ZLABRD (3lapack)

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).