1ZLARZT(1) LAPACK routine (version 3.1) ZLARZT(1)
2
6 ZLARZT - the triangular factor T of a complex block reflector H of
7 order > n, which is defined as a product of k elementary reflectors
8
10 SUBROUTINE ZLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
11 CHARACTER DIRECT, STOREV
13 INTEGER K, LDT, LDV, N
15 COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
17
19 ZLARZT 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.
21 If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
23 If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
25 If STOREV = 'C', the vector which defines the elementary reflector H(i)
27 is stored in the i-th column of the array V, and
28 H = I - V * T * V'
30 If STOREV = 'R', the vector which defines the elementary reflector H(i)
32 is stored in the i-th row of the array V, and
33 H = I - V' * T * V
35 Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
37
40 DIRECT (input) CHARACTER*1
41 Specifies the order in which the elementary reflectors are mul‐
42 tiplied to form the block reflector:
43 = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
44 = 'B': H = H(k) . . . H(2) H(1) (Backward)
45 STOREV (input) CHARACTER*1
47 Specifies how the vectors which define the elementary reflec‐
48 tors are stored (see also Further Details):
49 = 'R': rowwise
50 N (input) INTEGER
52 The order of the block reflector H. N >= 0.
53 K (input) INTEGER
55 The order of the triangular factor T (= the number of elemen‐
56 tary reflectors). K >= 1.
57 V (input/output) COMPLEX*16 array, dimension
59 (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V.
60 See further details.
61 LDV (input) INTEGER
63 The leading dimension of the array V. If STOREV = 'C', LDV >=
64 max(1,N); if STOREV = 'R', LDV >= K.
65 TAU (input) COMPLEX*16 array, dimension (K)
67 TAU(i) must contain the scalar factor of the elementary reflec‐
68 tor H(i).
69 T (output) COMPLEX*16 array, dimension (LDT,K)
71 The k by k triangular factor T of the block reflector. If
72 DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
73 lower triangular. The rest of the array is not used.
74 LDT (input) INTEGER
76 The leading dimension of the array T. LDT >= K.
77
79 Based on contributions by
80 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
81 The shape of the matrix V and the storage of the vectors which define
83 the H(i) is best illustrated by the following example with n = 5 and k
84 = 3. The elements equal to 1 are not stored; the corresponding array
85 elements are modified but restored on exit. The rest of the array is
86 not used.
87 DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
89 ______V_____
91 ( v1 v2 v3 ) / ( v1 v2
92 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
93 V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
94 ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
95 ( v1 v2 v3 )
96 . . .
97 . . .
98 1 . .
99 1 .
100 1
101 DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
103 ______V_____
105 1 /
106 . 1 ( 1 . . . . v1 v1 v1 v1 v1 )
107 . . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
108 . . . ( . . 1 . . v3 v3 v3 v3 v3 )
109 . . .
110 ( v1 v2 v3 )
111 ( v1 v2 v3 )
112 V = ( v1 v2 v3 )
113 ( v1 v2 v3 )
114 ( v1 v2 v3 )
115 LAPACK routine (version 3.1) November 2006 ZLARZT(1)