LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ 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 184 of file slarzt.f.

185 *
186 * -- LAPACK computational routine --
187 * -- LAPACK is a software package provided by Univ. of Tennessee, --
188 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189 *
190 * .. Scalar Arguments ..
191  CHARACTER DIRECT, STOREV
192  INTEGER K, LDT, LDV, N
193 * ..
194 * .. Array Arguments ..
195  REAL T( LDT, * ), TAU( * ), V( LDV, * )
196 * ..
197 *
198 * =====================================================================
199 *
200 * .. Parameters ..
201  REAL ZERO
202  parameter( zero = 0.0e+0 )
203 * ..
204 * .. Local Scalars ..
205  INTEGER I, INFO, J
206 * ..
207 * .. External Subroutines ..
208  EXTERNAL sgemv, strmv, xerbla
209 * ..
210 * .. External Functions ..
211  LOGICAL LSAME
212  EXTERNAL lsame
213 * ..
214 * .. Executable Statements ..
215 *
216 * Check for currently supported options
217 *
218  info = 0
219  IF( .NOT.lsame( direct, 'B' ) ) THEN
220  info = -1
221  ELSE IF( .NOT.lsame( storev, 'R' ) ) THEN
222  info = -2
223  END IF
224  IF( info.NE.0 ) THEN
225  CALL xerbla( 'SLARZT', -info )
226  RETURN
227  END IF
228 *
229  DO 20 i = k, 1, -1
230  IF( tau( i ).EQ.zero ) THEN
231 *
232 * H(i) = I
233 *
234  DO 10 j = i, k
235  t( j, i ) = zero
236  10 CONTINUE
237  ELSE
238 *
239 * general case
240 *
241  IF( i.LT.k ) THEN
242 *
243 * T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)**T
244 *
245  CALL sgemv( 'No transpose', k-i, n, -tau( i ),
246  $ v( i+1, 1 ), ldv, v( i, 1 ), ldv, zero,
247  $ t( i+1, i ), 1 )
248 *
249 * T(i+1:k,i) = T(i+1:k,i+1:k) * T(i+1:k,i)
250 *
251  CALL strmv( 'Lower', 'No transpose', 'Non-unit', k-i,
252  $ t( i+1, i+1 ), ldt, t( i+1, i ), 1 )
253  END IF
254  t( i, i ) = tau( i )
255  END IF
256  20 CONTINUE
257  RETURN
258 *
259 * End of SLARZT
260 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine strmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
STRMV
Definition: strmv.f:147
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
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