DLARZT  -  form  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 DLARZT( DIRECT,  STOREV,  N, K, V, LDV, TAU, T,
                          LDT )

           CHARACTER      DIRECT, STOREV

           INTEGER        K, LDT, LDV, N

           DOUBLE         PRECISION T( LDT, * ),  TAU(  *  ),  V(
                          LDV, * )


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'

       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 sup­
       ported.


ARGUMENTS

       DIRECT  (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 sup­
               ported yet)
               = 'B': H = H(k) . . . H(2) H(1) (Backward)

       STOREV  (input) CHARACTER*1
               Specifies how the vectors which define the elemen­
               tary  reflectors  are  stored  (see  also  Further
               Details):


       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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (K)
               TAU(i) must contain the scalar factor of the  ele­
               mentary reflector H(i).

       T       (output) DOUBLE PRECISION array, dimension (LDT,K)
               The k by  k  triangular  factor  T  of  the  block
               reflector.   If  DIRECT = 'F', T is upper triangu­
               lar; 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 )


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