CXML
        	    
LAPACK version 3.0
zlabrd(3)

PURPOSE
  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

SYNTAX
  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,
		     * )

DESCRIPTION
  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).