LAPACK  3.11.0
LAPACK: Linear Algebra PACKage
slasyf.f
1 *> \brief \b SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal pivoting method.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SLASYF + dependencies
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11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slasyf.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slasyf.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, KB, LDA, LDW, N, NB
26 * ..
27 * .. Array Arguments ..
28 * INTEGER IPIV( * )
29 * REAL A( LDA, * ), W( LDW, * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> SLASYF computes a partial factorization of a real symmetric matrix A
39 *> using the Bunch-Kaufman diagonal pivoting method. The partial
40 *> factorization has the form:
41 *>
42 *> A = ( I U12 ) ( A11 0 ) ( I 0 ) if UPLO = 'U', or:
43 *> ( 0 U22 ) ( 0 D ) ( U12**T U22**T )
44 *>
45 *> A = ( L11 0 ) ( D 0 ) ( L11**T L21**T ) if UPLO = 'L'
46 *> ( L21 I ) ( 0 A22 ) ( 0 I )
47 *>
48 *> where the order of D is at most NB. The actual order is returned in
49 *> the argument KB, and is either NB or NB-1, or N if N <= NB.
50 *>
51 *> SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code
52 *> (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
53 *> A22 (if UPLO = 'L').
54 *> \endverbatim
55 *
56 * Arguments:
57 * ==========
58 *
59 *> \param[in] UPLO
60 *> \verbatim
61 *> UPLO is CHARACTER*1
62 *> Specifies whether the upper or lower triangular part of the
63 *> symmetric matrix A is stored:
64 *> = 'U': Upper triangular
65 *> = 'L': Lower triangular
66 *> \endverbatim
67 *>
68 *> \param[in] N
69 *> \verbatim
70 *> N is INTEGER
71 *> The order of the matrix A. N >= 0.
72 *> \endverbatim
73 *>
74 *> \param[in] NB
75 *> \verbatim
76 *> NB is INTEGER
77 *> The maximum number of columns of the matrix A that should be
78 *> factored. NB should be at least 2 to allow for 2-by-2 pivot
79 *> blocks.
80 *> \endverbatim
81 *>
82 *> \param[out] KB
83 *> \verbatim
84 *> KB is INTEGER
85 *> The number of columns of A that were actually factored.
86 *> KB is either NB-1 or NB, or N if N <= NB.
87 *> \endverbatim
88 *>
89 *> \param[in,out] A
90 *> \verbatim
91 *> A is REAL array, dimension (LDA,N)
92 *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
93 *> n-by-n upper triangular part of A contains the upper
94 *> triangular part of the matrix A, and the strictly lower
95 *> triangular part of A is not referenced. If UPLO = 'L', the
96 *> leading n-by-n lower triangular part of A contains the lower
97 *> triangular part of the matrix A, and the strictly upper
98 *> triangular part of A is not referenced.
99 *> On exit, A contains details of the partial factorization.
100 *> \endverbatim
101 *>
102 *> \param[in] LDA
103 *> \verbatim
104 *> LDA is INTEGER
105 *> The leading dimension of the array A. LDA >= max(1,N).
106 *> \endverbatim
107 *>
108 *> \param[out] IPIV
109 *> \verbatim
110 *> IPIV is INTEGER array, dimension (N)
111 *> Details of the interchanges and the block structure of D.
112 *>
113 *> If UPLO = 'U':
114 *> Only the last KB elements of IPIV are set.
115 *>
116 *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
117 *> interchanged and D(k,k) is a 1-by-1 diagonal block.
118 *>
119 *> If IPIV(k) = IPIV(k-1) < 0, then rows and columns
120 *> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
121 *> is a 2-by-2 diagonal block.
122 *>
123 *> If UPLO = 'L':
124 *> Only the first KB elements of IPIV are set.
125 *>
126 *> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
127 *> interchanged and D(k,k) is a 1-by-1 diagonal block.
128 *>
129 *> If IPIV(k) = IPIV(k+1) < 0, then rows and columns
130 *> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1)
131 *> is a 2-by-2 diagonal block.
132 *> \endverbatim
133 *>
134 *> \param[out] W
135 *> \verbatim
136 *> W is REAL array, dimension (LDW,NB)
137 *> \endverbatim
138 *>
139 *> \param[in] LDW
140 *> \verbatim
141 *> LDW is INTEGER
142 *> The leading dimension of the array W. LDW >= max(1,N).
143 *> \endverbatim
144 *>
145 *> \param[out] INFO
146 *> \verbatim
147 *> INFO is INTEGER
148 *> = 0: successful exit
149 *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
150 *> has been completed, but the block diagonal matrix D is
151 *> exactly singular.
152 *> \endverbatim
153 *
154 * Authors:
155 * ========
156 *
157 *> \author Univ. of Tennessee
158 *> \author Univ. of California Berkeley
159 *> \author Univ. of Colorado Denver
160 *> \author NAG Ltd.
161 *
162 *> \ingroup lahef
163 *
164 *> \par Contributors:
165 * ==================
166 *>
167 *> \verbatim
168 *>
169 *> November 2013, Igor Kozachenko,
170 *> Computer Science Division,
171 *> University of California, Berkeley
172 *> \endverbatim
173 *
174 * =====================================================================
175  SUBROUTINE slasyf( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
176 *
177 * -- LAPACK computational routine --
178 * -- LAPACK is a software package provided by Univ. of Tennessee, --
179 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180 *
181 * .. Scalar Arguments ..
182  CHARACTER UPLO
183  INTEGER INFO, KB, LDA, LDW, N, NB
184 * ..
185 * .. Array Arguments ..
186  INTEGER IPIV( * )
187  REAL A( lda, * ), W( ldw, * )
188 * ..
189 *
190 * =====================================================================
191 *
192 * .. Parameters ..
193  REAL ZERO, ONE
194  parameter( zero = 0.0e+0, one = 1.0e+0 )
195  REAL EIGHT, SEVTEN
196  parameter( eight = 8.0e+0, sevten = 17.0e+0 )
197 * ..
198 * .. Local Scalars ..
199  INTEGER IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
200  $ kstep, kw
201  REAL ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1,
202  $ rowmax, t
203 * ..
204 * .. External Functions ..
205  LOGICAL LSAME
206  INTEGER ISAMAX
207  EXTERNAL lsame, isamax
208 * ..
209 * .. External Subroutines ..
210  EXTERNAL scopy, sgemm, sgemv, sscal, sswap
211 * ..
212 * .. Intrinsic Functions ..
213  INTRINSIC abs, max, min, sqrt
214 * ..
215 * .. Executable Statements ..
216 *
217  info = 0
218 *
219 * Initialize ALPHA for use in choosing pivot block size.
220 *
221  alpha = ( one+sqrt( sevten ) ) / eight
222 *
223  IF( lsame( uplo, 'U' ) ) THEN
224 *
225 * Factorize the trailing columns of A using the upper triangle
226 * of A and working backwards, and compute the matrix W = U12*D
227 * for use in updating A11
228 *
229 * K is the main loop index, decreasing from N in steps of 1 or 2
230 *
231 * KW is the column of W which corresponds to column K of A
232 *
233  k = n
234  10 CONTINUE
235  kw = nb + k - n
236 *
237 * Exit from loop
238 *
239  IF( ( k.LE.n-nb+1 .AND. nb.LT.n ) .OR. k.LT.1 )
240  $ GO TO 30
241 *
242 * Copy column K of A to column KW of W and update it
243 *
244  CALL scopy( k, a( 1, k ), 1, w( 1, kw ), 1 )
245  IF( k.LT.n )
246  $ CALL sgemv( 'No transpose', k, n-k, -one, a( 1, k+1 ), lda,
247  $ w( k, kw+1 ), ldw, one, w( 1, kw ), 1 )
248 *
249  kstep = 1
250 *
251 * Determine rows and columns to be interchanged and whether
252 * a 1-by-1 or 2-by-2 pivot block will be used
253 *
254  absakk = abs( w( k, kw ) )
255 *
256 * IMAX is the row-index of the largest off-diagonal element in
257 * column K, and COLMAX is its absolute value.
258 * Determine both COLMAX and IMAX.
259 *
260  IF( k.GT.1 ) THEN
261  imax = isamax( k-1, w( 1, kw ), 1 )
262  colmax = abs( w( imax, kw ) )
263  ELSE
264  colmax = zero
265  END IF
266 *
267  IF( max( absakk, colmax ).EQ.zero ) THEN
268 *
269 * Column K is zero or underflow: set INFO and continue
270 *
271  IF( info.EQ.0 )
272  $ info = k
273  kp = k
274  ELSE
275  IF( absakk.GE.alpha*colmax ) THEN
276 *
277 * no interchange, use 1-by-1 pivot block
278 *
279  kp = k
280  ELSE
281 *
282 * Copy column IMAX to column KW-1 of W and update it
283 *
284  CALL scopy( imax, a( 1, imax ), 1, w( 1, kw-1 ), 1 )
285  CALL scopy( k-imax, a( imax, imax+1 ), lda,
286  $ w( imax+1, kw-1 ), 1 )
287  IF( k.LT.n )
288  $ CALL sgemv( 'No transpose', k, n-k, -one, a( 1, k+1 ),
289  $ lda, w( imax, kw+1 ), ldw, one,
290  $ w( 1, kw-1 ), 1 )
291 *
292 * JMAX is the column-index of the largest off-diagonal
293 * element in row IMAX, and ROWMAX is its absolute value
294 *
295  jmax = imax + isamax( k-imax, w( imax+1, kw-1 ), 1 )
296  rowmax = abs( w( jmax, kw-1 ) )
297  IF( imax.GT.1 ) THEN
298  jmax = isamax( imax-1, w( 1, kw-1 ), 1 )
299  rowmax = max( rowmax, abs( w( jmax, kw-1 ) ) )
300  END IF
301 *
302  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
303 *
304 * no interchange, use 1-by-1 pivot block
305 *
306  kp = k
307  ELSE IF( abs( w( imax, kw-1 ) ).GE.alpha*rowmax ) THEN
308 *
309 * interchange rows and columns K and IMAX, use 1-by-1
310 * pivot block
311 *
312  kp = imax
313 *
314 * copy column KW-1 of W to column KW of W
315 *
316  CALL scopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
317  ELSE
318 *
319 * interchange rows and columns K-1 and IMAX, use 2-by-2
320 * pivot block
321 *
322  kp = imax
323  kstep = 2
324  END IF
325  END IF
326 *
327 * ============================================================
328 *
329 * KK is the column of A where pivoting step stopped
330 *
331  kk = k - kstep + 1
332 *
333 * KKW is the column of W which corresponds to column KK of A
334 *
335  kkw = nb + kk - n
336 *
337 * Interchange rows and columns KP and KK.
338 * Updated column KP is already stored in column KKW of W.
339 *
340  IF( kp.NE.kk ) THEN
341 *
342 * Copy non-updated column KK to column KP of submatrix A
343 * at step K. No need to copy element into column K
344 * (or K and K-1 for 2-by-2 pivot) of A, since these columns
345 * will be later overwritten.
346 *
347  a( kp, kp ) = a( kk, kk )
348  CALL scopy( kk-1-kp, a( kp+1, kk ), 1, a( kp, kp+1 ),
349  $ lda )
350  IF( kp.GT.1 )
351  $ CALL scopy( kp-1, a( 1, kk ), 1, a( 1, kp ), 1 )
352 *
353 * Interchange rows KK and KP in last K+1 to N columns of A
354 * (columns K (or K and K-1 for 2-by-2 pivot) of A will be
355 * later overwritten). Interchange rows KK and KP
356 * in last KKW to NB columns of W.
357 *
358  IF( k.LT.n )
359  $ CALL sswap( n-k, a( kk, k+1 ), lda, a( kp, k+1 ),
360  $ lda )
361  CALL sswap( n-kk+1, w( kk, kkw ), ldw, w( kp, kkw ),
362  $ ldw )
363  END IF
364 *
365  IF( kstep.EQ.1 ) THEN
366 *
367 * 1-by-1 pivot block D(k): column kw of W now holds
368 *
369 * W(kw) = U(k)*D(k),
370 *
371 * where U(k) is the k-th column of U
372 *
373 * Store subdiag. elements of column U(k)
374 * and 1-by-1 block D(k) in column k of A.
375 * NOTE: Diagonal element U(k,k) is a UNIT element
376 * and not stored.
377 * A(k,k) := D(k,k) = W(k,kw)
378 * A(1:k-1,k) := U(1:k-1,k) = W(1:k-1,kw)/D(k,k)
379 *
380  CALL scopy( k, w( 1, kw ), 1, a( 1, k ), 1 )
381  r1 = one / a( k, k )
382  CALL sscal( k-1, r1, a( 1, k ), 1 )
383 *
384  ELSE
385 *
386 * 2-by-2 pivot block D(k): columns kw and kw-1 of W now hold
387 *
388 * ( W(kw-1) W(kw) ) = ( U(k-1) U(k) )*D(k)
389 *
390 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
391 * of U
392 *
393 * Store U(1:k-2,k-1) and U(1:k-2,k) and 2-by-2
394 * block D(k-1:k,k-1:k) in columns k-1 and k of A.
395 * NOTE: 2-by-2 diagonal block U(k-1:k,k-1:k) is a UNIT
396 * block and not stored.
397 * A(k-1:k,k-1:k) := D(k-1:k,k-1:k) = W(k-1:k,kw-1:kw)
398 * A(1:k-2,k-1:k) := U(1:k-2,k:k-1:k) =
399 * = W(1:k-2,kw-1:kw) * ( D(k-1:k,k-1:k)**(-1) )
400 *
401  IF( k.GT.2 ) THEN
402 *
403 * Compose the columns of the inverse of 2-by-2 pivot
404 * block D in the following way to reduce the number
405 * of FLOPS when we myltiply panel ( W(kw-1) W(kw) ) by
406 * this inverse
407 *
408 * D**(-1) = ( d11 d21 )**(-1) =
409 * ( d21 d22 )
410 *
411 * = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
412 * ( (-d21 ) ( d11 ) )
413 *
414 * = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
415 *
416 * * ( ( d22/d21 ) ( -1 ) ) =
417 * ( ( -1 ) ( d11/d21 ) )
418 *
419 * = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
420 * ( ( -1 ) ( D22 ) )
421 *
422 * = 1/d21 * T * ( ( D11 ) ( -1 ) )
423 * ( ( -1 ) ( D22 ) )
424 *
425 * = D21 * ( ( D11 ) ( -1 ) )
426 * ( ( -1 ) ( D22 ) )
427 *
428  d21 = w( k-1, kw )
429  d11 = w( k, kw ) / d21
430  d22 = w( k-1, kw-1 ) / d21
431  t = one / ( d11*d22-one )
432  d21 = t / d21
433 *
434 * Update elements in columns A(k-1) and A(k) as
435 * dot products of rows of ( W(kw-1) W(kw) ) and columns
436 * of D**(-1)
437 *
438  DO 20 j = 1, k - 2
439  a( j, k-1 ) = d21*( d11*w( j, kw-1 )-w( j, kw ) )
440  a( j, k ) = d21*( d22*w( j, kw )-w( j, kw-1 ) )
441  20 CONTINUE
442  END IF
443 *
444 * Copy D(k) to A
445 *
446  a( k-1, k-1 ) = w( k-1, kw-1 )
447  a( k-1, k ) = w( k-1, kw )
448  a( k, k ) = w( k, kw )
449 *
450  END IF
451 *
452  END IF
453 *
454 * Store details of the interchanges in IPIV
455 *
456  IF( kstep.EQ.1 ) THEN
457  ipiv( k ) = kp
458  ELSE
459  ipiv( k ) = -kp
460  ipiv( k-1 ) = -kp
461  END IF
462 *
463 * Decrease K and return to the start of the main loop
464 *
465  k = k - kstep
466  GO TO 10
467 *
468  30 CONTINUE
469 *
470 * Update the upper triangle of A11 (= A(1:k,1:k)) as
471 *
472 * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
473 *
474 * computing blocks of NB columns at a time
475 *
476  DO 50 j = ( ( k-1 ) / nb )*nb + 1, 1, -nb
477  jb = min( nb, k-j+1 )
478 *
479 * Update the upper triangle of the diagonal block
480 *
481  DO 40 jj = j, j + jb - 1
482  CALL sgemv( 'No transpose', jj-j+1, n-k, -one,
483  $ a( j, k+1 ), lda, w( jj, kw+1 ), ldw, one,
484  $ a( j, jj ), 1 )
485  40 CONTINUE
486 *
487 * Update the rectangular superdiagonal block
488 *
489  CALL sgemm( 'No transpose', 'Transpose', j-1, jb, n-k, -one,
490  $ a( 1, k+1 ), lda, w( j, kw+1 ), ldw, one,
491  $ a( 1, j ), lda )
492  50 CONTINUE
493 *
494 * Put U12 in standard form by partially undoing the interchanges
495 * in columns k+1:n looping backwards from k+1 to n
496 *
497  j = k + 1
498  60 CONTINUE
499 *
500 * Undo the interchanges (if any) of rows JJ and JP at each
501 * step J
502 *
503 * (Here, J is a diagonal index)
504  jj = j
505  jp = ipiv( j )
506  IF( jp.LT.0 ) THEN
507  jp = -jp
508 * (Here, J is a diagonal index)
509  j = j + 1
510  END IF
511 * (NOTE: Here, J is used to determine row length. Length N-J+1
512 * of the rows to swap back doesn't include diagonal element)
513  j = j + 1
514  IF( jp.NE.jj .AND. j.LE.n )
515  $ CALL sswap( n-j+1, a( jp, j ), lda, a( jj, j ), lda )
516  IF( j.LT.n )
517  $ GO TO 60
518 *
519 * Set KB to the number of columns factorized
520 *
521  kb = n - k
522 *
523  ELSE
524 *
525 * Factorize the leading columns of A using the lower triangle
526 * of A and working forwards, and compute the matrix W = L21*D
527 * for use in updating A22
528 *
529 * K is the main loop index, increasing from 1 in steps of 1 or 2
530 *
531  k = 1
532  70 CONTINUE
533 *
534 * Exit from loop
535 *
536  IF( ( k.GE.nb .AND. nb.LT.n ) .OR. k.GT.n )
537  $ GO TO 90
538 *
539 * Copy column K of A to column K of W and update it
540 *
541  CALL scopy( n-k+1, a( k, k ), 1, w( k, k ), 1 )
542  CALL sgemv( 'No transpose', n-k+1, k-1, -one, a( k, 1 ), lda,
543  $ w( k, 1 ), ldw, one, w( k, k ), 1 )
544 *
545  kstep = 1
546 *
547 * Determine rows and columns to be interchanged and whether
548 * a 1-by-1 or 2-by-2 pivot block will be used
549 *
550  absakk = abs( w( k, k ) )
551 *
552 * IMAX is the row-index of the largest off-diagonal element in
553 * column K, and COLMAX is its absolute value.
554 * Determine both COLMAX and IMAX.
555 *
556  IF( k.LT.n ) THEN
557  imax = k + isamax( n-k, w( k+1, k ), 1 )
558  colmax = abs( w( imax, k ) )
559  ELSE
560  colmax = zero
561  END IF
562 *
563  IF( max( absakk, colmax ).EQ.zero ) THEN
564 *
565 * Column K is zero or underflow: set INFO and continue
566 *
567  IF( info.EQ.0 )
568  $ info = k
569  kp = k
570  ELSE
571  IF( absakk.GE.alpha*colmax ) THEN
572 *
573 * no interchange, use 1-by-1 pivot block
574 *
575  kp = k
576  ELSE
577 *
578 * Copy column IMAX to column K+1 of W and update it
579 *
580  CALL scopy( imax-k, a( imax, k ), lda, w( k, k+1 ), 1 )
581  CALL scopy( n-imax+1, a( imax, imax ), 1, w( imax, k+1 ),
582  $ 1 )
583  CALL sgemv( 'No transpose', n-k+1, k-1, -one, a( k, 1 ),
584  $ lda, w( imax, 1 ), ldw, one, w( k, k+1 ), 1 )
585 *
586 * JMAX is the column-index of the largest off-diagonal
587 * element in row IMAX, and ROWMAX is its absolute value
588 *
589  jmax = k - 1 + isamax( imax-k, w( k, k+1 ), 1 )
590  rowmax = abs( w( jmax, k+1 ) )
591  IF( imax.LT.n ) THEN
592  jmax = imax + isamax( n-imax, w( imax+1, k+1 ), 1 )
593  rowmax = max( rowmax, abs( w( jmax, k+1 ) ) )
594  END IF
595 *
596  IF( absakk.GE.alpha*colmax*( colmax / rowmax ) ) THEN
597 *
598 * no interchange, use 1-by-1 pivot block
599 *
600  kp = k
601  ELSE IF( abs( w( imax, k+1 ) ).GE.alpha*rowmax ) THEN
602 *
603 * interchange rows and columns K and IMAX, use 1-by-1
604 * pivot block
605 *
606  kp = imax
607 *
608 * copy column K+1 of W to column K of W
609 *
610  CALL scopy( n-k+1, w( k, k+1 ), 1, w( k, k ), 1 )
611  ELSE
612 *
613 * interchange rows and columns K+1 and IMAX, use 2-by-2
614 * pivot block
615 *
616  kp = imax
617  kstep = 2
618  END IF
619  END IF
620 *
621 * ============================================================
622 *
623 * KK is the column of A where pivoting step stopped
624 *
625  kk = k + kstep - 1
626 *
627 * Interchange rows and columns KP and KK.
628 * Updated column KP is already stored in column KK of W.
629 *
630  IF( kp.NE.kk ) THEN
631 *
632 * Copy non-updated column KK to column KP of submatrix A
633 * at step K. No need to copy element into column K
634 * (or K and K+1 for 2-by-2 pivot) of A, since these columns
635 * will be later overwritten.
636 *
637  a( kp, kp ) = a( kk, kk )
638  CALL scopy( kp-kk-1, a( kk+1, kk ), 1, a( kp, kk+1 ),
639  $ lda )
640  IF( kp.LT.n )
641  $ CALL scopy( n-kp, a( kp+1, kk ), 1, a( kp+1, kp ), 1 )
642 *
643 * Interchange rows KK and KP in first K-1 columns of A
644 * (columns K (or K and K+1 for 2-by-2 pivot) of A will be
645 * later overwritten). Interchange rows KK and KP
646 * in first KK columns of W.
647 *
648  IF( k.GT.1 )
649  $ CALL sswap( k-1, a( kk, 1 ), lda, a( kp, 1 ), lda )
650  CALL sswap( kk, w( kk, 1 ), ldw, w( kp, 1 ), ldw )
651  END IF
652 *
653  IF( kstep.EQ.1 ) THEN
654 *
655 * 1-by-1 pivot block D(k): column k of W now holds
656 *
657 * W(k) = L(k)*D(k),
658 *
659 * where L(k) is the k-th column of L
660 *
661 * Store subdiag. elements of column L(k)
662 * and 1-by-1 block D(k) in column k of A.
663 * (NOTE: Diagonal element L(k,k) is a UNIT element
664 * and not stored)
665 * A(k,k) := D(k,k) = W(k,k)
666 * A(k+1:N,k) := L(k+1:N,k) = W(k+1:N,k)/D(k,k)
667 *
668  CALL scopy( n-k+1, w( k, k ), 1, a( k, k ), 1 )
669  IF( k.LT.n ) THEN
670  r1 = one / a( k, k )
671  CALL sscal( n-k, r1, a( k+1, k ), 1 )
672  END IF
673 *
674  ELSE
675 *
676 * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
677 *
678 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
679 *
680 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
681 * of L
682 *
683 * Store L(k+2:N,k) and L(k+2:N,k+1) and 2-by-2
684 * block D(k:k+1,k:k+1) in columns k and k+1 of A.
685 * (NOTE: 2-by-2 diagonal block L(k:k+1,k:k+1) is a UNIT
686 * block and not stored)
687 * A(k:k+1,k:k+1) := D(k:k+1,k:k+1) = W(k:k+1,k:k+1)
688 * A(k+2:N,k:k+1) := L(k+2:N,k:k+1) =
689 * = W(k+2:N,k:k+1) * ( D(k:k+1,k:k+1)**(-1) )
690 *
691  IF( k.LT.n-1 ) THEN
692 *
693 * Compose the columns of the inverse of 2-by-2 pivot
694 * block D in the following way to reduce the number
695 * of FLOPS when we myltiply panel ( W(k) W(k+1) ) by
696 * this inverse
697 *
698 * D**(-1) = ( d11 d21 )**(-1) =
699 * ( d21 d22 )
700 *
701 * = 1/(d11*d22-d21**2) * ( ( d22 ) (-d21 ) ) =
702 * ( (-d21 ) ( d11 ) )
703 *
704 * = 1/d21 * 1/((d11/d21)*(d22/d21)-1) *
705 *
706 * * ( ( d22/d21 ) ( -1 ) ) =
707 * ( ( -1 ) ( d11/d21 ) )
708 *
709 * = 1/d21 * 1/(D22*D11-1) * ( ( D11 ) ( -1 ) ) =
710 * ( ( -1 ) ( D22 ) )
711 *
712 * = 1/d21 * T * ( ( D11 ) ( -1 ) )
713 * ( ( -1 ) ( D22 ) )
714 *
715 * = D21 * ( ( D11 ) ( -1 ) )
716 * ( ( -1 ) ( D22 ) )
717 *
718  d21 = w( k+1, k )
719  d11 = w( k+1, k+1 ) / d21
720  d22 = w( k, k ) / d21
721  t = one / ( d11*d22-one )
722  d21 = t / d21
723 *
724 * Update elements in columns A(k) and A(k+1) as
725 * dot products of rows of ( W(k) W(k+1) ) and columns
726 * of D**(-1)
727 *
728  DO 80 j = k + 2, n
729  a( j, k ) = d21*( d11*w( j, k )-w( j, k+1 ) )
730  a( j, k+1 ) = d21*( d22*w( j, k+1 )-w( j, k ) )
731  80 CONTINUE
732  END IF
733 *
734 * Copy D(k) to A
735 *
736  a( k, k ) = w( k, k )
737  a( k+1, k ) = w( k+1, k )
738  a( k+1, k+1 ) = w( k+1, k+1 )
739 *
740  END IF
741 *
742  END IF
743 *
744 * Store details of the interchanges in IPIV
745 *
746  IF( kstep.EQ.1 ) THEN
747  ipiv( k ) = kp
748  ELSE
749  ipiv( k ) = -kp
750  ipiv( k+1 ) = -kp
751  END IF
752 *
753 * Increase K and return to the start of the main loop
754 *
755  k = k + kstep
756  GO TO 70
757 *
758  90 CONTINUE
759 *
760 * Update the lower triangle of A22 (= A(k:n,k:n)) as
761 *
762 * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
763 *
764 * computing blocks of NB columns at a time
765 *
766  DO 110 j = k, n, nb
767  jb = min( nb, n-j+1 )
768 *
769 * Update the lower triangle of the diagonal block
770 *
771  DO 100 jj = j, j + jb - 1
772  CALL sgemv( 'No transpose', j+jb-jj, k-1, -one,
773  $ a( jj, 1 ), lda, w( jj, 1 ), ldw, one,
774  $ a( jj, jj ), 1 )
775  100 CONTINUE
776 *
777 * Update the rectangular subdiagonal block
778 *
779  IF( j+jb.LE.n )
780  $ CALL sgemm( 'No transpose', 'Transpose', n-j-jb+1, jb,
781  $ k-1, -one, a( j+jb, 1 ), lda, w( j, 1 ), ldw,
782  $ one, a( j+jb, j ), lda )
783  110 CONTINUE
784 *
785 * Put L21 in standard form by partially undoing the interchanges
786 * of rows in columns 1:k-1 looping backwards from k-1 to 1
787 *
788  j = k - 1
789  120 CONTINUE
790 *
791 * Undo the interchanges (if any) of rows JJ and JP at each
792 * step J
793 *
794 * (Here, J is a diagonal index)
795  jj = j
796  jp = ipiv( j )
797  IF( jp.LT.0 ) THEN
798  jp = -jp
799 * (Here, J is a diagonal index)
800  j = j - 1
801  END IF
802 * (NOTE: Here, J is used to determine row length. Length J
803 * of the rows to swap back doesn't include diagonal element)
804  j = j - 1
805  IF( jp.NE.jj .AND. j.GE.1 )
806  $ CALL sswap( j, a( jp, 1 ), lda, a( jj, 1 ), lda )
807  IF( j.GT.1 )
808  $ GO TO 120
809 *
810 * Set KB to the number of columns factorized
811 *
812  kb = k - 1
813 *
814  END IF
815  RETURN
816 *
817 * End of SLASYF
818 *
819  END
sgemm
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
sswap
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
scopy
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
slasyf
subroutine slasyf(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
SLASYF computes a partial factorization of a real symmetric matrix using the Bunch-Kaufman diagonal p...
Definition: slasyf.f:176
sscal
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
sgemv
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156