eigenvalue, then
                                    the i-th column VR(i) of VR  is its
                                    corresponding eigenvector. If T(i:i+1,i:i+1)
                                    is a 2-by-2 block whose eigenvalues are
                                    complex-conjugate eigenvalues of T, then
                                    VR(i)+sqrt(-1)*VR(i+1) is the complex
                                    eigenvector corresponding to the eigenvalue
                                    with positive real part.
                   if howmny = 1, the matrix Q*X;
                   if howmny = 2, the right eigenvectors of T specified by
                                    select, stored consecutively in the columns
                                    of VR, in the same order as their
                                    eigenvalues.
                   A complex eigenvector corresponding to a complex eigenvalue
                   is stored in two consecutive columns, the first holding the
                   real part and the second the imaginary part.
                   If side = 2, VR is not referenced.

           m:      The number of columns in the arrays VL and/or VR actually
                   used to store the eigenvectors.
                   If howmny = 0 or 1, m is set to N.
                   Each selected real eigenvector occupies one column and each
                   selected complex eigenvector occupies two columns.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random (50, 50);
        $info = null;
        $tau = zeroes(50);
        $z= zeroes(1,1);
        gehrd($a, 1, 50, $tau, $info);
        hseqr($a,0,0,1,50,($wr=null),($wi=null),$z,$info);

       trevc ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   tgevc
         Signature: ([io,phys]A(n,n); int side();int howmny();[io,phys]B(n,n);int [phys]select(q);[io,phys]VL(m,r); [io,phys]VR(p,s);int [o,phys]m(); int [o,phys]info())

       Computes some or all of the right and/or left generalized eigenvectors of a pair of real
       upper triangular matrices (A,B).

       The right generalized eigenvector x and the left generalized eigenvector y of (A,B)
       corresponding to a generalized eigenvalue w are defined by:

               (A - wB) * x = 0  and  y**H * (A - wB) = 0
               where y**H denotes the conjugate tranpose of y.

       If an eigenvalue w is determined by zero diagonal elements of both A and B, a unit vector
       is returned as the corresponding eigenvector.

       If all eigenvectors are requested, the routine may either return the matrices X and/or Y
       of right or left eigenvectors of (A,B), or the products Z*X and/or Q*Y, where Z and Q are
       input orthogonal matrices.  If (A,B) was obtained from the generalized real-Schur
       factorization of an original pair of matrices

               (A0,B0) = (Q*A*Z**H,Q*B*Z**H),

       then Z*X and Q*Y are the matrices of right or left eigenvectors of A.

       A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal blocks.  Corresponding
       to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors;
       only one eigenvector of the pair is computed, namely the one corresponding to the
       eigenvalue with positive imaginary part.

           Arguments
           =========

           side:   = 0 : compute both right and left eigenvectors;
                   = 1 : compute right eigenvectors only;
                   = 2 : compute left eigenvectors only.

           howmny: = 0 : compute all right and/or left eigenvectors;
                   = 1 : compute all right and/or left eigenvectors, and
                          backtransform them using the input matrices supplied
                          in VR and/or VL;
                   = 2 : compute selected right and/or left eigenvectors,
                          specified by the logical array select.

           select: If howmny=2, select specifies the eigenvectors to be
                   computed.
                   If howmny=0 or 1, select is not referenced.
                   To select the real eigenvector corresponding to the real
                   eigenvalue w(j), select(j) must be set to TRUE  To select
                   the complex eigenvector corresponding to a complex conjugate
                   pair w(j) and w(j+1), either select(j) or select(j+1) must
                   be set to TRUE.

           A:      The upper quasi-triangular matrix A.

           B:      The upper triangular matrix B.  If A has a 2-by-2 diagonal
                   block, then the corresponding 2-by-2 block of B must be
                   diagonal with positive elements.

           VL:     On entry, if side = 2 or 0 and howmny = 1, VL must
                   contain an N-by-N matrix Q (usually the orthogonal matrix Q
                   of left Schur vectors returned by hgqez).
                   On exit, if side = 2 or 0, VL contains:
                   if howmny = 0, the matrix Y of left eigenvectors of (A,B);
                   if howmny = 1, the matrix Q*Y;
                   if howmny = 2, the left eigenvectors of (A,B) specified by
                               select, stored consecutively in the columns of
                               VL, in the same order as their eigenvalues.
                   If side = 1, VL is not referenced.

                   A complex eigenvector corresponding to a complex eigenvalue
                   is stored in two consecutive columns, the first holding the
                   real part, and the second the imaginary part.

           VR:     On entry, if side = 1 or 0 and howmny = 1, VR must
                   contain an N-by-N matrix Q (usually the orthogonal matrix Z
                   of right Schur vectors returned by hgeqz).
                   On exit, if side = 1 or 0, VR contains:
                   if howmny = 0, the matrix X of right eigenvectors of (A,B);
                   if howmny = 1, the matrix Z*X;
                   if howmny = 2, the right eigenvectors of (A,B) specified by
                               select, stored consecutively in the columns of
                               VR, in the same order as their eigenvalues.
                   If side = 2, VR is not referenced.

                   A complex eigenvector corresponding to a complex eigenvalue
                   is stored in two consecutive columns, the first holding the
                   real part and the second the imaginary part.

           M:      The number of columns in the arrays VL and/or VR actually
                   used to store the eigenvectors.  If howmny = 0 or 1, M
                   is set to N.  Each selected real eigenvector occupies one
                   column and each selected complex eigenvector occupies two
                   columns.

           info:   = 0:  successful exit.
                   < 0:  if info = -i, the i-th argument had an illegal value.
                   > 0:  the 2-by-2 block (info:info+1) does not have a complex
                         eigenvalue.
       =for example

        $a = random (50, 50);
        $info = null;
        $tau = zeroes(50);
        $z= zeroes(1,1);
        gehrd($a, 1, 50, $tau, $info);
        hseqr($a,0,0,1,50,($wr=null),($wi=null),$z,$info);

       tgevc ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gebal
         Signature: ([io,phys]A(n,n); int job(); int [o,phys]ilo();int [o,phys]ihi();[o,phys]scale(n); int [o,phys]info())

       Balances a general real matrix A.  This involves, first, permuting A by a similarity
       transformation to isolate eigenvalues in the first 1 to ilo-1 and last ihi+1 to N elements
       on the diagonal; and second, applying a diagonal similarity transformation to rows and
       columns ilo to ihi to make the rows and columns as close in norm as possible.  Both steps
       are optional.

       Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed
       eigenvalues and/or eigenvectors.

       Further Details ===============

       The permutations consist of row and column interchanges which put the matrix in the form

                      ( T1   X   Y  )
              P A P = (  0   B   Z  )
                      (  0   0   T2 )

              where T1 and T2 are upper triangular matrices whose eigenvalues lie
              along the diagonal.  The column indices ilo and ihi mark the starting
              and ending columns of the submatrix B. Balancing consists of applying
              a diagonal similarity transformation inv(D) * B * D to make the
              1-norms of each row of B and its corresponding column nearly equal.

       The output matrix is

              ( T1     X*D          Y    )
              (  0  inv(D)*B*D  inv(D)*Z ).
              (  0      0           T2   )

       Information about the permutations P and the diagonal matrix D is returned in the vector
       "scale".

           Arguments
           =========

           job:    Specifies the operations to be performed on A:
                   = 0:  none:  simply set ilo = 1, ihi = N, scale(I) = 1.0
                           for i = 1,...,N;
                   = 1:  permute only;
                   = 2:  scale only;
                   = 3:  both permute and scale.

           A:      On entry, the input matrix A.
                   On exit,  A is overwritten by the balanced matrix.
                   If job = 0, A is not referenced.
                   See Further Details.

           ilo:
           ihi:    ilo and ihi are set to integers such that on exit
                   A(i,j) = 0 if i > j and j = 1,...,ilo-1 or I = ihi+1,...,N.
                   If job = 0 or 2, ilo = 1 and ihi = N.

           scale:  Details of the permutations and scaling factors applied to
                   A.  If P(j) is the index of the row and column interchanged
                   with row and column j and D(j) is the scaling factor
                   applied to row and column j, then
                   scale(j) = P(j)    for j = 1,...,ilo-1
                            = D(j)    for j = ilo,...,ihi
                            = P(j)    for j = ihi+1,...,N.
                   The order in which the interchanges are made is N to ihi+1,
                   then 1 to ilo-1.

           info:   = 0:  successful exit.
                   < 0:  if info = -i, the i-th argument had an illegal value.

        $a = random (50, 50);
        $scale = zeroes(50);
        $info = null;
        $ilo = null;
        $ihi = null;
        gebal($a, $ilo, $ihi, $scale, $info);

       gebal ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gebak
         Signature: ([io,phys]A(n,m); int job(); int side();int [phys]ilo();int [phys]ihi();[phys]scale(n); int [o,phys]info())

       gebak forms the right or left eigenvectors of a real general matrix by backward
       transformation on the computed eigenvectors of the balanced matrix output by gebal.

           Arguments
           =========

           A:      On entry, the matrix of right or left eigenvectors to be
                   transformed, as returned by hsein or trevc.
                   On exit, A is overwritten by the transformed eigenvectors.

           job:    Specifies the type of backward transformation required:
                   = 0 , do nothing, return immediately;
                   = 1, do backward transformation for permutation only;
                   = 2, do backward transformation for scaling only;
                   = 3, do backward transformations for both permutation and
                          scaling.
                   job must be the same as the argument job supplied to gebal.

           side:   = 0:  V contains left eigenvectors.
                   = 1:  V contains right eigenvectors;

           ilo:
           ihi:    The integers ilo and ihi determined by gebal.
                   1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.
                   Here N is the the number of rows of the matrix A.

           scale:  Details of the permutation and scaling factors, as returned
                   by gebal.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value.

        $a = random (50, 50);
        $scale = zeroes(50);
        $info = null;
        $ilo = null;
        $ihi = null;
        gebal($a, $ilo, $ihi, $scale, $info);
        # Compute eigenvectors ($ev)
        gebak($ev, $ilo, $ihi, $scale, $info);

       gebak ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   lange
         Signature: ([phys]A(n,m); int norm(); [o]b())

       Computes the value of the one norm,  or the Frobenius norm, or the  infinity norm,  or the
       element of  largest absolute value  of a real matrix A.

           Description
           ===========

           returns the value

              lange  = ( max(abs(A(i,j))), norm = 0
                       (
                       ( norm1(A),         norm = 1
                       (
                       ( normI(A),         norm = 2
                       (
                       ( normF(A),         norm = 3

           where  norm1  denotes the  one norm of a matrix (maximum column sum),
           normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
           normF  denotes the  Frobenius norm of a matrix (square root of sum of
           squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.

           Arguments
           =========

           norm:   Specifies the value to be returned in lange as described
                   above.

           A:      The n by m matrix A.

        $a = random (float, 100, 100);
        $norm  = $a->lange(1);

       lange ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   lansy
         Signature: ([phys]A(n,n); int uplo(); int norm(); [o]b())

       Computes the value of the one norm,  or the Frobenius norm, or the  infinity norm,  or the
       element of  largest absolute value  of a real symmetric matrix A.

           Description
           ===========

           returns the value

              lansy  = ( max(abs(A(i,j))), norm = 0
                       (
                       ( norm1(A),         norm = 1
                       (
                       ( normI(A),         norm = 2
                       (
                       ( normF(A),         norm = 3

           where  norm1  denotes the  one norm of a matrix (maximum column sum),
           normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
           normF  denotes the  Frobenius norm of a matrix (square root of sum of
           squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.

           norm:   Specifies the value to be returned in lansy as described
                   above.

           uplo:   Specifies whether the upper or lower triangular part of the
                   symmetric matrix A is to be referenced.
                   = 0:  Upper triangular part of A is referenced
                   = 1:  Lower triangular part of A is referenced

           A:      The symmetric matrix A.  If uplo = 0, the leading n by n
                   upper triangular part of A contains the upper triangular part
                   of the matrix A, and the strictly lower triangular part of A
                   is not referenced.  If uplo = 1, the leading n by n lower
                   triangular part of A contains the lower triangular part of
                   the matrix A, and the strictly upper triangular part of A is
                   not referenced.

        # Assume $a is symmetric
        $a = random (float, 100, 100);
        $norm  = $a->lansy(1, 1);

       lansy ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   lantr
         Signature: ([phys]A(m,n);int uplo();int norm();int diag();[o]b())

       Computes the value of the one norm,  or the Frobenius norm, or the  infinity norm,  or the
       element of  largest absolute value  of a trapezoidal or triangular matrix A.

           Description
           ===========

           returns the value

              lantr  = ( max(abs(A(i,j))), norm = 0
                       (
                       ( norm1(A),         norm = 1
                       (
                       ( normI(A),         norm = 2
                       (
                       ( normF(A),         norm = 3

           where  norm1  denotes the  one norm of a matrix (maximum column sum),
           normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
           normF  denotes the  Frobenius norm of a matrix (square root of sum of
           squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.

           norm:   Specifies the value to be returned in lantr as described
                   above.

           uplo:   Specifies whether the matrix A is upper or lower trapezoidal.
                   = 0:  Upper triangular part of A is referenced
                   = 1:  Lower triangular part of A is referenced
                   Note that A is triangular instead of trapezoidal if M = N.

           diag:   Specifies whether or not the matrix A has unit diagonal.
                   = 0:  Non-unit diagonal
                   = 1:  Unit diagonal

           A:      The trapezoidal matrix A (A is triangular if m = n).
                   If uplo = 0, the leading m by n upper trapezoidal part of
                   the array A contains the upper trapezoidal matrix, and the
                   strictly lower triangular part of A is not referenced.
                   If uplo = 1, the leading m by n lower trapezoidal part of
                   the array A contains the lower trapezoidal matrix, and the
                   strictly upper triangular part of A is not referenced.  Note
                   that when diag = 1, the diagonal elements of A are not
                   referenced and are assumed to be one.

        # Assume $a is upper triangular
        $a = random (float, 100, 100);
        $norm  = $a->lantr(1, 1, 0);

       lantr ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   gemm
         Signature: ([phys]A(m,n); int transa(); int transb(); [phys]B(p,q);[phys]alpha(); [phys]beta(); [io,phys]C(r,s))

       Performs one of the matrix-matrix operations

               C := alpha*op( A )*op( B ) + beta*C,
               where  op( X ) is one of p( X ) = X   or   op( X ) = X',
               alpha and beta are scalars, and A, B and C are matrices, with op( A )
               an m by k matrix,  op( B )  a  k by n matrix and  C an m by n matrix.

           Parameters
           ==========
           transa:  On entry, transa specifies the form of op( A ) to be used in
                    the matrix multiplication as follows:
                       transa = 0,     op( A ) = A.
                       transa = 1,     op( A ) = A'.

           transb:  On entry, transb specifies the form of op( B ) to be used in
                    the matrix multiplication as follows:
                       transb = 0,     op( B ) = B.
                       transb = 1,     op( B ) = B'.

           alpha:   On entry, alpha specifies the scalar alpha.

           A:       Before entry with  transa = 0,  the leading  m by k
                    part of the array  A  must contain the matrix  A,  otherwise
                    the leading  k by m  part of the array  A  must contain  the
                    matrix A.

           B:       Before entry with  transb = 0,  the leading  k by n
                    part of the array  B  must contain the matrix  B,  otherwise
                    the leading  n by k  part of the array  B  must contain  the
                    matrix B.

           beta:    On entry,  beta  specifies the scalar  beta.  When  beta  is
                    supplied as zero then C need not be set on input.

           C:       Before entry, the leading  m by n  part of the array  C must
                    contain the matrix  C,  except when  beta  is zero, in which
                    case C need not be set on entry.
                    On exit, the array  C  is overwritten by the  m by n  matrix
                    ( alpha*op( A )*op( B ) + beta*C ).

        $a = random(5,4);
        $b = random(5,4);
        $alpha = pdl(0.5);
        $beta = pdl(0);
        $c = zeroes(5,5);
        gemm($a, 0, 1,$b, $alpha, $beta, $c);

       gemm ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   mmult
         Signature: ([phys]A(m,n); [phys]B(p,m); [o,phys]C(p,n))

       Blas matrix multiplication based on gemm

       mmult ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   crossprod
         Signature: ([phys]A(n,m); [phys]B(p,m); [o,phys]C(p,n))

       Blas matrix cross product based on gemm

       crossprod ignores the bad-value flag of the input piddles.  It will set the bad-value flag
       of all output piddles if the flag is set for any of the input piddles.

   syrk
         Signature: ([phys]A(m,n); int uplo(); int trans(); [phys]alpha(); [phys]beta(); [io,phys]C(p,p))

       Performs one of the symmetric rank k operations

               C := alpha*A*A' + beta*C,

       or

               C := alpha*A'*A + beta*C,

               where  alpha and beta  are scalars, C is an  n by n  symmetric matrix
               and  A  is an  n by k  matrix in the first case and a  k by n  matrix
               in the second case.

           Parameters
           ==========
           uplo:    On  entry,   uplo  specifies  whether  the  upper  or  lower
                    triangular  part  of the  array  C  is to be  referenced  as
                    follows:
                       uplo = 0 Only the  upper triangular part of  C
                                is to be referenced.
                       uplo = 1 Only the  lower triangular part of  C
                                is to be referenced.
                    Unchanged on exit.

           trans:   On entry,  trans  specifies the operation to be performed as
                    follows:
                       trans = 0       C := alpha*A*A' + beta*C.
                       trans = 1       C := alpha*A'*A + beta*C.

           alpha:   On entry, alpha specifies the scalar alpha.
                    Unchanged on exit.

           A:       Before entry with  trans = 0,  the  leading  n by k
                    part of the array  A  must contain the matrix  A,  otherwise
                    the leading  k by n  part of the array  A  must contain  the
                    matrix A.

           beta:    On entry, beta specifies the scalar beta.

           C:       Before entry  with  uplo = 0,  the leading  n by n
                    upper triangular part of the array C must contain the upper
                    triangular part  of the  symmetric matrix  and the strictly
                    lower triangular part of C is not referenced.  On exit, the
                    upper triangular part of the array  C is overwritten by the
                    upper triangular part of the updated matrix.
                    Before entry  with  uplo = 1,  the leading  n by n
                    lower triangular part of the array C must contain the lower
                    triangular part  of the  symmetric matrix  and the strictly
                    upper triangular part of C is not referenced.  On exit, the
                    lower triangular part of the array  C is overwritten by the
                    lower triangular part of the updated matrix.

        $a = random(5,4);
        $b = zeroes(5,5);
        $alpha = 1;
        $beta = 0;
        syrk ($a, 1,0,$alpha, $beta , $b);

       syrk ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   dot
         Signature: ([phys]a(n);int [phys]inca();[phys]b(m);int [phys]incb();[o,phys]c())

       Dot product of two vectors using Blas.

        $a = random(5);
        $b = random(5);
        $c = dot($a, 1, $b, 1)

       dot ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   axpy
         Signature: ([phys]a(n);int [phys]inca();[phys] alpha();[io,phys]b(m);int [phys]incb())

       Linear combination of vectors ax + b using Blas.  Returns result in b.

        $a = random(5);
        $b = random(5);
        axpy($a, 1, 12, $b, 1)

       axpy ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   nrm2
         Signature: ([phys]a(n);int [phys]inca();[o,phys]b())

       Euclidean norm of a vector using Blas.

        $a = random(5);
        $norm2 = norm2($a,1)

       nrm2 ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   asum
         Signature: ([phys]a(n);int [phys]inca();[o,phys]b())

       Sum of absolute values of a vector using Blas.

        $a = random(5);
        $absum = asum($a,1)

       asum ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   scal
         Signature: ([io,phys]a(n);int [phys]inca();[phys]scale())

       Scale a vector by a constant using Blas.

        $a = random(5);
        $a->scal(1, 0.5)

       scal ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   rot
         Signature: ([io,phys]a(n);int [phys]inca();[phys]c(); [phys]s();[io,phys]b(n);int [phys]incb())

       Applies plane rotation using Blas.

        $a = random(5);
        $b = random(5);
        rot($a, 1, 0.5, 0.7, $b, 1)

       rot ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   rotg
         Signature: ([io,phys]a();[io,phys]b();[o,phys]c(); [o,phys]s())

       Generates plane rotation using Blas.

        $a = sequence(4);
        rotg($a(0), $a(1),$a(2),$a(3))

       rotg ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   lasrt
         Signature: ([io,phys]d(n); int id();int [o,phys]info())

       Sort the numbers in d in increasing order (if id = 0) or in decreasing order (if id = 1 ).

       Use Quick Sort, reverting to Insertion sort on arrays of size <= 20. Dimension of stack
       limits N to about 2**32.

           Arguments
           =========

           id:     = 0: sort d in increasing order;
                   = 1: sort d in decreasing order.

           d:      On entry, the array to be sorted.
                   On exit, d has been sorted into increasing order
                   (d(1) <= ... <= d(N) ) or into decreasing order
                   (d(1) >= ... >= d(N) ), depending on id.

           info:   = 0:  successful exit
                   < 0:  if info = -i, the i-th argument had an illegal value

        $a = random(5);
        lasrt ($a, 0, ($info = null));

       lasrt ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   lacpy
         Signature: ([phys]A(m,n); int uplo(); [o,phys]B(p,n))

       Copies all or part of a two-dimensional matrix A to another matrix B.

           Arguments
           =========

           uplo:   Specifies the part of the matrix A to be copied to B.
                   = 0:      Upper triangular part
                   = 1:      Lower triangular part
                   Otherwise:  All of the matrix A

           A:      The m by n matrix A.  If uplo = 0, only the upper triangle
                   or trapezoid is accessed; if uplo = 1, only the lower
                   triangle or trapezoid is accessed.

           B:      On exit, B = A in the locations specified by uplo.

        $a = random(5,5);
        $b = zeroes($a);
        lacpy ($a, 0, $b);

       lacpy ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   laswp
         Signature: ([io,phys]A(m,n);int [phys]k1();int [phys] k2(); int [phys]ipiv(p);int [phys]inc())

       Performs a series of row interchanges on the matrix A.  One row interchange is initiated
       for each of rows k1 through k2 of A.  Doesn't use PDL indice (start = 1).

           Arguments
           =========

           A:      On entry, the matrix of column dimension N to which the row
                   interchanges will be applied.
                   On exit, the permuted matrix.

           k1:     The first element of ipiv for which a row interchange will
                   be done.

           k2:     The last element of ipiv for which a row interchange will
                   be done.

           ipiv:   The vector of pivot indices.  Only the elements in positions
                   k1 through k2 of ipiv are accessed.
                   ipiv(k) = l implies rows k and l are to be interchanged.

           inc:    The increment between successive values of ipiv.  If ipiv
                   is negative, the pivots are applied in reverse order.

        $a = random(5,5);
        # reverse row (col for PDL)
        $b = pdl([5,4,3,2,1]);
        $a->laswp(1,2,$b,1);

       laswp ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   lamch
         Signature: (cmach(); [o]precision())

       Determines precision machine parameters.  Works inplace.

           Arguments
           =========

           cmach:  Specifies the value to be returned by lamch:
                   = 0 LAMCH := eps
                   = 1 LAMCH := sfmin
                   = 2 LAMCH := base
                   = 3 LAMCH := eps*base
                   = 4 LAMCH := t
                   = 5 LAMCH := rnd
                   = 6 LAMCH := emin
                   = 7 LAMCH := rmin
                   = 8 LAMCH := emax
                   = 9 LAMCH := rmax

                   where

                   eps   = relative machine precision
                   sfmin = safe minimum, such that 1/sfmin does not overflow
                   base  = base of the machine
                   prec  = eps*base
                   t     = number of (base) digits in the mantissa
                   rnd   = 1.0 when rounding occurs in addition, 0.0 otherwise
                   emin  = minimum exponent before (gradual) underflow
                   rmin  = underflow threshold - base**(emin-1)
                   emax  = largest exponent before overflow
                   rmax  = overflow threshold  - (base**emax)*(1-eps)

        $a = lamch (0);
        print "EPS is $a for double\n";

       lamch ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   labad
         Signature: ([io,phys]small(); [io,phys]large())

       Takes as input the values computed by "lamch" for underflow and overflow, and returns the
       square root of each of these values if the log of large is sufficiently large.  This
       subroutine is intended to identify machines with a large exponent range, such as the
       Crays, and redefine the underflow and overflow limits to be the square roots of the values
       computed by "lamch".  This subroutine is needed because lamch does not compensate for poor
       arithmetic in the upper half of the exponent range, as is found on a Cray.

           Arguments
           =========

           small:  On entry, the underflow threshold as computed by lamch.
                   On exit, if LOG10(large) is sufficiently large, the square
                   root of small, otherwise unchanged.

           large:  On entry, the overflow threshold as computed by lamch.
                   On exit, if LOG10(large) is sufficiently large, the square
                   root of large, otherwise unchanged.

        $underflow = lamch(7);
        $overflow = lamch(9);
        labad ($underflow, $overflow);

       labad ignores the bad-value flag of the input piddles.  It will set the bad-value flag of
       all output piddles if the flag is set for any of the input piddles.

   tricpy
         Signature: (A(m,n);int uplo();[o] C(m,n))

       Copy triangular part to another matrix. If uplo == 0 copy upper triangular part.

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

   cplx_eigen
         Signature: (eigreval(n);eigimval(n); eigvec(n,p);int fortran();[o]cplx_val(q=2,n);[o]cplx_vec(r=2,n,p))

       Output complex eigen-values/vectors from eigen-values/vectors as computed by geev or
       geevx.  'fortran' means fortran storage type.

       cplx_eigen does not process bad values.  It will set the bad-value flag of all output
       piddles if the flag is set for any of the input piddles.

   augment
         Signature: (x(n); y(p);[o]out(q))

       Combine two pidlles into a single piddle.  This routine does backward and forward dataflow
       automatically.

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

   mstack
         Signature: (x(n,m);y(n,p);[o]out(n,q))

       Combine two pidlles into a single piddle.  This routine does backward and forward dataflow
       automatically.

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

   charpol
         Signature: ([phys]A(n,n);[phys,o]Y(n,n);[phys,o]out(p))

       Compute adjoint matrix and characteristic polynomial.

       charpol does not process bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

AUTHOR

       Copyright (C) Grégory Vanuxem 2005-2007.

       This library is free software; you can redistribute it and/or modify it under the terms of
       the artistic license as specified in the Artistic file.