slarzt()

subroutine slarzt	(	character	direct,
		character	storev,
		integer	n,
		integer	k,
		real, dimension( ldv, * )	v,
		integer	ldv,
		real, dimension( * )	tau,
		real, dimension( ldt, * )	t,
		integer	ldt )

SLARZT forms the triangular factor T of a block reflector H = I - vtvH.

Download SLARZT + dependencies [TGZ] [ZIP] [TXT]

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**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
!>
!> Currently, only STOREV = 'R' and DIRECT = 'B' are supported.
!> 

Parameters

[in]	DIRECT	!> DIRECT is 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) !>
[in]	STOREV	!> STOREV is CHARACTER*1 !> Specifies how the vectors which define the elementary !> reflectors are stored (see also Further Details): !> = 'C': columnwise (not supported yet) !> = 'R': rowwise !>
[in]	N	!> N is INTEGER !> The order of the block reflector H. N >= 0. !>
[in]	K	!> K is INTEGER !> The order of the triangular factor T (= the number of !> elementary reflectors). K >= 1. !>
[in,out]	V	!> V is REAL array, dimension !> (LDV,K) if STOREV = 'C' !> (LDV,N) if STOREV = 'R' !> The matrix V. See further details. !>
[in]	LDV	!> LDV is INTEGER !> The leading dimension of the array V. !> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. !>
[in]	TAU	!> TAU is REAL array, dimension (K) !> TAU(i) must contain the scalar factor of the elementary !> reflector H(i). !>
[out]	T	!> T is 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. !>
[in]	LDT	!> LDT is INTEGER !> The leading dimension of the array T. LDT >= K. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

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_____
!>         ( 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 )
!> 

Definition at line 182 of file slarzt.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          DIRECT, STOREV
      INTEGER            K, LDT, LDV, N
*     ..
*     .. Array Arguments ..
      REAL               T( LDT, * ), TAU( * ), V( LDV, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO
      parameter( zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, INFO, J
*     ..
*     .. External Subroutines ..
      EXTERNAL           sgemv, strmv, xerbla
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. Executable Statements ..
*
*     Check for currently supported options
*
      info = 0
      IF( .NOT.lsame( direct, 'B' ) ) THEN
         info = -1
      ELSE IF( .NOT.lsame( storev, 'R' ) ) THEN
         info = -2
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SLARZT', -info )
         RETURN
      END IF
*
      DO 20 i = k, 1, -1
         IF( tau( i ).EQ.zero ) THEN
*
*           H(i)  =  I
*
            DO 10 j = i, k
               t( j, i ) = zero
   10       CONTINUE
         ELSE
*
*           general case
*
            IF( i.LT.k ) THEN
*
*              T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)**T
*
               CALL sgemv( 'No transpose', k-i, n, -tau( i ),
     $                     v( i+1, 1 ), ldv, v( i, 1 ), ldv, zero,
     $                     t( i+1, i ), 1 )
*
*              T(i+1:k,i) = T(i+1:k,i+1:k) * T(i+1:k,i)
*
               CALL strmv( 'Lower', 'No transpose', 'Non-unit', k-i,
     $                     t( i+1, i+1 ), ldt, t( i+1, i ), 1 )
            END IF
            t( i, i ) = tau( i )
         END IF
   20 CONTINUE
      RETURN
*
*     End of SLARZT
*

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