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RedHat 9 (Linux i386) - man page for sgehrd (redhat section l)

SGEHRD(l)					)					SGEHRD(l)

NAME
       SGEHRD  - reduce a real general matrix A to upper Hessenberg form H by an orthogonal simi-
       larity 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( * )

PURPOSE
       SGEHRD reduces a real general matrix A to upper Hessenberg form H by an	orthogonal  simi-
       larity 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  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).  Elements
	       1:ILO-1 and IHI:N-1 of TAU are set to zero.

       WORK    (workspace/output) 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 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 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).

LAPACK version 3.0			   15 June 2000 				SGEHRD(l)


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