ZLARZT - Online Linux Manual Page

Section : 1
Updated : November 2008
Source : LAPACK routine (version 3.2)
Note : LAPACK routine (version 3.2)
NAMEZLARZT - forms the triangular factor T of a complex block reflector H of order > n, which is defined as a product of k elementary reflectors
SYNOPSISSUBROUTINE ZLARZT(  DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )  CHARACTER DIRECT, STOREV  INTEGER K, LDT, LDV, N  COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
PURPOSEZLARZT 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
Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
ARGUMENTSDIRECT (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, not supported yet)
= '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 DETAILSBased on contributions by

  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.
DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
                                            ______V_____

       ( 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 )

          . . .

          . . .

          1 . .

             1 .

                1
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 )
          . . .

       ( v1 v2 v3 )

       ( v1 v2 v3 )

   V = ( v1 v2 v3 )

       ( v1 v2 v3 )

       ( v1 v2 v3 )
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