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NAME
     dgebrd - reduce a general real M-by-N matrix A to  upper  or
     lower bidiagonal form B by an orthogonal transformation

SYNOPSIS
     SUBROUTINE DGEBRD( M, N, A, LDA, D,  E,  TAUQ,  TAUP,  WORK,
               LWORK, INFO )

     INTEGER INFO, LDA, LWORK, M, N

     DOUBLE PRECISION A( LDA, * ), D( * ), E( *  ),  TAUP(  *  ),
               TAUQ( * ), WORK( LWORK )



     #include <sunperf.h>

     void dgebrd(int m, int n, double *da, int  lda,  double  *d,
               double  *e, double *tauq, double *taup, int *info)
               ;

PURPOSE
     DGEBRD reduces a general real M-by-N matrix A  to  upper  or
     lower  bidiagonal  form  B  by an orthogonal transformation:
     Q**T * A * P = B.

     If m >= n, B is upper bidiagonal; if m < n, B is lower bidi-
     agonal.


ARGUMENTS
     M         (input) INTEGER
               The number of rows in the matrix A.  M >= 0.

     N         (input) INTEGER
               The number of columns in the matrix A.  N >= 0.

     A         (input/output) DOUBLE PRECISION  array,  dimension
               (LDA,N)
               On entry, the M-by-N general matrix to be reduced.
               On  exit,  if  m  >= n, the diagonal and the first
               superdiagonal are overwritten with the upper bidi-
               agonal  matrix B; the elements below the diagonal,
               with the  array  TAUQ,  represent  the  orthogonal
               matrix  Q  as  a product of elementary reflectors,
               and the elements above  the  first  superdiagonal,
               with  the  array  TAUP,  represent  the orthogonal
               matrix P as a product of elementary reflectors; if
               m  < n, the diagonal and the first subdiagonal are
               overwritten with the lower  bidiagonal  matrix  B;
               the elements below the first subdiagonal, with the
               array TAUQ, represent the orthogonal matrix Q as a
               product of elementary reflectors, and the elements
               above the diagonal, with the array TAUP, represent
               the orthogonal matrix P as a product of elementary
               reflectors.  See Further Details.  LDA     (input)
               INTEGER The leading dimension of the array A.  LDA
               >= max(1,M).

     D         (output)   DOUBLE   PRECISION   array,   dimension
               (min(M,N))
               The diagonal elements of the bidiagonal matrix  B:
               D(i) = A(i,i).

     E         (output)   DOUBLE   PRECISION   array,   dimension
               (min(M,N)-1)
               The off-diagonal elements of the bidiagonal matrix
               B:   if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-
               1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.

     TAUQ      (output)   DOUBLE   PRECISION   array    dimension
               (min(M,N))
               The scalar factors of  the  elementary  reflectors
               which  represent  the  orthogonal  matrix  Q.  See
               Further Details.  TAUP    (output)  DOUBLE  PRECI-
               SION  array,  dimension (min(M,N)) The scalar fac-
               tors of the elementary reflectors which  represent
               the  orthogonal  matrix  P.  See  Further Details.
               WORK    (workspace/output) DOUBLE PRECISION array,
               dimension  (LWORK)  On  exit, if INFO = 0, WORK(1)
               returns the optimal LWORK.

     LWORK     (input) INTEGER
               The  length  of  the   array   WORK.    LWORK   >=
               max(1,M,N).   For  optimum  performance  LWORK  >=
               (M+N)*NB, where NB is the optimal blocksize.

     INFO      (output) INTEGER
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal value.

FURTHER DETAILS
     The matrices Q and P are represented as products of  elemen-
     tary reflectors:

     If m >= n,

        Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

     Each H(i) and G(i) has the form:

        H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

     where tauq and taup are real scalars, and v and u  are  real
     vectors;  v(1:i-1)  = 0, v(i) = 1, and v(i+1:m) is stored on
     exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n)  is
     stored  on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and
     taup in TAUP(i).

     If m < n,

        Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)

     Each H(i) and G(i) has the form:

        H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'

     where tauq and taup are real scalars, and v and u  are  real
     vectors;  v(1:i)  = 0, v(i+1) = 1, and v(i+2:m) is stored on
     exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n)  is
     stored  on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and
     taup in TAUP(i).

     The contents of A on exit are illustrated by  the  following
     examples:

     m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

       (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
       (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
       (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
       (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
       (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
       (  v1  v2  v3  v4  v5 )

     where d and e denote diagonal and off-diagonal  elements  of
     B, vi denotes an element of the vector defining H(i), and ui
     an element of the vector defining G(i).

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