zgehrd(3)

NAME

ZGEHRD - reduce a complex general matrix A to upper Hes

senberg form H by a unitary similarity transformation

SYNOPSIS

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

PURPOSE

ZGEHRD 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). Elements 1:ILO-1 and IHI:N-1 of TAU are

set to zero.

WORK (workspace/output) COMPLEX*16 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 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

ZGEHRD(3)