.TH SGEHD2 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
SGEHD2 - a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation
.SH SYNOPSIS
.TP 19
SUBROUTINE SGEHD2(
N, ILO, IHI, A, LDA, TAU, WORK, INFO )
.TP 19
.ti +4
INTEGER
IHI, ILO, INFO, LDA, N
.TP 19
.ti +4
REAL
A( LDA, * ), TAU( * ), WORK( * )
.SH PURPOSE
SGEHD2 reduces a real general matrix A to upper Hessenberg form H by
an orthogonal similarity transformation: Q\(aq * A * Q = H .
.SH ARGUMENTS
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
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.
.TP 8
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).
.TP 8
TAU (output) REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).
.TP 8
WORK (workspace) REAL array, dimension (N)
.TP 8
INFO (output) INTEGER
= 0: successful exit.
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.SH FURTHER DETAILS
The matrix Q is represented as a product of (ihi-ilo) elementary
reflectors
.br
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
.br
Each H(i) has the form
.br
H(i) = I - tau * v * v\(aq
.br
where tau is a real scalar, and v is a real vector with
.br
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).
.br
The contents of A are illustrated by the following example, with
n = 7, ilo = 2 and ihi = 6:
.br
on entry, on exit,
.br
( 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).
.br