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dgehd2

DGEHD2(l)								 )								 DGEHD2(l)

NAME

       DGEHD2 - reduce a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation

SYNOPSIS

       SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )

	   INTEGER	  IHI, ILO, INFO, LDA, N

	   DOUBLE	  PRECISION A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

       DGEHD2 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 triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
	       normally set by a previous call to DGEBAL; otherwise they should be set to 1 and N respectively. See Further Details.

       A       (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N-1)
	       The scalar factors of the elementary reflectors (see Further Details).

       WORK    (workspace) DOUBLE PRECISION array, dimension (N)

       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  ele-
       ment of the vector defining H(i).

LAPACK version 3.0						   15 June 2000 							 DGEHD2(l)

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