DGEHD2(l) — LAPACK routine (version 2.0)

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 element of the vector defining H(i).