LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ clarft()

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

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

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

Purpose:
 CLARFT forms the triangular factor T of a complex 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**H

 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**H * T * V
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)
          = '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
          = '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]V
          V is COMPLEX 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 COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).
[out]T
          T is COMPLEX 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.
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 )

Definition at line 162 of file clarft.f.

163*
164* -- LAPACK auxiliary routine --
165* -- LAPACK is a software package provided by Univ. of Tennessee, --
166* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167*
168* .. Scalar Arguments ..
169 CHARACTER DIRECT, STOREV
170 INTEGER K, LDT, LDV, N
171* ..
172* .. Array Arguments ..
173 COMPLEX T( LDT, * ), TAU( * ), V( LDV, * )
174* ..
175*
176* =====================================================================
177*
178* .. Parameters ..
179 COMPLEX ONE, ZERO
180 parameter( one = ( 1.0e+0, 0.0e+0 ),
181 $ zero = ( 0.0e+0, 0.0e+0 ) )
182* ..
183* .. Local Scalars ..
184 INTEGER I, J, PREVLASTV, LASTV
185* ..
186* .. External Subroutines ..
187 EXTERNAL cgemm, cgemv, ctrmv
188* ..
189* .. External Functions ..
190 LOGICAL LSAME
191 EXTERNAL lsame
192* ..
193* .. Executable Statements ..
194*
195* Quick return if possible
196*
197 IF( n.EQ.0 )
198 $ RETURN
199*
200 IF( lsame( direct, 'F' ) ) THEN
201 prevlastv = n
202 DO i = 1, k
203 prevlastv = max( prevlastv, i )
204 IF( tau( i ).EQ.zero ) THEN
205*
206* H(i) = I
207*
208 DO j = 1, i
209 t( j, i ) = zero
210 END DO
211 ELSE
212*
213* general case
214*
215 IF( lsame( storev, 'C' ) ) THEN
216* Skip any trailing zeros.
217 DO lastv = n, i+1, -1
218 IF( v( lastv, i ).NE.zero ) EXIT
219 END DO
220 DO j = 1, i-1
221 t( j, i ) = -tau( i ) * conjg( v( i , j ) )
222 END DO
223 j = min( lastv, prevlastv )
224*
225* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**H * V(i:j,i)
226*
227 CALL cgemv( 'Conjugate transpose', j-i, i-1,
228 $ -tau( i ), v( i+1, 1 ), ldv,
229 $ v( i+1, i ), 1,
230 $ one, t( 1, i ), 1 )
231 ELSE
232* Skip any trailing zeros.
233 DO lastv = n, i+1, -1
234 IF( v( i, lastv ).NE.zero ) EXIT
235 END DO
236 DO j = 1, i-1
237 t( j, i ) = -tau( i ) * v( j , i )
238 END DO
239 j = min( lastv, prevlastv )
240*
241* T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**H
242*
243 CALL cgemm( 'N', 'C', i-1, 1, j-i, -tau( i ),
244 $ v( 1, i+1 ), ldv, v( i, i+1 ), ldv,
245 $ one, t( 1, i ), ldt )
246 END IF
247*
248* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
249*
250 CALL ctrmv( 'Upper', 'No transpose', 'Non-unit', i-1, t,
251 $ ldt, t( 1, i ), 1 )
252 t( i, i ) = tau( i )
253 IF( i.GT.1 ) THEN
254 prevlastv = max( prevlastv, lastv )
255 ELSE
256 prevlastv = lastv
257 END IF
258 END IF
259 END DO
260 ELSE
261 prevlastv = 1
262 DO i = k, 1, -1
263 IF( tau( i ).EQ.zero ) THEN
264*
265* H(i) = I
266*
267 DO j = i, k
268 t( j, i ) = zero
269 END DO
270 ELSE
271*
272* general case
273*
274 IF( i.LT.k ) THEN
275 IF( lsame( storev, 'C' ) ) THEN
276* Skip any leading zeros.
277 DO lastv = 1, i-1
278 IF( v( lastv, i ).NE.zero ) EXIT
279 END DO
280 DO j = i+1, k
281 t( j, i ) = -tau( i ) * conjg( v( n-k+i , j ) )
282 END DO
283 j = max( lastv, prevlastv )
284*
285* T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**H * V(j:n-k+i,i)
286*
287 CALL cgemv( 'Conjugate transpose', n-k+i-j, k-i,
288 $ -tau( i ), v( j, i+1 ), ldv, v( j, i ),
289 $ 1, one, t( i+1, i ), 1 )
290 ELSE
291* Skip any leading zeros.
292 DO lastv = 1, i-1
293 IF( v( i, lastv ).NE.zero ) EXIT
294 END DO
295 DO j = i+1, k
296 t( j, i ) = -tau( i ) * v( j, n-k+i )
297 END DO
298 j = max( lastv, prevlastv )
299*
300* T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**H
301*
302 CALL cgemm( 'N', 'C', k-i, 1, n-k+i-j, -tau( i ),
303 $ v( i+1, j ), ldv, v( i, j ), ldv,
304 $ one, t( i+1, i ), ldt )
305 END IF
306*
307* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
308*
309 CALL ctrmv( 'Lower', 'No transpose', 'Non-unit', k-i,
310 $ t( i+1, i+1 ), ldt, t( i+1, i ), 1 )
311 IF( i.GT.1 ) THEN
312 prevlastv = min( prevlastv, lastv )
313 ELSE
314 prevlastv = lastv
315 END IF
316 END IF
317 t( i, i ) = tau( i )
318 END IF
319 END DO
320 END IF
321 RETURN
322*
323* End of CLARFT
324*
cgemm
subroutine cgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
CGEMM
Definition cgemm.f:188
cgemv
subroutine cgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
CGEMV
Definition cgemv.f:160
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
ctrmv
subroutine ctrmv(uplo, trans, diag, n, a, lda, x, incx)
CTRMV
Definition ctrmv.f:147
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