1CLARFT(1) LAPACK auxiliary routine (version 3.2) CLARFT(1)
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6 CLARFT - forms the triangular factor T of a complex block reflector H
7 of order n, which is defined as a product of k elementary reflectors
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10 SUBROUTINE CLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
11 CHARACTER DIRECT, STOREV
13 INTEGER K, LDT, LDV, N
15 COMPLEX T( LDT, * ), TAU( * ), V( LDV, * )
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19 CLARFT forms the triangular factor T of a complex block reflector H of
20 order n, which is defined as a product of k elementary reflectors. If
21 DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If
22 DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If
23 STOREV = 'C', the vector which defines the elementary reflector H(i) is
24 stored in the i-th column of the array V, and
25 H = I - V * T * V'
26 If STOREV = 'R', the vector which defines the elementary reflector H(i)
27 is stored in the i-th row of the array V, and
28 H = I - V' * T * V
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31 DIRECT (input) CHARACTER*1
32 Specifies the order in which the elementary reflectors are mul‐
33 tiplied to form the block reflector:
34 = 'F': H = H(1) H(2) . . . H(k) (Forward)
35 = 'B': H = H(k) . . . H(2) H(1) (Backward)
36 STOREV (input) CHARACTER*1
38 Specifies how the vectors which define the elementary reflec‐
39 tors are stored (see also Further Details):
40 = 'R': rowwise
41 N (input) INTEGER
43 The order of the block reflector H. N >= 0.
44 K (input) INTEGER
46 The order of the triangular factor T (= the number of elemen‐
47 tary reflectors). K >= 1.
48 V (input/output) COMPLEX array, dimension
50 (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V.
51 See further details.
52 LDV (input) INTEGER
54 The leading dimension of the array V. If STOREV = 'C', LDV >=
55 max(1,N); if STOREV = 'R', LDV >= K.
56 TAU (input) COMPLEX array, dimension (K)
58 TAU(i) must contain the scalar factor of the elementary reflec‐
59 tor H(i).
60 T (output) COMPLEX array, dimension (LDT,K)
62 The k by k triangular factor T of the block reflector. If
63 DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
64 lower triangular. The rest of the array is not used.
65 LDT (input) INTEGER
67 The leading dimension of the array T. LDT >= K.
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70 The shape of the matrix V and the storage of the vectors which define
71 the H(i) is best illustrated by the following example with n = 5 and k
72 = 3. The elements equal to 1 are not stored; the corresponding array
73 elements are modified but restored on exit. The rest of the array is
74 not used.
75 DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
76 V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
77 ( v1 1 ) ( 1 v2 v2 v2 )
78 ( v1 v2 1 ) ( 1 v3 v3 )
79 ( v1 v2 v3 )
80 ( v1 v2 v3 )
81 DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
82 V = ( v1 v2 v3 ) V = ( v1 v1 1 )
83 ( v1 v2 v3 ) ( v2 v2 v2 1 )
84 ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
85 ( 1 v3 )
86 ( 1 )
87 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 CLARFT(1)