.TH ZLARFT 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME ZLARFT - 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 ZLARFT( 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*16 T( LDT, * ), TAU( * ), V( LDV, * ) .SH 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 = \(aqF\(aq, H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = \(aqB\(aq, H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = \(aqC\(aq, 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\(aq .br If STOREV = \(aqR\(aq, 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\(aq * T * V .br .SH ARGUMENTS .TP 8 DIRECT (input) CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: .br = \(aqF\(aq: H = H(1) H(2) . . . H(k) (Forward) .br = \(aqB\(aq: 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 = \(aqR\(aq: 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*16 array, dimension (LDV,K) if STOREV = \(aqC\(aq (LDV,N) if STOREV = \(aqR\(aq The matrix V. See further details. .TP 8 LDV (input) INTEGER The leading dimension of the array V. If STOREV = \(aqC\(aq, LDV >= max(1,N); if STOREV = \(aqR\(aq, LDV >= K. .TP 8 TAU (input) COMPLEX*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i). .TP 8 T (output) COMPLEX*16 array, dimension (LDT,K) The k by k triangular factor T of the block reflector. If DIRECT = \(aqF\(aq, T is upper triangular; if DIRECT = \(aqB\(aq, 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 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 = \(aqF\(aq and STOREV = \(aqC\(aq: DIRECT = \(aqF\(aq and STOREV = \(aqR\(aq: V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ( v1 1 ) ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) .br ( v1 v2 v3 ) .br DIRECT = \(aqB\(aq and STOREV = \(aqC\(aq: DIRECT = \(aqB\(aq and STOREV = \(aqR\(aq: 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 ) .br ( 1 ) .br