1: *> \brief \b DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is relatively isolated.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLARRF + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarrf.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarrf.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarrf.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
22: * W, WGAP, WERR,
23: * SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
24: * DPLUS, LPLUS, WORK, INFO )
25: *
26: * .. Scalar Arguments ..
27: * INTEGER CLSTRT, CLEND, INFO, N
28: * DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
29: * ..
30: * .. Array Arguments ..
31: * DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ),
32: * $ LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> Given the initial representation L D L^T and its cluster of close
42: *> eigenvalues (in a relative measure), W( CLSTRT ), W( CLSTRT+1 ), ...
43: *> W( CLEND ), DLARRF finds a new relatively robust representation
44: *> L D L^T - SIGMA I = L(+) D(+) L(+)^T such that at least one of the
45: *> eigenvalues of L(+) D(+) L(+)^T is relatively isolated.
46: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] N
52: *> \verbatim
53: *> N is INTEGER
54: *> The order of the matrix (subblock, if the matrix split).
55: *> \endverbatim
56: *>
57: *> \param[in] D
58: *> \verbatim
59: *> D is DOUBLE PRECISION array, dimension (N)
60: *> The N diagonal elements of the diagonal matrix D.
61: *> \endverbatim
62: *>
63: *> \param[in] L
64: *> \verbatim
65: *> L is DOUBLE PRECISION array, dimension (N-1)
66: *> The (N-1) subdiagonal elements of the unit bidiagonal
67: *> matrix L.
68: *> \endverbatim
69: *>
70: *> \param[in] LD
71: *> \verbatim
72: *> LD is DOUBLE PRECISION array, dimension (N-1)
73: *> The (N-1) elements L(i)*D(i).
74: *> \endverbatim
75: *>
76: *> \param[in] CLSTRT
77: *> \verbatim
78: *> CLSTRT is INTEGER
79: *> The index of the first eigenvalue in the cluster.
80: *> \endverbatim
81: *>
82: *> \param[in] CLEND
83: *> \verbatim
84: *> CLEND is INTEGER
85: *> The index of the last eigenvalue in the cluster.
86: *> \endverbatim
87: *>
88: *> \param[in] W
89: *> \verbatim
90: *> W is DOUBLE PRECISION array, dimension
91: *> dimension is >= (CLEND-CLSTRT+1)
92: *> The eigenvalue APPROXIMATIONS of L D L^T in ascending order.
93: *> W( CLSTRT ) through W( CLEND ) form the cluster of relatively
94: *> close eigenalues.
95: *> \endverbatim
96: *>
97: *> \param[in,out] WGAP
98: *> \verbatim
99: *> WGAP is DOUBLE PRECISION array, dimension
100: *> dimension is >= (CLEND-CLSTRT+1)
101: *> The separation from the right neighbor eigenvalue in W.
102: *> \endverbatim
103: *>
104: *> \param[in] WERR
105: *> \verbatim
106: *> WERR is DOUBLE PRECISION array, dimension
107: *> dimension is >= (CLEND-CLSTRT+1)
108: *> WERR contain the semiwidth of the uncertainty
109: *> interval of the corresponding eigenvalue APPROXIMATION in W
110: *> \endverbatim
111: *>
112: *> \param[in] SPDIAM
113: *> \verbatim
114: *> SPDIAM is DOUBLE PRECISION
115: *> estimate of the spectral diameter obtained from the
116: *> Gerschgorin intervals
117: *> \endverbatim
118: *>
119: *> \param[in] CLGAPL
120: *> \verbatim
121: *> CLGAPL is DOUBLE PRECISION
122: *> \endverbatim
123: *>
124: *> \param[in] CLGAPR
125: *> \verbatim
126: *> CLGAPR is DOUBLE PRECISION
127: *> absolute gap on each end of the cluster.
128: *> Set by the calling routine to protect against shifts too close
129: *> to eigenvalues outside the cluster.
130: *> \endverbatim
131: *>
132: *> \param[in] PIVMIN
133: *> \verbatim
134: *> PIVMIN is DOUBLE PRECISION
135: *> The minimum pivot allowed in the Sturm sequence.
136: *> \endverbatim
137: *>
138: *> \param[out] SIGMA
139: *> \verbatim
140: *> SIGMA is DOUBLE PRECISION
141: *> The shift used to form L(+) D(+) L(+)^T.
142: *> \endverbatim
143: *>
144: *> \param[out] DPLUS
145: *> \verbatim
146: *> DPLUS is DOUBLE PRECISION array, dimension (N)
147: *> The N diagonal elements of the diagonal matrix D(+).
148: *> \endverbatim
149: *>
150: *> \param[out] LPLUS
151: *> \verbatim
152: *> LPLUS is DOUBLE PRECISION array, dimension (N-1)
153: *> The first (N-1) elements of LPLUS contain the subdiagonal
154: *> elements of the unit bidiagonal matrix L(+).
155: *> \endverbatim
156: *>
157: *> \param[out] WORK
158: *> \verbatim
159: *> WORK is DOUBLE PRECISION array, dimension (2*N)
160: *> Workspace.
161: *> \endverbatim
162: *>
163: *> \param[out] INFO
164: *> \verbatim
165: *> INFO is INTEGER
166: *> Signals processing OK (=0) or failure (=1)
167: *> \endverbatim
168: *
169: * Authors:
170: * ========
171: *
172: *> \author Univ. of Tennessee
173: *> \author Univ. of California Berkeley
174: *> \author Univ. of Colorado Denver
175: *> \author NAG Ltd.
176: *
177: *> \ingroup OTHERauxiliary
178: *
179: *> \par Contributors:
180: * ==================
181: *>
182: *> Beresford Parlett, University of California, Berkeley, USA \n
183: *> Jim Demmel, University of California, Berkeley, USA \n
184: *> Inderjit Dhillon, University of Texas, Austin, USA \n
185: *> Osni Marques, LBNL/NERSC, USA \n
186: *> Christof Voemel, University of California, Berkeley, USA
187: *
188: * =====================================================================
189: SUBROUTINE DLARRF( N, D, L, LD, CLSTRT, CLEND,
190: $ W, WGAP, WERR,
191: $ SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
192: $ DPLUS, LPLUS, WORK, INFO )
193: *
194: * -- LAPACK auxiliary routine --
195: * -- LAPACK is a software package provided by Univ. of Tennessee, --
196: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
197: *
198: * .. Scalar Arguments ..
199: INTEGER CLSTRT, CLEND, INFO, N
200: DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
201: * ..
202: * .. Array Arguments ..
203: DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ),
204: $ LPLUS( * ), W( * ), WGAP( * ), WERR( * ), WORK( * )
205: * ..
206: *
207: * =====================================================================
208: *
209: * .. Parameters ..
210: DOUBLE PRECISION FOUR, MAXGROWTH1, MAXGROWTH2, ONE, QUART, TWO
211: PARAMETER ( ONE = 1.0D0, TWO = 2.0D0, FOUR = 4.0D0,
212: $ QUART = 0.25D0,
213: $ MAXGROWTH1 = 8.D0,
214: $ MAXGROWTH2 = 8.D0 )
215: * ..
216: * .. Local Scalars ..
217: LOGICAL DORRR1, FORCER, NOFAIL, SAWNAN1, SAWNAN2, TRYRRR1
218: INTEGER I, INDX, KTRY, KTRYMAX, SLEFT, SRIGHT, SHIFT
219: PARAMETER ( KTRYMAX = 1, SLEFT = 1, SRIGHT = 2 )
220: DOUBLE PRECISION AVGAP, BESTSHIFT, CLWDTH, EPS, FACT, FAIL,
221: $ FAIL2, GROWTHBOUND, LDELTA, LDMAX, LSIGMA,
222: $ MAX1, MAX2, MINGAP, OLDP, PROD, RDELTA, RDMAX,
223: $ RRR1, RRR2, RSIGMA, S, SMLGROWTH, TMP, ZNM2
224: * ..
225: * .. External Functions ..
226: LOGICAL DISNAN
227: DOUBLE PRECISION DLAMCH
228: EXTERNAL DISNAN, DLAMCH
229: * ..
230: * .. External Subroutines ..
231: EXTERNAL DCOPY
232: * ..
233: * .. Intrinsic Functions ..
234: INTRINSIC ABS
235: * ..
236: * .. Executable Statements ..
237: *
238: INFO = 0
239: *
240: * Quick return if possible
241: *
242: IF( N.LE.0 ) THEN
243: RETURN
244: END IF
245: *
246: FACT = DBLE(2**KTRYMAX)
247: EPS = DLAMCH( 'Precision' )
248: SHIFT = 0
249: FORCER = .FALSE.
250:
251:
252: * Note that we cannot guarantee that for any of the shifts tried,
253: * the factorization has a small or even moderate element growth.
254: * There could be Ritz values at both ends of the cluster and despite
255: * backing off, there are examples where all factorizations tried
256: * (in IEEE mode, allowing zero pivots & infinities) have INFINITE
257: * element growth.
258: * For this reason, we should use PIVMIN in this subroutine so that at
259: * least the L D L^T factorization exists. It can be checked afterwards
260: * whether the element growth caused bad residuals/orthogonality.
261:
262: * Decide whether the code should accept the best among all
263: * representations despite large element growth or signal INFO=1
264: * Setting NOFAIL to .FALSE. for quick fix for bug 113
265: NOFAIL = .FALSE.
266: *
267:
268: * Compute the average gap length of the cluster
269: CLWDTH = ABS(W(CLEND)-W(CLSTRT)) + WERR(CLEND) + WERR(CLSTRT)
270: AVGAP = CLWDTH / DBLE(CLEND-CLSTRT)
271: MINGAP = MIN(CLGAPL, CLGAPR)
272: * Initial values for shifts to both ends of cluster
273: LSIGMA = MIN(W( CLSTRT ),W( CLEND )) - WERR( CLSTRT )
274: RSIGMA = MAX(W( CLSTRT ),W( CLEND )) + WERR( CLEND )
275:
276: * Use a small fudge to make sure that we really shift to the outside
277: LSIGMA = LSIGMA - ABS(LSIGMA)* FOUR * EPS
278: RSIGMA = RSIGMA + ABS(RSIGMA)* FOUR * EPS
279:
280: * Compute upper bounds for how much to back off the initial shifts
281: LDMAX = QUART * MINGAP + TWO * PIVMIN
282: RDMAX = QUART * MINGAP + TWO * PIVMIN
283:
284: LDELTA = MAX(AVGAP,WGAP( CLSTRT ))/FACT
285: RDELTA = MAX(AVGAP,WGAP( CLEND-1 ))/FACT
286: *
287: * Initialize the record of the best representation found
288: *
289: S = DLAMCH( 'S' )
290: SMLGROWTH = ONE / S
291: FAIL = DBLE(N-1)*MINGAP/(SPDIAM*EPS)
292: FAIL2 = DBLE(N-1)*MINGAP/(SPDIAM*SQRT(EPS))
293: BESTSHIFT = LSIGMA
294: *
295: * while (KTRY <= KTRYMAX)
296: KTRY = 0
297: GROWTHBOUND = MAXGROWTH1*SPDIAM
298:
299: 5 CONTINUE
300: SAWNAN1 = .FALSE.
301: SAWNAN2 = .FALSE.
302: * Ensure that we do not back off too much of the initial shifts
303: LDELTA = MIN(LDMAX,LDELTA)
304: RDELTA = MIN(RDMAX,RDELTA)
305:
306: * Compute the element growth when shifting to both ends of the cluster
307: * accept the shift if there is no element growth at one of the two ends
308:
309: * Left end
310: S = -LSIGMA
311: DPLUS( 1 ) = D( 1 ) + S
312: IF(ABS(DPLUS(1)).LT.PIVMIN) THEN
313: DPLUS(1) = -PIVMIN
314: * Need to set SAWNAN1 because refined RRR test should not be used
315: * in this case
316: SAWNAN1 = .TRUE.
317: ENDIF
318: MAX1 = ABS( DPLUS( 1 ) )
319: DO 6 I = 1, N - 1
320: LPLUS( I ) = LD( I ) / DPLUS( I )
321: S = S*LPLUS( I )*L( I ) - LSIGMA
322: DPLUS( I+1 ) = D( I+1 ) + S
323: IF(ABS(DPLUS(I+1)).LT.PIVMIN) THEN
324: DPLUS(I+1) = -PIVMIN
325: * Need to set SAWNAN1 because refined RRR test should not be used
326: * in this case
327: SAWNAN1 = .TRUE.
328: ENDIF
329: MAX1 = MAX( MAX1,ABS(DPLUS(I+1)) )
330: 6 CONTINUE
331: SAWNAN1 = SAWNAN1 .OR. DISNAN( MAX1 )
332:
333: IF( FORCER .OR.
334: $ (MAX1.LE.GROWTHBOUND .AND. .NOT.SAWNAN1 ) ) THEN
335: SIGMA = LSIGMA
336: SHIFT = SLEFT
337: GOTO 100
338: ENDIF
339:
340: * Right end
341: S = -RSIGMA
342: WORK( 1 ) = D( 1 ) + S
343: IF(ABS(WORK(1)).LT.PIVMIN) THEN
344: WORK(1) = -PIVMIN
345: * Need to set SAWNAN2 because refined RRR test should not be used
346: * in this case
347: SAWNAN2 = .TRUE.
348: ENDIF
349: MAX2 = ABS( WORK( 1 ) )
350: DO 7 I = 1, N - 1
351: WORK( N+I ) = LD( I ) / WORK( I )
352: S = S*WORK( N+I )*L( I ) - RSIGMA
353: WORK( I+1 ) = D( I+1 ) + S
354: IF(ABS(WORK(I+1)).LT.PIVMIN) THEN
355: WORK(I+1) = -PIVMIN
356: * Need to set SAWNAN2 because refined RRR test should not be used
357: * in this case
358: SAWNAN2 = .TRUE.
359: ENDIF
360: MAX2 = MAX( MAX2,ABS(WORK(I+1)) )
361: 7 CONTINUE
362: SAWNAN2 = SAWNAN2 .OR. DISNAN( MAX2 )
363:
364: IF( FORCER .OR.
365: $ (MAX2.LE.GROWTHBOUND .AND. .NOT.SAWNAN2 ) ) THEN
366: SIGMA = RSIGMA
367: SHIFT = SRIGHT
368: GOTO 100
369: ENDIF
370: * If we are at this point, both shifts led to too much element growth
371:
372: * Record the better of the two shifts (provided it didn't lead to NaN)
373: IF(SAWNAN1.AND.SAWNAN2) THEN
374: * both MAX1 and MAX2 are NaN
375: GOTO 50
376: ELSE
377: IF( .NOT.SAWNAN1 ) THEN
378: INDX = 1
379: IF(MAX1.LE.SMLGROWTH) THEN
380: SMLGROWTH = MAX1
381: BESTSHIFT = LSIGMA
382: ENDIF
383: ENDIF
384: IF( .NOT.SAWNAN2 ) THEN
385: IF(SAWNAN1 .OR. MAX2.LE.MAX1) INDX = 2
386: IF(MAX2.LE.SMLGROWTH) THEN
387: SMLGROWTH = MAX2
388: BESTSHIFT = RSIGMA
389: ENDIF
390: ENDIF
391: ENDIF
392:
393: * If we are here, both the left and the right shift led to
394: * element growth. If the element growth is moderate, then
395: * we may still accept the representation, if it passes a
396: * refined test for RRR. This test supposes that no NaN occurred.
397: * Moreover, we use the refined RRR test only for isolated clusters.
398: IF((CLWDTH.LT.MINGAP/DBLE(128)) .AND.
399: $ (MIN(MAX1,MAX2).LT.FAIL2)
400: $ .AND.(.NOT.SAWNAN1).AND.(.NOT.SAWNAN2)) THEN
401: DORRR1 = .TRUE.
402: ELSE
403: DORRR1 = .FALSE.
404: ENDIF
405: TRYRRR1 = .TRUE.
406: IF( TRYRRR1 .AND. DORRR1 ) THEN
407: IF(INDX.EQ.1) THEN
408: TMP = ABS( DPLUS( N ) )
409: ZNM2 = ONE
410: PROD = ONE
411: OLDP = ONE
412: DO 15 I = N-1, 1, -1
413: IF( PROD .LE. EPS ) THEN
414: PROD =
415: $ ((DPLUS(I+1)*WORK(N+I+1))/(DPLUS(I)*WORK(N+I)))*OLDP
416: ELSE
417: PROD = PROD*ABS(WORK(N+I))
418: END IF
419: OLDP = PROD
420: ZNM2 = ZNM2 + PROD**2
421: TMP = MAX( TMP, ABS( DPLUS( I ) * PROD ))
422: 15 CONTINUE
423: RRR1 = TMP/( SPDIAM * SQRT( ZNM2 ) )
424: IF (RRR1.LE.MAXGROWTH2) THEN
425: SIGMA = LSIGMA
426: SHIFT = SLEFT
427: GOTO 100
428: ENDIF
429: ELSE IF(INDX.EQ.2) THEN
430: TMP = ABS( WORK( N ) )
431: ZNM2 = ONE
432: PROD = ONE
433: OLDP = ONE
434: DO 16 I = N-1, 1, -1
435: IF( PROD .LE. EPS ) THEN
436: PROD = ((WORK(I+1)*LPLUS(I+1))/(WORK(I)*LPLUS(I)))*OLDP
437: ELSE
438: PROD = PROD*ABS(LPLUS(I))
439: END IF
440: OLDP = PROD
441: ZNM2 = ZNM2 + PROD**2
442: TMP = MAX( TMP, ABS( WORK( I ) * PROD ))
443: 16 CONTINUE
444: RRR2 = TMP/( SPDIAM * SQRT( ZNM2 ) )
445: IF (RRR2.LE.MAXGROWTH2) THEN
446: SIGMA = RSIGMA
447: SHIFT = SRIGHT
448: GOTO 100
449: ENDIF
450: END IF
451: ENDIF
452:
453: 50 CONTINUE
454:
455: IF (KTRY.LT.KTRYMAX) THEN
456: * If we are here, both shifts failed also the RRR test.
457: * Back off to the outside
458: LSIGMA = MAX( LSIGMA - LDELTA,
459: $ LSIGMA - LDMAX)
460: RSIGMA = MIN( RSIGMA + RDELTA,
461: $ RSIGMA + RDMAX )
462: LDELTA = TWO * LDELTA
463: RDELTA = TWO * RDELTA
464: KTRY = KTRY + 1
465: GOTO 5
466: ELSE
467: * None of the representations investigated satisfied our
468: * criteria. Take the best one we found.
469: IF((SMLGROWTH.LT.FAIL).OR.NOFAIL) THEN
470: LSIGMA = BESTSHIFT
471: RSIGMA = BESTSHIFT
472: FORCER = .TRUE.
473: GOTO 5
474: ELSE
475: INFO = 1
476: RETURN
477: ENDIF
478: END IF
479:
480: 100 CONTINUE
481: IF (SHIFT.EQ.SLEFT) THEN
482: ELSEIF (SHIFT.EQ.SRIGHT) THEN
483: * store new L and D back into DPLUS, LPLUS
484: CALL DCOPY( N, WORK, 1, DPLUS, 1 )
485: CALL DCOPY( N-1, WORK(N+1), 1, LPLUS, 1 )
486: ENDIF
487:
488: RETURN
489: *
490: * End of DLARRF
491: *
492: END
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