zlarft (l) - Linux Manuals

zlarft: forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary reflectors

Command to display zlarft manual in Linux: $ man l zlarft

NAME

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

SYNOPSIS

SUBROUTINE ZLARFT(: DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
: CHARACTER DIRECT, STOREV
: INTEGER K, LDT, LDV, N
: COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )

PURPOSE

ZLARFT 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 = aqFaq, H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = aqBaq, H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = aqCaq, the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and

H = I - V * T * Vaq
If STOREV = aqRaq, the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and

H = I - Vaq * T * V

ARGUMENTS

DIRECT (input) CHARACTER*1: Specifies the order in which the elementary reflectors are multiplied to form the block reflector:
= aqFaq: H = H(1) H(2) . . . H(k) (Forward)
= aqBaq: H = H(k) . . . H(2) H(1) (Backward)
STOREV (input) CHARACTER*1: Specifies how the vectors which define the elementary reflectors are stored (see also Further Details):
= aqRaq: rowwise
N (input) INTEGER: The order of the block reflector H. N >= 0.
K (input) INTEGER: The order of the triangular factor T (= the number of elementary reflectors). K >= 1.
V (input/output) COMPLEX*16 array, dimension: (LDV,K) if STOREV = aqCaq (LDV,N) if STOREV = aqRaq The matrix V. See further details.
LDV (input) INTEGER: The leading dimension of the array V. If STOREV = aqCaq, LDV >= max(1,N); if STOREV = aqRaq, LDV >= K.
TAU (input) COMPLEX*16 array, dimension (K): TAU(i) must contain the scalar factor of the elementary reflector H(i).
T (output) COMPLEX*16 array, dimension (LDT,K): The k by k triangular factor T of the block reflector. If DIRECT = aqFaq, T is upper triangular; if DIRECT = aqBaq, T is lower triangular. The rest of the array is not used.
LDT (input) INTEGER: The leading dimension of the array T. LDT >= K.

FURTHER DETAILS

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.
DIRECT = aqFaq and STOREV = aqCaq: DIRECT = aqFaq and STOREV = aqRaq:
       V = (  1       )                 V = (  1 v1 v1 v1 v1 )
           ( v1  1    )                     (     1 v2 v2 v2 )
           ( v1 v2  1 )                     (        1 v3 v3 )
           ( v1 v2 v3 )

           ( v1 v2 v3 )
DIRECT = aqBaq and STOREV = aqCaq: DIRECT = aqBaq and STOREV = aqRaq:
       V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
           ( v1 v2 v3 )                     ( v2 v2 v2  1    )
           (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
           (     1 v3 )

           (        1 )