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 multiplied 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 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

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 )