1: *> \brief \b DLASYF_RK computes a partial factorization of a real symmetric indefinite matrix using 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: *
8: *> \htmlonly
9: *> Download DLASYF_RK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlasyf_rk.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlasyf_rk.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlasyf_rk.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
22: * INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER UPLO
26: * INTEGER INFO, KB, LDA, LDW, N, NB
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IPIV( * )
30: * DOUBLE PRECISION A( LDA, * ), E( * ), W( LDW, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *> DLASYF_RK 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: *> DLASYF_RK is an auxiliary routine called by DSYTRF_RK. 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 DOUBLE PRECISION array, dimension (LDA,N)
92: *> On entry, the symmetric matrix A.
93: *> If UPLO = 'U': the leading N-by-N upper triangular part
94: *> of A contains the upper triangular part of the matrix A,
95: *> and the strictly lower triangular part of A is not
96: *> referenced.
97: *>
98: *> If UPLO = 'L': the leading N-by-N lower triangular part
99: *> of A contains the lower triangular part of the matrix A,
100: *> and the strictly upper triangular part of A is not
101: *> referenced.
102: *>
103: *> On exit, contains:
104: *> a) ONLY diagonal elements of the symmetric block diagonal
105: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
106: *> (superdiagonal (or subdiagonal) elements of D
107: *> are stored on exit in array E), and
108: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
109: *> If UPLO = 'L': factor L in the subdiagonal part of A.
110: *> \endverbatim
111: *>
112: *> \param[in] LDA
113: *> \verbatim
114: *> LDA is INTEGER
115: *> The leading dimension of the array A. LDA >= max(1,N).
116: *> \endverbatim
117: *>
118: *> \param[out] E
119: *> \verbatim
120: *> E is DOUBLE PRECISION array, dimension (N)
121: *> On exit, contains the superdiagonal (or subdiagonal)
122: *> elements of the symmetric block diagonal matrix D
123: *> with 1-by-1 or 2-by-2 diagonal blocks, where
124: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
125: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
126: *>
127: *> NOTE: For 1-by-1 diagonal block D(k), where
128: *> 1 <= k <= N, the element E(k) is set to 0 in both
129: *> UPLO = 'U' or UPLO = 'L' cases.
130: *> \endverbatim
131: *>
132: *> \param[out] IPIV
133: *> \verbatim
134: *> IPIV is INTEGER array, dimension (N)
135: *> IPIV describes the permutation matrix P in the factorization
136: *> of matrix A as follows. The absolute value of IPIV(k)
137: *> represents the index of row and column that were
138: *> interchanged with the k-th row and column. The value of UPLO
139: *> describes the order in which the interchanges were applied.
140: *> Also, the sign of IPIV represents the block structure of
141: *> the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
142: *> diagonal blocks which correspond to 1 or 2 interchanges
143: *> at each factorization step.
144: *>
145: *> If UPLO = 'U',
146: *> ( in factorization order, k decreases from N to 1 ):
147: *> a) A single positive entry IPIV(k) > 0 means:
148: *> D(k,k) is a 1-by-1 diagonal block.
149: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
150: *> interchanged in the submatrix A(1:N,N-KB+1:N);
151: *> If IPIV(k) = k, no interchange occurred.
152: *>
153: *>
154: *> b) A pair of consecutive negative entries
155: *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
156: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
157: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
158: *> 1) If -IPIV(k) != k, rows and columns
159: *> k and -IPIV(k) were interchanged
160: *> in the matrix A(1:N,N-KB+1:N).
161: *> If -IPIV(k) = k, no interchange occurred.
162: *> 2) If -IPIV(k-1) != k-1, rows and columns
163: *> k-1 and -IPIV(k-1) were interchanged
164: *> in the submatrix A(1:N,N-KB+1:N).
165: *> If -IPIV(k-1) = k-1, no interchange occurred.
166: *>
167: *> c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
168: *>
169: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
170: *>
171: *> If UPLO = 'L',
172: *> ( in factorization order, k increases from 1 to N ):
173: *> a) A single positive entry IPIV(k) > 0 means:
174: *> D(k,k) is a 1-by-1 diagonal block.
175: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
176: *> interchanged in the submatrix A(1:N,1:KB).
177: *> If IPIV(k) = k, no interchange occurred.
178: *>
179: *> b) A pair of consecutive negative entries
180: *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
181: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
182: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
183: *> 1) If -IPIV(k) != k, rows and columns
184: *> k and -IPIV(k) were interchanged
185: *> in the submatrix A(1:N,1:KB).
186: *> If -IPIV(k) = k, no interchange occurred.
187: *> 2) If -IPIV(k+1) != k+1, rows and columns
188: *> k-1 and -IPIV(k-1) were interchanged
189: *> in the submatrix A(1:N,1:KB).
190: *> If -IPIV(k+1) = k+1, no interchange occurred.
191: *>
192: *> c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
193: *>
194: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
195: *> \endverbatim
196: *>
197: *> \param[out] W
198: *> \verbatim
199: *> W is DOUBLE PRECISION array, dimension (LDW,NB)
200: *> \endverbatim
201: *>
202: *> \param[in] LDW
203: *> \verbatim
204: *> LDW is INTEGER
205: *> The leading dimension of the array W. LDW >= max(1,N).
206: *> \endverbatim
207: *>
208: *> \param[out] INFO
209: *> \verbatim
210: *> INFO is INTEGER
211: *> = 0: successful exit
212: *>
213: *> < 0: If INFO = -k, the k-th argument had an illegal value
214: *>
215: *> > 0: If INFO = k, the matrix A is singular, because:
216: *> If UPLO = 'U': column k in the upper
217: *> triangular part of A contains all zeros.
218: *> If UPLO = 'L': column k in the lower
219: *> triangular part of A contains all zeros.
220: *>
221: *> Therefore D(k,k) is exactly zero, and superdiagonal
222: *> elements of column k of U (or subdiagonal elements of
223: *> column k of L ) are all zeros. The factorization has
224: *> been completed, but the block diagonal matrix D is
225: *> exactly singular, and division by zero will occur if
226: *> it is used to solve a system of equations.
227: *>
228: *> NOTE: INFO only stores the first occurrence of
229: *> a singularity, any subsequent occurrence of singularity
230: *> is not stored in INFO even though the factorization
231: *> always completes.
232: *> \endverbatim
233: *
234: * Authors:
235: * ========
236: *
237: *> \author Univ. of Tennessee
238: *> \author Univ. of California Berkeley
239: *> \author Univ. of Colorado Denver
240: *> \author NAG Ltd.
241: *
242: *> \ingroup doubleSYcomputational
243: *
244: *> \par Contributors:
245: * ==================
246: *>
247: *> \verbatim
248: *>
249: *> December 2016, Igor Kozachenko,
250: *> Computer Science Division,
251: *> University of California, Berkeley
252: *>
253: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
254: *> School of Mathematics,
255: *> University of Manchester
256: *>
257: *> \endverbatim
258: *
259: * =====================================================================
260: SUBROUTINE DLASYF_RK( UPLO, N, NB, KB, A, LDA, E, IPIV, W, LDW,
261: $ INFO )
262: *
263: * -- LAPACK computational routine --
264: * -- LAPACK is a software package provided by Univ. of Tennessee, --
265: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
266: *
267: * .. Scalar Arguments ..
268: CHARACTER UPLO
269: INTEGER INFO, KB, LDA, LDW, N, NB
270: * ..
271: * .. Array Arguments ..
272: INTEGER IPIV( * )
273: DOUBLE PRECISION A( LDA, * ), E( * ), W( LDW, * )
274: * ..
275: *
276: * =====================================================================
277: *
278: * .. Parameters ..
279: DOUBLE PRECISION ZERO, ONE
280: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
281: DOUBLE PRECISION EIGHT, SEVTEN
282: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
283: * ..
284: * .. Local Scalars ..
285: LOGICAL DONE
286: INTEGER IMAX, ITEMP, J, JB, JJ, JMAX, K, KK, KW, KKW,
287: $ KP, KSTEP, P, II
288: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
289: $ DTEMP, R1, ROWMAX, T, SFMIN
290: * ..
291: * .. External Functions ..
292: LOGICAL LSAME
293: INTEGER IDAMAX
294: DOUBLE PRECISION DLAMCH
295: EXTERNAL LSAME, IDAMAX, DLAMCH
296: * ..
297: * .. External Subroutines ..
298: EXTERNAL DCOPY, DGEMM, DGEMV, DSCAL, DSWAP
299: * ..
300: * .. Intrinsic Functions ..
301: INTRINSIC ABS, MAX, MIN, SQRT
302: * ..
303: * .. Executable Statements ..
304: *
305: INFO = 0
306: *
307: * Initialize ALPHA for use in choosing pivot block size.
308: *
309: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
310: *
311: * Compute machine safe minimum
312: *
313: SFMIN = DLAMCH( 'S' )
314: *
315: IF( LSAME( UPLO, 'U' ) ) THEN
316: *
317: * Factorize the trailing columns of A using the upper triangle
318: * of A and working backwards, and compute the matrix W = U12*D
319: * for use in updating A11
320: *
321: * Initialize the first entry of array E, where superdiagonal
322: * elements of D are stored
323: *
324: E( 1 ) = ZERO
325: *
326: * K is the main loop index, decreasing from N in steps of 1 or 2
327: *
328: K = N
329: 10 CONTINUE
330: *
331: * KW is the column of W which corresponds to column K of A
332: *
333: KW = NB + K - N
334: *
335: * Exit from loop
336: *
337: IF( ( K.LE.N-NB+1 .AND. NB.LT.N ) .OR. K.LT.1 )
338: $ GO TO 30
339: *
340: KSTEP = 1
341: P = K
342: *
343: * Copy column K of A to column KW of W and update it
344: *
345: CALL DCOPY( K, A( 1, K ), 1, W( 1, KW ), 1 )
346: IF( K.LT.N )
347: $ CALL DGEMV( 'No transpose', K, N-K, -ONE, A( 1, K+1 ),
348: $ LDA, W( K, KW+1 ), LDW, ONE, W( 1, KW ), 1 )
349: *
350: * Determine rows and columns to be interchanged and whether
351: * a 1-by-1 or 2-by-2 pivot block will be used
352: *
353: ABSAKK = ABS( W( K, KW ) )
354: *
355: * IMAX is the row-index of the largest off-diagonal element in
356: * column K, and COLMAX is its absolute value.
357: * Determine both COLMAX and IMAX.
358: *
359: IF( K.GT.1 ) THEN
360: IMAX = IDAMAX( K-1, W( 1, KW ), 1 )
361: COLMAX = ABS( W( IMAX, KW ) )
362: ELSE
363: COLMAX = ZERO
364: END IF
365: *
366: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
367: *
368: * Column K is zero or underflow: set INFO and continue
369: *
370: IF( INFO.EQ.0 )
371: $ INFO = K
372: KP = K
373: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
374: *
375: * Set E( K ) to zero
376: *
377: IF( K.GT.1 )
378: $ E( K ) = ZERO
379: *
380: ELSE
381: *
382: * ============================================================
383: *
384: * Test for interchange
385: *
386: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
387: * (used to handle NaN and Inf)
388: *
389: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
390: *
391: * no interchange, use 1-by-1 pivot block
392: *
393: KP = K
394: *
395: ELSE
396: *
397: DONE = .FALSE.
398: *
399: * Loop until pivot found
400: *
401: 12 CONTINUE
402: *
403: * Begin pivot search loop body
404: *
405: *
406: * Copy column IMAX to column KW-1 of W and update it
407: *
408: CALL DCOPY( IMAX, A( 1, IMAX ), 1, W( 1, KW-1 ), 1 )
409: CALL DCOPY( K-IMAX, A( IMAX, IMAX+1 ), LDA,
410: $ W( IMAX+1, KW-1 ), 1 )
411: *
412: IF( K.LT.N )
413: $ CALL DGEMV( 'No transpose', K, N-K, -ONE,
414: $ A( 1, K+1 ), LDA, W( IMAX, KW+1 ), LDW,
415: $ ONE, W( 1, KW-1 ), 1 )
416: *
417: * JMAX is the column-index of the largest off-diagonal
418: * element in row IMAX, and ROWMAX is its absolute value.
419: * Determine both ROWMAX and JMAX.
420: *
421: IF( IMAX.NE.K ) THEN
422: JMAX = IMAX + IDAMAX( K-IMAX, W( IMAX+1, KW-1 ),
423: $ 1 )
424: ROWMAX = ABS( W( JMAX, KW-1 ) )
425: ELSE
426: ROWMAX = ZERO
427: END IF
428: *
429: IF( IMAX.GT.1 ) THEN
430: ITEMP = IDAMAX( IMAX-1, W( 1, KW-1 ), 1 )
431: DTEMP = ABS( W( ITEMP, KW-1 ) )
432: IF( DTEMP.GT.ROWMAX ) THEN
433: ROWMAX = DTEMP
434: JMAX = ITEMP
435: END IF
436: END IF
437: *
438: * Equivalent to testing for
439: * ABS( W( IMAX, KW-1 ) ).GE.ALPHA*ROWMAX
440: * (used to handle NaN and Inf)
441: *
442: IF( .NOT.(ABS( W( IMAX, KW-1 ) ).LT.ALPHA*ROWMAX ) )
443: $ THEN
444: *
445: * interchange rows and columns K and IMAX,
446: * use 1-by-1 pivot block
447: *
448: KP = IMAX
449: *
450: * copy column KW-1 of W to column KW of W
451: *
452: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
453: *
454: DONE = .TRUE.
455: *
456: * Equivalent to testing for ROWMAX.EQ.COLMAX,
457: * (used to handle NaN and Inf)
458: *
459: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
460: $ THEN
461: *
462: * interchange rows and columns K-1 and IMAX,
463: * use 2-by-2 pivot block
464: *
465: KP = IMAX
466: KSTEP = 2
467: DONE = .TRUE.
468: ELSE
469: *
470: * Pivot not found: set params and repeat
471: *
472: P = IMAX
473: COLMAX = ROWMAX
474: IMAX = JMAX
475: *
476: * Copy updated JMAXth (next IMAXth) column to Kth of W
477: *
478: CALL DCOPY( K, W( 1, KW-1 ), 1, W( 1, KW ), 1 )
479: *
480: END IF
481: *
482: * End pivot search loop body
483: *
484: IF( .NOT. DONE ) GOTO 12
485: *
486: END IF
487: *
488: * ============================================================
489: *
490: KK = K - KSTEP + 1
491: *
492: * KKW is the column of W which corresponds to column KK of A
493: *
494: KKW = NB + KK - N
495: *
496: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
497: *
498: * Copy non-updated column K to column P
499: *
500: CALL DCOPY( K-P, A( P+1, K ), 1, A( P, P+1 ), LDA )
501: CALL DCOPY( P, A( 1, K ), 1, A( 1, P ), 1 )
502: *
503: * Interchange rows K and P in last N-K+1 columns of A
504: * and last N-K+2 columns of W
505: *
506: CALL DSWAP( N-K+1, A( K, K ), LDA, A( P, K ), LDA )
507: CALL DSWAP( N-KK+1, W( K, KKW ), LDW, W( P, KKW ), LDW )
508: END IF
509: *
510: * Updated column KP is already stored in column KKW of W
511: *
512: IF( KP.NE.KK ) THEN
513: *
514: * Copy non-updated column KK to column KP
515: *
516: A( KP, K ) = A( KK, K )
517: CALL DCOPY( K-1-KP, A( KP+1, KK ), 1, A( KP, KP+1 ),
518: $ LDA )
519: CALL DCOPY( KP, A( 1, KK ), 1, A( 1, KP ), 1 )
520: *
521: * Interchange rows KK and KP in last N-KK+1 columns
522: * of A and W
523: *
524: CALL DSWAP( N-KK+1, A( KK, KK ), LDA, A( KP, KK ), LDA )
525: CALL DSWAP( N-KK+1, W( KK, KKW ), LDW, W( KP, KKW ),
526: $ LDW )
527: END IF
528: *
529: IF( KSTEP.EQ.1 ) THEN
530: *
531: * 1-by-1 pivot block D(k): column KW of W now holds
532: *
533: * W(k) = U(k)*D(k)
534: *
535: * where U(k) is the k-th column of U
536: *
537: * Store U(k) in column k of A
538: *
539: CALL DCOPY( K, W( 1, KW ), 1, A( 1, K ), 1 )
540: IF( K.GT.1 ) THEN
541: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
542: R1 = ONE / A( K, K )
543: CALL DSCAL( K-1, R1, A( 1, K ), 1 )
544: ELSE IF( A( K, K ).NE.ZERO ) THEN
545: DO 14 II = 1, K - 1
546: A( II, K ) = A( II, K ) / A( K, K )
547: 14 CONTINUE
548: END IF
549: *
550: * Store the superdiagonal element of D in array E
551: *
552: E( K ) = ZERO
553: *
554: END IF
555: *
556: ELSE
557: *
558: * 2-by-2 pivot block D(k): columns KW and KW-1 of W now
559: * hold
560: *
561: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
562: *
563: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
564: * of U
565: *
566: IF( K.GT.2 ) THEN
567: *
568: * Store U(k) and U(k-1) in columns k and k-1 of A
569: *
570: D12 = W( K-1, KW )
571: D11 = W( K, KW ) / D12
572: D22 = W( K-1, KW-1 ) / D12
573: T = ONE / ( D11*D22-ONE )
574: DO 20 J = 1, K - 2
575: A( J, K-1 ) = T*( (D11*W( J, KW-1 )-W( J, KW ) ) /
576: $ D12 )
577: A( J, K ) = T*( ( D22*W( J, KW )-W( J, KW-1 ) ) /
578: $ D12 )
579: 20 CONTINUE
580: END IF
581: *
582: * Copy diagonal elements of D(K) to A,
583: * copy superdiagonal element of D(K) to E(K) and
584: * ZERO out superdiagonal entry of A
585: *
586: A( K-1, K-1 ) = W( K-1, KW-1 )
587: A( K-1, K ) = ZERO
588: A( K, K ) = W( K, KW )
589: E( K ) = W( K-1, KW )
590: E( K-1 ) = ZERO
591: *
592: END IF
593: *
594: * End column K is nonsingular
595: *
596: END IF
597: *
598: * Store details of the interchanges in IPIV
599: *
600: IF( KSTEP.EQ.1 ) THEN
601: IPIV( K ) = KP
602: ELSE
603: IPIV( K ) = -P
604: IPIV( K-1 ) = -KP
605: END IF
606: *
607: * Decrease K and return to the start of the main loop
608: *
609: K = K - KSTEP
610: GO TO 10
611: *
612: 30 CONTINUE
613: *
614: * Update the upper triangle of A11 (= A(1:k,1:k)) as
615: *
616: * A11 := A11 - U12*D*U12**T = A11 - U12*W**T
617: *
618: * computing blocks of NB columns at a time
619: *
620: DO 50 J = ( ( K-1 ) / NB )*NB + 1, 1, -NB
621: JB = MIN( NB, K-J+1 )
622: *
623: * Update the upper triangle of the diagonal block
624: *
625: DO 40 JJ = J, J + JB - 1
626: CALL DGEMV( 'No transpose', JJ-J+1, N-K, -ONE,
627: $ A( J, K+1 ), LDA, W( JJ, KW+1 ), LDW, ONE,
628: $ A( J, JJ ), 1 )
629: 40 CONTINUE
630: *
631: * Update the rectangular superdiagonal block
632: *
633: IF( J.GE.2 )
634: $ CALL DGEMM( 'No transpose', 'Transpose', J-1, JB,
635: $ N-K, -ONE, A( 1, K+1 ), LDA, W( J, KW+1 ),
636: $ LDW, ONE, A( 1, J ), LDA )
637: 50 CONTINUE
638: *
639: * Set KB to the number of columns factorized
640: *
641: KB = N - K
642: *
643: ELSE
644: *
645: * Factorize the leading columns of A using the lower triangle
646: * of A and working forwards, and compute the matrix W = L21*D
647: * for use in updating A22
648: *
649: * Initialize the unused last entry of the subdiagonal array E.
650: *
651: E( N ) = ZERO
652: *
653: * K is the main loop index, increasing from 1 in steps of 1 or 2
654: *
655: K = 1
656: 70 CONTINUE
657: *
658: * Exit from loop
659: *
660: IF( ( K.GE.NB .AND. NB.LT.N ) .OR. K.GT.N )
661: $ GO TO 90
662: *
663: KSTEP = 1
664: P = K
665: *
666: * Copy column K of A to column K of W and update it
667: *
668: CALL DCOPY( N-K+1, A( K, K ), 1, W( K, K ), 1 )
669: IF( K.GT.1 )
670: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE, A( K, 1 ),
671: $ LDA, W( K, 1 ), LDW, ONE, W( K, K ), 1 )
672: *
673: * Determine rows and columns to be interchanged and whether
674: * a 1-by-1 or 2-by-2 pivot block will be used
675: *
676: ABSAKK = ABS( W( K, K ) )
677: *
678: * IMAX is the row-index of the largest off-diagonal element in
679: * column K, and COLMAX is its absolute value.
680: * Determine both COLMAX and IMAX.
681: *
682: IF( K.LT.N ) THEN
683: IMAX = K + IDAMAX( N-K, W( K+1, K ), 1 )
684: COLMAX = ABS( W( IMAX, K ) )
685: ELSE
686: COLMAX = ZERO
687: END IF
688: *
689: IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN
690: *
691: * Column K is zero or underflow: set INFO and continue
692: *
693: IF( INFO.EQ.0 )
694: $ INFO = K
695: KP = K
696: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
697: *
698: * Set E( K ) to zero
699: *
700: IF( K.LT.N )
701: $ E( K ) = ZERO
702: *
703: ELSE
704: *
705: * ============================================================
706: *
707: * Test for interchange
708: *
709: * Equivalent to testing for ABSAKK.GE.ALPHA*COLMAX
710: * (used to handle NaN and Inf)
711: *
712: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
713: *
714: * no interchange, use 1-by-1 pivot block
715: *
716: KP = K
717: *
718: ELSE
719: *
720: DONE = .FALSE.
721: *
722: * Loop until pivot found
723: *
724: 72 CONTINUE
725: *
726: * Begin pivot search loop body
727: *
728: *
729: * Copy column IMAX to column K+1 of W and update it
730: *
731: CALL DCOPY( IMAX-K, A( IMAX, K ), LDA, W( K, K+1 ), 1)
732: CALL DCOPY( N-IMAX+1, A( IMAX, IMAX ), 1,
733: $ W( IMAX, K+1 ), 1 )
734: IF( K.GT.1 )
735: $ CALL DGEMV( 'No transpose', N-K+1, K-1, -ONE,
736: $ A( K, 1 ), LDA, W( IMAX, 1 ), LDW,
737: $ ONE, W( K, K+1 ), 1 )
738: *
739: * JMAX is the column-index of the largest off-diagonal
740: * element in row IMAX, and ROWMAX is its absolute value.
741: * Determine both ROWMAX and JMAX.
742: *
743: IF( IMAX.NE.K ) THEN
744: JMAX = K - 1 + IDAMAX( IMAX-K, W( K, K+1 ), 1 )
745: ROWMAX = ABS( W( JMAX, K+1 ) )
746: ELSE
747: ROWMAX = ZERO
748: END IF
749: *
750: IF( IMAX.LT.N ) THEN
751: ITEMP = IMAX + IDAMAX( N-IMAX, W( IMAX+1, K+1 ), 1)
752: DTEMP = ABS( W( ITEMP, K+1 ) )
753: IF( DTEMP.GT.ROWMAX ) THEN
754: ROWMAX = DTEMP
755: JMAX = ITEMP
756: END IF
757: END IF
758: *
759: * Equivalent to testing for
760: * ABS( W( IMAX, K+1 ) ).GE.ALPHA*ROWMAX
761: * (used to handle NaN and Inf)
762: *
763: IF( .NOT.( ABS( W( IMAX, K+1 ) ).LT.ALPHA*ROWMAX ) )
764: $ THEN
765: *
766: * interchange rows and columns K and IMAX,
767: * use 1-by-1 pivot block
768: *
769: KP = IMAX
770: *
771: * copy column K+1 of W to column K of W
772: *
773: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
774: *
775: DONE = .TRUE.
776: *
777: * Equivalent to testing for ROWMAX.EQ.COLMAX,
778: * (used to handle NaN and Inf)
779: *
780: ELSE IF( ( P.EQ.JMAX ) .OR. ( ROWMAX.LE.COLMAX ) )
781: $ THEN
782: *
783: * interchange rows and columns K+1 and IMAX,
784: * use 2-by-2 pivot block
785: *
786: KP = IMAX
787: KSTEP = 2
788: DONE = .TRUE.
789: ELSE
790: *
791: * Pivot not found: set params and repeat
792: *
793: P = IMAX
794: COLMAX = ROWMAX
795: IMAX = JMAX
796: *
797: * Copy updated JMAXth (next IMAXth) column to Kth of W
798: *
799: CALL DCOPY( N-K+1, W( K, K+1 ), 1, W( K, K ), 1 )
800: *
801: END IF
802: *
803: * End pivot search loop body
804: *
805: IF( .NOT. DONE ) GOTO 72
806: *
807: END IF
808: *
809: * ============================================================
810: *
811: KK = K + KSTEP - 1
812: *
813: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
814: *
815: * Copy non-updated column K to column P
816: *
817: CALL DCOPY( P-K, A( K, K ), 1, A( P, K ), LDA )
818: CALL DCOPY( N-P+1, A( P, K ), 1, A( P, P ), 1 )
819: *
820: * Interchange rows K and P in first K columns of A
821: * and first K+1 columns of W
822: *
823: CALL DSWAP( K, A( K, 1 ), LDA, A( P, 1 ), LDA )
824: CALL DSWAP( KK, W( K, 1 ), LDW, W( P, 1 ), LDW )
825: END IF
826: *
827: * Updated column KP is already stored in column KK of W
828: *
829: IF( KP.NE.KK ) THEN
830: *
831: * Copy non-updated column KK to column KP
832: *
833: A( KP, K ) = A( KK, K )
834: CALL DCOPY( KP-K-1, A( K+1, KK ), 1, A( KP, K+1 ), LDA )
835: CALL DCOPY( N-KP+1, A( KP, KK ), 1, A( KP, KP ), 1 )
836: *
837: * Interchange rows KK and KP in first KK columns of A and W
838: *
839: CALL DSWAP( KK, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
840: CALL DSWAP( KK, W( KK, 1 ), LDW, W( KP, 1 ), LDW )
841: END IF
842: *
843: IF( KSTEP.EQ.1 ) THEN
844: *
845: * 1-by-1 pivot block D(k): column k of W now holds
846: *
847: * W(k) = L(k)*D(k)
848: *
849: * where L(k) is the k-th column of L
850: *
851: * Store L(k) in column k of A
852: *
853: CALL DCOPY( N-K+1, W( K, K ), 1, A( K, K ), 1 )
854: IF( K.LT.N ) THEN
855: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
856: R1 = ONE / A( K, K )
857: CALL DSCAL( N-K, R1, A( K+1, K ), 1 )
858: ELSE IF( A( K, K ).NE.ZERO ) THEN
859: DO 74 II = K + 1, N
860: A( II, K ) = A( II, K ) / A( K, K )
861: 74 CONTINUE
862: END IF
863: *
864: * Store the subdiagonal element of D in array E
865: *
866: E( K ) = ZERO
867: *
868: END IF
869: *
870: ELSE
871: *
872: * 2-by-2 pivot block D(k): columns k and k+1 of W now hold
873: *
874: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
875: *
876: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
877: * of L
878: *
879: IF( K.LT.N-1 ) THEN
880: *
881: * Store L(k) and L(k+1) in columns k and k+1 of A
882: *
883: D21 = W( K+1, K )
884: D11 = W( K+1, K+1 ) / D21
885: D22 = W( K, K ) / D21
886: T = ONE / ( D11*D22-ONE )
887: DO 80 J = K + 2, N
888: A( J, K ) = T*( ( D11*W( J, K )-W( J, K+1 ) ) /
889: $ D21 )
890: A( J, K+1 ) = T*( ( D22*W( J, K+1 )-W( J, K ) ) /
891: $ D21 )
892: 80 CONTINUE
893: END IF
894: *
895: * Copy diagonal elements of D(K) to A,
896: * copy subdiagonal element of D(K) to E(K) and
897: * ZERO out subdiagonal entry of A
898: *
899: A( K, K ) = W( K, K )
900: A( K+1, K ) = ZERO
901: A( K+1, K+1 ) = W( K+1, K+1 )
902: E( K ) = W( K+1, K )
903: E( K+1 ) = ZERO
904: *
905: END IF
906: *
907: * End column K is nonsingular
908: *
909: END IF
910: *
911: * Store details of the interchanges in IPIV
912: *
913: IF( KSTEP.EQ.1 ) THEN
914: IPIV( K ) = KP
915: ELSE
916: IPIV( K ) = -P
917: IPIV( K+1 ) = -KP
918: END IF
919: *
920: * Increase K and return to the start of the main loop
921: *
922: K = K + KSTEP
923: GO TO 70
924: *
925: 90 CONTINUE
926: *
927: * Update the lower triangle of A22 (= A(k:n,k:n)) as
928: *
929: * A22 := A22 - L21*D*L21**T = A22 - L21*W**T
930: *
931: * computing blocks of NB columns at a time
932: *
933: DO 110 J = K, N, NB
934: JB = MIN( NB, N-J+1 )
935: *
936: * Update the lower triangle of the diagonal block
937: *
938: DO 100 JJ = J, J + JB - 1
939: CALL DGEMV( 'No transpose', J+JB-JJ, K-1, -ONE,
940: $ A( JJ, 1 ), LDA, W( JJ, 1 ), LDW, ONE,
941: $ A( JJ, JJ ), 1 )
942: 100 CONTINUE
943: *
944: * Update the rectangular subdiagonal block
945: *
946: IF( J+JB.LE.N )
947: $ CALL DGEMM( 'No transpose', 'Transpose', N-J-JB+1, JB,
948: $ K-1, -ONE, A( J+JB, 1 ), LDA, W( J, 1 ),
949: $ LDW, ONE, A( J+JB, J ), LDA )
950: 110 CONTINUE
951: *
952: * Set KB to the number of columns factorized
953: *
954: KB = K - 1
955: *
956: END IF
957: *
958: RETURN
959: *
960: * End of DLASYF_RK
961: *
962: END
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