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### LAPACK-3: forms the triangular factor T of a complex block reflector H of order n, which is defined as a product of k elementary

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## NAME

LAPACK-3 - 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 = '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 H = I - V * T * V' If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and H = I - V' * T * V## ARGUMENTS

DIRECT (input) CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': 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): = 'R': 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 = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details. LDV (input) INTEGER The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', 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 = 'F', T is upper triangular; if DIRECT = 'B', 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 = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': 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 = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': 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 ) LAPACK auxiliary routine (versionMarch 2011 LAPACK-3(3)

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