SLARFT (3lapack)

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 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

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)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

  STOREV  (input) CHARACTER*1
          Specifies how the vectors which define the elementary 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 elementary reflector
          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

  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 )