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: *> \ingroup doubleSYcomputational
212: *
213: *> \par Further Details:
214: * =====================
215: *>
216: *> \verbatim
217: *> TODO: put further details
218: *> \endverbatim
219: *
220: *> \par Contributors:
221: * ==================
222: *>
223: *> \verbatim
224: *>
225: *> December 2016, Igor Kozachenko,
226: *> Computer Science Division,
227: *> University of California, Berkeley
228: *>
229: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
230: *> School of Mathematics,
231: *> University of Manchester
232: *>
233: *> 01-01-96 - Based on modifications by
234: *> J. Lewis, Boeing Computer Services Company
235: *> A. Petitet, Computer Science Dept.,
236: *> Univ. of Tenn., Knoxville abd , USA
237: *> \endverbatim
238: *
239: * =====================================================================
240: SUBROUTINE DSYTF2_RK( UPLO, N, A, LDA, E, IPIV, INFO )
241: *
242: * -- LAPACK computational routine --
243: * -- LAPACK is a software package provided by Univ. of Tennessee, --
244: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
245: *
246: * .. Scalar Arguments ..
247: CHARACTER UPLO
248: INTEGER INFO, LDA, N
249: * ..
250: * .. Array Arguments ..
251: INTEGER IPIV( * )
252: DOUBLE PRECISION A( LDA, * ), E( * )
253: * ..
254: *
255: * =====================================================================
256: *
257: * .. Parameters ..
258: DOUBLE PRECISION ZERO, ONE
259: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
260: DOUBLE PRECISION EIGHT, SEVTEN
261: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
262: * ..
263: * .. Local Scalars ..
264: LOGICAL UPPER, DONE
265: INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
266: $ P, II
267: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
268: $ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
269: * ..
270: * .. External Functions ..
271: LOGICAL LSAME
272: INTEGER IDAMAX
273: DOUBLE PRECISION DLAMCH
274: EXTERNAL LSAME, IDAMAX, DLAMCH
275: * ..
276: * .. External Subroutines ..
277: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
278: * ..
279: * .. Intrinsic Functions ..
280: INTRINSIC ABS, MAX, SQRT
281: * ..
282: * .. Executable Statements ..
283: *
284: * Test the input parameters.
285: *
286: INFO = 0
287: UPPER = LSAME( UPLO, 'U' )
288: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
289: INFO = -1
290: ELSE IF( N.LT.0 ) THEN
291: INFO = -2
292: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
293: INFO = -4
294: END IF
295: IF( INFO.NE.0 ) THEN
296: CALL XERBLA( 'DSYTF2_RK', -INFO )
297: RETURN
298: END IF
299: *
300: * Initialize ALPHA for use in choosing pivot block size.
301: *
302: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
303: *
304: * Compute machine safe minimum
305: *
306: SFMIN = DLAMCH( 'S' )
307: *
308: IF( UPPER ) THEN
309: *
310: * Factorize A as U*D*U**T using the upper triangle of A
311: *
312: * Initialize the first entry of array E, where superdiagonal
313: * elements of D are stored
314: *
315: E( 1 ) = ZERO
316: *
317: * K is the main loop index, decreasing from N to 1 in steps of
318: * 1 or 2
319: *
320: K = N
321: 10 CONTINUE
322: *
323: * If K < 1, exit from loop
324: *
325: IF( K.LT.1 )
326: $ GO TO 34
327: KSTEP = 1
328: P = K
329: *
330: * Determine rows and columns to be interchanged and whether
331: * a 1-by-1 or 2-by-2 pivot block will be used
332: *
333: ABSAKK = ABS( A( K, K ) )
334: *
335: * IMAX is the row-index of the largest off-diagonal element in
336: * column K, and COLMAX is its absolute value.
337: * Determine both COLMAX and IMAX.
338: *
339: IF( K.GT.1 ) THEN
340: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
341: COLMAX = ABS( A( IMAX, K ) )
342: ELSE
343: COLMAX = ZERO
344: END IF
345: *
346: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) ) THEN
347: *
348: * Column K is zero or underflow: set INFO and continue
349: *
350: IF( INFO.EQ.0 )
351: $ INFO = K
352: KP = K
353: *
354: * Set E( K ) to zero
355: *
356: IF( K.GT.1 )
357: $ E( K ) = ZERO
358: *
359: ELSE
360: *
361: * Test for interchange
362: *
363: * Equivalent to testing for (used to handle NaN and Inf)
364: * ABSAKK.GE.ALPHA*COLMAX
365: *
366: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
367: *
368: * no interchange,
369: * use 1-by-1 pivot block
370: *
371: KP = K
372: ELSE
373: *
374: DONE = .FALSE.
375: *
376: * Loop until pivot found
377: *
378: 12 CONTINUE
379: *
380: * Begin pivot search loop body
381: *
382: * JMAX is the column-index of the largest off-diagonal
383: * element in row IMAX, and ROWMAX is its absolute value.
384: * Determine both ROWMAX and JMAX.
385: *
386: IF( IMAX.NE.K ) THEN
387: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
388: $ LDA )
389: ROWMAX = ABS( A( IMAX, JMAX ) )
390: ELSE
391: ROWMAX = ZERO
392: END IF
393: *
394: IF( IMAX.GT.1 ) THEN
395: ITEMP = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
396: DTEMP = ABS( A( ITEMP, IMAX ) )
397: IF( DTEMP.GT.ROWMAX ) THEN
398: ROWMAX = DTEMP
399: JMAX = ITEMP
400: END IF
401: END IF
402: *
403: * Equivalent to testing for (used to handle NaN and Inf)
404: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
405: *
406: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
407: $ THEN
408: *
409: * interchange rows and columns K and IMAX,
410: * use 1-by-1 pivot block
411: *
412: KP = IMAX
413: DONE = .TRUE.
414: *
415: * Equivalent to testing for ROWMAX .EQ. COLMAX,
416: * used to handle NaN and Inf
417: *
418: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
419: *
420: * interchange rows and columns K+1 and IMAX,
421: * use 2-by-2 pivot block
422: *
423: KP = IMAX
424: KSTEP = 2
425: DONE = .TRUE.
426: ELSE
427: *
428: * Pivot NOT found, set variables and repeat
429: *
430: P = IMAX
431: COLMAX = ROWMAX
432: IMAX = JMAX
433: END IF
434: *
435: * End pivot search loop body
436: *
437: IF( .NOT. DONE ) GOTO 12
438: *
439: END IF
440: *
441: * Swap TWO rows and TWO columns
442: *
443: * First swap
444: *
445: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
446: *
447: * Interchange rows and column K and P in the leading
448: * submatrix A(1:k,1:k) if we have a 2-by-2 pivot
449: *
450: IF( P.GT.1 )
451: $ CALL DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
452: IF( P.LT.(K-1) )
453: $ CALL DSWAP( K-P-1, A( P+1, K ), 1, A( P, P+1 ),
454: $ LDA )
455: T = A( K, K )
456: A( K, K ) = A( P, P )
457: A( P, P ) = T
458: *
459: * Convert upper triangle of A into U form by applying
460: * the interchanges in columns k+1:N.
461: *
462: IF( K.LT.N )
463: $ CALL DSWAP( N-K, A( K, K+1 ), LDA, A( P, K+1 ), LDA )
464: *
465: END IF
466: *
467: * Second swap
468: *
469: KK = K - KSTEP + 1
470: IF( KP.NE.KK ) THEN
471: *
472: * Interchange rows and columns KK and KP in the leading
473: * submatrix A(1:k,1:k)
474: *
475: IF( KP.GT.1 )
476: $ CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
477: IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
478: $ CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
479: $ LDA )
480: T = A( KK, KK )
481: A( KK, KK ) = A( KP, KP )
482: A( KP, KP ) = T
483: IF( KSTEP.EQ.2 ) THEN
484: T = A( K-1, K )
485: A( K-1, K ) = A( KP, K )
486: A( KP, K ) = T
487: END IF
488: *
489: * Convert upper triangle of A into U form by applying
490: * the interchanges in columns k+1:N.
491: *
492: IF( K.LT.N )
493: $ CALL DSWAP( N-K, A( KK, K+1 ), LDA, A( KP, K+1 ),
494: $ LDA )
495: *
496: END IF
497: *
498: * Update the leading submatrix
499: *
500: IF( KSTEP.EQ.1 ) THEN
501: *
502: * 1-by-1 pivot block D(k): column k now holds
503: *
504: * W(k) = U(k)*D(k)
505: *
506: * where U(k) is the k-th column of U
507: *
508: IF( K.GT.1 ) THEN
509: *
510: * Perform a rank-1 update of A(1:k-1,1:k-1) and
511: * store U(k) in column k
512: *
513: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
514: *
515: * Perform a rank-1 update of A(1:k-1,1:k-1) as
516: * A := A - U(k)*D(k)*U(k)**T
517: * = A - W(k)*1/D(k)*W(k)**T
518: *
519: D11 = ONE / A( K, K )
520: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
521: *
522: * Store U(k) in column k
523: *
524: CALL DSCAL( K-1, D11, A( 1, K ), 1 )
525: ELSE
526: *
527: * Store L(k) in column K
528: *
529: D11 = A( K, K )
530: DO 16 II = 1, K - 1
531: A( II, K ) = A( II, K ) / D11
532: 16 CONTINUE
533: *
534: * Perform a rank-1 update of A(k+1:n,k+1:n) as
535: * A := A - U(k)*D(k)*U(k)**T
536: * = A - W(k)*(1/D(k))*W(k)**T
537: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
538: *
539: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
540: END IF
541: *
542: * Store the superdiagonal element of D in array E
543: *
544: E( K ) = ZERO
545: *
546: END IF
547: *
548: ELSE
549: *
550: * 2-by-2 pivot block D(k): columns k and k-1 now hold
551: *
552: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
553: *
554: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
555: * of U
556: *
557: * Perform a rank-2 update of A(1:k-2,1:k-2) as
558: *
559: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
560: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
561: *
562: * and store L(k) and L(k+1) in columns k and k+1
563: *
564: IF( K.GT.2 ) THEN
565: *
566: D12 = A( K-1, K )
567: D22 = A( K-1, K-1 ) / D12
568: D11 = A( K, K ) / D12
569: T = ONE / ( D11*D22-ONE )
570: *
571: DO 30 J = K - 2, 1, -1
572: *
573: WKM1 = T*( D11*A( J, K-1 )-A( J, K ) )
574: WK = T*( D22*A( J, K )-A( J, K-1 ) )
575: *
576: DO 20 I = J, 1, -1
577: A( I, J ) = A( I, J ) - (A( I, K ) / D12 )*WK -
578: $ ( A( I, K-1 ) / D12 )*WKM1
579: 20 CONTINUE
580: *
581: * Store U(k) and U(k-1) in cols k and k-1 for row J
582: *
583: A( J, K ) = WK / D12
584: A( J, K-1 ) = WKM1 / D12
585: *
586: 30 CONTINUE
587: *
588: END IF
589: *
590: * Copy superdiagonal elements of D(K) to E(K) and
591: * ZERO out superdiagonal entry of A
592: *
593: E( K ) = A( K-1, K )
594: E( K-1 ) = ZERO
595: A( K-1, K ) = ZERO
596: *
597: END IF
598: *
599: * End column K is nonsingular
600: *
601: END IF
602: *
603: * Store details of the interchanges in IPIV
604: *
605: IF( KSTEP.EQ.1 ) THEN
606: IPIV( K ) = KP
607: ELSE
608: IPIV( K ) = -P
609: IPIV( K-1 ) = -KP
610: END IF
611: *
612: * Decrease K and return to the start of the main loop
613: *
614: K = K - KSTEP
615: GO TO 10
616: *
617: 34 CONTINUE
618: *
619: ELSE
620: *
621: * Factorize A as L*D*L**T using the lower triangle of A
622: *
623: * Initialize the unused last entry of the subdiagonal array E.
624: *
625: E( N ) = ZERO
626: *
627: * K is the main loop index, increasing from 1 to N in steps of
628: * 1 or 2
629: *
630: K = 1
631: 40 CONTINUE
632: *
633: * If K > N, exit from loop
634: *
635: IF( K.GT.N )
636: $ GO TO 64
637: KSTEP = 1
638: P = K
639: *
640: * Determine rows and columns to be interchanged and whether
641: * a 1-by-1 or 2-by-2 pivot block will be used
642: *
643: ABSAKK = ABS( A( K, K ) )
644: *
645: * IMAX is the row-index of the largest off-diagonal element in
646: * column K, and COLMAX is its absolute value.
647: * Determine both COLMAX and IMAX.
648: *
649: IF( K.LT.N ) THEN
650: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
651: COLMAX = ABS( A( IMAX, K ) )
652: ELSE
653: COLMAX = ZERO
654: END IF
655: *
656: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
657: *
658: * Column K is zero or underflow: set INFO and continue
659: *
660: IF( INFO.EQ.0 )
661: $ INFO = K
662: KP = K
663: *
664: * Set E( K ) to zero
665: *
666: IF( K.LT.N )
667: $ E( K ) = ZERO
668: *
669: ELSE
670: *
671: * Test for interchange
672: *
673: * Equivalent to testing for (used to handle NaN and Inf)
674: * ABSAKK.GE.ALPHA*COLMAX
675: *
676: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
677: *
678: * no interchange, use 1-by-1 pivot block
679: *
680: KP = K
681: *
682: ELSE
683: *
684: DONE = .FALSE.
685: *
686: * Loop until pivot found
687: *
688: 42 CONTINUE
689: *
690: * Begin pivot search loop body
691: *
692: * JMAX is the column-index of the largest off-diagonal
693: * element in row IMAX, and ROWMAX is its absolute value.
694: * Determine both ROWMAX and JMAX.
695: *
696: IF( IMAX.NE.K ) THEN
697: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
698: ROWMAX = ABS( A( IMAX, JMAX ) )
699: ELSE
700: ROWMAX = ZERO
701: END IF
702: *
703: IF( IMAX.LT.N ) THEN
704: ITEMP = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
705: $ 1 )
706: DTEMP = ABS( A( ITEMP, IMAX ) )
707: IF( DTEMP.GT.ROWMAX ) THEN
708: ROWMAX = DTEMP
709: JMAX = ITEMP
710: END IF
711: END IF
712: *
713: * Equivalent to testing for (used to handle NaN and Inf)
714: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
715: *
716: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
717: $ THEN
718: *
719: * interchange rows and columns K and IMAX,
720: * use 1-by-1 pivot block
721: *
722: KP = IMAX
723: DONE = .TRUE.
724: *
725: * Equivalent to testing for ROWMAX .EQ. COLMAX,
726: * used to handle NaN and Inf
727: *
728: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
729: *
730: * interchange rows and columns K+1 and IMAX,
731: * use 2-by-2 pivot block
732: *
733: KP = IMAX
734: KSTEP = 2
735: DONE = .TRUE.
736: ELSE
737: *
738: * Pivot NOT found, set variables and repeat
739: *
740: P = IMAX
741: COLMAX = ROWMAX
742: IMAX = JMAX
743: END IF
744: *
745: * End pivot search loop body
746: *
747: IF( .NOT. DONE ) GOTO 42
748: *
749: END IF
750: *
751: * Swap TWO rows and TWO columns
752: *
753: * First swap
754: *
755: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
756: *
757: * Interchange rows and column K and P in the trailing
758: * submatrix A(k:n,k:n) if we have a 2-by-2 pivot
759: *
760: IF( P.LT.N )
761: $ CALL DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
762: IF( P.GT.(K+1) )
763: $ CALL DSWAP( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
764: T = A( K, K )
765: A( K, K ) = A( P, P )
766: A( P, P ) = T
767: *
768: * Convert lower triangle of A into L form by applying
769: * the interchanges in columns 1:k-1.
770: *
771: IF ( K.GT.1 )
772: $ CALL DSWAP( K-1, A( K, 1 ), LDA, A( P, 1 ), LDA )
773: *
774: END IF
775: *
776: * Second swap
777: *
778: KK = K + KSTEP - 1
779: IF( KP.NE.KK ) THEN
780: *
781: * Interchange rows and columns KK and KP in the trailing
782: * submatrix A(k:n,k:n)
783: *
784: IF( KP.LT.N )
785: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
786: IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
787: $ CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
788: $ LDA )
789: T = A( KK, KK )
790: A( KK, KK ) = A( KP, KP )
791: A( KP, KP ) = T
792: IF( KSTEP.EQ.2 ) THEN
793: T = A( K+1, K )
794: A( K+1, K ) = A( KP, K )
795: A( KP, K ) = T
796: END IF
797: *
798: * Convert lower triangle of A into L form by applying
799: * the interchanges in columns 1:k-1.
800: *
801: IF ( K.GT.1 )
802: $ CALL DSWAP( K-1, A( KK, 1 ), LDA, A( KP, 1 ), LDA )
803: *
804: END IF
805: *
806: * Update the trailing submatrix
807: *
808: IF( KSTEP.EQ.1 ) THEN
809: *
810: * 1-by-1 pivot block D(k): column k now holds
811: *
812: * W(k) = L(k)*D(k)
813: *
814: * where L(k) is the k-th column of L
815: *
816: IF( K.LT.N ) THEN
817: *
818: * Perform a rank-1 update of A(k+1:n,k+1:n) and
819: * store L(k) in column k
820: *
821: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
822: *
823: * Perform a rank-1 update of A(k+1:n,k+1:n) as
824: * A := A - L(k)*D(k)*L(k)**T
825: * = A - W(k)*(1/D(k))*W(k)**T
826: *
827: D11 = ONE / A( K, K )
828: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
829: $ A( K+1, K+1 ), LDA )
830: *
831: * Store L(k) in column k
832: *
833: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
834: ELSE
835: *
836: * Store L(k) in column k
837: *
838: D11 = A( K, K )
839: DO 46 II = K + 1, N
840: A( II, K ) = A( II, K ) / D11
841: 46 CONTINUE
842: *
843: * Perform a rank-1 update of A(k+1:n,k+1:n) as
844: * A := A - L(k)*D(k)*L(k)**T
845: * = A - W(k)*(1/D(k))*W(k)**T
846: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
847: *
848: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
849: $ A( K+1, K+1 ), LDA )
850: END IF
851: *
852: * Store the subdiagonal element of D in array E
853: *
854: E( K ) = ZERO
855: *
856: END IF
857: *
858: ELSE
859: *
860: * 2-by-2 pivot block D(k): columns k and k+1 now hold
861: *
862: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
863: *
864: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
865: * of L
866: *
867: *
868: * Perform a rank-2 update of A(k+2:n,k+2:n) as
869: *
870: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
871: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
872: *
873: * and store L(k) and L(k+1) in columns k and k+1
874: *
875: IF( K.LT.N-1 ) THEN
876: *
877: D21 = A( K+1, K )
878: D11 = A( K+1, K+1 ) / D21
879: D22 = A( K, K ) / D21
880: T = ONE / ( D11*D22-ONE )
881: *
882: DO 60 J = K + 2, N
883: *
884: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
885: *
886: WK = T*( D11*A( J, K )-A( J, K+1 ) )
887: WKP1 = T*( D22*A( J, K+1 )-A( J, K ) )
888: *
889: * Perform a rank-2 update of A(k+2:n,k+2:n)
890: *
891: DO 50 I = J, N
892: A( I, J ) = A( I, J ) - ( A( I, K ) / D21 )*WK -
893: $ ( A( I, K+1 ) / D21 )*WKP1
894: 50 CONTINUE
895: *
896: * Store L(k) and L(k+1) in cols k and k+1 for row J
897: *
898: A( J, K ) = WK / D21
899: A( J, K+1 ) = WKP1 / D21
900: *
901: 60 CONTINUE
902: *
903: END IF
904: *
905: * Copy subdiagonal elements of D(K) to E(K) and
906: * ZERO out subdiagonal entry of A
907: *
908: E( K ) = A( K+1, K )
909: E( K+1 ) = ZERO
910: A( K+1, K ) = ZERO
911: *
912: END IF
913: *
914: * End column K is nonsingular
915: *
916: END IF
917: *
918: * Store details of the interchanges in IPIV
919: *
920: IF( KSTEP.EQ.1 ) THEN
921: IPIV( K ) = KP
922: ELSE
923: IPIV( K ) = -P
924: IPIV( K+1 ) = -KP
925: END IF
926: *
927: * Increase K and return to the start of the main loop
928: *
929: K = K + KSTEP
930: GO TO 40
931: *
932: 64 CONTINUE
933: *
934: END IF
935: *
936: RETURN
937: *
938: * End of DSYTF2_RK
939: *
940: END
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