1: *> \brief \b DSYTF2_ROOK computes the factorization of a real symmetric indefinite matrix using the bounded Bunch-Kaufman ("rook") diagonal pivoting method (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_ROOK + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytf2_rook.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytf2_rook.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytf2_rook.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTF2_ROOK( UPLO, N, A, LDA, 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, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DSYTF2_ROOK computes the factorization of a real symmetric matrix A
39: *> using the bounded Bunch-Kaufman ("rook") diagonal pivoting method:
40: *>
41: *> A = U*D*U**T or A = L*D*L**T
42: *>
43: *> where U (or L) is a product of permutation and unit upper (lower)
44: *> triangular matrices, U**T is the transpose of U, and D is symmetric and
45: *> block 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: *> \endverbatim
49: *
50: * Arguments:
51: * ==========
52: *
53: *> \param[in] UPLO
54: *> \verbatim
55: *> UPLO is CHARACTER*1
56: *> Specifies whether the upper or lower triangular part of the
57: *> symmetric matrix A is stored:
58: *> = 'U': Upper triangular
59: *> = 'L': Lower triangular
60: *> \endverbatim
61: *>
62: *> \param[in] N
63: *> \verbatim
64: *> N is INTEGER
65: *> The order of the matrix A. N >= 0.
66: *> \endverbatim
67: *>
68: *> \param[in,out] A
69: *> \verbatim
70: *> A is DOUBLE PRECISION array, dimension (LDA,N)
71: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
72: *> n-by-n upper triangular part of A contains the upper
73: *> triangular part of the matrix A, and the strictly lower
74: *> triangular part of A is not referenced. If UPLO = 'L', the
75: *> leading n-by-n lower triangular part of A contains the lower
76: *> triangular part of the matrix A, and the strictly upper
77: *> triangular part of A is not referenced.
78: *>
79: *> On exit, the block diagonal matrix D and the multipliers used
80: *> to obtain the factor U or L (see below for further details).
81: *> \endverbatim
82: *>
83: *> \param[in] LDA
84: *> \verbatim
85: *> LDA is INTEGER
86: *> The leading dimension of the array A. LDA >= max(1,N).
87: *> \endverbatim
88: *>
89: *> \param[out] IPIV
90: *> \verbatim
91: *> IPIV is INTEGER array, dimension (N)
92: *> Details of the interchanges and the block structure of D.
93: *>
94: *> If UPLO = 'U':
95: *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
96: *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
97: *>
98: *> If IPIV(k) < 0 and IPIV(k-1) < 0, then rows and
99: *> columns k and -IPIV(k) were interchanged and rows and
100: *> columns k-1 and -IPIV(k-1) were inerchaged,
101: *> D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
102: *>
103: *> If UPLO = 'L':
104: *> If IPIV(k) > 0, then rows and columns k and IPIV(k)
105: *> were interchanged and D(k,k) is a 1-by-1 diagonal block.
106: *>
107: *> If IPIV(k) < 0 and IPIV(k+1) < 0, then rows and
108: *> columns k and -IPIV(k) were interchanged and rows and
109: *> columns k+1 and -IPIV(k+1) were inerchaged,
110: *> D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
111: *> \endverbatim
112: *>
113: *> \param[out] INFO
114: *> \verbatim
115: *> INFO is INTEGER
116: *> = 0: successful exit
117: *> < 0: if INFO = -k, the k-th argument had an illegal value
118: *> > 0: if INFO = k, D(k,k) is exactly zero. The factorization
119: *> has been completed, but the block diagonal matrix D is
120: *> exactly singular, and division by zero will occur if it
121: *> is used to solve a system of equations.
122: *> \endverbatim
123: *
124: * Authors:
125: * ========
126: *
127: *> \author Univ. of Tennessee
128: *> \author Univ. of California Berkeley
129: *> \author Univ. of Colorado Denver
130: *> \author NAG Ltd.
131: *
132: *> \ingroup doubleSYcomputational
133: *
134: *> \par Further Details:
135: * =====================
136: *>
137: *> \verbatim
138: *>
139: *> If UPLO = 'U', then A = U*D*U**T, where
140: *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
141: *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
142: *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
143: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
144: *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
145: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
146: *>
147: *> ( I v 0 ) k-s
148: *> U(k) = ( 0 I 0 ) s
149: *> ( 0 0 I ) n-k
150: *> k-s s n-k
151: *>
152: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
153: *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
154: *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
155: *>
156: *> If UPLO = 'L', then A = L*D*L**T, where
157: *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
158: *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
159: *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
160: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
161: *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
162: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
163: *>
164: *> ( I 0 0 ) k-1
165: *> L(k) = ( 0 I 0 ) s
166: *> ( 0 v I ) n-k-s+1
167: *> k-1 s n-k-s+1
168: *>
169: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
170: *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
171: *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
172: *> \endverbatim
173: *
174: *> \par Contributors:
175: * ==================
176: *>
177: *> \verbatim
178: *>
179: *> November 2013, Igor Kozachenko,
180: *> Computer Science Division,
181: *> University of California, Berkeley
182: *>
183: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
184: *> School of Mathematics,
185: *> University of Manchester
186: *>
187: *> 01-01-96 - Based on modifications by
188: *> J. Lewis, Boeing Computer Services Company
189: *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA
190: *> \endverbatim
191: *
192: * =====================================================================
193: SUBROUTINE DSYTF2_ROOK( UPLO, N, A, LDA, IPIV, INFO )
194: *
195: * -- LAPACK computational routine --
196: * -- LAPACK is a software package provided by Univ. of Tennessee, --
197: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
198: *
199: * .. Scalar Arguments ..
200: CHARACTER UPLO
201: INTEGER INFO, LDA, N
202: * ..
203: * .. Array Arguments ..
204: INTEGER IPIV( * )
205: DOUBLE PRECISION A( LDA, * )
206: * ..
207: *
208: * =====================================================================
209: *
210: * .. Parameters ..
211: DOUBLE PRECISION ZERO, ONE
212: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
213: DOUBLE PRECISION EIGHT, SEVTEN
214: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
215: * ..
216: * .. Local Scalars ..
217: LOGICAL UPPER, DONE
218: INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
219: $ P, II
220: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
221: $ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
222: * ..
223: * .. External Functions ..
224: LOGICAL LSAME
225: INTEGER IDAMAX
226: DOUBLE PRECISION DLAMCH
227: EXTERNAL LSAME, IDAMAX, DLAMCH
228: * ..
229: * .. External Subroutines ..
230: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
231: * ..
232: * .. Intrinsic Functions ..
233: INTRINSIC ABS, MAX, SQRT
234: * ..
235: * .. Executable Statements ..
236: *
237: * Test the input parameters.
238: *
239: INFO = 0
240: UPPER = LSAME( UPLO, 'U' )
241: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
242: INFO = -1
243: ELSE IF( N.LT.0 ) THEN
244: INFO = -2
245: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
246: INFO = -4
247: END IF
248: IF( INFO.NE.0 ) THEN
249: CALL XERBLA( 'DSYTF2_ROOK', -INFO )
250: RETURN
251: END IF
252: *
253: * Initialize ALPHA for use in choosing pivot block size.
254: *
255: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
256: *
257: * Compute machine safe minimum
258: *
259: SFMIN = DLAMCH( 'S' )
260: *
261: IF( UPPER ) THEN
262: *
263: * Factorize A as U*D*U**T using the upper triangle of A
264: *
265: * K is the main loop index, decreasing from N to 1 in steps of
266: * 1 or 2
267: *
268: K = N
269: 10 CONTINUE
270: *
271: * If K < 1, exit from loop
272: *
273: IF( K.LT.1 )
274: $ GO TO 70
275: KSTEP = 1
276: P = K
277: *
278: * Determine rows and columns to be interchanged and whether
279: * a 1-by-1 or 2-by-2 pivot block will be used
280: *
281: ABSAKK = ABS( A( K, K ) )
282: *
283: * IMAX is the row-index of the largest off-diagonal element in
284: * column K, and COLMAX is its absolute value.
285: * Determine both COLMAX and IMAX.
286: *
287: IF( K.GT.1 ) THEN
288: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
289: COLMAX = ABS( A( IMAX, K ) )
290: ELSE
291: COLMAX = ZERO
292: END IF
293: *
294: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) ) THEN
295: *
296: * Column K is zero or underflow: set INFO and continue
297: *
298: IF( INFO.EQ.0 )
299: $ INFO = K
300: KP = K
301: ELSE
302: *
303: * Test for interchange
304: *
305: * Equivalent to testing for (used to handle NaN and Inf)
306: * ABSAKK.GE.ALPHA*COLMAX
307: *
308: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
309: *
310: * no interchange,
311: * use 1-by-1 pivot block
312: *
313: KP = K
314: ELSE
315: *
316: DONE = .FALSE.
317: *
318: * Loop until pivot found
319: *
320: 12 CONTINUE
321: *
322: * Begin pivot search loop body
323: *
324: * JMAX is the column-index of the largest off-diagonal
325: * element in row IMAX, and ROWMAX is its absolute value.
326: * Determine both ROWMAX and JMAX.
327: *
328: IF( IMAX.NE.K ) THEN
329: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
330: $ LDA )
331: ROWMAX = ABS( A( IMAX, JMAX ) )
332: ELSE
333: ROWMAX = ZERO
334: END IF
335: *
336: IF( IMAX.GT.1 ) THEN
337: ITEMP = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
338: DTEMP = ABS( A( ITEMP, IMAX ) )
339: IF( DTEMP.GT.ROWMAX ) THEN
340: ROWMAX = DTEMP
341: JMAX = ITEMP
342: END IF
343: END IF
344: *
345: * Equivalent to testing for (used to handle NaN and Inf)
346: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
347: *
348: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
349: $ THEN
350: *
351: * interchange rows and columns K and IMAX,
352: * use 1-by-1 pivot block
353: *
354: KP = IMAX
355: DONE = .TRUE.
356: *
357: * Equivalent to testing for ROWMAX .EQ. COLMAX,
358: * used to handle NaN and Inf
359: *
360: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
361: *
362: * interchange rows and columns K+1 and IMAX,
363: * use 2-by-2 pivot block
364: *
365: KP = IMAX
366: KSTEP = 2
367: DONE = .TRUE.
368: ELSE
369: *
370: * Pivot NOT found, set variables and repeat
371: *
372: P = IMAX
373: COLMAX = ROWMAX
374: IMAX = JMAX
375: END IF
376: *
377: * End pivot search loop body
378: *
379: IF( .NOT. DONE ) GOTO 12
380: *
381: END IF
382: *
383: * Swap TWO rows and TWO columns
384: *
385: * First swap
386: *
387: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
388: *
389: * Interchange rows and column K and P in the leading
390: * submatrix A(1:k,1:k) if we have a 2-by-2 pivot
391: *
392: IF( P.GT.1 )
393: $ CALL DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
394: IF( P.LT.(K-1) )
395: $ CALL DSWAP( K-P-1, A( P+1, K ), 1, A( P, P+1 ),
396: $ LDA )
397: T = A( K, K )
398: A( K, K ) = A( P, P )
399: A( P, P ) = T
400: END IF
401: *
402: * Second swap
403: *
404: KK = K - KSTEP + 1
405: IF( KP.NE.KK ) THEN
406: *
407: * Interchange rows and columns KK and KP in the leading
408: * submatrix A(1:k,1:k)
409: *
410: IF( KP.GT.1 )
411: $ CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
412: IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
413: $ CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
414: $ LDA )
415: T = A( KK, KK )
416: A( KK, KK ) = A( KP, KP )
417: A( KP, KP ) = T
418: IF( KSTEP.EQ.2 ) THEN
419: T = A( K-1, K )
420: A( K-1, K ) = A( KP, K )
421: A( KP, K ) = T
422: END IF
423: END IF
424: *
425: * Update the leading submatrix
426: *
427: IF( KSTEP.EQ.1 ) THEN
428: *
429: * 1-by-1 pivot block D(k): column k now holds
430: *
431: * W(k) = U(k)*D(k)
432: *
433: * where U(k) is the k-th column of U
434: *
435: IF( K.GT.1 ) THEN
436: *
437: * Perform a rank-1 update of A(1:k-1,1:k-1) and
438: * store U(k) in column k
439: *
440: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
441: *
442: * Perform a rank-1 update of A(1:k-1,1:k-1) as
443: * A := A - U(k)*D(k)*U(k)**T
444: * = A - W(k)*1/D(k)*W(k)**T
445: *
446: D11 = ONE / A( K, K )
447: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
448: *
449: * Store U(k) in column k
450: *
451: CALL DSCAL( K-1, D11, A( 1, K ), 1 )
452: ELSE
453: *
454: * Store L(k) in column K
455: *
456: D11 = A( K, K )
457: DO 16 II = 1, K - 1
458: A( II, K ) = A( II, K ) / D11
459: 16 CONTINUE
460: *
461: * Perform a rank-1 update of A(k+1:n,k+1:n) as
462: * A := A - U(k)*D(k)*U(k)**T
463: * = A - W(k)*(1/D(k))*W(k)**T
464: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
465: *
466: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
467: END IF
468: END IF
469: *
470: ELSE
471: *
472: * 2-by-2 pivot block D(k): columns k and k-1 now hold
473: *
474: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
475: *
476: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
477: * of U
478: *
479: * Perform a rank-2 update of A(1:k-2,1:k-2) as
480: *
481: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
482: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
483: *
484: * and store L(k) and L(k+1) in columns k and k+1
485: *
486: IF( K.GT.2 ) THEN
487: *
488: D12 = A( K-1, K )
489: D22 = A( K-1, K-1 ) / D12
490: D11 = A( K, K ) / D12
491: T = ONE / ( D11*D22-ONE )
492: *
493: DO 30 J = K - 2, 1, -1
494: *
495: WKM1 = T*( D11*A( J, K-1 )-A( J, K ) )
496: WK = T*( D22*A( J, K )-A( J, K-1 ) )
497: *
498: DO 20 I = J, 1, -1
499: A( I, J ) = A( I, J ) - (A( I, K ) / D12 )*WK -
500: $ ( A( I, K-1 ) / D12 )*WKM1
501: 20 CONTINUE
502: *
503: * Store U(k) and U(k-1) in cols k and k-1 for row J
504: *
505: A( J, K ) = WK / D12
506: A( J, K-1 ) = WKM1 / D12
507: *
508: 30 CONTINUE
509: *
510: END IF
511: *
512: END IF
513: END IF
514: *
515: * Store details of the interchanges in IPIV
516: *
517: IF( KSTEP.EQ.1 ) THEN
518: IPIV( K ) = KP
519: ELSE
520: IPIV( K ) = -P
521: IPIV( K-1 ) = -KP
522: END IF
523: *
524: * Decrease K and return to the start of the main loop
525: *
526: K = K - KSTEP
527: GO TO 10
528: *
529: ELSE
530: *
531: * Factorize A as L*D*L**T using the lower triangle of A
532: *
533: * K is the main loop index, increasing from 1 to N in steps of
534: * 1 or 2
535: *
536: K = 1
537: 40 CONTINUE
538: *
539: * If K > N, exit from loop
540: *
541: IF( K.GT.N )
542: $ GO TO 70
543: KSTEP = 1
544: P = K
545: *
546: * Determine rows and columns to be interchanged and whether
547: * a 1-by-1 or 2-by-2 pivot block will be used
548: *
549: ABSAKK = ABS( A( K, K ) )
550: *
551: * IMAX is the row-index of the largest off-diagonal element in
552: * column K, and COLMAX is its absolute value.
553: * Determine both COLMAX and IMAX.
554: *
555: IF( K.LT.N ) THEN
556: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
557: COLMAX = ABS( A( IMAX, K ) )
558: ELSE
559: COLMAX = ZERO
560: END IF
561: *
562: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
563: *
564: * Column K is zero or underflow: set INFO and continue
565: *
566: IF( INFO.EQ.0 )
567: $ INFO = K
568: KP = K
569: ELSE
570: *
571: * Test for interchange
572: *
573: * Equivalent to testing for (used to handle NaN and Inf)
574: * ABSAKK.GE.ALPHA*COLMAX
575: *
576: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
577: *
578: * no interchange, use 1-by-1 pivot block
579: *
580: KP = K
581: ELSE
582: *
583: DONE = .FALSE.
584: *
585: * Loop until pivot found
586: *
587: 42 CONTINUE
588: *
589: * Begin pivot search loop body
590: *
591: * JMAX is the column-index of the largest off-diagonal
592: * element in row IMAX, and ROWMAX is its absolute value.
593: * Determine both ROWMAX and JMAX.
594: *
595: IF( IMAX.NE.K ) THEN
596: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
597: ROWMAX = ABS( A( IMAX, JMAX ) )
598: ELSE
599: ROWMAX = ZERO
600: END IF
601: *
602: IF( IMAX.LT.N ) THEN
603: ITEMP = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
604: $ 1 )
605: DTEMP = ABS( A( ITEMP, IMAX ) )
606: IF( DTEMP.GT.ROWMAX ) THEN
607: ROWMAX = DTEMP
608: JMAX = ITEMP
609: END IF
610: END IF
611: *
612: * Equivalent to testing for (used to handle NaN and Inf)
613: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
614: *
615: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
616: $ THEN
617: *
618: * interchange rows and columns K and IMAX,
619: * use 1-by-1 pivot block
620: *
621: KP = IMAX
622: DONE = .TRUE.
623: *
624: * Equivalent to testing for ROWMAX .EQ. COLMAX,
625: * used to handle NaN and Inf
626: *
627: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
628: *
629: * interchange rows and columns K+1 and IMAX,
630: * use 2-by-2 pivot block
631: *
632: KP = IMAX
633: KSTEP = 2
634: DONE = .TRUE.
635: ELSE
636: *
637: * Pivot NOT found, set variables and repeat
638: *
639: P = IMAX
640: COLMAX = ROWMAX
641: IMAX = JMAX
642: END IF
643: *
644: * End pivot search loop body
645: *
646: IF( .NOT. DONE ) GOTO 42
647: *
648: END IF
649: *
650: * Swap TWO rows and TWO columns
651: *
652: * First swap
653: *
654: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
655: *
656: * Interchange rows and column K and P in the trailing
657: * submatrix A(k:n,k:n) if we have a 2-by-2 pivot
658: *
659: IF( P.LT.N )
660: $ CALL DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
661: IF( P.GT.(K+1) )
662: $ CALL DSWAP( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
663: T = A( K, K )
664: A( K, K ) = A( P, P )
665: A( P, P ) = T
666: END IF
667: *
668: * Second swap
669: *
670: KK = K + KSTEP - 1
671: IF( KP.NE.KK ) THEN
672: *
673: * Interchange rows and columns KK and KP in the trailing
674: * submatrix A(k:n,k:n)
675: *
676: IF( KP.LT.N )
677: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
678: IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
679: $ CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
680: $ LDA )
681: T = A( KK, KK )
682: A( KK, KK ) = A( KP, KP )
683: A( KP, KP ) = T
684: IF( KSTEP.EQ.2 ) THEN
685: T = A( K+1, K )
686: A( K+1, K ) = A( KP, K )
687: A( KP, K ) = T
688: END IF
689: END IF
690: *
691: * Update the trailing submatrix
692: *
693: IF( KSTEP.EQ.1 ) THEN
694: *
695: * 1-by-1 pivot block D(k): column k now holds
696: *
697: * W(k) = L(k)*D(k)
698: *
699: * where L(k) is the k-th column of L
700: *
701: IF( K.LT.N ) THEN
702: *
703: * Perform a rank-1 update of A(k+1:n,k+1:n) and
704: * store L(k) in column k
705: *
706: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
707: *
708: * Perform a rank-1 update of A(k+1:n,k+1:n) as
709: * A := A - L(k)*D(k)*L(k)**T
710: * = A - W(k)*(1/D(k))*W(k)**T
711: *
712: D11 = ONE / A( K, K )
713: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
714: $ A( K+1, K+1 ), LDA )
715: *
716: * Store L(k) in column k
717: *
718: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
719: ELSE
720: *
721: * Store L(k) in column k
722: *
723: D11 = A( K, K )
724: DO 46 II = K + 1, N
725: A( II, K ) = A( II, K ) / D11
726: 46 CONTINUE
727: *
728: * Perform a rank-1 update of A(k+1:n,k+1:n) as
729: * A := A - L(k)*D(k)*L(k)**T
730: * = A - W(k)*(1/D(k))*W(k)**T
731: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
732: *
733: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
734: $ A( K+1, K+1 ), LDA )
735: END IF
736: END IF
737: *
738: ELSE
739: *
740: * 2-by-2 pivot block D(k): columns k and k+1 now hold
741: *
742: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
743: *
744: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
745: * of L
746: *
747: *
748: * Perform a rank-2 update of A(k+2:n,k+2:n) as
749: *
750: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
751: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
752: *
753: * and store L(k) and L(k+1) in columns k and k+1
754: *
755: IF( K.LT.N-1 ) THEN
756: *
757: D21 = A( K+1, K )
758: D11 = A( K+1, K+1 ) / D21
759: D22 = A( K, K ) / D21
760: T = ONE / ( D11*D22-ONE )
761: *
762: DO 60 J = K + 2, N
763: *
764: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
765: *
766: WK = T*( D11*A( J, K )-A( J, K+1 ) )
767: WKP1 = T*( D22*A( J, K+1 )-A( J, K ) )
768: *
769: * Perform a rank-2 update of A(k+2:n,k+2:n)
770: *
771: DO 50 I = J, N
772: A( I, J ) = A( I, J ) - ( A( I, K ) / D21 )*WK -
773: $ ( A( I, K+1 ) / D21 )*WKP1
774: 50 CONTINUE
775: *
776: * Store L(k) and L(k+1) in cols k and k+1 for row J
777: *
778: A( J, K ) = WK / D21
779: A( J, K+1 ) = WKP1 / D21
780: *
781: 60 CONTINUE
782: *
783: END IF
784: *
785: END IF
786: END IF
787: *
788: * Store details of the interchanges in IPIV
789: *
790: IF( KSTEP.EQ.1 ) THEN
791: IPIV( K ) = KP
792: ELSE
793: IPIV( K ) = -P
794: IPIV( K+1 ) = -KP
795: END IF
796: *
797: * Increase K and return to the start of the main loop
798: *
799: K = K + KSTEP
800: GO TO 40
801: *
802: END IF
803: *
804: 70 CONTINUE
805: *
806: RETURN
807: *
808: * End of DSYTF2_ROOK
809: *
810: END
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