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ZGEHRD(l)					)					ZGEHRD(l)

       ZGEHRD  -  reduce a complex general matrix A to upper Hessenberg form H by a unitary simi-
       larity transformation



	   COMPLEX*16	  A( LDA, * ), TAU( * ), WORK( * )

       ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by a unitary similar-
       ity transformation: Q' * A * Q = H .

       N       (input) INTEGER
	       The order of the matrix A.  N >= 0.

       ILO     (input) INTEGER
	       IHI	(input)  INTEGER It is assumed that A is already upper triangular in rows
	       and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a  previous  call
	       to  ZGEBAL;  otherwise  they  should  be  set to 1 and N respectively. See Further

       A       (input/output) COMPLEX*16 array, dimension (LDA,N)
	       On entry, the N-by-N general matrix to be reduced.  On exit,  the  upper  triangle
	       and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H,
	       and the elements below the first subdiagonal, with the array  TAU,  represent  the
	       unitary	matrix Q as a product of elementary reflectors. See Further Details.  LDA
	       (input) INTEGER The leading dimension of the array A.  LDA >= max(1,N).

       TAU     (output) COMPLEX*16 array, dimension (N-1)
	       The scalar factors of the elementary reflectors (see  Further  Details).  Elements
	       1:ILO-1 and IHI:N-1 of TAU are set to zero.

       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,N).  For optimum performance LWORK
	       >= 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.

       The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

	  Q = H(ilo) H(ilo+1) . . . H(ihi-1).

       Each H(i) has the form

	  H(i) = I - tau * v * v'

       where tau is a complex scalar, and v is a complex vector with v(1:i) = 0, v(i+1) =  1  and
       v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in A(i+2:ihi,i), and tau in TAU(i).

       The  contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi =

       on entry,			on exit,

       ( a   a	 a   a	 a   a	 a )	(  a   a   h   h   h   h   a ) (     a	 a    a    a	a
       a  )	(	a   h	h   h	h   a ) (     a   a   a   a   a   a )	 (	h   h	h
       h   h   h ) (	 a   a	 a   a	 a   a )    (	   v2  h   h   h   h   h )  (	   a	a
       a    a	 a    a  )     (       v2  v3  h   h   h   h ) (     a	 a   a	 a   a	 a )	(
       v2  v3  v4  h   h   h ) (			 a )	(			   a )

       where a denotes an element of the original matrix A, h denotes a modified element  of  the
       upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).

LAPACK version 3.0			   15 June 2000 				ZGEHRD(l)
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