sgebrd(3F)

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

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

         INTEGER        INFO, LDA, LWORK, M, N

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

PURPOSE
     SGEBRD 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 bidiagonal.


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 bidiagonal 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 orthogonal matrix P.
             See Further Details.  WORK    (workspace/output) REAL 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 illegal value.

FURTHER DETAILS
     The matrices Q and P are represented as products of elementary
     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).