CLARZT - form 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 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 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 sup
ported.
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, 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
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
reflector. If DIRECT = 'F', T is upper triangu
lar; 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
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
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 )
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