.TH DLARZT 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME DLARZT - the triangular factor T of a real block reflector H of order > n, which is defined as a product of k elementary reflectors .SH SYNOPSIS .TP 19 SUBROUTINE DLARZT( 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 DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * ) .SH PURPOSE DLARZT forms the triangular factor T of a real 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 Currently, only STOREV = \(aqR\(aq and DIRECT = \(aqB\(aq 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 = \(aqF\(aq: H = H(1) H(2) . . . H(k) (Forward, not supported yet) .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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i). .TP 8 T (output) DOUBLE PRECISION 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 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 = \(aqF\(aq and STOREV = \(aqC\(aq: DIRECT = \(aqF\(aq and STOREV = \(aqR\(aq: ______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 = \(aqB\(aq and STOREV = \(aqC\(aq: DIRECT = \(aqB\(aq and STOREV = \(aqR\(aq: ______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