CXML
        	    
LAPACK version 3.0
zgebd2(3)

PURPOSE
  ZGEBD2 - reduce a complex general m by n matrix A to upper or lower real
  bidiagonal form B by a unitary transformation

SYNTAX
  SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )

      INTEGER	     INFO, LDA, M, N

      DOUBLE	     PRECISION D( * ), E( * )

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

DESCRIPTION
  ZGEBD2 reduces a complex general m by n matrix A to upper or lower real
  bidiagonal form B by a unitary transformation: Q' * 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) COMPLEX*16 array, dimension
	  (max(M,N))

  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, 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).