LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ slarft()

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

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

Purpose:
``` SLARFT 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```
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 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.```
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 slarft.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 REAL T( LDT, * ), TAU( * ), V( LDV, * )
174* ..
175*
176* =====================================================================
177*
178* .. Parameters ..
179 REAL ONE, ZERO
180 parameter( one = 1.0e+0, zero = 0.0e+0 )
181* ..
182* .. Local Scalars ..
183 INTEGER I, J, PREVLASTV, LASTV
184* ..
185* .. External Subroutines ..
186 EXTERNAL sgemv, strmv
187* ..
188* .. External Functions ..
189 LOGICAL LSAME
190 EXTERNAL lsame
191* ..
192* .. Executable Statements ..
193*
194* Quick return if possible
195*
196 IF( n.EQ.0 )
197 \$ RETURN
198*
199 IF( lsame( direct, 'F' ) ) THEN
200 prevlastv = n
201 DO i = 1, k
202 prevlastv = max( i, prevlastv )
203 IF( tau( i ).EQ.zero ) THEN
204*
205* H(i) = I
206*
207 DO j = 1, i
208 t( j, i ) = zero
209 END DO
210 ELSE
211*
212* general case
213*
214 IF( lsame( storev, 'C' ) ) THEN
215* Skip any trailing zeros.
216 DO lastv = n, i+1, -1
217 IF( v( lastv, i ).NE.zero ) EXIT
218 END DO
219 DO j = 1, i-1
220 t( j, i ) = -tau( i ) * v( i , j )
221 END DO
222 j = min( lastv, prevlastv )
223*
224* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**T * V(i:j,i)
225*
226 CALL sgemv( 'Transpose', j-i, i-1, -tau( i ),
227 \$ v( i+1, 1 ), ldv, v( i+1, i ), 1, one,
228 \$ t( 1, i ), 1 )
229 ELSE
230* Skip any trailing zeros.
231 DO lastv = n, i+1, -1
232 IF( v( i, lastv ).NE.zero ) EXIT
233 END DO
234 DO j = 1, i-1
235 t( j, i ) = -tau( i ) * v( j , i )
236 END DO
237 j = min( lastv, prevlastv )
238*
239* T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**T
240*
241 CALL sgemv( 'No transpose', i-1, j-i, -tau( i ),
242 \$ v( 1, i+1 ), ldv, v( i, i+1 ), ldv,
243 \$ one, t( 1, i ), 1 )
244 END IF
245*
246* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
247*
248 CALL strmv( 'Upper', 'No transpose', 'Non-unit', i-1, t,
249 \$ ldt, t( 1, i ), 1 )
250 t( i, i ) = tau( i )
251 IF( i.GT.1 ) THEN
252 prevlastv = max( prevlastv, lastv )
253 ELSE
254 prevlastv = lastv
255 END IF
256 END IF
257 END DO
258 ELSE
259 prevlastv = 1
260 DO i = k, 1, -1
261 IF( tau( i ).EQ.zero ) THEN
262*
263* H(i) = I
264*
265 DO j = i, k
266 t( j, i ) = zero
267 END DO
268 ELSE
269*
270* general case
271*
272 IF( i.LT.k ) THEN
273 IF( lsame( storev, 'C' ) ) THEN
275 DO lastv = 1, i-1
276 IF( v( lastv, i ).NE.zero ) EXIT
277 END DO
278 DO j = i+1, k
279 t( j, i ) = -tau( i ) * v( n-k+i , j )
280 END DO
281 j = max( lastv, prevlastv )
282*
283* T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**T * V(j:n-k+i,i)
284*
285 CALL sgemv( 'Transpose', n-k+i-j, k-i, -tau( i ),
286 \$ v( j, i+1 ), ldv, v( j, i ), 1, one,
287 \$ t( i+1, i ), 1 )
288 ELSE
290 DO lastv = 1, i-1
291 IF( v( i, lastv ).NE.zero ) EXIT
292 END DO
293 DO j = i+1, k
294 t( j, i ) = -tau( i ) * v( j, n-k+i )
295 END DO
296 j = max( lastv, prevlastv )
297*
298* T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**T
299*
300 CALL sgemv( 'No transpose', k-i, n-k+i-j,
301 \$ -tau( i ), v( i+1, j ), ldv, v( i, j ), ldv,
302 \$ one, t( i+1, i ), 1 )
303 END IF
304*
305* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
306*
307 CALL strmv( 'Lower', 'No transpose', 'Non-unit', k-i,
308 \$ t( i+1, i+1 ), ldt, t( i+1, i ), 1 )
309 IF( i.GT.1 ) THEN
310 prevlastv = min( prevlastv, lastv )
311 ELSE
312 prevlastv = lastv
313 END IF
314 END IF
315 t( i, i ) = tau( i )
316 END IF
317 END DO
318 END IF
319 RETURN
320*
321* End of SLARFT
322*
subroutine sgemv(trans, m, n, alpha, a, lda, x, incx, beta, y, incy)
SGEMV
Definition sgemv.f:158
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine strmv(uplo, trans, diag, n, a, lda, x, incx)
STRMV
Definition strmv.f:147
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