CLARFT(3) forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors

## SYNOPSIS

SUBROUTINE CLARFT(
DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )

CHARACTER DIRECT, STOREV

INTEGER K, LDT, LDV, N

COMPLEX T( LDT, * ), TAU( * ), V( LDV, * )

## PURPOSE

CLARFT forms the triangular factor T of a complex 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

## ARGUMENTS

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)
= '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) 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 elementary reflector H(i).
T (output) COMPLEX 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.

## FURTHER DETAILS

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 = (  1       )                 V = (  1 v1 v1 v1 v1 )
( v1  1    )                     (     1 v2 v2 v2 )
( v1 v2  1 )                     (        1 v3 v3 )
( v1 v2 v3 )

( v1 v2 v3 )
DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
( v1 v2 v3 )                     ( v2 v2 v2  1    )
(  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
(     1 v3 )

(        1 )