001:       SUBROUTINE SLASYF( UPLO, N, NB, KB, A, LDA, IPIV, W, LDW, INFO )
002: *
003: *  -- LAPACK routine (version 3.2) --
004: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
005: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
006: *     November 2006
007: *
008: *     .. Scalar Arguments ..
009:       CHARACTER          UPLO
010:       INTEGER            INFO, KB, LDA, LDW, N, NB
011: *     ..
012: *     .. Array Arguments ..
013:       INTEGER            IPIV( * )
014:       REAL               A( LDA, * ), W( LDW, * )
015: *     ..
016: *
017: *  Purpose
018: *  =======
019: *
020: *  SLASYF computes a partial factorization of a real symmetric matrix A
021: *  using the Bunch-Kaufman diagonal pivoting method. The partial
022: *  factorization has the form:
023: *
024: *  A  =  ( I  U12 ) ( A11  0  ) (  I    0   )  if UPLO = 'U', or:
025: *        ( 0  U22 ) (  0   D  ) ( U12' U22' )
026: *
027: *  A  =  ( L11  0 ) (  D   0  ) ( L11' L21' )  if UPLO = 'L'
028: *        ( L21  I ) (  0  A22 ) (  0    I   )
029: *
030: *  where the order of D is at most NB. The actual order is returned in
031: *  the argument KB, and is either NB or NB-1, or N if N <= NB.
032: *
033: *  SLASYF is an auxiliary routine called by SSYTRF. It uses blocked code
034: *  (calling Level 3 BLAS) to update the submatrix A11 (if UPLO = 'U') or
035: *  A22 (if UPLO = 'L').
036: *
037: *  Arguments
038: *  =========
039: *
040: *  UPLO    (input) CHARACTER*1
041: *          Specifies whether the upper or lower triangular part of the
042: *          symmetric matrix A is stored:
043: *          = 'U':  Upper triangular
044: *          = 'L':  Lower triangular
045: *
046: *  N       (input) INTEGER
047: *          The order of the matrix A.  N >= 0.
048: *
049: *  NB      (input) INTEGER
050: *          The maximum number of columns of the matrix A that should be
051: *          factored.  NB should be at least 2 to allow for 2-by-2 pivot
052: *          blocks.
053: *
054: *  KB      (output) INTEGER
055: *          The number of columns of A that were actually factored.
056: *          KB is either NB-1 or NB, or N if N <= NB.
057: *
058: *  A       (input/output) REAL array, dimension (LDA,N)
059: *          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
060: *          n-by-n upper triangular part of A contains the upper
061: *          triangular part of the matrix A, and the strictly lower
062: *          triangular part of A is not referenced.  If UPLO = 'L', the
063: *          leading n-by-n lower triangular part of A contains the lower
064: *          triangular part of the matrix A, and the strictly upper
065: *          triangular part of A is not referenced.
066: *          On exit, A contains details of the partial factorization.
067: *
068: *  LDA     (input) INTEGER
069: *          The leading dimension of the array A.  LDA >= max(1,N).
070: *
071: *  IPIV    (output) INTEGER array, dimension (N)
072: *          Details of the interchanges and the block structure of D.
073: *          If UPLO = 'U', only the last KB elements of IPIV are set;
074: *          if UPLO = 'L', only the first KB elements are set.
075: *
076: *          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
077: *          interchanged and D(k,k) is a 1-by-1 diagonal block.
078: *          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
079: *          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
080: *          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
081: *          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
082: *          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
083: *
084: *  W       (workspace) REAL array, dimension (LDW,NB)
085: *
086: *  LDW     (input) INTEGER
087: *          The leading dimension of the array W.  LDW >= max(1,N).
088: *
089: *  INFO    (output) INTEGER
090: *          = 0: successful exit
091: *          > 0: if INFO = k, D(k,k) is exactly zero.  The factorization
092: *               has been completed, but the block diagonal matrix D is
093: *               exactly singular.
094: *
095: *  =====================================================================
096: *
097: *     .. Parameters ..
098:       REAL               ZERO, ONE
099:       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
100:       REAL               EIGHT, SEVTEN
101:       PARAMETER          ( EIGHT = 8.0E+0, SEVTEN = 17.0E+0 )
102: *     ..
103: *     .. Local Scalars ..
104:       INTEGER            IMAX, J, JB, JJ, JMAX, JP, K, KK, KKW, KP,
105:      $                   KSTEP, KW
106:       REAL               ABSAKK, ALPHA, COLMAX, D11, D21, D22, R1,
107:      $                   ROWMAX, T
108: *     ..
109: *     .. External Functions ..
110:       LOGICAL            LSAME
111:       INTEGER            ISAMAX
112:       EXTERNAL           LSAME, ISAMAX
113: *     ..
114: *     .. External Subroutines ..
115:       EXTERNAL           SCOPY, SGEMM, SGEMV, SSCAL, SSWAP
116: *     ..
117: *     .. Intrinsic Functions ..
118:       INTRINSIC          ABS, MAX, MIN, SQRT
119: *     ..
120: *     .. Executable Statements ..
121: *
122:       INFO = 0
123: *
124: *     Initialize ALPHA for use in choosing pivot block size.
125: *
126:       ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
127: *
128:       IF( LSAME( UPLO, 'U' ) ) THEN
129: *
130: *        Factorize the trailing columns of A using the upper triangle
131: *        of A and working backwards, and compute the matrix W = U12*D
132: *        for use in updating A11
133: *
134: *        K is the main loop index, decreasing from N in steps of 1 or 2
135: *
136: *        KW is the column of W which corresponds to column K of A
137: *
138:          K = N
139:    10    CONTINUE
140:          KW = NB + K - N
141: *
142: *        Exit from loop
143: *
144:          IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
145:      $      GO TO 30
146: *
147: *        Copy column K of A to column KW of W and update it
148: *
149:          CALL SCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
150:          IF( K.LT.N )
151:      $      CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ), LDA,
152:      $                  W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
153: *
154:          KSTEP = 1
155: *
156: *        Determine rows and columns to be interchanged and whether
157: *        a 1-by-1 or 2-by-2 pivot block will be used
158: *
159:          ABSAKK = ABS( W( K, KW ) )
160: *
161: *        IMAX is the row-index of the largest off-diagonal element in
162: *        column K, and COLMAX is its absolute value
163: *
164:          IF( K.GT.1 ) THEN
165:             IMAX = ISAMAX( K-1, W( 1, KW ), 1 )
166:             COLMAX = ABS( W( IMAX, KW ) )
167:          ELSE
168:             COLMAX = ZERO
169:          END IF
170: *
171:          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
172: *
173: *           Column K is zero: set INFO and continue
174: *
175:             IF( INFO.EQ.0 )
176:      $         INFO = K
177:             KP = K
178:          ELSE
179:             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
180: *
181: *              no interchange, use 1-by-1 pivot block
182: *
183:                KP = K
184:             ELSE
185: *
186: *              Copy column IMAX to column KW-1 of W and update it
187: *
188:                CALL SCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
189:                CALL SCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
190:      $                     W( IMAX+1, KW-1 ), 1 )
191:                IF( K.LT.N )
192:      $            CALL SGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
193:      $                        LDA, W( IMAX, KW+1 ), LDW, ONE,
194:      $                        W( 1, KW-1 ), 1 )
195: *
196: *              JMAX is the column-index of the largest off-diagonal
197: *              element in row IMAX, and ROWMAX is its absolute value
198: *
199:                JMAX = IMAX + ISAMAX( K-IMAX, W( IMAX+1, KW-1 ), 1 )
200:                ROWMAX = ABS( W( JMAX, KW-1 ) )
201:                IF( IMAX.GT.1 ) THEN
202:                   JMAX = ISAMAX( IMAX-1, W( 1, KW-1 ), 1 )
203:                   ROWMAX = MAX( ROWMAX, ABS( W( JMAX, KW-1 ) ) )
204:                END IF
205: *
206:                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
207: *
208: *                 no interchange, use 1-by-1 pivot block
209: *
210:                   KP = K
211:                ELSE IF( ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX ) THEN
212: *
213: *                 interchange rows and columns K and IMAX, use 1-by-1
214: *                 pivot block
215: *
216:                   KP = IMAX
217: *
218: *                 copy column KW-1 of W to column KW
219: *
220:                   CALL SCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
221:                ELSE
222: *
223: *                 interchange rows and columns K-1 and IMAX, use 2-by-2
224: *                 pivot block
225: *
226:                   KP = IMAX
227:                   KSTEP = 2
228:                END IF
229:             END IF
230: *
231:             KK = K - KSTEP + 1
232:             KKW = NB + KK - N
233: *
234: *           Updated column KP is already stored in column KKW of W
235: *
236:             IF( KP.NE.KK ) THEN
237: *
238: *              Copy non-updated column KK to column KP
239: *
240:                A( KP, K ) = A( KK, K )
241:                CALL SCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
242:      $                     LDA )
243:                CALL SCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
244: *
245: *              Interchange rows KK and KP in last KK columns of A and W
246: *
247:                CALL SSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
248:                CALL SSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
249:      $                     LDW )
250:             END IF
251: *
252:             IF( KSTEP.EQ.1 ) THEN
253: *
254: *              1-by-1 pivot block D(k): column KW of W now holds
255: *
256: *              W(k) = U(k)*D(k)
257: *
258: *              where U(k) is the k-th column of U
259: *
260: *              Store U(k) in column k of A
261: *
262:                CALL SCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
263:                R1 = ONE / A( K, K )
264:                CALL SSCAL( K-1, R1, A( 1, K ), 1 )
265:             ELSE
266: *
267: *              2-by-2 pivot block D(k): columns KW and KW-1 of W now
268: *              hold
269: *
270: *              ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
271: *
272: *              where U(k) and U(k-1) are the k-th and (k-1)-th columns
273: *              of U
274: *
275:                IF( K.GT.2 ) THEN
276: *
277: *                 Store U(k) and U(k-1) in columns k and k-1 of A
278: *
279:                   D21 = W( K-1, KW )
280:                   D11 = W( K, KW ) / D21
281:                   D22 = W( K-1, KW-1 ) / D21
282:                   T = ONE / ( D11*D22-ONE )
283:                   D21 = T / D21
284:                   DO 20 J = 1, K - 2
285:                      A( J, K-1 ) = D21*( D11*W( J, KW-1 )-W( J, KW ) )
286:                      A( J, K ) = D21*( D22*W( J, KW )-W( J, KW-1 ) )
287:    20             CONTINUE
288:                END IF
289: *
290: *              Copy D(k) to A
291: *
292:                A( K-1, K-1 ) = W( K-1, KW-1 )
293:                A( K-1, K ) = W( K-1, KW )
294:                A( K, K ) = W( K, KW )
295:             END IF
296:          END IF
297: *
298: *        Store details of the interchanges in IPIV
299: *
300:          IF( KSTEP.EQ.1 ) THEN
301:             IPIV( K ) = KP
302:          ELSE
303:             IPIV( K ) = -KP
304:             IPIV( K-1 ) = -KP
305:          END IF
306: *
307: *        Decrease K and return to the start of the main loop
308: *
309:          K = K - KSTEP
310:          GO TO 10
311: *
312:    30    CONTINUE
313: *
314: *        Update the upper triangle of A11 (= A(1:k,1:k)) as
315: *
316: *        A11 := A11 - U12*D*U12' = A11 - U12*W'
317: *
318: *        computing blocks of NB columns at a time
319: *
320:          DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
321:             JB = MIN( NB, K-J+1 )
322: *
323: *           Update the upper triangle of the diagonal block
324: *
325:             DO 40 JJ = J, J + JB - 1
326:                CALL SGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
327:      $                     A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
328:      $                     A( J, JJ ), 1 )
329:    40       CONTINUE
330: *
331: *           Update the rectangular superdiagonal block
332: *
333:             CALL SGEMM( 'No transpose', 'Transpose', J-1, JB, N-K, -ONE,
334:      $                  A( 1, K+1 ), LDA, W( J, KW+1 ), LDW, ONE,
335:      $                  A( 1, J ), LDA )
336:    50    CONTINUE
337: *
338: *        Put U12 in standard form by partially undoing the interchanges
339: *        in columns k+1:n
340: *
341:          J = K + 1
342:    60    CONTINUE
343:          JJ = J
344:          JP = IPIV( J )
345:          IF( JP.LT.0 ) THEN
346:             JP = -JP
347:             J = J + 1
348:          END IF
349:          J = J + 1
350:          IF( JP.NE.JJ .AND. J.LE.N )
351:      $      CALL SSWAP( N-J+1, A( JP, J ), LDA, A( JJ, J ), LDA )
352:          IF( J.LE.N )
353:      $      GO TO 60
354: *
355: *        Set KB to the number of columns factorized
356: *
357:          KB = N - K
358: *
359:       ELSE
360: *
361: *        Factorize the leading columns of A using the lower triangle
362: *        of A and working forwards, and compute the matrix W = L21*D
363: *        for use in updating A22
364: *
365: *        K is the main loop index, increasing from 1 in steps of 1 or 2
366: *
367:          K = 1
368:    70    CONTINUE
369: *
370: *        Exit from loop
371: *
372:          IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
373:      $      GO TO 90
374: *
375: *        Copy column K of A to column K of W and update it
376: *
377:          CALL SCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
378:          CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ), LDA,
379:      $               W( K, 1 ), LDW, ONE, W( K, K ), 1 )
380: *
381:          KSTEP = 1
382: *
383: *        Determine rows and columns to be interchanged and whether
384: *        a 1-by-1 or 2-by-2 pivot block will be used
385: *
386:          ABSAKK = ABS( W( K, K ) )
387: *
388: *        IMAX is the row-index of the largest off-diagonal element in
389: *        column K, and COLMAX is its absolute value
390: *
391:          IF( K.LT.N ) THEN
392:             IMAX = K + ISAMAX( N-K, W( K+1, K ), 1 )
393:             COLMAX = ABS( W( IMAX, K ) )
394:          ELSE
395:             COLMAX = ZERO
396:          END IF
397: *
398:          IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
399: *
400: *           Column K is zero: set INFO and continue
401: *
402:             IF( INFO.EQ.0 )
403:      $         INFO = K
404:             KP = K
405:          ELSE
406:             IF( ABSAKK.GE.ALPHA*COLMAX ) THEN
407: *
408: *              no interchange, use 1-by-1 pivot block
409: *
410:                KP = K
411:             ELSE
412: *
413: *              Copy column IMAX to column K+1 of W and update it
414: *
415:                CALL SCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1 )
416:                CALL SCOPY( N-IMAX+1, A( IMAX, IMAX ), 1, W( IMAX, K+1 ),
417:      $                     1 )
418:                CALL SGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
419:      $                     LDA, W( IMAX, 1 ), LDW, ONE, W( K, K+1 ), 1 )
420: *
421: *              JMAX is the column-index of the largest off-diagonal
422: *              element in row IMAX, and ROWMAX is its absolute value
423: *
424:                JMAX = K - 1 + ISAMAX( IMAX-K, W( K, K+1 ), 1 )
425:                ROWMAX = ABS( W( JMAX, K+1 ) )
426:                IF( IMAX.LT.N ) THEN
427:                   JMAX = IMAX + ISAMAX( N-IMAX, W( IMAX+1, K+1 ), 1 )
428:                   ROWMAX = MAX( ROWMAX, ABS( W( JMAX, K+1 ) ) )
429:                END IF
430: *
431:                IF( ABSAKK.GE.ALPHA*COLMAX*( COLMAX / ROWMAX ) ) THEN
432: *
433: *                 no interchange, use 1-by-1 pivot block
434: *
435:                   KP = K
436:                ELSE IF( ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX ) THEN
437: *
438: *                 interchange rows and columns K and IMAX, use 1-by-1
439: *                 pivot block
440: *
441:                   KP = IMAX
442: *
443: *                 copy column K+1 of W to column K
444: *
445:                   CALL SCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
446:                ELSE
447: *
448: *                 interchange rows and columns K+1 and IMAX, use 2-by-2
449: *                 pivot block
450: *
451:                   KP = IMAX
452:                   KSTEP = 2
453:                END IF
454:             END IF
455: *
456:             KK = K + KSTEP - 1
457: *
458: *           Updated column KP is already stored in column KK of W
459: *
460:             IF( KP.NE.KK ) THEN
461: *
462: *              Copy non-updated column KK to column KP
463: *
464:                A( KP, K ) = A( KK, K )
465:                CALL SCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
466:                CALL SCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
467: *
468: *              Interchange rows KK and KP in first KK columns of A and W
469: *
470:                CALL SSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
471:                CALL SSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
472:             END IF
473: *
474:             IF( KSTEP.EQ.1 ) THEN
475: *
476: *              1-by-1 pivot block D(k): column k of W now holds
477: *
478: *              W(k) = L(k)*D(k)
479: *
480: *              where L(k) is the k-th column of L
481: *
482: *              Store L(k) in column k of A
483: *
484:                CALL SCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
485:                IF( K.LT.N ) THEN
486:                   R1 = ONE / A( K, K )
487:                   CALL SSCAL( N-K, R1, A( K+1, K ), 1 )
488:                END IF
489:             ELSE
490: *
491: *              2-by-2 pivot block D(k): columns k and k+1 of W now hold
492: *
493: *              ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
494: *
495: *              where L(k) and L(k+1) are the k-th and (k+1)-th columns
496: *              of L
497: *
498:                IF( K.LT.N-1 ) THEN
499: *
500: *                 Store L(k) and L(k+1) in columns k and k+1 of A
501: *
502:                   D21 = W( K+1, K )
503:                   D11 = W( K+1, K+1 ) / D21
504:                   D22 = W( K, K ) / D21
505:                   T = ONE / ( D11*D22-ONE )
506:                   D21 = T / D21
507:                   DO 80 J = K + 2, N
508:                      A( J, K ) = D21*( D11*W( J, K )-W( J, K+1 ) )
509:                      A( J, K+1 ) = D21*( D22*W( J, K+1 )-W( J, K ) )
510:    80             CONTINUE
511:                END IF
512: *
513: *              Copy D(k) to A
514: *
515:                A( K, K ) = W( K, K )
516:                A( K+1, K ) = W( K+1, K )
517:                A( K+1, K+1 ) = W( K+1, K+1 )
518:             END IF
519:          END IF
520: *
521: *        Store details of the interchanges in IPIV
522: *
523:          IF( KSTEP.EQ.1 ) THEN
524:             IPIV( K ) = KP
525:          ELSE
526:             IPIV( K ) = -KP
527:             IPIV( K+1 ) = -KP
528:          END IF
529: *
530: *        Increase K and return to the start of the main loop
531: *
532:          K = K + KSTEP
533:          GO TO 70
534: *
535:    90    CONTINUE
536: *
537: *        Update the lower triangle of A22 (= A(k:n,k:n)) as
538: *
539: *        A22 := A22 - L21*D*L21' = A22 - L21*W'
540: *
541: *        computing blocks of NB columns at a time
542: *
543:          DO 110 J = K, N, NB
544:             JB = MIN( NB, N-J+1 )
545: *
546: *           Update the lower triangle of the diagonal block
547: *
548:             DO 100 JJ = J, J + JB - 1
549:                CALL SGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
550:      $                     A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
551:      $                     A( JJ, JJ ), 1 )
552:   100       CONTINUE
553: *
554: *           Update the rectangular subdiagonal block
555: *
556:             IF( J+JB.LE.N )
557:      $         CALL SGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
558:      $                     K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ), LDW,
559:      $                     ONE, A( J+JB, J ), LDA )
560:   110    CONTINUE
561: *
562: *        Put L21 in standard form by partially undoing the interchanges
563: *        in columns 1:k-1
564: *
565:          J = K - 1
566:   120    CONTINUE
567:          JJ = J
568:          JP = IPIV( J )
569:          IF( JP.LT.0 ) THEN
570:             JP = -JP
571:             J = J - 1
572:          END IF
573:          J = J - 1
574:          IF( JP.NE.JJ .AND. J.GE.1 )
575:      $      CALL SSWAP( J, A( JP, 1 ), LDA, A( JJ, 1 ), LDA )
576:          IF( J.GE.1 )
577:      $      GO TO 120
578: *
579: *        Set KB to the number of columns factorized
580: *
581:          KB = K - 1
582: *
583:       END IF
584:       RETURN
585: *
586: *     End of SLASYF
587: *
588:       END
589: