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SGEHD2(3F)							    SGEHD2(3F)


NAME    [Toc]    [Back]

     SGEHD2 - reduce a real general matrix A to	upper Hessenberg form H	by an
     orthogonal	similarity transformation

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	SGEHD2(	N, ILO,	IHI, A,	LDA, TAU, WORK,	INFO )

	 INTEGER	IHI, ILO, INFO,	LDA, N

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

PURPOSE    [Toc]    [Back]

     SGEHD2 reduces a real general matrix A to upper Hessenberg	form H by an
     orthogonal	similarity transformation:  Q' * A * Q = H .

ARGUMENTS    [Toc]    [Back]

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

     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).

     WORK    (workspace) REAL 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



									Page 1






SGEHD2(3F)							    SGEHD2(3F)



	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).
SGEHD2(3F)							    SGEHD2(3F)


NAME    [Toc]    [Back]

     SGEHD2 - reduce a real general matrix A to	upper Hessenberg form H	by an
     orthogonal	similarity transformation

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	SGEHD2(	N, ILO,	IHI, A,	LDA, TAU, WORK,	INFO )

	 INTEGER	IHI, ILO, INFO,	LDA, N

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

PURPOSE    [Toc]    [Back]

     SGEHD2 reduces a real general matrix A to upper Hessenberg	form H by an
     orthogonal	similarity transformation:  Q' * A * Q = H .

ARGUMENTS    [Toc]    [Back]

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

     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).

     WORK    (workspace) REAL 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



									Page 1






SGEHD2(3F)							    SGEHD2(3F)



	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).


									PPPPaaaaggggeeee 2222
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