ZLARFT(3F) ZLARFT(3F)
ZLARFT - form the triangular factor T of a complex block reflector H of
order n, which is defined as a product of k elementary reflectors
SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
ZLARFT 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
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.
Page 1
ZLARFT(3F) ZLARFT(3F)
V (input/output) COMPLEX*16 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*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary reflector
H(i).
T (output) COMPLEX*16 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 )
ZLARFT(3F) ZLARFT(3F)
ZLARFT - form the triangular factor T of a complex block reflector H of
order n, which is defined as a product of k elementary reflectors
SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
ZLARFT 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
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.
Page 1
ZLARFT(3F) ZLARFT(3F)
V (input/output) COMPLEX*16 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*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary reflector
H(i).
T (output) COMPLEX*16 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 )
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