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1.1 bertrand 1: *> \brief \b DSYTF2_RK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS2 unblocked algorithm).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSYTF2_RK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2_rk.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytf2_rk.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytf2_rk.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, N
26: * ..
27: * .. Array Arguments ..
28: * INTEGER IPIV( * )
29: * DOUBLE PRECISION A( LDA, * ), E ( * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *> DSYTF2_RK computes the factorization of a real symmetric matrix A
38: *> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
39: *>
40: *> A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),
41: *>
42: *> where U (or L) is unit upper (or lower) triangular matrix,
43: *> U**T (or L**T) is the transpose of U (or L), P is a permutation
44: *> matrix, P**T is the transpose of P, and D is symmetric and block
45: *> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
46: *>
47: *> This is the unblocked version of the algorithm, calling Level 2 BLAS.
48: *> For more information see Further Details section.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] UPLO
55: *> \verbatim
56: *> UPLO is CHARACTER*1
57: *> Specifies whether the upper or lower triangular part of the
58: *> symmetric matrix A is stored:
59: *> = 'U': Upper triangular
60: *> = 'L': Lower triangular
61: *> \endverbatim
62: *>
63: *> \param[in] N
64: *> \verbatim
65: *> N is INTEGER
66: *> The order of the matrix A. N >= 0.
67: *> \endverbatim
68: *>
69: *> \param[in,out] A
70: *> \verbatim
71: *> A is DOUBLE PRECISION array, dimension (LDA,N)
72: *> On entry, the symmetric matrix A.
73: *> If UPLO = 'U': the leading N-by-N upper triangular part
74: *> of A contains the upper triangular part of the matrix A,
75: *> and the strictly lower triangular part of A is not
76: *> referenced.
77: *>
78: *> If UPLO = 'L': the leading N-by-N lower triangular part
79: *> of A contains the lower triangular part of the matrix A,
80: *> and the strictly upper triangular part of A is not
81: *> referenced.
82: *>
83: *> On exit, contains:
84: *> a) ONLY diagonal elements of the symmetric block diagonal
85: *> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
86: *> (superdiagonal (or subdiagonal) elements of D
87: *> are stored on exit in array E), and
88: *> b) If UPLO = 'U': factor U in the superdiagonal part of A.
89: *> If UPLO = 'L': factor L in the subdiagonal part of A.
90: *> \endverbatim
91: *>
92: *> \param[in] LDA
93: *> \verbatim
94: *> LDA is INTEGER
95: *> The leading dimension of the array A. LDA >= max(1,N).
96: *> \endverbatim
97: *>
98: *> \param[out] E
99: *> \verbatim
100: *> E is DOUBLE PRECISION array, dimension (N)
101: *> On exit, contains the superdiagonal (or subdiagonal)
102: *> elements of the symmetric block diagonal matrix D
103: *> with 1-by-1 or 2-by-2 diagonal blocks, where
104: *> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
105: *> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
106: *>
107: *> NOTE: For 1-by-1 diagonal block D(k), where
108: *> 1 <= k <= N, the element E(k) is set to 0 in both
109: *> UPLO = 'U' or UPLO = 'L' cases.
110: *> \endverbatim
111: *>
112: *> \param[out] IPIV
113: *> \verbatim
114: *> IPIV is INTEGER array, dimension (N)
115: *> IPIV describes the permutation matrix P in the factorization
116: *> of matrix A as follows. The absolute value of IPIV(k)
117: *> represents the index of row and column that were
118: *> interchanged with the k-th row and column. The value of UPLO
119: *> describes the order in which the interchanges were applied.
120: *> Also, the sign of IPIV represents the block structure of
121: *> the symmetric block diagonal matrix D with 1-by-1 or 2-by-2
122: *> diagonal blocks which correspond to 1 or 2 interchanges
123: *> at each factorization step. For more info see Further
124: *> Details section.
125: *>
126: *> If UPLO = 'U',
127: *> ( in factorization order, k decreases from N to 1 ):
128: *> a) A single positive entry IPIV(k) > 0 means:
129: *> D(k,k) is a 1-by-1 diagonal block.
130: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
131: *> interchanged in the matrix A(1:N,1:N);
132: *> If IPIV(k) = k, no interchange occurred.
133: *>
134: *> b) A pair of consecutive negative entries
135: *> IPIV(k) < 0 and IPIV(k-1) < 0 means:
136: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
137: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
138: *> 1) If -IPIV(k) != k, rows and columns
139: *> k and -IPIV(k) were interchanged
140: *> in the matrix A(1:N,1:N).
141: *> If -IPIV(k) = k, no interchange occurred.
142: *> 2) If -IPIV(k-1) != k-1, rows and columns
143: *> k-1 and -IPIV(k-1) were interchanged
144: *> in the matrix A(1:N,1:N).
145: *> If -IPIV(k-1) = k-1, no interchange occurred.
146: *>
147: *> c) In both cases a) and b), always ABS( IPIV(k) ) <= k.
148: *>
149: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
150: *>
151: *> If UPLO = 'L',
152: *> ( in factorization order, k increases from 1 to N ):
153: *> a) A single positive entry IPIV(k) > 0 means:
154: *> D(k,k) is a 1-by-1 diagonal block.
155: *> If IPIV(k) != k, rows and columns k and IPIV(k) were
156: *> interchanged in the matrix A(1:N,1:N).
157: *> If IPIV(k) = k, no interchange occurred.
158: *>
159: *> b) A pair of consecutive negative entries
160: *> IPIV(k) < 0 and IPIV(k+1) < 0 means:
161: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
162: *> (NOTE: negative entries in IPIV appear ONLY in pairs).
163: *> 1) If -IPIV(k) != k, rows and columns
164: *> k and -IPIV(k) were interchanged
165: *> in the matrix A(1:N,1:N).
166: *> If -IPIV(k) = k, no interchange occurred.
167: *> 2) If -IPIV(k+1) != k+1, rows and columns
168: *> k-1 and -IPIV(k-1) were interchanged
169: *> in the matrix A(1:N,1:N).
170: *> If -IPIV(k+1) = k+1, no interchange occurred.
171: *>
172: *> c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
173: *>
174: *> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
175: *> \endverbatim
176: *>
177: *> \param[out] INFO
178: *> \verbatim
179: *> INFO is INTEGER
180: *> = 0: successful exit
181: *>
182: *> < 0: If INFO = -k, the k-th argument had an illegal value
183: *>
184: *> > 0: If INFO = k, the matrix A is singular, because:
185: *> If UPLO = 'U': column k in the upper
186: *> triangular part of A contains all zeros.
187: *> If UPLO = 'L': column k in the lower
188: *> triangular part of A contains all zeros.
189: *>
190: *> Therefore D(k,k) is exactly zero, and superdiagonal
191: *> elements of column k of U (or subdiagonal elements of
192: *> column k of L ) are all zeros. The factorization has
193: *> been completed, but the block diagonal matrix D is
194: *> exactly singular, and division by zero will occur if
195: *> it is used to solve a system of equations.
196: *>
197: *> NOTE: INFO only stores the first occurrence of
198: *> a singularity, any subsequent occurrence of singularity
199: *> is not stored in INFO even though the factorization
200: *> always completes.
201: *> \endverbatim
202: *
203: * Authors:
204: * ========
205: *
206: *> \author Univ. of Tennessee
207: *> \author Univ. of California Berkeley
208: *> \author Univ. of Colorado Denver
209: *> \author NAG Ltd.
210: *
211: *> \date December 2016
212: *
213: *> \ingroup doubleSYcomputational
214: *
215: *> \par Further Details:
216: * =====================
217: *>
218: *> \verbatim
219: *> TODO: put further details
220: *> \endverbatim
221: *
222: *> \par Contributors:
223: * ==================
224: *>
225: *> \verbatim
226: *>
227: *> December 2016, Igor Kozachenko,
228: *> Computer Science Division,
229: *> University of California, Berkeley
230: *>
231: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
232: *> School of Mathematics,
233: *> University of Manchester
234: *>
235: *> 01-01-96 - Based on modifications by
236: *> J. Lewis, Boeing Computer Services Company
237: *> A. Petitet, Computer Science Dept.,
238: *> Univ. of Tenn., Knoxville abd , USA
239: *> \endverbatim
240: *
241: * =====================================================================
242: SUBROUTINE DSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
243: *
244: * -- LAPACK computational routine (version 3.7.0) --
245: * -- LAPACK is a software package provided by Univ. of Tennessee, --
246: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
247: * December 2016
248: *
249: * .. Scalar Arguments ..
250: CHARACTER UPLO
251: INTEGER INFO, LDA, N
252: * ..
253: * .. Array Arguments ..
254: INTEGER IPIV( * )
255: DOUBLE PRECISION A( LDA, * ), E( * )
256: * ..
257: *
258: * =====================================================================
259: *
260: * .. Parameters ..
261: DOUBLE PRECISION ZERO, ONE
262: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
263: DOUBLE PRECISION EIGHT, SEVTEN
264: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
265: * ..
266: * .. Local Scalars ..
267: LOGICAL UPPER, DONE
268: INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
269: $ P, II
270: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
271: $ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
272: * ..
273: * .. External Functions ..
274: LOGICAL LSAME
275: INTEGER IDAMAX
276: DOUBLE PRECISION DLAMCH
277: EXTERNAL LSAME, IDAMAX, DLAMCH
278: * ..
279: * .. External Subroutines ..
280: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
281: * ..
282: * .. Intrinsic Functions ..
283: INTRINSIC ABS, MAX, SQRT
284: * ..
285: * .. Executable Statements ..
286: *
287: * Test the input parameters.
288: *
289: INFO = 0
290: UPPER = LSAME( UPLO, 'U' )
291: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
292: INFO = -1
293: ELSE IF( N.LT.0 ) THEN
294: INFO = -2
295: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
296: INFO = -4
297: END IF
298: IF( INFO.NE.0 ) THEN
299: CALL XERBLA( 'DSYTF2_RK', -INFO )
300: RETURN
301: END IF
302: *
303: * Initialize ALPHA for use in choosing pivot block size.
304: *
305: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
306: *
307: * Compute machine safe minimum
308: *
309: SFMIN = DLAMCH( 'S' )
310: *
311: IF( UPPER ) THEN
312: *
313: * Factorize A as U*D*U**T using the upper triangle of A
314: *
1.4 ! bertrand 315: * Initialize the first entry of array E, where superdiagonal
1.1 bertrand 316: * elements of D are stored
317: *
318: E( 1 ) = ZERO
319: *
320: * K is the main loop index, decreasing from N to 1 in steps of
321: * 1 or 2
322: *
323: K = N
324: 10 CONTINUE
325: *
326: * If K < 1, exit from loop
327: *
328: IF( K.LT.1 )
329: $ GO TO 34
330: KSTEP = 1
331: P = K
332: *
333: * Determine rows and columns to be interchanged and whether
334: * a 1-by-1 or 2-by-2 pivot block will be used
335: *
336: ABSAKK = ABS( A( K, K ) )
337: *
338: * IMAX is the row-index of the largest off-diagonal element in
339: * column K, and COLMAX is its absolute value.
340: * Determine both COLMAX and IMAX.
341: *
342: IF( K.GT.1 ) THEN
343: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
344: COLMAX = ABS( A( IMAX, K ) )
345: ELSE
346: COLMAX = ZERO
347: END IF
348: *
349: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) ) THEN
350: *
351: * Column K is zero or underflow: set INFO and continue
352: *
353: IF( INFO.EQ.0 )
354: $ INFO = K
355: KP = K
356: *
357: * Set E( K ) to zero
358: *
359: IF( K.GT.1 )
360: $ E( K ) = ZERO
361: *
362: ELSE
363: *
364: * Test for interchange
365: *
366: * Equivalent to testing for (used to handle NaN and Inf)
367: * ABSAKK.GE.ALPHA*COLMAX
368: *
369: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
370: *
371: * no interchange,
372: * use 1-by-1 pivot block
373: *
374: KP = K
375: ELSE
376: *
377: DONE = .FALSE.
378: *
379: * Loop until pivot found
380: *
381: 12 CONTINUE
382: *
383: * Begin pivot search loop body
384: *
385: * JMAX is the column-index of the largest off-diagonal
386: * element in row IMAX, and ROWMAX is its absolute value.
387: * Determine both ROWMAX and JMAX.
388: *
389: IF( IMAX.NE.K ) THEN
390: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
391: $ LDA )
392: ROWMAX = ABS( A( IMAX, JMAX ) )
393: ELSE
394: ROWMAX = ZERO
395: END IF
396: *
397: IF( IMAX.GT.1 ) THEN
398: ITEMP = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
399: DTEMP = ABS( A( ITEMP, IMAX ) )
400: IF( DTEMP.GT.ROWMAX ) THEN
401: ROWMAX = DTEMP
402: JMAX = ITEMP
403: END IF
404: END IF
405: *
406: * Equivalent to testing for (used to handle NaN and Inf)
407: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
408: *
409: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
410: $ THEN
411: *
412: * interchange rows and columns K and IMAX,
413: * use 1-by-1 pivot block
414: *
415: KP = IMAX
416: DONE = .TRUE.
417: *
418: * Equivalent to testing for ROWMAX .EQ. COLMAX,
419: * used to handle NaN and Inf
420: *
421: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
422: *
423: * interchange rows and columns K+1 and IMAX,
424: * use 2-by-2 pivot block
425: *
426: KP = IMAX
427: KSTEP = 2
428: DONE = .TRUE.
429: ELSE
430: *
431: * Pivot NOT found, set variables and repeat
432: *
433: P = IMAX
434: COLMAX = ROWMAX
435: IMAX = JMAX
436: END IF
437: *
438: * End pivot search loop body
439: *
440: IF( .NOT. DONE ) GOTO 12
441: *
442: END IF
443: *
444: * Swap TWO rows and TWO columns
445: *
446: * First swap
447: *
448: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
449: *
450: * Interchange rows and column K and P in the leading
451: * submatrix A(1:k,1:k) if we have a 2-by-2 pivot
452: *
453: IF( P.GT.1 )
454: $ CALL DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
455: IF( P.LT.(K-1) )
456: $ CALL DSWAP( K-P-1, A( P+1, K ), 1, A( P, P+1 ),
457: $ LDA )
458: T = A( K, K )
459: A( K, K ) = A( P, P )
460: A( P, P ) = T
461: *
462: * Convert upper triangle of A into U form by applying
463: * the interchanges in columns k+1:N.
464: *
465: IF( K.LT.N )
466: $ CALL DSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA )
467: *
468: END IF
469: *
470: * Second swap
471: *
472: KK = K - KSTEP + 1
473: IF( KP.NE.KK ) THEN
474: *
475: * Interchange rows and columns KK and KP in the leading
476: * submatrix A(1:k,1:k)
477: *
478: IF( KP.GT.1 )
479: $ CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
480: IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
481: $ CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
482: $ LDA )
483: T = A( KK, KK )
484: A( KK, KK ) = A( KP, KP )
485: A( KP, KP ) = T
486: IF( KSTEP.EQ.2 ) THEN
487: T = A( K-1, K )
488: A( K-1, K ) = A( KP, K )
489: A( KP, K ) = T
490: END IF
491: *
492: * Convert upper triangle of A into U form by applying
493: * the interchanges in columns k+1:N.
494: *
495: IF( K.LT.N )
496: $ CALL DSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
497: $ LDA )
498: *
499: END IF
500: *
501: * Update the leading submatrix
502: *
503: IF( KSTEP.EQ.1 ) THEN
504: *
505: * 1-by-1 pivot block D(k): column k now holds
506: *
507: * W(k) = U(k)*D(k)
508: *
509: * where U(k) is the k-th column of U
510: *
511: IF( K.GT.1 ) THEN
512: *
513: * Perform a rank-1 update of A(1:k-1,1:k-1) and
514: * store U(k) in column k
515: *
516: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
517: *
518: * Perform a rank-1 update of A(1:k-1,1:k-1) as
519: * A := A - U(k)*D(k)*U(k)**T
520: * = A - W(k)*1/D(k)*W(k)**T
521: *
522: D11 = ONE / A( K, K )
523: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
524: *
525: * Store U(k) in column k
526: *
527: CALL DSCAL( K-1, D11, A( 1, K ), 1 )
528: ELSE
529: *
530: * Store L(k) in column K
531: *
532: D11 = A( K, K )
533: DO 16 II = 1, K - 1
534: A( II, K ) = A( II, K ) / D11
535: 16 CONTINUE
536: *
537: * Perform a rank-1 update of A(k+1:n,k+1:n) as
538: * A := A - U(k)*D(k)*U(k)**T
539: * = A - W(k)*(1/D(k))*W(k)**T
540: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
541: *
542: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
543: END IF
544: *
545: * Store the superdiagonal element of D in array E
546: *
547: E( K ) = ZERO
548: *
549: END IF
550: *
551: ELSE
552: *
553: * 2-by-2 pivot block D(k): columns k and k-1 now hold
554: *
555: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
556: *
557: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
558: * of U
559: *
560: * Perform a rank-2 update of A(1:k-2,1:k-2) as
561: *
562: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
563: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
564: *
565: * and store L(k) and L(k+1) in columns k and k+1
566: *
567: IF( K.GT.2 ) THEN
568: *
569: D12 = A( K-1, K )
570: D22 = A( K-1, K-1 ) / D12
571: D11 = A( K, K ) / D12
572: T = ONE / ( D11*D22-ONE )
573: *
574: DO 30 J = K - 2, 1, -1
575: *
576: WKM1 = T*( D11*A( J, K-1 )-A( J, K ) )
577: WK = T*( D22*A( J, K )-A( J, K-1 ) )
578: *
579: DO 20 I = J, 1, -1
580: A( I, J ) = A( I, J ) - (A( I, K ) / D12 )*WK -
581: $ ( A( I, K-1 ) / D12 )*WKM1
582: 20 CONTINUE
583: *
584: * Store U(k) and U(k-1) in cols k and k-1 for row J
585: *
586: A( J, K ) = WK / D12
587: A( J, K-1 ) = WKM1 / D12
588: *
589: 30 CONTINUE
590: *
591: END IF
592: *
593: * Copy superdiagonal elements of D(K) to E(K) and
594: * ZERO out superdiagonal entry of A
595: *
596: E( K ) = A( K-1, K )
597: E( K-1 ) = ZERO
598: A( K-1, K ) = ZERO
599: *
600: END IF
601: *
602: * End column K is nonsingular
603: *
604: END IF
605: *
606: * Store details of the interchanges in IPIV
607: *
608: IF( KSTEP.EQ.1 ) THEN
609: IPIV( K ) = KP
610: ELSE
611: IPIV( K ) = -P
612: IPIV( K-1 ) = -KP
613: END IF
614: *
615: * Decrease K and return to the start of the main loop
616: *
617: K = K - KSTEP
618: GO TO 10
619: *
620: 34 CONTINUE
621: *
622: ELSE
623: *
624: * Factorize A as L*D*L**T using the lower triangle of A
625: *
1.4 ! bertrand 626: * Initialize the unused last entry of the subdiagonal array E.
1.1 bertrand 627: *
628: E( N ) = ZERO
629: *
630: * K is the main loop index, increasing from 1 to N in steps of
631: * 1 or 2
632: *
633: K = 1
634: 40 CONTINUE
635: *
636: * If K > N, exit from loop
637: *
638: IF( K.GT.N )
639: $ GO TO 64
640: KSTEP = 1
641: P = K
642: *
643: * Determine rows and columns to be interchanged and whether
644: * a 1-by-1 or 2-by-2 pivot block will be used
645: *
646: ABSAKK = ABS( A( K, K ) )
647: *
648: * IMAX is the row-index of the largest off-diagonal element in
649: * column K, and COLMAX is its absolute value.
650: * Determine both COLMAX and IMAX.
651: *
652: IF( K.LT.N ) THEN
653: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
654: COLMAX = ABS( A( IMAX, K ) )
655: ELSE
656: COLMAX = ZERO
657: END IF
658: *
659: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
660: *
661: * Column K is zero or underflow: set INFO and continue
662: *
663: IF( INFO.EQ.0 )
664: $ INFO = K
665: KP = K
666: *
667: * Set E( K ) to zero
668: *
669: IF( K.LT.N )
670: $ E( K ) = ZERO
671: *
672: ELSE
673: *
674: * Test for interchange
675: *
676: * Equivalent to testing for (used to handle NaN and Inf)
677: * ABSAKK.GE.ALPHA*COLMAX
678: *
679: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
680: *
681: * no interchange, use 1-by-1 pivot block
682: *
683: KP = K
684: *
685: ELSE
686: *
687: DONE = .FALSE.
688: *
689: * Loop until pivot found
690: *
691: 42 CONTINUE
692: *
693: * Begin pivot search loop body
694: *
695: * JMAX is the column-index of the largest off-diagonal
696: * element in row IMAX, and ROWMAX is its absolute value.
697: * Determine both ROWMAX and JMAX.
698: *
699: IF( IMAX.NE.K ) THEN
700: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
701: ROWMAX = ABS( A( IMAX, JMAX ) )
702: ELSE
703: ROWMAX = ZERO
704: END IF
705: *
706: IF( IMAX.LT.N ) THEN
707: ITEMP = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
708: $ 1 )
709: DTEMP = ABS( A( ITEMP, IMAX ) )
710: IF( DTEMP.GT.ROWMAX ) THEN
711: ROWMAX = DTEMP
712: JMAX = ITEMP
713: END IF
714: END IF
715: *
716: * Equivalent to testing for (used to handle NaN and Inf)
717: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
718: *
719: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
720: $ THEN
721: *
722: * interchange rows and columns K and IMAX,
723: * use 1-by-1 pivot block
724: *
725: KP = IMAX
726: DONE = .TRUE.
727: *
728: * Equivalent to testing for ROWMAX .EQ. COLMAX,
729: * used to handle NaN and Inf
730: *
731: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
732: *
733: * interchange rows and columns K+1 and IMAX,
734: * use 2-by-2 pivot block
735: *
736: KP = IMAX
737: KSTEP = 2
738: DONE = .TRUE.
739: ELSE
740: *
741: * Pivot NOT found, set variables and repeat
742: *
743: P = IMAX
744: COLMAX = ROWMAX
745: IMAX = JMAX
746: END IF
747: *
748: * End pivot search loop body
749: *
750: IF( .NOT. DONE ) GOTO 42
751: *
752: END IF
753: *
754: * Swap TWO rows and TWO columns
755: *
756: * First swap
757: *
758: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
759: *
760: * Interchange rows and column K and P in the trailing
761: * submatrix A(k:n,k:n) if we have a 2-by-2 pivot
762: *
763: IF( P.LT.N )
764: $ CALL DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
765: IF( P.GT.(K+1) )
766: $ CALL DSWAP( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
767: T = A( K, K )
768: A( K, K ) = A( P, P )
769: A( P, P ) = T
770: *
771: * Convert lower triangle of A into L form by applying
772: * the interchanges in columns 1:k-1.
773: *
774: IF ( K.GT.1 )
775: $ CALL DSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
776: *
777: END IF
778: *
779: * Second swap
780: *
781: KK = K + KSTEP - 1
782: IF( KP.NE.KK ) THEN
783: *
784: * Interchange rows and columns KK and KP in the trailing
785: * submatrix A(k:n,k:n)
786: *
787: IF( KP.LT.N )
788: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
789: IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
790: $ CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
791: $ LDA )
792: T = A( KK, KK )
793: A( KK, KK ) = A( KP, KP )
794: A( KP, KP ) = T
795: IF( KSTEP.EQ.2 ) THEN
796: T = A( K+1, K )
797: A( K+1, K ) = A( KP, K )
798: A( KP, K ) = T
799: END IF
800: *
801: * Convert lower triangle of A into L form by applying
802: * the interchanges in columns 1:k-1.
803: *
804: IF ( K.GT.1 )
805: $ CALL DSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
806: *
807: END IF
808: *
809: * Update the trailing submatrix
810: *
811: IF( KSTEP.EQ.1 ) THEN
812: *
813: * 1-by-1 pivot block D(k): column k now holds
814: *
815: * W(k) = L(k)*D(k)
816: *
817: * where L(k) is the k-th column of L
818: *
819: IF( K.LT.N ) THEN
820: *
821: * Perform a rank-1 update of A(k+1:n,k+1:n) and
822: * store L(k) in column k
823: *
824: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
825: *
826: * Perform a rank-1 update of A(k+1:n,k+1:n) as
827: * A := A - L(k)*D(k)*L(k)**T
828: * = A - W(k)*(1/D(k))*W(k)**T
829: *
830: D11 = ONE / A( K, K )
831: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
832: $ A( K+1, K+1 ), LDA )
833: *
834: * Store L(k) in column k
835: *
836: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
837: ELSE
838: *
839: * Store L(k) in column k
840: *
841: D11 = A( K, K )
842: DO 46 II = K + 1, N
843: A( II, K ) = A( II, K ) / D11
844: 46 CONTINUE
845: *
846: * Perform a rank-1 update of A(k+1:n,k+1:n) as
847: * A := A - L(k)*D(k)*L(k)**T
848: * = A - W(k)*(1/D(k))*W(k)**T
849: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
850: *
851: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
852: $ A( K+1, K+1 ), LDA )
853: END IF
854: *
855: * Store the subdiagonal element of D in array E
856: *
857: E( K ) = ZERO
858: *
859: END IF
860: *
861: ELSE
862: *
863: * 2-by-2 pivot block D(k): columns k and k+1 now hold
864: *
865: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
866: *
867: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
868: * of L
869: *
870: *
871: * Perform a rank-2 update of A(k+2:n,k+2:n) as
872: *
873: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
874: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
875: *
876: * and store L(k) and L(k+1) in columns k and k+1
877: *
878: IF( K.LT.N-1 ) THEN
879: *
880: D21 = A( K+1, K )
881: D11 = A( K+1, K+1 ) / D21
882: D22 = A( K, K ) / D21
883: T = ONE / ( D11*D22-ONE )
884: *
885: DO 60 J = K + 2, N
886: *
887: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
888: *
889: WK = T*( D11*A( J, K )-A( J, K+1 ) )
890: WKP1 = T*( D22*A( J, K+1 )-A( J, K ) )
891: *
892: * Perform a rank-2 update of A(k+2:n,k+2:n)
893: *
894: DO 50 I = J, N
895: A( I, J ) = A( I, J ) - ( A( I, K ) / D21 )*WK -
896: $ ( A( I, K+1 ) / D21 )*WKP1
897: 50 CONTINUE
898: *
899: * Store L(k) and L(k+1) in cols k and k+1 for row J
900: *
901: A( J, K ) = WK / D21
902: A( J, K+1 ) = WKP1 / D21
903: *
904: 60 CONTINUE
905: *
906: END IF
907: *
908: * Copy subdiagonal elements of D(K) to E(K) and
909: * ZERO out subdiagonal entry of A
910: *
911: E( K ) = A( K+1, K )
912: E( K+1 ) = ZERO
913: A( K+1, K ) = ZERO
914: *
915: END IF
916: *
917: * End column K is nonsingular
918: *
919: END IF
920: *
921: * Store details of the interchanges in IPIV
922: *
923: IF( KSTEP.EQ.1 ) THEN
924: IPIV( K ) = KP
925: ELSE
926: IPIV( K ) = -P
927: IPIV( K+1 ) = -KP
928: END IF
929: *
930: * Increase K and return to the start of the main loop
931: *
932: K = K + KSTEP
933: GO TO 40
934: *
935: 64 CONTINUE
936: *
937: END IF
938: *
939: RETURN
940: *
941: * End of DSYTF2_RK
942: *
943: END