clarzt(3)

NAME

CLARZT - form the triangular factor T of a complex block

reflector H of order > n, which is defined as a product of k ele

mentary reflectors

SYNOPSIS

SUBROUTINE CLARZT( DIRECT, STOREV, N, K, V, LDV,  TAU,  T,
LDT )
    CHARACTER      DIRECT, STOREV
    INTEGER        K, LDT, LDV, N
    COMPLEX        T( LDT, * ), TAU( * ), V( LDV, * )

PURPOSE

CLARZT forms the triangular factor T of a complex block

reflector H of order > n, which is defined as a product of k ele

mentary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k)

and T is upper triangular;

If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower

triangular.

If STOREV = 'C', the vector which defines the elementary

reflector H(i) is stored in the i-th column of the array V, and

H = I - V * T * V'

If STOREV = 'R', the vector which defines the elementary

reflector H(i) is stored in the i-th row of the array V, and

H = I - V' * T * V

Currently, only STOREV = 'R' and DIRECT = 'B' are support

ed.

ARGUMENTS

DIRECT (input) CHARACTER*1

Specifies the order in which the elementary re

flectors are multiplied to form the block reflector:
= 'F': H = H(1) H(2) . . . H(k) (Forward, not sup

ported yet)
= 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV (input) CHARACTER*1

Specifies how the vectors which define the elemen

tary reflectors are stored (see also Further Details):
= 'R': rowwise

N (input) INTEGER

The order of the block reflector H. N >= 0.

K (input) INTEGER

The order of the triangular factor T (= the number

of elementary reflectors). K >= 1.

V (input/output) COMPLEX array, dimension

(LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R'

The matrix V. See further details.

LDV (input) INTEGER

The leading dimension of the array V. If STOREV =

'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

TAU (input) COMPLEX array, dimension (K)

TAU(i) must contain the scalar factor of the ele

mentary reflector H(i).

T (output) COMPLEX array, dimension (LDT,K)

The k by k triangular factor T of the block re

flector. If DIRECT = 'F', T is upper triangular; if DIRECT =

'B', T is lower triangular. The rest of the array is not used.

LDT (input) INTEGER

The leading dimension of the array T. LDT >= K.

FURTHER DETAILS

Based on contributions by

A. Petitet, Computer Science Dept., Univ. of Tenn.,

Knoxville, USA

The shape of the matrix V and the storage of the vectors

which define the H(i) is best illustrated by the following exam

ple with n = 5 and k = 3. The elements equal to 1 are not stored;

the corresponding array elements are modified but restored on ex

it. The rest of the array is not used.

DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and

STOREV = 'R':

V

( v1 v2 v3 ) /

( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )

V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2

. . . 1 )

( v1 v2 v3 ) ( v3 v3 v3 v3 v3

. . 1 )
( v1 v2 v3 )

. . .
. . .
1 . .

1 .

1

DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and

STOREV = 'R':

V

1 /

. 1 ( 1 . . . . v1 v1 v1 v1 v1 )
. . 1 ( . 1 . . . v2 v2

v2 v2 v2 )
. . . ( . . 1 . . v3 v3

v3 v3 v3 )
. . .

( v1 v2 v3 )
( v1 v2 v3 )

V = ( v1 v2 v3 )

( v1 v2 v3 )
( v1 v2 v3 )

LAPACK version 3.0 15 June 2000

CLARZT(3)