ZLARFT(3)      LAPACK routine of NEC Numeric Library Collection      ZLARFT(3)



NAME
       ZLARFT

SYNOPSIS
       SUBROUTINE ZLARFT (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)



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**H

            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**H * T * V




ARGUMENTS
           DIRECT    (input)
                     DIRECT is 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)
                     STOREV is CHARACTER*1
                     Specifies how the vectors which define the elementary
                     reflectors are stored (see also Further Details):
                     = 'C': columnwise
                     = 'R': rowwise

           N         (input)
                     N is INTEGER
                     The order of the block reflector H. N >= 0.

           K         (input)
                     K is INTEGER
                     The order of the triangular factor T (= the number of
                     elementary reflectors). K >= 1.

           V         (input)
                     V is COMPLEX*16 array, dimension
                                          (LDV,K) if STOREV = 'C'
                                          (LDV,N) if STOREV = 'R'
                     The matrix V. See further details.

           LDV       (input)
                     LDV is INTEGER
                     The leading dimension of the array V.
                     If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

           TAU       (input)
                     TAU is COMPLEX*16 array, dimension (K)
                     TAU(i) must contain the scalar factor of the elementary
                     reflector H(i).

           T         (output)
                     T is 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)
                     LDT is 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.

             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 routine                  31 October 2017                      ZLARFT(3)