1ZLARZT(1) LAPACK routine (version 3.2) ZLARZT(1)
2
6 ZLARZT - 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
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; 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
29 Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
30
32 DIRECT (input) CHARACTER*1
33 Specifies the order in which the elementary reflectors are mul‐
34 tiplied to form the block reflector:
35 = 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
36 = 'B': H = H(k) . . . H(2) H(1) (Backward)
37 STOREV (input) CHARACTER*1
39 Specifies how the vectors which define the elementary reflec‐
40 tors are stored (see also Further Details):
41 = 'R': rowwise
42 N (input) INTEGER
44 The order of the block reflector H. N >= 0.
45 K (input) INTEGER
47 The order of the triangular factor T (= the number of elemen‐
48 tary reflectors). K >= 1.
49 V (input/output) COMPLEX*16 array, dimension
51 (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V.
52 See further details.
53 LDV (input) INTEGER
55 The leading dimension of the array V. If STOREV = 'C', LDV >=
56 max(1,N); if STOREV = 'R', LDV >= K.
57 TAU (input) COMPLEX*16 array, dimension (K)
59 TAU(i) must contain the scalar factor of the elementary reflec‐
60 tor H(i).
61 T (output) COMPLEX*16 array, dimension (LDT,K)
63 The k by k triangular factor T of the block reflector. If
64 DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
65 lower triangular. The rest of the array is not used.
66 LDT (input) INTEGER
68 The leading dimension of the array T. LDT >= K.
69
71 Based on contributions by
72 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
73 The shape of the matrix V and the storage of the vectors which define
74 the H(i) is best illustrated by the following example with n = 5 and k
75 = 3. The elements equal to 1 are not stored; the corresponding array
76 elements are modified but restored on exit. The rest of the array is
77 not used.
78 DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
79 ______V_____
80 ( v1 v2 v3 ) / ( v1 v2
81 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
82 V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
83 ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
84 ( v1 v2 v3 )
85 . . .
86 . . .
87 1 . .
88 1 .
89 1
90 DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
91 ______V_____
92 1 /
93 . 1 ( 1 . . . . v1 v1 v1 v1 v1 )
94 . . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
95 . . . ( . . 1 . . v3 v3 v3 v3 v3 )
96 . . .
97 ( v1 v2 v3 )
98 ( v1 v2 v3 )
99 V = ( v1 v2 v3 )
100 ( v1 v2 v3 )
101 ( v1 v2 v3 )
102 LAPACK routine (version 3.2) November 2008 ZLARZT(1)