zgehd2(3)

NAME

ZGEHD2 - reduce a complex general matrix A to upper Hes

senberg form H by a unitary similarity transformation

SYNOPSIS

SUBROUTINE ZGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
    INTEGER        IHI, ILO, INFO, LDA, N
    COMPLEX*16     A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

ZGEHD2 reduces a complex general matrix A to upper Hessen

berg form H by a unitary 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 ZGEBAL; other

wise they should be set to 1 and N respectively. See Further De

tails.

A (input/output) COMPLEX*16 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

unitary matrix Q as a product of elementary reflectors. See Fur

ther Details. LDA (input) INTEGER The leading dimension of

the array A. LDA >= max(1,N).

TAU (output) COMPLEX*16 array, dimension (N-1)

The scalar factors of the elementary reflectors

(see Further Details).

WORK (workspace) COMPLEX*16 array, dimension (N)

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal value.

FURTHER DETAILS

The matrix Q is represented as a product of (ihi-ilo) ele

mentary 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 complex scalar, and v is a complex 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 exam

ple, 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 de

notes 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

ZGEHD2(3)