zlabrd(3F)

NAME
     ZLABRD - reduce the first NB rows and columns of a complex general m by n
     matrix A to upper or lower real bidiagonal form by a unitary
     transformation Q' * A * P, and returns the matrices X and Y which are
     needed to apply the transformation to the unreduced part of A

SYNOPSIS
     SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )

         INTEGER        LDA, LDX, LDY, M, N, NB

         DOUBLE         PRECISION D( * ), E( * )

         COMPLEX*16     A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y(
                        LDY, * )

PURPOSE
     ZLABRD reduces the first NB rows and columns of a complex general m by n
     matrix A to upper or lower real bidiagonal form by a unitary
     transformation Q' * A * P, and returns the matrices X and Y which are
     needed to apply the transformation to the unreduced part of A.

     If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
     bidiagonal form.

     This is an auxiliary routine called by ZGEBRD


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

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

     NB      (input) INTEGER
             The number of leading rows and columns of A to be reduced.

     A       (input/output) COMPLEX*16 array, dimension (LDA,N)
             On entry, the m by n general matrix to be reduced.  On exit, the
             first NB rows and columns of the matrix are overwritten; the rest
             of the array is unchanged.  If m >= n, elements on and below the
             diagonal in the first NB columns, with the array TAUQ, represent
             the unitary matrix Q as a product of elementary reflectors; and
             elements above the diagonal in the first NB rows, with the array
             TAUP, represent the unitary matrix P as a product of elementary
             reflectors.  If m < n, elements below the diagonal in the first
             NB columns, with the array TAUQ, represent the unitary matrix Q
             as a product of elementary reflectors, and elements on and above
             the diagonal in the first NB rows, 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 (NB)
             The diagonal elements of the first NB rows and columns of the
             reduced matrix.  D(i) = A(i,i).

     E       (output) DOUBLE PRECISION array, dimension (NB)
             The off-diagonal elements of the first NB rows and columns of the
             reduced matrix.

     TAUQ    (output) COMPLEX*16 array dimension (NB)
             The scalar factors of the elementary reflectors which represent
             the unitary matrix Q. See Further Details.  TAUP    (output)
             COMPLEX*16 array, dimension (NB) The scalar factors of the
             elementary reflectors which represent the unitary matrix P. See
             Further Details.  X       (output) COMPLEX*16 array, dimension
             (LDX,NB) The m-by-nb matrix X required to update the unreduced
             part of A.

     LDX     (input) INTEGER
             The leading dimension of the array X. LDX >= max(1,M).

     Y       (output) COMPLEX*16 array, dimension (LDY,NB)
             The n-by-nb matrix Y required to update the unreduced part of A.

     LDY     (output) INTEGER
             The leading dimension of the array Y. LDY >= max(1,N).

FURTHER DETAILS
     The matrices Q and P are represented as products of elementary
     reflectors:

        Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

     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.

     If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
     A(i:m,i); u(1:i) = 0, u(i+1) = 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).

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

     The elements of the vectors v and u together form the m-by-nb matrix V
     and the nb-by-n matrix U' which are needed, with X and Y, to apply the
     transformation to the unreduced part of the matrix, using a block update
     of the form:  A := A - V*Y' - X*U'.
     The contents of A on exit are illustrated by the following examples with
     nb = 2:

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

       (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
       (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
       (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
       (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
       (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
       (  v1  v2  a   a   a  )

     where a denotes an element of the original matrix which is unchanged, vi
     denotes an element of the vector defining H(i), and ui an element of the
     vector defining G(i).