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.TH CLARZT l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
CLARZT - form the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors
.SH SYNOPSIS
.TP 19
SUBROUTINE CLARZT(
DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
.TP 19
.ti +4
CHARACTER
DIRECT, STOREV
.TP 19
.ti +4
INTEGER
K, LDT, LDV, N
.TP 19
.ti +4
COMPLEX
T( LDT, * ), TAU( * ), V( LDV, * )
.SH PURPOSE
CLARZT 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
.br
H = I - V * T * V'
.br
If STOREV = 'R', the vector which defines the elementary reflector
H(i) is stored in the i-th row of the array V, and
.br
H = I - V' * T * V
.br
Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
.SH ARGUMENTS
.TP 8
DIRECT (input) CHARACTER*1
Specifies the order in which the elementary reflectors are
multiplied to form the block reflector:
.br
= 'F': H = H(1) H(2) . . . H(k) (Forward, not supported yet)
.br
= 'B': H = H(k) . . . H(2) H(1) (Backward)
.TP 8
STOREV (input) CHARACTER*1
Specifies how the vectors which define the elementary
reflectors are stored (see also Further Details):
.br
= 'R': rowwise
.TP 8
N (input) INTEGER
The order of the block reflector H. N >= 0.
.TP 8
K (input) INTEGER
The order of the triangular factor T (= the number of
elementary reflectors). K >= 1.
.TP 8
V (input/output) COMPLEX array, dimension
(LDV,K) if STOREV = 'C'
(LDV,N) if STOREV = 'R'
The matrix V. See further details.
.TP 8
LDV (input) INTEGER
The leading dimension of the array V.
If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
.TP 8
TAU (input) COMPLEX array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i).
.TP 8
T (output) COMPLEX 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.
.TP 8
LDT (input) INTEGER
The leading dimension of the array T. LDT >= K.
.SH FURTHER DETAILS
Based on contributions by
.br
A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
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.
.br
DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
______V_____
.br
( v1 v2 v3 ) / \
( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 )
V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 )
( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 )
( v1 v2 v3 )
.br
. . .
.br
. . .
.br
1 . .
.br
1 .
.br
1
.br
DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
______V_____
1 / \
. 1 ( 1 . . . . v1 v1 v1 v1 v1 )
. . 1 ( . 1 . . . v2 v2 v2 v2 v2 )
. . . ( . . 1 . . v3 v3 v3 v3 v3 )
. . .
.br
( v1 v2 v3 )
.br
( v1 v2 v3 )
.br
V = ( v1 v2 v3 )
.br
( v1 v2 v3 )
.br
( v1 v2 v3 )
.br
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