LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ 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

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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*
subroutine zgemm(transa, transb, m, n, k, alpha, a, lda, b, ldb, beta, c, ldc)
ZGEMM
Definition zgemm.f:188
subroutine zgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
ZGEMV
Definition zgemv.f:160
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine ztrmv(uplo, trans, diag, n, a, lda, x, incx)
ZTRMV
Definition ztrmv.f:147
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