Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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**H * 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**H * 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     (input) 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**H  and G(i) = I - taup * u * u**H
        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**H 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**H - X*U**H.
        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).

 LAPACK auxiliary routine (version 3.3.1)   April 2011                            ZLABRD(3lapack)