DLARZT - form the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors

SUBROUTINE DLARZT(

DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )

CHARACTER DIRECT, STOREV

INTEGER K, LDT, LDV, N

DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )

DLARZT forms the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary 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 supported.

DIRECT (input) CHARACTER*1

Specifies the order in which the elementary reflectors are multiplied to form the block reflector:

= 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)

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

STOREV (input) CHARACTER*1

Specifies how the vectors which define the elementary 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (K)

TAU(i) must contain the scalar factor of the elementary reflector H(i).

T (output) DOUBLE PRECISION array, dimension (LDT,K)

The k by k triangular factor T of the block reflector. 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.

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 example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. 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 )