LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ zlarft()

 subroutine zlarft ( character DIRECT, character STOREV, integer N, integer K, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) TAU, complex*16, dimension( ldt, * ) T, integer LDT )

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

Purpose:
``` ZLARFT 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*16 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*16 array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i).``` [out] T ``` T is 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.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= K.```
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 zlarft.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*16 T( LDT, * ), TAU( * ), V( LDV, * )
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  COMPLEX*16 ONE, ZERO
180  parameter( one = ( 1.0d+0, 0.0d+0 ),
181  \$ zero = ( 0.0d+0, 0.0d+0 ) )
182 * ..
183 * .. Local Scalars ..
184  INTEGER I, J, PREVLASTV, LASTV
185 * ..
186 * .. External Subroutines ..
187  EXTERNAL zgemv, ztrmv, zgemm
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 zgemv( 'Conjugate transpose', j-i, i-1,
228  \$ -tau( i ), v( i+1, 1 ), ldv,
229  \$ v( i+1, i ), 1, one, t( 1, i ), 1 )
230  ELSE
231 * Skip any trailing zeros.
232  DO lastv = n, i+1, -1
233  IF( v( i, lastv ).NE.zero ) EXIT
234  END DO
235  DO j = 1, i-1
236  t( j, i ) = -tau( i ) * v( j , i )
237  END DO
238  j = min( lastv, prevlastv )
239 *
240 * T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**H
241 *
242  CALL zgemm( 'N', 'C', i-1, 1, j-i, -tau( i ),
243  \$ v( 1, i+1 ), ldv, v( i, i+1 ), ldv,
244  \$ one, t( 1, i ), ldt )
245  END IF
246 *
247 * T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
248 *
249  CALL ztrmv( 'Upper', 'No transpose', 'Non-unit', i-1, t,
250  \$ ldt, t( 1, i ), 1 )
251  t( i, i ) = tau( i )
252  IF( i.GT.1 ) THEN
253  prevlastv = max( prevlastv, lastv )
254  ELSE
255  prevlastv = lastv
256  END IF
257  END IF
258  END DO
259  ELSE
260  prevlastv = 1
261  DO i = k, 1, -1
262  IF( tau( i ).EQ.zero ) THEN
263 *
264 * H(i) = I
265 *
266  DO j = i, k
267  t( j, i ) = zero
268  END DO
269  ELSE
270 *
271 * general case
272 *
273  IF( i.LT.k ) THEN
274  IF( lsame( storev, 'C' ) ) THEN
275 * Skip any leading zeros.
276  DO lastv = 1, i-1
277  IF( v( lastv, i ).NE.zero ) EXIT
278  END DO
279  DO j = i+1, k
280  t( j, i ) = -tau( i ) * conjg( v( n-k+i , j ) )
281  END DO
282  j = max( lastv, prevlastv )
283 *
284 * T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**H * V(j:n-k+i,i)
285 *
286  CALL zgemv( 'Conjugate transpose', n-k+i-j, k-i,
287  \$ -tau( i ), v( j, i+1 ), ldv, v( j, i ),
288  \$ 1, one, t( i+1, i ), 1 )
289  ELSE
290 * Skip any leading zeros.
291  DO lastv = 1, i-1
292  IF( v( i, lastv ).NE.zero ) EXIT
293  END DO
294  DO j = i+1, k
295  t( j, i ) = -tau( i ) * v( j, n-k+i )
296  END DO
297  j = max( lastv, prevlastv )
298 *
299 * T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**H
300 *
301  CALL zgemm( 'N', 'C', k-i, 1, n-k+i-j, -tau( i ),
302  \$ v( i+1, j ), ldv, v( i, j ), ldv,
303  \$ one, t( i+1, i ), ldt )
304  END IF
305 *
306 * T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
307 *
308  CALL ztrmv( 'Lower', 'No transpose', 'Non-unit', k-i,
309  \$ t( i+1, i+1 ), ldt, t( i+1, i ), 1 )
310  IF( i.GT.1 ) THEN
311  prevlastv = min( prevlastv, lastv )
312  ELSE
313  prevlastv = lastv
314  END IF
315  END IF
316  t( i, i ) = tau( i )
317  END IF
318  END DO
319  END IF
320  RETURN
321 *
322 * End of ZLARFT
323 *
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ztrmv(UPLO, TRANS, DIAG, N, A, LDA, X, INCX)
ZTRMV
Definition: ztrmv.f:147
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
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