SLARZT(3)                  MathKeisan LAPACK routine                 SLARZT(3)



NAME
       SLARZT - the triangular factor T of a real block reflector H of order >
       n, which is defined as a product of k elementary reflectors

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

           CHARACTER      DIRECT, STOREV

           INTEGER        K, LDT, LDV, N

           REAL           T( LDT, * ), TAU( * ), V( LDV, * )

PURPOSE
       SLARZT 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'

       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.


ARGUMENTS
       DIRECT  (input) CHARACTER*1
               Specifies the order in which the elementary reflectors are mul-
               tiplied 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  reflec-
               tors 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 elemen-
               tary reflectors). K >= 1.

       V       (input/output) REAL 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) REAL array, dimension (K)
               TAU(i) must contain the scalar factor of the elementary reflec-
               tor H(i).

       T       (output) REAL 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 DETAILS
       Based 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 )




 LAPACK routine (version 3.1)    November 2006                       SLARZT(3)