1ZLARFT(1)           LAPACK auxiliary routine (version 3.1)           ZLARFT(1)
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NAME

6       ZLARFT  -  the  triangular  factor  T of a complex block reflector H of
7       order n, which is defined as a product of k elementary reflectors
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SYNOPSIS

10       SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
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12           CHARACTER      DIRECT, STOREV
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14           INTEGER        K, LDT, LDV, N
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16           COMPLEX*16     T( LDT, * ), TAU( * ), V( LDV, * )
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PURPOSE

19       ZLARFT 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.
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22       If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
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24       If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
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26       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
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29          H  =  I - V * T * V'
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31       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
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34          H  =  I - V' * T * V
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ARGUMENTS

38       DIRECT  (input) CHARACTER*1
39               Specifies the order in which the elementary reflectors are mul‐
40               tiplied to form the block reflector:
41               = 'F': H = H(1) H(2) . . . H(k) (Forward)
42               = 'B': H = H(k) . . . H(2) H(1) (Backward)
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44       STOREV  (input) CHARACTER*1
45               Specifies how the vectors which define the  elementary  reflec‐
46               tors are stored (see also Further Details):
47               = 'R': rowwise
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49       N       (input) INTEGER
50               The order of the block reflector H. N >= 0.
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52       K       (input) INTEGER
53               The  order  of the triangular factor T (= the number of elemen‐
54               tary reflectors). K >= 1.
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56       V       (input/output) COMPLEX*16 array, dimension
57               (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The  matrix  V.
58               See further details.
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60       LDV     (input) INTEGER
61               The  leading dimension of the array V.  If STOREV = 'C', LDV >=
62               max(1,N); if STOREV = 'R', LDV >= K.
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64       TAU     (input) COMPLEX*16 array, dimension (K)
65               TAU(i) must contain the scalar factor of the elementary reflec‐
66               tor H(i).
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68       T       (output) COMPLEX*16 array, dimension (LDT,K)
69               The  k  by  k  triangular  factor T of the block reflector.  If
70               DIRECT = 'F', T is upper triangular; if  DIRECT  =  'B',  T  is
71               lower triangular. The rest of the array is not used.
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73       LDT     (input) INTEGER
74               The leading dimension of the array T. LDT >= K.
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FURTHER DETAILS

77       The  shape  of the matrix V and the storage of the vectors which define
78       the H(i) is best illustrated by the following example with n = 5 and  k
79       =  3.  The  elements equal to 1 are not stored; the corresponding array
80       elements are modified but restored on exit. The rest of  the  array  is
81       not used.
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83       DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':
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85                    V = (  1       )                 V = (  1 v1 v1 v1 v1 )
86                        ( v1  1    )                     (     1 v2 v2 v2 )
87                        ( v1 v2  1 )                     (        1 v3 v3 )
88                        ( v1 v2 v3 )
89                        ( v1 v2 v3 )
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91       DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':
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93                    V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
94                        ( v1 v2 v3 )                     ( v2 v2 v2  1    )
95                        (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
96                        (     1 v3 )
97                        (        1 )
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102 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       ZLARFT(1)