LAPACK  3.7.1
LAPACK: Linear Algebra PACKage
slasyf_rook.f
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1 *> \brief \b SLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method.
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SLASYF_ROOK( 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_ROOK computes a partial factorization of a real symmetric
39 *> matrix A using the bounded Bunch-Kaufman ("rook") diagonal
40 *> pivoting method. The partial 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_ROOK is an auxiliary routine called by SSYTRF_ROOK. It uses
52 *> blocked code (calling Level 3 BLAS) to update the submatrix
53 *> A11 (if UPLO = 'U') or 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) < 0 and IPIV(k-1) < 0, then rows and
120 *> columns k and -IPIV(k) were interchanged and rows and
121 *> columns k-1 and -IPIV(k-1) were inerchaged,
122 *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
123 *>
124 *> If UPLO = 'L':
125 *> Only the first KB elements of IPIV are set.
126 *>
127 *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
128 *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
129 *>
130 *> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
131 *> columns k and -IPIV(k) were interchanged and rows and
132 *> columns k+1 and -IPIV(k+1) were inerchaged,
133 *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
134 *> \endverbatim
135 *>
136 *> \param[out] W
137 *> \verbatim
138 *> W is REAL array, dimension (LDW,NB)
139 *> \endverbatim
140 *>
141 *> \param[in] LDW
142 *> \verbatim
143 *> LDW is INTEGER
144 *> The leading dimension of the array W. LDW >= max(1,N).
145 *> \endverbatim
146 *>
147 *> \param[out] INFO
148 *> \verbatim
149 *> INFO is INTEGER
150 *> = 0: successful exit
151 *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
152 *> has been completed, but the block diagonal matrix D is
153 *> exactly singular.
154 *> \endverbatim
155 *
156 * Authors:
157 * ========
158 *
159 *> \author Univ. of Tennessee
160 *> \author Univ. of California Berkeley
161 *> \author Univ. of Colorado Denver
162 *> \author NAG Ltd.
163 *
164 *> \date November 2013
165 *
166 *> \ingroup realSYcomputational
167 *
168 *> \par Contributors:
169 * ==================
170 *>
171 *> \verbatim
172 *>
173 *> November 2013, Igor Kozachenko,
174 *> Computer Science Division,
175 *> University of California, Berkeley
176 *>
177 *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
178 *> School of Mathematics,
179 *> University of Manchester
180 *>
181 *> \endverbatim
182 *
183 * =====================================================================
184  SUBROUTINE slasyf_rook( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW,
185  $ INFO )
186 *
187 * -- LAPACK computational routine (version 3.5.0) --
188 * -- LAPACK is a software package provided by Univ. of Tennessee, --
189 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
190 * November 2013
191 *
192 * .. Scalar Arguments ..
193  CHARACTER UPLO
194  INTEGER INFO, KB, LDA, LDW, N, NB
195 * ..
196 * .. Array Arguments ..
197  INTEGER IPIV( * )
198  REAL A( lda, * ), W( ldw, * )
199 * ..
200 *
201 * =====================================================================
202 *
203 * .. Parameters ..
204  REAL ZERO, ONE
205  parameter( zero = 0.0e+0, one = 1.0e+0 )
206  REAL EIGHT, SEVTEN
207  parameter( eight = 8.0e+0, sevten = 17.0e+0 )
208 * ..
209 * .. Local Scalars ..
210  LOGICAL DONE
211  INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, JP1, JP2, K, KK,
212  $ kw, kkw, kp, kstep, p, ii
213 
214  REAL ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
215  $ stemp, r1, rowmax, t, sfmin
216 * ..
217 * .. External Functions ..
218  LOGICAL LSAME
219  INTEGER ISAMAX
220  REAL SLAMCH
221  EXTERNAL lsame, isamax, slamch
222 * ..
223 * .. External Subroutines ..
224  EXTERNAL scopy, sgemm, sgemv, sscal, sswap
225 * ..
226 * .. Intrinsic Functions ..
227  INTRINSIC abs, max, min, sqrt
228 * ..
229 * .. Executable Statements ..
230 *
231  info = 0
232 *
233 * Initialize ALPHA for use in choosing pivot block size.
234 *
235  alpha = ( one+sqrt( sevten ) ) / eight
236 *
237 * Compute machine safe minimum
238 *
239  sfmin = slamch( 'S' )
240 *
241  IF( lsame( uplo, 'U' ) ) THEN
242 *
243 * Factorize the trailing columns of A using the upper triangle
244 * of A and working backwards, and compute the matrix W = U12*D
245 * for use in updating A11
246 *
247 * K is the main loop index, decreasing from N in steps of 1 or 2
248 *
249  k = n
250  10 CONTINUE
251 *
252 * KW is the column of W which corresponds to column K of A
253 *
254  kw = nb + k - n
255 *
256 * Exit from loop
257 *
258  IF( ( k.LE.n-nb+1 .AND. nb.LT.n ) .OR. k.LT.1 )
259  $ GO TO 30
260 *
261  kstep = 1
262  p = k
263 *
264 * Copy column K of A to column KW of W and update it
265 *
266  CALL scopy( k, a( 1, k ), 1, w( 1, kw ), 1 )
267  IF( k.LT.n )
268  $ CALL sgemv( 'No transpose', k, n-k, -one, a( 1, k+1 ),
269  $ lda, w( k, kw+1 ), ldw, one, w( 1, kw ), 1 )
270 *
271 * Determine rows and columns to be interchanged and whether
272 * a 1-by-1 or 2-by-2 pivot block will be used
273 *
274  absakk = abs( w( k, kw ) )
275 *
276 * IMAX is the row-index of the largest off-diagonal element in
277 * column K, and COLMAX is its absolute value.
278 * Determine both COLMAX and IMAX.
279 *
280  IF( k.GT.1 ) THEN
281  imax = isamax( k-1, w( 1, kw ), 1 )
282  colmax = abs( w( imax, kw ) )
283  ELSE
284  colmax = zero
285  END IF
286 *
287  IF( max( absakk, colmax ).EQ.zero ) THEN
288 *
289 * Column K is zero or underflow: set INFO and continue
290 *
291  IF( info.EQ.0 )
292  $ info = k
293  kp = k
294  CALL scopy( k, w( 1, kw ), 1, a( 1, k ), 1 )
295  ELSE
296 *
297 * ============================================================
298 *
299 * Test for interchange
300 *
301 * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
302 * (used to handle NaN and Inf)
303 *
304  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
305 *
306 * no interchange, use 1-by-1 pivot block
307 *
308  kp = k
309 *
310  ELSE
311 *
312  done = .false.
313 *
314 * Loop until pivot found
315 *
316  12 CONTINUE
317 *
318 * Begin pivot search loop body
319 *
320 *
321 * Copy column IMAX to column KW-1 of W and update it
322 *
323  CALL scopy( imax, a( 1, imax ), 1, w( 1, kw-1 ), 1 )
324  CALL scopy( k-imax, a( imax, imax+1 ), lda,
325  $ w( imax+1, kw-1 ), 1 )
326 *
327  IF( k.LT.n )
328  $ CALL sgemv( 'No transpose', k, n-k, -one,
329  $ a( 1, k+1 ), lda, w( imax, kw+1 ), ldw,
330  $ one, w( 1, kw-1 ), 1 )
331 *
332 * JMAX is the column-index of the largest off-diagonal
333 * element in row IMAX, and ROWMAX is its absolute value.
334 * Determine both ROWMAX and JMAX.
335 *
336  IF( imax.NE.k ) THEN
337  jmax = imax + isamax( k-imax, w( imax+1, kw-1 ),
338  $ 1 )
339  rowmax = abs( w( jmax, kw-1 ) )
340  ELSE
341  rowmax = zero
342  END IF
343 *
344  IF( imax.GT.1 ) THEN
345  itemp = isamax( imax-1, w( 1, kw-1 ), 1 )
346  stemp = abs( w( itemp, kw-1 ) )
347  IF( stemp.GT.rowmax ) THEN
348  rowmax = stemp
349  jmax = itemp
350  END IF
351  END IF
352 *
353 * Equivalent to testing for
354 * ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
355 * (used to handle NaN and Inf)
356 *
357  IF( .NOT.(abs( w( imax, kw-1 ) ).LT.alpha*rowmax ) )
358  $ THEN
359 *
360 * interchange rows and columns K and IMAX,
361 * use 1-by-1 pivot block
362 *
363  kp = imax
364 *
365 * copy column KW-1 of W to column KW of W
366 *
367  CALL scopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
368 *
369  done = .true.
370 *
371 * Equivalent to testing for ROWMAX.EQ.COLMAX,
372 * (used to handle NaN and Inf)
373 *
374  ELSE IF( ( p.EQ.jmax ) .OR. ( rowmax.LE.colmax ) )
375  $ THEN
376 *
377 * interchange rows and columns K-1 and IMAX,
378 * use 2-by-2 pivot block
379 *
380  kp = imax
381  kstep = 2
382  done = .true.
383  ELSE
384 *
385 * Pivot not found: set params and repeat
386 *
387  p = imax
388  colmax = rowmax
389  imax = jmax
390 *
391 * Copy updated JMAXth (next IMAXth) column to Kth of W
392 *
393  CALL scopy( k, w( 1, kw-1 ), 1, w( 1, kw ), 1 )
394 *
395  END IF
396 *
397 * End pivot search loop body
398 *
399  IF( .NOT. done ) GOTO 12
400 *
401  END IF
402 *
403 * ============================================================
404 *
405  kk = k - kstep + 1
406 *
407 * KKW is the column of W which corresponds to column KK of A
408 *
409  kkw = nb + kk - n
410 *
411  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
412 *
413 * Copy non-updated column K to column P
414 *
415  CALL scopy( k-p, a( p+1, k ), 1, a( p, p+1 ), lda )
416  CALL scopy( p, a( 1, k ), 1, a( 1, p ), 1 )
417 *
418 * Interchange rows K and P in last N-K+1 columns of A
419 * and last N-K+2 columns of W
420 *
421  CALL sswap( n-k+1, a( k, k ), lda, a( p, k ), lda )
422  CALL sswap( n-kk+1, w( k, kkw ), ldw, w( p, kkw ), ldw )
423  END IF
424 *
425 * Updated column KP is already stored in column KKW of W
426 *
427  IF( kp.NE.kk ) THEN
428 *
429 * Copy non-updated column KK to column KP
430 *
431  a( kp, k ) = a( kk, k )
432  CALL scopy( k-1-kp, a( kp+1, kk ), 1, a( kp, kp+1 ),
433  $ lda )
434  CALL scopy( kp, a( 1, kk ), 1, a( 1, kp ), 1 )
435 *
436 * Interchange rows KK and KP in last N-KK+1 columns
437 * of A and W
438 *
439  CALL sswap( n-kk+1, a( kk, kk ), lda, a( kp, kk ), lda )
440  CALL sswap( n-kk+1, w( kk, kkw ), ldw, w( kp, kkw ),
441  $ ldw )
442  END IF
443 *
444  IF( kstep.EQ.1 ) THEN
445 *
446 * 1-by-1 pivot block D(k): column KW of W now holds
447 *
448 * W(k) = U(k)*D(k)
449 *
450 * where U(k) is the k-th column of U
451 *
452 * Store U(k) in column k of A
453 *
454  CALL scopy( k, w( 1, kw ), 1, a( 1, k ), 1 )
455  IF( k.GT.1 ) THEN
456  IF( abs( a( k, k ) ).GE.sfmin ) THEN
457  r1 = one / a( k, k )
458  CALL sscal( k-1, r1, a( 1, k ), 1 )
459  ELSE IF( a( k, k ).NE.zero ) THEN
460  DO 14 ii = 1, k - 1
461  a( ii, k ) = a( ii, k ) / a( k, k )
462  14 CONTINUE
463  END IF
464  END IF
465 *
466  ELSE
467 *
468 * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
469 * hold
470 *
471 * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
472 *
473 * where U(k) and U(k-1) are the k-th and (k-1)-th columns
474 * of U
475 *
476  IF( k.GT.2 ) THEN
477 *
478 * Store U(k) and U(k-1) in columns k and k-1 of A
479 *
480  d12 = w( k-1, kw )
481  d11 = w( k, kw ) / d12
482  d22 = w( k-1, kw-1 ) / d12
483  t = one / ( d11*d22-one )
484  DO 20 j = 1, k - 2
485  a( j, k-1 ) = t*( (d11*w( j, kw-1 )-w( j, kw ) ) /
486  $ d12 )
487  a( j, k ) = t*( ( d22*w( j, kw )-w( j, kw-1 ) ) /
488  $ d12 )
489  20 CONTINUE
490  END IF
491 *
492 * Copy D(k) to A
493 *
494  a( k-1, k-1 ) = w( k-1, kw-1 )
495  a( k-1, k ) = w( k-1, kw )
496  a( k, k ) = w( k, kw )
497  END IF
498  END IF
499 *
500 * Store details of the interchanges in IPIV
501 *
502  IF( kstep.EQ.1 ) THEN
503  ipiv( k ) = kp
504  ELSE
505  ipiv( k ) = -p
506  ipiv( k-1 ) = -kp
507  END IF
508 *
509 * Decrease K and return to the start of the main loop
510 *
511  k = k - kstep
512  GO TO 10
513 *
514  30 CONTINUE
515 *
516 * Update the upper triangle of A11 (= A(1:k,1:k)) as
517 *
518 * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
519 *
520 * computing blocks of NB columns at a time
521 *
522  DO 50 j = ( ( k-1 ) / nb )*nb + 1, 1, -nb
523  jb = min( nb, k-j+1 )
524 *
525 * Update the upper triangle of the diagonal block
526 *
527  DO 40 jj = j, j + jb - 1
528  CALL sgemv( 'No transpose', jj-j+1, n-k, -one,
529  $ a( j, k+1 ), lda, w( jj, kw+1 ), ldw, one,
530  $ a( j, jj ), 1 )
531  40 CONTINUE
532 *
533 * Update the rectangular superdiagonal block
534 *
535  IF( j.GE.2 )
536  $ CALL sgemm( 'No transpose', 'Transpose', j-1, jb,
537  $ n-k, -one, a( 1, k+1 ), lda, w( j, kw+1 ), ldw,
538  $ one, a( 1, j ), lda )
539  50 CONTINUE
540 *
541 * Put U12 in standard form by partially undoing the interchanges
542 * in columns k+1:n
543 *
544  j = k + 1
545  60 CONTINUE
546 *
547  kstep = 1
548  jp1 = 1
549  jj = j
550  jp2 = ipiv( j )
551  IF( jp2.LT.0 ) THEN
552  jp2 = -jp2
553  j = j + 1
554  jp1 = -ipiv( j )
555  kstep = 2
556  END IF
557 *
558  j = j + 1
559  IF( jp2.NE.jj .AND. j.LE.n )
560  $ CALL sswap( n-j+1, a( jp2, j ), lda, a( jj, j ), lda )
561  jj = j - 1
562  IF( jp1.NE.jj .AND. kstep.EQ.2 )
563  $ CALL sswap( n-j+1, a( jp1, j ), lda, a( jj, j ), lda )
564  IF( j.LE.n )
565  $ GO TO 60
566 *
567 * Set KB to the number of columns factorized
568 *
569  kb = n - k
570 *
571  ELSE
572 *
573 * Factorize the leading columns of A using the lower triangle
574 * of A and working forwards, and compute the matrix W = L21*D
575 * for use in updating A22
576 *
577 * K is the main loop index, increasing from 1 in steps of 1 or 2
578 *
579  k = 1
580  70 CONTINUE
581 *
582 * Exit from loop
583 *
584  IF( ( k.GE.nb .AND. nb.LT.n ) .OR. k.GT.n )
585  $ GO TO 90
586 *
587  kstep = 1
588  p = k
589 *
590 * Copy column K of A to column K of W and update it
591 *
592  CALL scopy( n-k+1, a( k, k ), 1, w( k, k ), 1 )
593  IF( k.GT.1 )
594  $ CALL sgemv( 'No transpose', n-k+1, k-1, -one, a( k, 1 ),
595  $ lda, w( k, 1 ), ldw, one, w( k, k ), 1 )
596 *
597 * Determine rows and columns to be interchanged and whether
598 * a 1-by-1 or 2-by-2 pivot block will be used
599 *
600  absakk = abs( w( k, k ) )
601 *
602 * IMAX is the row-index of the largest off-diagonal element in
603 * column K, and COLMAX is its absolute value.
604 * Determine both COLMAX and IMAX.
605 *
606  IF( k.LT.n ) THEN
607  imax = k + isamax( n-k, w( k+1, k ), 1 )
608  colmax = abs( w( imax, k ) )
609  ELSE
610  colmax = zero
611  END IF
612 *
613  IF( max( absakk, colmax ).EQ.zero ) THEN
614 *
615 * Column K is zero or underflow: set INFO and continue
616 *
617  IF( info.EQ.0 )
618  $ info = k
619  kp = k
620  CALL scopy( n-k+1, w( k, k ), 1, a( k, k ), 1 )
621  ELSE
622 *
623 * ============================================================
624 *
625 * Test for interchange
626 *
627 * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
628 * (used to handle NaN and Inf)
629 *
630  IF( .NOT.( absakk.LT.alpha*colmax ) ) THEN
631 *
632 * no interchange, use 1-by-1 pivot block
633 *
634  kp = k
635 *
636  ELSE
637 *
638  done = .false.
639 *
640 * Loop until pivot found
641 *
642  72 CONTINUE
643 *
644 * Begin pivot search loop body
645 *
646 *
647 * Copy column IMAX to column K+1 of W and update it
648 *
649  CALL scopy( imax-k, a( imax, k ), lda, w( k, k+1 ), 1)
650  CALL scopy( n-imax+1, a( imax, imax ), 1,
651  $ w( imax, k+1 ), 1 )
652  IF( k.GT.1 )
653  $ CALL sgemv( 'No transpose', n-k+1, k-1, -one,
654  $ a( k, 1 ), lda, w( imax, 1 ), ldw,
655  $ one, w( k, k+1 ), 1 )
656 *
657 * JMAX is the column-index of the largest off-diagonal
658 * element in row IMAX, and ROWMAX is its absolute value.
659 * Determine both ROWMAX and JMAX.
660 *
661  IF( imax.NE.k ) THEN
662  jmax = k - 1 + isamax( imax-k, w( k, k+1 ), 1 )
663  rowmax = abs( w( jmax, k+1 ) )
664  ELSE
665  rowmax = zero
666  END IF
667 *
668  IF( imax.LT.n ) THEN
669  itemp = imax + isamax( n-imax, w( imax+1, k+1 ), 1)
670  stemp = abs( w( itemp, k+1 ) )
671  IF( stemp.GT.rowmax ) THEN
672  rowmax = stemp
673  jmax = itemp
674  END IF
675  END IF
676 *
677 * Equivalent to testing for
678 * ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
679 * (used to handle NaN and Inf)
680 *
681  IF( .NOT.( abs( w( imax, k+1 ) ).LT.alpha*rowmax ) )
682  $ THEN
683 *
684 * interchange rows and columns K and IMAX,
685 * use 1-by-1 pivot block
686 *
687  kp = imax
688 *
689 * copy column K+1 of W to column K of W
690 *
691  CALL scopy( n-k+1, w( k, k+1 ), 1, w( k, k ), 1 )
692 *
693  done = .true.
694 *
695 * Equivalent to testing for ROWMAX.EQ.COLMAX,
696 * (used to handle NaN and Inf)
697 *
698  ELSE IF( ( p.EQ.jmax ) .OR. ( rowmax.LE.colmax ) )
699  $ THEN
700 *
701 * interchange rows and columns K+1 and IMAX,
702 * use 2-by-2 pivot block
703 *
704  kp = imax
705  kstep = 2
706  done = .true.
707  ELSE
708 *
709 * Pivot not found: set params and repeat
710 *
711  p = imax
712  colmax = rowmax
713  imax = jmax
714 *
715 * Copy updated JMAXth (next IMAXth) column to Kth of W
716 *
717  CALL scopy( n-k+1, w( k, k+1 ), 1, w( k, k ), 1 )
718 *
719  END IF
720 *
721 * End pivot search loop body
722 *
723  IF( .NOT. done ) GOTO 72
724 *
725  END IF
726 *
727 * ============================================================
728 *
729  kk = k + kstep - 1
730 *
731  IF( ( kstep.EQ.2 ) .AND. ( p.NE.k ) ) THEN
732 *
733 * Copy non-updated column K to column P
734 *
735  CALL scopy( p-k, a( k, k ), 1, a( p, k ), lda )
736  CALL scopy( n-p+1, a( p, k ), 1, a( p, p ), 1 )
737 *
738 * Interchange rows K and P in first K columns of A
739 * and first K+1 columns of W
740 *
741  CALL sswap( k, a( k, 1 ), lda, a( p, 1 ), lda )
742  CALL sswap( kk, w( k, 1 ), ldw, w( p, 1 ), ldw )
743  END IF
744 *
745 * Updated column KP is already stored in column KK of W
746 *
747  IF( kp.NE.kk ) THEN
748 *
749 * Copy non-updated column KK to column KP
750 *
751  a( kp, k ) = a( kk, k )
752  CALL scopy( kp-k-1, a( k+1, kk ), 1, a( kp, k+1 ), lda )
753  CALL scopy( n-kp+1, a( kp, kk ), 1, a( kp, kp ), 1 )
754 *
755 * Interchange rows KK and KP in first KK columns of A and W
756 *
757  CALL sswap( kk, a( kk, 1 ), lda, a( kp, 1 ), lda )
758  CALL sswap( kk, w( kk, 1 ), ldw, w( kp, 1 ), ldw )
759  END IF
760 *
761  IF( kstep.EQ.1 ) THEN
762 *
763 * 1-by-1 pivot block D(k): column k of W now holds
764 *
765 * W(k) = L(k)*D(k)
766 *
767 * where L(k) is the k-th column of L
768 *
769 * Store L(k) in column k of A
770 *
771  CALL scopy( n-k+1, w( k, k ), 1, a( k, k ), 1 )
772  IF( k.LT.n ) THEN
773  IF( abs( a( k, k ) ).GE.sfmin ) THEN
774  r1 = one / a( k, k )
775  CALL sscal( n-k, r1, a( k+1, k ), 1 )
776  ELSE IF( a( k, k ).NE.zero ) THEN
777  DO 74 ii = k + 1, n
778  a( ii, k ) = a( ii, k ) / a( k, k )
779  74 CONTINUE
780  END IF
781  END IF
782 *
783  ELSE
784 *
785 * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
786 *
787 * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
788 *
789 * where L(k) and L(k+1) are the k-th and (k+1)-th columns
790 * of L
791 *
792  IF( k.LT.n-1 ) THEN
793 *
794 * Store L(k) and L(k+1) in columns k and k+1 of A
795 *
796  d21 = w( k+1, k )
797  d11 = w( k+1, k+1 ) / d21
798  d22 = w( k, k ) / d21
799  t = one / ( d11*d22-one )
800  DO 80 j = k + 2, n
801  a( j, k ) = t*( ( d11*w( j, k )-w( j, k+1 ) ) /
802  $ d21 )
803  a( j, k+1 ) = t*( ( d22*w( j, k+1 )-w( j, k ) ) /
804  $ d21 )
805  80 CONTINUE
806  END IF
807 *
808 * Copy D(k) to A
809 *
810  a( k, k ) = w( k, k )
811  a( k+1, k ) = w( k+1, k )
812  a( k+1, k+1 ) = w( k+1, k+1 )
813  END IF
814  END IF
815 *
816 * Store details of the interchanges in IPIV
817 *
818  IF( kstep.EQ.1 ) THEN
819  ipiv( k ) = kp
820  ELSE
821  ipiv( k ) = -p
822  ipiv( k+1 ) = -kp
823  END IF
824 *
825 * Increase K and return to the start of the main loop
826 *
827  k = k + kstep
828  GO TO 70
829 *
830  90 CONTINUE
831 *
832 * Update the lower triangle of A22 (= A(k:n,k:n)) as
833 *
834 * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
835 *
836 * computing blocks of NB columns at a time
837 *
838  DO 110 j = k, n, nb
839  jb = min( nb, n-j+1 )
840 *
841 * Update the lower triangle of the diagonal block
842 *
843  DO 100 jj = j, j + jb - 1
844  CALL sgemv( 'No transpose', j+jb-jj, k-1, -one,
845  $ a( jj, 1 ), lda, w( jj, 1 ), ldw, one,
846  $ a( jj, jj ), 1 )
847  100 CONTINUE
848 *
849 * Update the rectangular subdiagonal block
850 *
851  IF( j+jb.LE.n )
852  $ CALL sgemm( 'No transpose', 'Transpose', n-j-jb+1, jb,
853  $ k-1, -one, a( j+jb, 1 ), lda, w( j, 1 ), ldw,
854  $ one, a( j+jb, j ), lda )
855  110 CONTINUE
856 *
857 * Put L21 in standard form by partially undoing the interchanges
858 * in columns 1:k-1
859 *
860  j = k - 1
861  120 CONTINUE
862 *
863  kstep = 1
864  jp1 = 1
865  jj = j
866  jp2 = ipiv( j )
867  IF( jp2.LT.0 ) THEN
868  jp2 = -jp2
869  j = j - 1
870  jp1 = -ipiv( j )
871  kstep = 2
872  END IF
873 *
874  j = j - 1
875  IF( jp2.NE.jj .AND. j.GE.1 )
876  $ CALL sswap( j, a( jp2, 1 ), lda, a( jj, 1 ), lda )
877  jj = j + 1
878  IF( jp1.NE.jj .AND. kstep.EQ.2 )
879  $ CALL sswap( j, a( jp1, 1 ), lda, a( jj, 1 ), lda )
880  IF( j.GE.1 )
881  $ GO TO 120
882 *
883 * Set KB to the number of columns factorized
884 *
885  kb = k - 1
886 *
887  END IF
888  RETURN
889 *
890 * End of SLASYF_ROOK
891 *
892  END
slasyf_rook
subroutine slasyf_rook(UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO)
SLASYF_ROOK computes a partial factorization of a real symmetric matrix using the bounded Bunch-Kaufm...
Definition: slasyf_rook.f:186
sgemm
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
sgemv
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:158
sscal
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81
sswap
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:84
scopy
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84