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CGEHRD(l)			       )			     CGEHRD(l)

NAME
       CGEHRD  -  reduce a complex general matrix A to upper Hessenberg form H
       by a unitary similarity transformation

SYNOPSIS
       SUBROUTINE CGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )

	   INTEGER	  IHI, ILO, INFO, LDA, LWORK, N

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

PURPOSE
       CGEHRD reduces a complex general matrix A to upper Hessenberg form H by
       a unitary similarity transformation: Q' * A * Q = H .

ARGUMENTS
       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 CGEBAL; otherwise they
	       should be set to 1 and N respectively. See Further Details.

       A       (input/output) COMPLEX 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 overwrit‐
	       ten with the upper Hessenberg matrix H, and the elements	 below
	       the  first  subdiagonal, with the array TAU, represent the uni‐
	       tary matrix Q as a product of elementary reflectors.  See  Fur‐
	       ther Details.  LDA     (input) INTEGER The leading dimension of
	       the array A.  LDA >= max(1,N).

       TAU     (output) COMPLEX 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 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.

FURTHER DETAILS
       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 = 6:

       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 modi‐
       fied element of the upper Hessenberg matrix H, and vi denotes  an  ele‐
       ment of the vector defining H(i).

LAPACK version 3.0		 15 June 2000			     CGEHRD(l)
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