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NAME
     sgebd2 - reduce a real general m by n matrix A to  upper  or
     lower bidiagonal form B by an orthogonal transformation

SYNOPSIS
     SUBROUTINE SGEBD2( M, N, A, LDA, D,  E,  TAUQ,  TAUP,  WORK,
               INFO )

     INTEGER INFO, LDA, M, N

     REAL A( LDA, * ), D( * ), E( * ), TAUP(  *  ),  TAUQ(  *  ),
               WORK( * )



     #include <sunperf.h>

     void sgebd2(int m, int n, float  *sa,  int  lda,  float  *d,
               float *e, float *tauq, float *taup, int *info) ;

PURPOSE
     SGEBD2 reduces a real general m by n matrix A  to  upper  or
     lower  bidiagonal form B by an orthogonal transformation: Q'
     * 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) REAL 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) REAL array, dimension (min(M,N))
               The diagonal elements of the bidiagonal matrix  B:
               D(i) = A(i,i).

     E         (output) REAL 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) REAL array dimension (min(M,N))
               The scalar factors of  the  elementary  reflectors
               which  represent  the  orthogonal  matrix  Q.  See
               Further Details.   TAUP     (output)  REAL  array,
               dimension  (min(M,N))  The  scalar  factors of the
               elementary reflectors which represent the orthogo-
               nal   matrix   P.   See   Further  Details.   WORK
               (workspace) REAL array, dimension (max(M,N))

     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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