LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ slasyf_aa()

 subroutine slasyf_aa ( character UPLO, integer J1, integer M, integer NB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) IPIV, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WORK )

SLASYF_AA

Purpose:
``` DLATRF_AA factorizes a panel of a real symmetric matrix A using
the Aasen's algorithm. The panel consists of a set of NB rows of A
when UPLO is U, or a set of NB columns when UPLO is L.

In order to factorize the panel, the Aasen's algorithm requires the
last row, or column, of the previous panel. The first row, or column,
of A is set to be the first row, or column, of an identity matrix,
which is used to factorize the first panel.

The resulting J-th row of U, or J-th column of L, is stored in the
(J-1)-th row, or column, of A (without the unit diagonals), while
the diagonal and subdiagonal of A are overwritten by those of T.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] J1 ``` J1 is INTEGER The location of the first row, or column, of the panel within the submatrix of A, passed to this routine, e.g., when called by SSYTRF_AA, for the first panel, J1 is 1, while for the remaining panels, J1 is 2.``` [in] M ``` M is INTEGER The dimension of the submatrix. M >= 0.``` [in] NB ``` NB is INTEGER The dimension of the panel to be facotorized.``` [in,out] A ``` A is REAL array, dimension (LDA,M) for the first panel, while dimension (LDA,M+1) for the remaining panels. On entry, A contains the last row, or column, of the previous panel, and the trailing submatrix of A to be factorized, except for the first panel, only the panel is passed. On exit, the leading panel is factorized.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] IPIV ``` IPIV is INTEGER array, dimension (M) Details of the row and column interchanges, the row and column k were interchanged with the row and column IPIV(k).``` [in,out] H ` H is REAL workspace, dimension (LDH,NB).` [in] LDH ``` LDH is INTEGER The leading dimension of the workspace H. LDH >= max(1,M).``` [out] WORK ` WORK is REAL workspace, dimension (M).`

Definition at line 142 of file slasyf_aa.f.

144 *
145 * -- LAPACK computational routine --
146 * -- LAPACK is a software package provided by Univ. of Tennessee, --
147 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
148 *
149  IMPLICIT NONE
150 *
151 * .. Scalar Arguments ..
152  CHARACTER UPLO
153  INTEGER M, NB, J1, LDA, LDH
154 * ..
155 * .. Array Arguments ..
156  INTEGER IPIV( * )
157  REAL A( LDA, * ), H( LDH, * ), WORK( * )
158 * ..
159 *
160 * =====================================================================
161 * .. Parameters ..
162  REAL ZERO, ONE
163  parameter( zero = 0.0e+0, one = 1.0e+0 )
164 *
165 * .. Local Scalars ..
166  INTEGER J, K, K1, I1, I2, MJ
167  REAL PIV, ALPHA
168 * ..
169 * .. External Functions ..
170  LOGICAL LSAME
171  INTEGER ISAMAX, ILAENV
172  EXTERNAL lsame, ilaenv, isamax
173 * ..
174 * .. External Subroutines ..
175  EXTERNAL saxpy, sgemv, sscal, scopy, sswap, slaset,
176  \$ xerbla
177 * ..
178 * .. Intrinsic Functions ..
179  INTRINSIC max
180 * ..
181 * .. Executable Statements ..
182 *
183  j = 1
184 *
185 * K1 is the first column of the panel to be factorized
186 * i.e., K1 is 2 for the first block column, and 1 for the rest of the blocks
187 *
188  k1 = (2-j1)+1
189 *
190  IF( lsame( uplo, 'U' ) ) THEN
191 *
192 * .....................................................
193 * Factorize A as U**T*D*U using the upper triangle of A
194 * .....................................................
195 *
196  10 CONTINUE
197  IF ( j.GT.min(m, nb) )
198  \$ GO TO 20
199 *
200 * K is the column to be factorized
201 * when being called from SSYTRF_AA,
202 * > for the first block column, J1 is 1, hence J1+J-1 is J,
203 * > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
204 *
205  k = j1+j-1
206  IF( j.EQ.m ) THEN
207 *
208 * Only need to compute T(J, J)
209 *
210  mj = 1
211  ELSE
212  mj = m-j+1
213  END IF
214 *
215 * H(J:M, J) := A(J, J:M) - H(J:M, 1:(J-1)) * L(J1:(J-1), J),
216 * where H(J:M, J) has been initialized to be A(J, J:M)
217 *
218  IF( k.GT.2 ) THEN
219 *
220 * K is the column to be factorized
221 * > for the first block column, K is J, skipping the first two
222 * columns
223 * > for the rest of the columns, K is J+1, skipping only the
224 * first column
225 *
226  CALL sgemv( 'No transpose', mj, j-k1,
227  \$ -one, h( j, k1 ), ldh,
228  \$ a( 1, j ), 1,
229  \$ one, h( j, j ), 1 )
230  END IF
231 *
232 * Copy H(i:M, i) into WORK
233 *
234  CALL scopy( mj, h( j, j ), 1, work( 1 ), 1 )
235 *
236  IF( j.GT.k1 ) THEN
237 *
238 * Compute WORK := WORK - L(J-1, J:M) * T(J-1,J),
239 * where A(J-1, J) stores T(J-1, J) and A(J-2, J:M) stores U(J-1, J:M)
240 *
241  alpha = -a( k-1, j )
242  CALL saxpy( mj, alpha, a( k-2, j ), lda, work( 1 ), 1 )
243  END IF
244 *
245 * Set A(J, J) = T(J, J)
246 *
247  a( k, j ) = work( 1 )
248 *
249  IF( j.LT.m ) THEN
250 *
251 * Compute WORK(2:M) = T(J, J) L(J, (J+1):M)
252 * where A(J, J) stores T(J, J) and A(J-1, (J+1):M) stores U(J, (J+1):M)
253 *
254  IF( k.GT.1 ) THEN
255  alpha = -a( k, j )
256  CALL saxpy( m-j, alpha, a( k-1, j+1 ), lda,
257  \$ work( 2 ), 1 )
258  ENDIF
259 *
260 * Find max(|WORK(2:M)|)
261 *
262  i2 = isamax( m-j, work( 2 ), 1 ) + 1
263  piv = work( i2 )
264 *
265 * Apply symmetric pivot
266 *
267  IF( (i2.NE.2) .AND. (piv.NE.0) ) THEN
268 *
269 * Swap WORK(I1) and WORK(I2)
270 *
271  i1 = 2
272  work( i2 ) = work( i1 )
273  work( i1 ) = piv
274 *
275 * Swap A(I1, I1+1:M) with A(I1+1:M, I2)
276 *
277  i1 = i1+j-1
278  i2 = i2+j-1
279  CALL sswap( i2-i1-1, a( j1+i1-1, i1+1 ), lda,
280  \$ a( j1+i1, i2 ), 1 )
281 *
282 * Swap A(I1, I2+1:M) with A(I2, I2+1:M)
283 *
284  IF( i2.LT.m )
285  \$ CALL sswap( m-i2, a( j1+i1-1, i2+1 ), lda,
286  \$ a( j1+i2-1, i2+1 ), lda )
287 *
288 * Swap A(I1, I1) with A(I2,I2)
289 *
290  piv = a( i1+j1-1, i1 )
291  a( j1+i1-1, i1 ) = a( j1+i2-1, i2 )
292  a( j1+i2-1, i2 ) = piv
293 *
294 * Swap H(I1, 1:J1) with H(I2, 1:J1)
295 *
296  CALL sswap( i1-1, h( i1, 1 ), ldh, h( i2, 1 ), ldh )
297  ipiv( i1 ) = i2
298 *
299  IF( i1.GT.(k1-1) ) THEN
300 *
301 * Swap L(1:I1-1, I1) with L(1:I1-1, I2),
302 * skipping the first column
303 *
304  CALL sswap( i1-k1+1, a( 1, i1 ), 1,
305  \$ a( 1, i2 ), 1 )
306  END IF
307  ELSE
308  ipiv( j+1 ) = j+1
309  ENDIF
310 *
311 * Set A(J, J+1) = T(J, J+1)
312 *
313  a( k, j+1 ) = work( 2 )
314 *
315  IF( j.LT.nb ) THEN
316 *
317 * Copy A(J+1:M, J+1) into H(J:M, J),
318 *
319  CALL scopy( m-j, a( k+1, j+1 ), lda,
320  \$ h( j+1, j+1 ), 1 )
321  END IF
322 *
323 * Compute L(J+2, J+1) = WORK( 3:M ) / T(J, J+1),
324 * where A(J, J+1) = T(J, J+1) and A(J+2:M, J) = L(J+2:M, J+1)
325 *
326  IF( j.LT.(m-1) ) THEN
327  IF( a( k, j+1 ).NE.zero ) THEN
328  alpha = one / a( k, j+1 )
329  CALL scopy( m-j-1, work( 3 ), 1, a( k, j+2 ), lda )
330  CALL sscal( m-j-1, alpha, a( k, j+2 ), lda )
331  ELSE
332  CALL slaset( 'Full', 1, m-j-1, zero, zero,
333  \$ a( k, j+2 ), lda)
334  END IF
335  END IF
336  END IF
337  j = j + 1
338  GO TO 10
339  20 CONTINUE
340 *
341  ELSE
342 *
343 * .....................................................
344 * Factorize A as L*D*L**T using the lower triangle of A
345 * .....................................................
346 *
347  30 CONTINUE
348  IF( j.GT.min( m, nb ) )
349  \$ GO TO 40
350 *
351 * K is the column to be factorized
352 * when being called from SSYTRF_AA,
353 * > for the first block column, J1 is 1, hence J1+J-1 is J,
354 * > for the rest of the columns, J1 is 2, and J1+J-1 is J+1,
355 *
356  k = j1+j-1
357  IF( j.EQ.m ) THEN
358 *
359 * Only need to compute T(J, J)
360 *
361  mj = 1
362  ELSE
363  mj = m-j+1
364  END IF
365 *
366 * H(J:M, J) := A(J:M, J) - H(J:M, 1:(J-1)) * L(J, J1:(J-1))^T,
367 * where H(J:M, J) has been initialized to be A(J:M, J)
368 *
369  IF( k.GT.2 ) THEN
370 *
371 * K is the column to be factorized
372 * > for the first block column, K is J, skipping the first two
373 * columns
374 * > for the rest of the columns, K is J+1, skipping only the
375 * first column
376 *
377  CALL sgemv( 'No transpose', mj, j-k1,
378  \$ -one, h( j, k1 ), ldh,
379  \$ a( j, 1 ), lda,
380  \$ one, h( j, j ), 1 )
381  END IF
382 *
383 * Copy H(J:M, J) into WORK
384 *
385  CALL scopy( mj, h( j, j ), 1, work( 1 ), 1 )
386 *
387  IF( j.GT.k1 ) THEN
388 *
389 * Compute WORK := WORK - L(J:M, J-1) * T(J-1,J),
390 * where A(J-1, J) = T(J-1, J) and A(J, J-2) = L(J, J-1)
391 *
392  alpha = -a( j, k-1 )
393  CALL saxpy( mj, alpha, a( j, k-2 ), 1, work( 1 ), 1 )
394  END IF
395 *
396 * Set A(J, J) = T(J, J)
397 *
398  a( j, k ) = work( 1 )
399 *
400  IF( j.LT.m ) THEN
401 *
402 * Compute WORK(2:M) = T(J, J) L((J+1):M, J)
403 * where A(J, J) = T(J, J) and A((J+1):M, J-1) = L((J+1):M, J)
404 *
405  IF( k.GT.1 ) THEN
406  alpha = -a( j, k )
407  CALL saxpy( m-j, alpha, a( j+1, k-1 ), 1,
408  \$ work( 2 ), 1 )
409  ENDIF
410 *
411 * Find max(|WORK(2:M)|)
412 *
413  i2 = isamax( m-j, work( 2 ), 1 ) + 1
414  piv = work( i2 )
415 *
416 * Apply symmetric pivot
417 *
418  IF( (i2.NE.2) .AND. (piv.NE.0) ) THEN
419 *
420 * Swap WORK(I1) and WORK(I2)
421 *
422  i1 = 2
423  work( i2 ) = work( i1 )
424  work( i1 ) = piv
425 *
426 * Swap A(I1+1:M, I1) with A(I2, I1+1:M)
427 *
428  i1 = i1+j-1
429  i2 = i2+j-1
430  CALL sswap( i2-i1-1, a( i1+1, j1+i1-1 ), 1,
431  \$ a( i2, j1+i1 ), lda )
432 *
433 * Swap A(I2+1:M, I1) with A(I2+1:M, I2)
434 *
435  IF( i2.LT.m )
436  \$ CALL sswap( m-i2, a( i2+1, j1+i1-1 ), 1,
437  \$ a( i2+1, j1+i2-1 ), 1 )
438 *
439 * Swap A(I1, I1) with A(I2, I2)
440 *
441  piv = a( i1, j1+i1-1 )
442  a( i1, j1+i1-1 ) = a( i2, j1+i2-1 )
443  a( i2, j1+i2-1 ) = piv
444 *
445 * Swap H(I1, I1:J1) with H(I2, I2:J1)
446 *
447  CALL sswap( i1-1, h( i1, 1 ), ldh, h( i2, 1 ), ldh )
448  ipiv( i1 ) = i2
449 *
450  IF( i1.GT.(k1-1) ) THEN
451 *
452 * Swap L(1:I1-1, I1) with L(1:I1-1, I2),
453 * skipping the first column
454 *
455  CALL sswap( i1-k1+1, a( i1, 1 ), lda,
456  \$ a( i2, 1 ), lda )
457  END IF
458  ELSE
459  ipiv( j+1 ) = j+1
460  ENDIF
461 *
462 * Set A(J+1, J) = T(J+1, J)
463 *
464  a( j+1, k ) = work( 2 )
465 *
466  IF( j.LT.nb ) THEN
467 *
468 * Copy A(J+1:M, J+1) into H(J+1:M, J),
469 *
470  CALL scopy( m-j, a( j+1, k+1 ), 1,
471  \$ h( j+1, j+1 ), 1 )
472  END IF
473 *
474 * Compute L(J+2, J+1) = WORK( 3:M ) / T(J, J+1),
475 * where A(J, J+1) = T(J, J+1) and A(J+2:M, J) = L(J+2:M, J+1)
476 *
477  IF( j.LT.(m-1) ) THEN
478  IF( a( j+1, k ).NE.zero ) THEN
479  alpha = one / a( j+1, k )
480  CALL scopy( m-j-1, work( 3 ), 1, a( j+2, k ), 1 )
481  CALL sscal( m-j-1, alpha, a( j+2, k ), 1 )
482  ELSE
483  CALL slaset( 'Full', m-j-1, 1, zero, zero,
484  \$ a( j+2, k ), lda )
485  END IF
486  END IF
487  END IF
488  j = j + 1
489  GO TO 30
490  40 CONTINUE
491  END IF
492  RETURN
493 *
494 * End of SLASYF_AA
495 *
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
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