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: *> \date November 2013
133: *
134: *> \ingroup doubleSYcomputational
135: *
136: *> \par Further Details:
137: * =====================
138: *>
139: *> \verbatim
140: *>
141: *> If UPLO = 'U', then A = U*D*U**T, where
142: *> U = P(n)*U(n)* ... *P(k)U(k)* ...,
143: *> i.e., U is a product of terms P(k)*U(k), where k decreases from n to
144: *> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
145: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
146: *> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
147: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
148: *>
149: *> ( I v 0 ) k-s
150: *> U(k) = ( 0 I 0 ) s
151: *> ( 0 0 I ) n-k
152: *> k-s s n-k
153: *>
154: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
155: *> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
156: *> and A(k,k), and v overwrites A(1:k-2,k-1:k).
157: *>
158: *> If UPLO = 'L', then A = L*D*L**T, where
159: *> L = P(1)*L(1)* ... *P(k)*L(k)* ...,
160: *> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
161: *> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
162: *> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
163: *> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
164: *> that if the diagonal block D(k) is of order s (s = 1 or 2), then
165: *>
166: *> ( I 0 0 ) k-1
167: *> L(k) = ( 0 I 0 ) s
168: *> ( 0 v I ) n-k-s+1
169: *> k-1 s n-k-s+1
170: *>
171: *> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
172: *> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
173: *> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
174: *> \endverbatim
175: *
176: *> \par Contributors:
177: * ==================
178: *>
179: *> \verbatim
180: *>
181: *> November 2013, Igor Kozachenko,
182: *> Computer Science Division,
183: *> University of California, Berkeley
184: *>
185: *> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
186: *> School of Mathematics,
187: *> University of Manchester
188: *>
189: *> 01-01-96 - Based on modifications by
190: *> J. Lewis, Boeing Computer Services Company
191: *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville abd , USA
192: *> \endverbatim
193: *
194: * =====================================================================
195: SUBROUTINE DSYTF2_ROOK( UPLO, N, A, LDA, IPIV, INFO )
196: *
197: * -- LAPACK computational routine (version 3.5.0) --
198: * -- LAPACK is a software package provided by Univ. of Tennessee, --
199: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
200: * November 2013
201: *
202: * .. Scalar Arguments ..
203: CHARACTER UPLO
204: INTEGER INFO, LDA, N
205: * ..
206: * .. Array Arguments ..
207: INTEGER IPIV( * )
208: DOUBLE PRECISION A( LDA, * )
209: * ..
210: *
211: * =====================================================================
212: *
213: * .. Parameters ..
214: DOUBLE PRECISION ZERO, ONE
215: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
216: DOUBLE PRECISION EIGHT, SEVTEN
217: PARAMETER ( EIGHT = 8.0D+0, SEVTEN = 17.0D+0 )
218: * ..
219: * .. Local Scalars ..
220: LOGICAL UPPER, DONE
221: INTEGER I, IMAX, J, JMAX, ITEMP, K, KK, KP, KSTEP,
222: $ P, II
223: DOUBLE PRECISION ABSAKK, ALPHA, COLMAX, D11, D12, D21, D22,
224: $ ROWMAX, DTEMP, T, WK, WKM1, WKP1, SFMIN
225: * ..
226: * .. External Functions ..
227: LOGICAL LSAME
228: INTEGER IDAMAX
229: DOUBLE PRECISION DLAMCH
230: EXTERNAL LSAME, IDAMAX, DLAMCH
231: * ..
232: * .. External Subroutines ..
233: EXTERNAL DSCAL, DSWAP, DSYR, XERBLA
234: * ..
235: * .. Intrinsic Functions ..
236: INTRINSIC ABS, MAX, SQRT
237: * ..
238: * .. Executable Statements ..
239: *
240: * Test the input parameters.
241: *
242: INFO = 0
243: UPPER = LSAME( UPLO, 'U' )
244: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
245: INFO = -1
246: ELSE IF( N.LT.0 ) THEN
247: INFO = -2
248: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
249: INFO = -4
250: END IF
251: IF( INFO.NE.0 ) THEN
252: CALL XERBLA( 'DSYTF2_ROOK', -INFO )
253: RETURN
254: END IF
255: *
256: * Initialize ALPHA for use in choosing pivot block size.
257: *
258: ALPHA = ( ONE+SQRT( SEVTEN ) ) / EIGHT
259: *
260: * Compute machine safe minimum
261: *
262: SFMIN = DLAMCH( 'S' )
263: *
264: IF( UPPER ) THEN
265: *
266: * Factorize A as U*D*U**T using the upper triangle of A
267: *
268: * K is the main loop index, decreasing from N to 1 in steps of
269: * 1 or 2
270: *
271: K = N
272: 10 CONTINUE
273: *
274: * If K < 1, exit from loop
275: *
276: IF( K.LT.1 )
277: $ GO TO 70
278: KSTEP = 1
279: P = K
280: *
281: * Determine rows and columns to be interchanged and whether
282: * a 1-by-1 or 2-by-2 pivot block will be used
283: *
284: ABSAKK = ABS( A( K, K ) )
285: *
286: * IMAX is the row-index of the largest off-diagonal element in
287: * column K, and COLMAX is its absolute value.
288: * Determine both COLMAX and IMAX.
289: *
290: IF( K.GT.1 ) THEN
291: IMAX = IDAMAX( K-1, A( 1, K ), 1 )
292: COLMAX = ABS( A( IMAX, K ) )
293: ELSE
294: COLMAX = ZERO
295: END IF
296: *
297: IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) ) THEN
298: *
299: * Column K is zero or underflow: set INFO and continue
300: *
301: IF( INFO.EQ.0 )
302: $ INFO = K
303: KP = K
304: ELSE
305: *
306: * Test for interchange
307: *
308: * Equivalent to testing for (used to handle NaN and Inf)
309: * ABSAKK.GE.ALPHA*COLMAX
310: *
311: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
312: *
313: * no interchange,
314: * use 1-by-1 pivot block
315: *
316: KP = K
317: ELSE
318: *
319: DONE = .FALSE.
320: *
321: * Loop until pivot found
322: *
323: 12 CONTINUE
324: *
325: * Begin pivot search loop body
326: *
327: * JMAX is the column-index of the largest off-diagonal
328: * element in row IMAX, and ROWMAX is its absolute value.
329: * Determine both ROWMAX and JMAX.
330: *
331: IF( IMAX.NE.K ) THEN
332: JMAX = IMAX + IDAMAX( K-IMAX, A( IMAX, IMAX+1 ),
333: $ LDA )
334: ROWMAX = ABS( A( IMAX, JMAX ) )
335: ELSE
336: ROWMAX = ZERO
337: END IF
338: *
339: IF( IMAX.GT.1 ) THEN
340: ITEMP = IDAMAX( IMAX-1, A( 1, IMAX ), 1 )
341: DTEMP = ABS( A( ITEMP, IMAX ) )
342: IF( DTEMP.GT.ROWMAX ) THEN
343: ROWMAX = DTEMP
344: JMAX = ITEMP
345: END IF
346: END IF
347: *
348: * Equivalent to testing for (used to handle NaN and Inf)
349: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
350: *
351: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
352: $ THEN
353: *
354: * interchange rows and columns K and IMAX,
355: * use 1-by-1 pivot block
356: *
357: KP = IMAX
358: DONE = .TRUE.
359: *
360: * Equivalent to testing for ROWMAX .EQ. COLMAX,
361: * used to handle NaN and Inf
362: *
363: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
364: *
365: * interchange rows and columns K+1 and IMAX,
366: * use 2-by-2 pivot block
367: *
368: KP = IMAX
369: KSTEP = 2
370: DONE = .TRUE.
371: ELSE
372: *
373: * Pivot NOT found, set variables and repeat
374: *
375: P = IMAX
376: COLMAX = ROWMAX
377: IMAX = JMAX
378: END IF
379: *
380: * End pivot search loop body
381: *
382: IF( .NOT. DONE ) GOTO 12
383: *
384: END IF
385: *
386: * Swap TWO rows and TWO columns
387: *
388: * First swap
389: *
390: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
391: *
392: * Interchange rows and column K and P in the leading
393: * submatrix A(1:k,1:k) if we have a 2-by-2 pivot
394: *
395: IF( P.GT.1 )
396: $ CALL DSWAP( P-1, A( 1, K ), 1, A( 1, P ), 1 )
397: IF( P.LT.(K-1) )
398: $ CALL DSWAP( K-P-1, A( P+1, K ), 1, A( P, P+1 ),
399: $ LDA )
400: T = A( K, K )
401: A( K, K ) = A( P, P )
402: A( P, P ) = T
403: END IF
404: *
405: * Second swap
406: *
407: KK = K - KSTEP + 1
408: IF( KP.NE.KK ) THEN
409: *
410: * Interchange rows and columns KK and KP in the leading
411: * submatrix A(1:k,1:k)
412: *
413: IF( KP.GT.1 )
414: $ CALL DSWAP( KP-1, A( 1, KK ), 1, A( 1, KP ), 1 )
415: IF( ( KK.GT.1 ) .AND. ( KP.LT.(KK-1) ) )
416: $ CALL DSWAP( KK-KP-1, A( KP+1, KK ), 1, A( KP, KP+1 ),
417: $ LDA )
418: T = A( KK, KK )
419: A( KK, KK ) = A( KP, KP )
420: A( KP, KP ) = T
421: IF( KSTEP.EQ.2 ) THEN
422: T = A( K-1, K )
423: A( K-1, K ) = A( KP, K )
424: A( KP, K ) = T
425: END IF
426: END IF
427: *
428: * Update the leading submatrix
429: *
430: IF( KSTEP.EQ.1 ) THEN
431: *
432: * 1-by-1 pivot block D(k): column k now holds
433: *
434: * W(k) = U(k)*D(k)
435: *
436: * where U(k) is the k-th column of U
437: *
438: IF( K.GT.1 ) THEN
439: *
440: * Perform a rank-1 update of A(1:k-1,1:k-1) and
441: * store U(k) in column k
442: *
443: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
444: *
445: * Perform a rank-1 update of A(1:k-1,1:k-1) as
446: * A := A - U(k)*D(k)*U(k)**T
447: * = A - W(k)*1/D(k)*W(k)**T
448: *
449: D11 = ONE / A( K, K )
450: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
451: *
452: * Store U(k) in column k
453: *
454: CALL DSCAL( K-1, D11, A( 1, K ), 1 )
455: ELSE
456: *
457: * Store L(k) in column K
458: *
459: D11 = A( K, K )
460: DO 16 II = 1, K - 1
461: A( II, K ) = A( II, K ) / D11
462: 16 CONTINUE
463: *
464: * Perform a rank-1 update of A(k+1:n,k+1:n) as
465: * A := A - U(k)*D(k)*U(k)**T
466: * = A - W(k)*(1/D(k))*W(k)**T
467: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
468: *
469: CALL DSYR( UPLO, K-1, -D11, A( 1, K ), 1, A, LDA )
470: END IF
471: END IF
472: *
473: ELSE
474: *
475: * 2-by-2 pivot block D(k): columns k and k-1 now hold
476: *
477: * ( W(k-1) W(k) ) = ( U(k-1) U(k) )*D(k)
478: *
479: * where U(k) and U(k-1) are the k-th and (k-1)-th columns
480: * of U
481: *
482: * Perform a rank-2 update of A(1:k-2,1:k-2) as
483: *
484: * A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**T
485: * = A - ( ( A(k-1)A(k) )*inv(D(k)) ) * ( A(k-1)A(k) )**T
486: *
487: * and store L(k) and L(k+1) in columns k and k+1
488: *
489: IF( K.GT.2 ) THEN
490: *
491: D12 = A( K-1, K )
492: D22 = A( K-1, K-1 ) / D12
493: D11 = A( K, K ) / D12
494: T = ONE / ( D11*D22-ONE )
495: *
496: DO 30 J = K - 2, 1, -1
497: *
498: WKM1 = T*( D11*A( J, K-1 )-A( J, K ) )
499: WK = T*( D22*A( J, K )-A( J, K-1 ) )
500: *
501: DO 20 I = J, 1, -1
502: A( I, J ) = A( I, J ) - (A( I, K ) / D12 )*WK -
503: $ ( A( I, K-1 ) / D12 )*WKM1
504: 20 CONTINUE
505: *
506: * Store U(k) and U(k-1) in cols k and k-1 for row J
507: *
508: A( J, K ) = WK / D12
509: A( J, K-1 ) = WKM1 / D12
510: *
511: 30 CONTINUE
512: *
513: END IF
514: *
515: END IF
516: END IF
517: *
518: * Store details of the interchanges in IPIV
519: *
520: IF( KSTEP.EQ.1 ) THEN
521: IPIV( K ) = KP
522: ELSE
523: IPIV( K ) = -P
524: IPIV( K-1 ) = -KP
525: END IF
526: *
527: * Decrease K and return to the start of the main loop
528: *
529: K = K - KSTEP
530: GO TO 10
531: *
532: ELSE
533: *
534: * Factorize A as L*D*L**T using the lower triangle of A
535: *
536: * K is the main loop index, increasing from 1 to N in steps of
537: * 1 or 2
538: *
539: K = 1
540: 40 CONTINUE
541: *
542: * If K > N, exit from loop
543: *
544: IF( K.GT.N )
545: $ GO TO 70
546: KSTEP = 1
547: P = K
548: *
549: * Determine rows and columns to be interchanged and whether
550: * a 1-by-1 or 2-by-2 pivot block will be used
551: *
552: ABSAKK = ABS( A( K, K ) )
553: *
554: * IMAX is the row-index of the largest off-diagonal element in
555: * column K, and COLMAX is its absolute value.
556: * Determine both COLMAX and IMAX.
557: *
558: IF( K.LT.N ) THEN
559: IMAX = K + IDAMAX( N-K, A( K+1, K ), 1 )
560: COLMAX = ABS( A( IMAX, K ) )
561: ELSE
562: COLMAX = ZERO
563: END IF
564: *
565: IF( ( MAX( ABSAKK, COLMAX ).EQ.ZERO ) ) THEN
566: *
567: * Column K is zero or underflow: set INFO and continue
568: *
569: IF( INFO.EQ.0 )
570: $ INFO = K
571: KP = K
572: ELSE
573: *
574: * Test for interchange
575: *
576: * Equivalent to testing for (used to handle NaN and Inf)
577: * ABSAKK.GE.ALPHA*COLMAX
578: *
579: IF( .NOT.( ABSAKK.LT.ALPHA*COLMAX ) ) THEN
580: *
581: * no interchange, use 1-by-1 pivot block
582: *
583: KP = K
584: ELSE
585: *
586: DONE = .FALSE.
587: *
588: * Loop until pivot found
589: *
590: 42 CONTINUE
591: *
592: * Begin pivot search loop body
593: *
594: * JMAX is the column-index of the largest off-diagonal
595: * element in row IMAX, and ROWMAX is its absolute value.
596: * Determine both ROWMAX and JMAX.
597: *
598: IF( IMAX.NE.K ) THEN
599: JMAX = K - 1 + IDAMAX( IMAX-K, A( IMAX, K ), LDA )
600: ROWMAX = ABS( A( IMAX, JMAX ) )
601: ELSE
602: ROWMAX = ZERO
603: END IF
604: *
605: IF( IMAX.LT.N ) THEN
606: ITEMP = IMAX + IDAMAX( N-IMAX, A( IMAX+1, IMAX ),
607: $ 1 )
608: DTEMP = ABS( A( ITEMP, IMAX ) )
609: IF( DTEMP.GT.ROWMAX ) THEN
610: ROWMAX = DTEMP
611: JMAX = ITEMP
612: END IF
613: END IF
614: *
615: * Equivalent to testing for (used to handle NaN and Inf)
616: * ABS( A( IMAX, IMAX ) ).GE.ALPHA*ROWMAX
617: *
618: IF( .NOT.( ABS( A( IMAX, IMAX ) ).LT.ALPHA*ROWMAX ) )
619: $ THEN
620: *
621: * interchange rows and columns K and IMAX,
622: * use 1-by-1 pivot block
623: *
624: KP = IMAX
625: DONE = .TRUE.
626: *
627: * Equivalent to testing for ROWMAX .EQ. COLMAX,
628: * used to handle NaN and Inf
629: *
630: ELSE IF( ( P.EQ.JMAX ).OR.( ROWMAX.LE.COLMAX ) ) THEN
631: *
632: * interchange rows and columns K+1 and IMAX,
633: * use 2-by-2 pivot block
634: *
635: KP = IMAX
636: KSTEP = 2
637: DONE = .TRUE.
638: ELSE
639: *
640: * Pivot NOT found, set variables and repeat
641: *
642: P = IMAX
643: COLMAX = ROWMAX
644: IMAX = JMAX
645: END IF
646: *
647: * End pivot search loop body
648: *
649: IF( .NOT. DONE ) GOTO 42
650: *
651: END IF
652: *
653: * Swap TWO rows and TWO columns
654: *
655: * First swap
656: *
657: IF( ( KSTEP.EQ.2 ) .AND. ( P.NE.K ) ) THEN
658: *
659: * Interchange rows and column K and P in the trailing
660: * submatrix A(k:n,k:n) if we have a 2-by-2 pivot
661: *
662: IF( P.LT.N )
663: $ CALL DSWAP( N-P, A( P+1, K ), 1, A( P+1, P ), 1 )
664: IF( P.GT.(K+1) )
665: $ CALL DSWAP( P-K-1, A( K+1, K ), 1, A( P, K+1 ), LDA )
666: T = A( K, K )
667: A( K, K ) = A( P, P )
668: A( P, P ) = T
669: END IF
670: *
671: * Second swap
672: *
673: KK = K + KSTEP - 1
674: IF( KP.NE.KK ) THEN
675: *
676: * Interchange rows and columns KK and KP in the trailing
677: * submatrix A(k:n,k:n)
678: *
679: IF( KP.LT.N )
680: $ CALL DSWAP( N-KP, A( KP+1, KK ), 1, A( KP+1, KP ), 1 )
681: IF( ( KK.LT.N ) .AND. ( KP.GT.(KK+1) ) )
682: $ CALL DSWAP( KP-KK-1, A( KK+1, KK ), 1, A( KP, KK+1 ),
683: $ LDA )
684: T = A( KK, KK )
685: A( KK, KK ) = A( KP, KP )
686: A( KP, KP ) = T
687: IF( KSTEP.EQ.2 ) THEN
688: T = A( K+1, K )
689: A( K+1, K ) = A( KP, K )
690: A( KP, K ) = T
691: END IF
692: END IF
693: *
694: * Update the trailing submatrix
695: *
696: IF( KSTEP.EQ.1 ) THEN
697: *
698: * 1-by-1 pivot block D(k): column k now holds
699: *
700: * W(k) = L(k)*D(k)
701: *
702: * where L(k) is the k-th column of L
703: *
704: IF( K.LT.N ) THEN
705: *
706: * Perform a rank-1 update of A(k+1:n,k+1:n) and
707: * store L(k) in column k
708: *
709: IF( ABS( A( K, K ) ).GE.SFMIN ) THEN
710: *
711: * Perform a rank-1 update of A(k+1:n,k+1:n) as
712: * A := A - L(k)*D(k)*L(k)**T
713: * = A - W(k)*(1/D(k))*W(k)**T
714: *
715: D11 = ONE / A( K, K )
716: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
717: $ A( K+1, K+1 ), LDA )
718: *
719: * Store L(k) in column k
720: *
721: CALL DSCAL( N-K, D11, A( K+1, K ), 1 )
722: ELSE
723: *
724: * Store L(k) in column k
725: *
726: D11 = A( K, K )
727: DO 46 II = K + 1, N
728: A( II, K ) = A( II, K ) / D11
729: 46 CONTINUE
730: *
731: * Perform a rank-1 update of A(k+1:n,k+1:n) as
732: * A := A - L(k)*D(k)*L(k)**T
733: * = A - W(k)*(1/D(k))*W(k)**T
734: * = A - (W(k)/D(k))*(D(k))*(W(k)/D(K))**T
735: *
736: CALL DSYR( UPLO, N-K, -D11, A( K+1, K ), 1,
737: $ A( K+1, K+1 ), LDA )
738: END IF
739: END IF
740: *
741: ELSE
742: *
743: * 2-by-2 pivot block D(k): columns k and k+1 now hold
744: *
745: * ( W(k) W(k+1) ) = ( L(k) L(k+1) )*D(k)
746: *
747: * where L(k) and L(k+1) are the k-th and (k+1)-th columns
748: * of L
749: *
750: *
751: * Perform a rank-2 update of A(k+2:n,k+2:n) as
752: *
753: * A := A - ( L(k) L(k+1) ) * D(k) * ( L(k) L(k+1) )**T
754: * = A - ( ( A(k)A(k+1) )*inv(D(k) ) * ( A(k)A(k+1) )**T
755: *
756: * and store L(k) and L(k+1) in columns k and k+1
757: *
758: IF( K.LT.N-1 ) THEN
759: *
760: D21 = A( K+1, K )
761: D11 = A( K+1, K+1 ) / D21
762: D22 = A( K, K ) / D21
763: T = ONE / ( D11*D22-ONE )
764: *
765: DO 60 J = K + 2, N
766: *
767: * Compute D21 * ( W(k)W(k+1) ) * inv(D(k)) for row J
768: *
769: WK = T*( D11*A( J, K )-A( J, K+1 ) )
770: WKP1 = T*( D22*A( J, K+1 )-A( J, K ) )
771: *
772: * Perform a rank-2 update of A(k+2:n,k+2:n)
773: *
774: DO 50 I = J, N
775: A( I, J ) = A( I, J ) - ( A( I, K ) / D21 )*WK -
776: $ ( A( I, K+1 ) / D21 )*WKP1
777: 50 CONTINUE
778: *
779: * Store L(k) and L(k+1) in cols k and k+1 for row J
780: *
781: A( J, K ) = WK / D21
782: A( J, K+1 ) = WKP1 / D21
783: *
784: 60 CONTINUE
785: *
786: END IF
787: *
788: END IF
789: END IF
790: *
791: * Store details of the interchanges in IPIV
792: *
793: IF( KSTEP.EQ.1 ) THEN
794: IPIV( K ) = KP
795: ELSE
796: IPIV( K ) = -P
797: IPIV( K+1 ) = -KP
798: END IF
799: *
800: * Increase K and return to the start of the main loop
801: *
802: K = K + KSTEP
803: GO TO 40
804: *
805: END IF
806: *
807: 70 CONTINUE
808: *
809: RETURN
810: *
811: * End of DSYTF2_ROOK
812: *
813: END
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