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slarft

 NAME
      SLARFT - 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 SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT
                         )

          CHARACTER      DIRECT, STOREV

          INTEGER        K, LDT, LDV, N

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

 PURPOSE
      SLARFT forms the triangular factor T of a real block reflec-
      tor H of order n, which is defined as a product of k elemen-
      tary 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

 ARGUMENTS
      DIRECT  (input) CHARACTER*1
              Specifies the order in which the elementary reflec-
              tors 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) CHARACTER*1
              Specifies how the vectors which define the elemen-
              tary 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) 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 elemen-
              tary reflector H(i).

      T       (output) REAL array, dimension (LDT,K)
              The k by k triangular factor T of the block reflec-
              tor.  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
      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 = (  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 )