LARFT - 5.2 English - 68552

Triangular factor of a block reflector.

Purpose:

    Formation of 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

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.

                  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)

Parameters:

• direct – [in]

  direct is char*

  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 – [in]

  storev is char*

  Specifies how the vectors which define the elementary reflectors are stored (see also Further Details):

  = ‘C’: columnwise

  = ‘R’: rowwise

• n – [in]

  n is integer*

  The order of the block reflector H. n >= 0.

• k – [in]

  k is integer*

  The order of the triangular factor T (= the number of elementary reflectors). K >= 1.

• v – [in]

  v is float/double/COMPLEX/COMPLEX*16 array, dimension

  (ldv,k) if storev = ‘C’

  (ldv,k) if storev = ‘R’

  The matrix v. See further details.

• ldv – [in]

  ldv is integer*

  The leading dimension of the array V.

  If storev = ‘C’, ldv >= fla_max(1,n); if storev = ‘R’, ldv >= k.

• tau – [in]

  tau is float/double/COMPLEX/COMPLEX*16 array, dimension (k)

  tau(i) must contain the scalar factor of the elementary reflector H(i).

• t – [out]

  t is float/double/COMPLEX/COMPLEX*16 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 – [in]

  ldt is integer*

  The leading dimension of the array T. ldt >= k.