zgebd2(3F)

NAME
     ZGEBD2 - reduce a complex general m by n matrix A to upper or lower real
     bidiagonal form B by a unitary transformation

SYNOPSIS
     SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

         INTEGER        INFO, LDA, M, N

         DOUBLE         PRECISION D( * ), E( * )

         COMPLEX*16     A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )

PURPOSE
     ZGEBD2 reduces a complex general m by n matrix A to upper or lower real
     bidiagonal form B by a unitary transformation: Q' * 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) COMPLEX*16 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 unitary matrix Q as
             a product of elementary reflectors, and the elements above the
             first superdiagonal, with the array TAUP, represent the unitary
             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 unitary matrix Q as a product
             of elementary reflectors, and the elements above the diagonal,
             with the array TAUP, represent the unitary 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) COMPLEX*16 array dimension (min(M,N))
             The scalar factors of the elementary reflectors which represent
             the unitary matrix Q. See Further Details.  TAUP    (output)
             COMPLEX*16 array, dimension (min(M,N)) The scalar factors of the
             elementary reflectors which represent the unitary matrix P. See
             Further Details.  WORK    (workspace) COMPLEX*16 array, dimension
             (max(M,N))

     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 complex scalars, and v and u are complex 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 complex scalars, v and u are complex 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).