CGEBRD(l)		LAPACK routine (version	1.1)		    CGEBRD(l)

NAME
  CGEBRD - reduce a general complex M-by-N matrix A to upper or	lower bidiag-
  onal form B by a unitary transformation

SYNOPSIS

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

      INTEGER	     INFO, LDA,	LWORK, M, N

      REAL	     D(	* ), E(	* )

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

PURPOSE
  CGEBRD reduces a general complex M-by-N matrix A to upper or lower bidiago-
  nal 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 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 superdiago-
	  nal, with the	array TAUP, represent the unitary matrix P as a	pro-
	  duct 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 reflec-
	  tors,	and the	elements above the diagonal, with the array TAUP,
	  represent the	unitary	matrix P as a product of elementary reflec-
	  tors.	 See Further Details.  LDA     (input) INTEGER The leading
	  dimension of the array A.  LDA >= max(1,M).

  D	  (output) REAL	array, dimension (min(M,N))
	  The diagonal elements	of the bidiagonal matrix B: D(i) = A(i,i).

  E	  (output) REAL	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 array dimension (min(M,N))
	  The scalar factors of	the elementary reflectors which	represent the
	  unitary matrix Q. See	Further	Details.  TAUP	  (output) COMPLEX
	  array, dimension (min(M,N)) The scalar factors of the	elementary
	  reflectors which represent the unitary matrix	P. See Further
	  Details.  WORK    (workspace)	COMPLEX	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.

  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 defin-
  ing G(i).