CXML
        	    
LAPACK version 3.0
zgebrd(3)

PURPOSE
  ZGEBRD - reduce a general complex M-by-N matrix A to upper or lower
  bidiagonal form B by a unitary transformation

SYNTAX
  SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )

      INTEGER	     INFO, LDA, LWORK, M, N

      DOUBLE	     PRECISION D( * ), E( * )

      COMPLEX*16     A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )

DESCRIPTION
  ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
  bidiagonal form B by a unitary transformation: Q**H * A * P = B.  If m >=
  n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

ARGUMENTS
  M	  (input) INTEGER
	  The number of rows in the matrix A.  M >= 0.

  N	  (input) INTEGER
	  The number of columns in the matrix A.  N >= 0.

  A	  (input/output) COMPLEX*16 array, dimension (LDA,N)
	  On entry, the M-by-N general matrix to be reduced.  On exit, if m
	  >= n, the diagonal and the first superdiagonal are overwritten with
	  the upper bidiagonal matrix B; the elements below the diagonal,
	  with the array TAUQ, represent the unitary matrix Q as a product of
	  elementary reflectors, and the elements above the first
	  superdiagonal, with the array TAUP, represent the unitary matrix P
	  as a product of elementary reflectors; if m < n, the diagonal and
	  the first subdiagonal are overwritten with the lower bidiagonal
	  matrix B; the elements below the first subdiagonal, with the array
	  TAUQ, represent the unitary matrix Q as a product of elementary
	  reflectors, and the elements above the diagonal, 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 (min(M,N))
	  The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).

  E	  (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
	  The off-diagonal elements of the bidiagonal matrix B: if m >= n,
	  E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for
	  i = 1,2,...,m-1.

  TAUQ	  (output) COMPLEX*16 array dimension (min(M,N))
	  The scalar factors of the elementary reflectors which represent the
	  unitary matrix Q. See Further Details.  TAUP	  (output) COMPLEX*16
	  array, dimension (min(M,N)) The scalar factors of the elementary
	  reflectors which represent the unitary matrix P. See Further
	  Details.  WORK    (workspace/output) COMPLEX*16 array, dimension
	  (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK	  (input) INTEGER
	  The length of the array WORK.	 LWORK >= max(1,M,N).  For optimum
	  performance LWORK >= (M+N)*NB, where NB is the optimal blocksize.

	  If LWORK = -1, then a workspace query is assumed; the routine only
	  calculates the optimal size of the WORK array, returns this value
	  as the first entry of the WORK array, and no error message related
	  to LWORK is issued by XERBLA.

  INFO	  (output) INTEGER
	  = 0:	successful exit.
	  < 0:	if INFO = -i, the i-th argument had an illegal value.

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

  If m >= n,

     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)

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

  If m < n,

     Q = H(1) H(2) . . . H(m-1)	 and  P = G(1) G(2) . . . G(m)

  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;
  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
  u(1:i-1) = 0, u(i) = 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).

  The contents of A on exit are illustrated by the following examples:

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

    (  d   e   u1  u1  u1 )	      (	 d   u1	 u1  u1	 u1  u1 )
    (  v1  d   e   u2  u2 )	      (	 e   d	 u2  u2	 u2  u2 )
    (  v1  v2  d   e   u3 )	      (	 v1  e	 d   u3	 u3  u3 )
    (  v1  v2  v3  d   e  )	      (	 v1  v2	 e   d	 u4  u4 )
    (  v1  v2  v3  v4  d  )	      (	 v1  v2	 v3  e	 d   u5 )
    (  v1  v2  v3  v4  v5 )

  where d and e denote diagonal and off-diagonal elements of B, vi denotes an
  element of the vector defining H(i), and ui an element of the vector
  defining G(i).