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

M

          M is INTEGER
          The number of rows in the matrix A.

N

          N is INTEGER
          The number of columns in the matrix A.

NB

          NB is INTEGER
          The number of leading rows and columns of A to be reduced.

A

          A is 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

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is 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

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

TAUQ

          TAUQ is COMPLEX*16 array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q. See Further Details.

TAUP

          TAUP is COMPLEX*16 array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix P. See Further Details.

X

          X is COMPLEX*16 array, dimension (LDX,NB)
          The m-by-nb matrix X required to update the unreduced part
          of A.

LDX

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

Y

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

LDY

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

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  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).