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



NAME
       DLARZT

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



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 = '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**T

            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 * T * V

            Currently, only STOREV = 'R' and DIRECT = 'B' are supported.




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, not supported yet)
                     = '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                        (not supported yet)
                     = '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/output)
                     V is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (K)
                     TAU(i) must contain the scalar factor of the elementary
                     reflector H(i).

           T         (output)
                     T is DOUBLE PRECISION 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; 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 )



LAPACK routine                  31 October 2017                      DLARZT(3)