SGEHRD(l)		LAPACK routine (version	1.1)		    SGEHRD(l)

NAME
  SGEHRD - reduce a real general matrix	A to upper Hessenberg form H by	an
  orthogonal similarity	transformation

SYNOPSIS

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

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

      REAL	     A(	LDA, * ), TAU( * ), WORK( LWORK	)

PURPOSE
  SGEHRD reduces a real	general	matrix A to upper Hessenberg form H by an
  orthogonal 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	tri-
	  angular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
	  normally set by a previous call to SGEBAL; otherwise they should be
	  set to 1 and N respectively. See Further Details.  If	N > 0,

  A	  (input/output) REAL 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 orthogonal 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) REAL	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) REAL 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 per-
	  formance LWORK >= 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 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 real scalar, and v is a real 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 modified
  element of the upper Hessenberg matrix H, and	vi denotes an element of the
  vector defining H(i).