LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
dlarrf.f
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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
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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 *> \date June 2016
178 *
179 *> \ingroup OTHERauxiliary
180 *
181 *> \par Contributors:
182 * ==================
183 *>
184 *> Beresford Parlett, University of California, Berkeley, USA \n
185 *> Jim Demmel, University of California, Berkeley, USA \n
186 *> Inderjit Dhillon, University of Texas, Austin, USA \n
187 *> Osni Marques, LBNL/NERSC, USA \n
188 *> Christof Voemel, University of California, Berkeley, USA
189 *
190 * =====================================================================
191  SUBROUTINE dlarrf( N, D, L, LD, CLSTRT, CLEND,
192  $ W, WGAP, WERR,
193  $ SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA,
194  $ DPLUS, LPLUS, WORK, INFO )
195 *
196 * -- LAPACK auxiliary routine (version 3.7.1) --
197 * -- LAPACK is a software package provided by Univ. of Tennessee, --
198 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
199 * June 2016
200 *
201 * .. Scalar Arguments ..
202  INTEGER CLSTRT, CLEND, INFO, N
203  DOUBLE PRECISION CLGAPL, CLGAPR, PIVMIN, SIGMA, SPDIAM
204 * ..
205 * .. Array Arguments ..
206  DOUBLE PRECISION D( * ), DPLUS( * ), L( * ), LD( * ),
207  $ lplus( * ), w( * ), wgap( * ), werr( * ), work( * )
208 * ..
209 *
210 * =====================================================================
211 *
212 * .. Parameters ..
213  DOUBLE PRECISION FOUR, MAXGROWTH1, MAXGROWTH2, ONE, QUART, TWO
214  parameter( one = 1.0d0, two = 2.0d0, four = 4.0d0,
215  $ quart = 0.25d0,
216  $ maxgrowth1 = 8.d0,
217  $ maxgrowth2 = 8.d0 )
218 * ..
219 * .. Local Scalars ..
220  LOGICAL DORRR1, FORCER, NOFAIL, SAWNAN1, SAWNAN2, TRYRRR1
221  INTEGER I, INDX, KTRY, KTRYMAX, SLEFT, SRIGHT, SHIFT
222  parameter( ktrymax = 1, sleft = 1, sright = 2 )
223  DOUBLE PRECISION AVGAP, BESTSHIFT, CLWDTH, EPS, FACT, FAIL,
224  $ fail2, growthbound, ldelta, ldmax, lsigma,
225  $ max1, max2, mingap, oldp, prod, rdelta, rdmax,
226  $ rrr1, rrr2, rsigma, s, smlgrowth, tmp, znm2
227 * ..
228 * .. External Functions ..
229  LOGICAL DISNAN
230  DOUBLE PRECISION DLAMCH
231  EXTERNAL disnan, dlamch
232 * ..
233 * .. External Subroutines ..
234  EXTERNAL dcopy
235 * ..
236 * .. Intrinsic Functions ..
237  INTRINSIC abs
238 * ..
239 * .. Executable Statements ..
240 *
241  info = 0
242 *
243 * Quick return if possible
244 *
245  IF( n.LE.0 ) THEN
246  RETURN
247  END IF
248 *
249  fact = dble(2**ktrymax)
250  eps = dlamch( 'Precision' )
251  shift = 0
252  forcer = .false.
253 
254 
255 * Note that we cannot guarantee that for any of the shifts tried,
256 * the factorization has a small or even moderate element growth.
257 * There could be Ritz values at both ends of the cluster and despite
258 * backing off, there are examples where all factorizations tried
259 * (in IEEE mode, allowing zero pivots & infinities) have INFINITE
260 * element growth.
261 * For this reason, we should use PIVMIN in this subroutine so that at
262 * least the L D L^T factorization exists. It can be checked afterwards
263 * whether the element growth caused bad residuals/orthogonality.
264 
265 * Decide whether the code should accept the best among all
266 * representations despite large element growth or signal INFO=1
267 * Setting NOFAIL to .FALSE. for quick fix for bug 113
268  nofail = .false.
269 *
270 
271 * Compute the average gap length of the cluster
272  clwdth = abs(w(clend)-w(clstrt)) + werr(clend) + werr(clstrt)
273  avgap = clwdth / dble(clend-clstrt)
274  mingap = min(clgapl, clgapr)
275 * Initial values for shifts to both ends of cluster
276  lsigma = min(w( clstrt ),w( clend )) - werr( clstrt )
277  rsigma = max(w( clstrt ),w( clend )) + werr( clend )
278 
279 * Use a small fudge to make sure that we really shift to the outside
280  lsigma = lsigma - abs(lsigma)* four * eps
281  rsigma = rsigma + abs(rsigma)* four * eps
282 
283 * Compute upper bounds for how much to back off the initial shifts
284  ldmax = quart * mingap + two * pivmin
285  rdmax = quart * mingap + two * pivmin
286 
287  ldelta = max(avgap,wgap( clstrt ))/fact
288  rdelta = max(avgap,wgap( clend-1 ))/fact
289 *
290 * Initialize the record of the best representation found
291 *
292  s = dlamch( 'S' )
293  smlgrowth = one / s
294  fail = dble(n-1)*mingap/(spdiam*eps)
295  fail2 = dble(n-1)*mingap/(spdiam*sqrt(eps))
296  bestshift = lsigma
297 *
298 * while (KTRY <= KTRYMAX)
299  ktry = 0
300  growthbound = maxgrowth1*spdiam
301 
302  5 CONTINUE
303  sawnan1 = .false.
304  sawnan2 = .false.
305 * Ensure that we do not back off too much of the initial shifts
306  ldelta = min(ldmax,ldelta)
307  rdelta = min(rdmax,rdelta)
308 
309 * Compute the element growth when shifting to both ends of the cluster
310 * accept the shift if there is no element growth at one of the two ends
311 
312 * Left end
313  s = -lsigma
314  dplus( 1 ) = d( 1 ) + s
315  IF(abs(dplus(1)).LT.pivmin) THEN
316  dplus(1) = -pivmin
317 * Need to set SAWNAN1 because refined RRR test should not be used
318 * in this case
319  sawnan1 = .true.
320  ENDIF
321  max1 = abs( dplus( 1 ) )
322  DO 6 i = 1, n - 1
323  lplus( i ) = ld( i ) / dplus( i )
324  s = s*lplus( i )*l( i ) - lsigma
325  dplus( i+1 ) = d( i+1 ) + s
326  IF(abs(dplus(i+1)).LT.pivmin) THEN
327  dplus(i+1) = -pivmin
328 * Need to set SAWNAN1 because refined RRR test should not be used
329 * in this case
330  sawnan1 = .true.
331  ENDIF
332  max1 = max( max1,abs(dplus(i+1)) )
333  6 CONTINUE
334  sawnan1 = sawnan1 .OR. disnan( max1 )
335 
336  IF( forcer .OR.
337  $ (max1.LE.growthbound .AND. .NOT.sawnan1 ) ) THEN
338  sigma = lsigma
339  shift = sleft
340  GOTO 100
341  ENDIF
342 
343 * Right end
344  s = -rsigma
345  work( 1 ) = d( 1 ) + s
346  IF(abs(work(1)).LT.pivmin) THEN
347  work(1) = -pivmin
348 * Need to set SAWNAN2 because refined RRR test should not be used
349 * in this case
350  sawnan2 = .true.
351  ENDIF
352  max2 = abs( work( 1 ) )
353  DO 7 i = 1, n - 1
354  work( n+i ) = ld( i ) / work( i )
355  s = s*work( n+i )*l( i ) - rsigma
356  work( i+1 ) = d( i+1 ) + s
357  IF(abs(work(i+1)).LT.pivmin) THEN
358  work(i+1) = -pivmin
359 * Need to set SAWNAN2 because refined RRR test should not be used
360 * in this case
361  sawnan2 = .true.
362  ENDIF
363  max2 = max( max2,abs(work(i+1)) )
364  7 CONTINUE
365  sawnan2 = sawnan2 .OR. disnan( max2 )
366 
367  IF( forcer .OR.
368  $ (max2.LE.growthbound .AND. .NOT.sawnan2 ) ) THEN
369  sigma = rsigma
370  shift = sright
371  GOTO 100
372  ENDIF
373 * If we are at this point, both shifts led to too much element growth
374 
375 * Record the better of the two shifts (provided it didn't lead to NaN)
376  IF(sawnan1.AND.sawnan2) THEN
377 * both MAX1 and MAX2 are NaN
378  GOTO 50
379  ELSE
380  IF( .NOT.sawnan1 ) THEN
381  indx = 1
382  IF(max1.LE.smlgrowth) THEN
383  smlgrowth = max1
384  bestshift = lsigma
385  ENDIF
386  ENDIF
387  IF( .NOT.sawnan2 ) THEN
388  IF(sawnan1 .OR. max2.LE.max1) indx = 2
389  IF(max2.LE.smlgrowth) THEN
390  smlgrowth = max2
391  bestshift = rsigma
392  ENDIF
393  ENDIF
394  ENDIF
395 
396 * If we are here, both the left and the right shift led to
397 * element growth. If the element growth is moderate, then
398 * we may still accept the representation, if it passes a
399 * refined test for RRR. This test supposes that no NaN occurred.
400 * Moreover, we use the refined RRR test only for isolated clusters.
401  IF((clwdth.LT.mingap/dble(128)) .AND.
402  $ (min(max1,max2).LT.fail2)
403  $ .AND.(.NOT.sawnan1).AND.(.NOT.sawnan2)) THEN
404  dorrr1 = .true.
405  ELSE
406  dorrr1 = .false.
407  ENDIF
408  tryrrr1 = .true.
409  IF( tryrrr1 .AND. dorrr1 ) THEN
410  IF(indx.EQ.1) THEN
411  tmp = abs( dplus( n ) )
412  znm2 = one
413  prod = one
414  oldp = one
415  DO 15 i = n-1, 1, -1
416  IF( prod .LE. eps ) THEN
417  prod =
418  $ ((dplus(i+1)*work(n+i+1))/(dplus(i)*work(n+i)))*oldp
419  ELSE
420  prod = prod*abs(work(n+i))
421  END IF
422  oldp = prod
423  znm2 = znm2 + prod**2
424  tmp = max( tmp, abs( dplus( i ) * prod ))
425  15 CONTINUE
426  rrr1 = tmp/( spdiam * sqrt( znm2 ) )
427  IF (rrr1.LE.maxgrowth2) THEN
428  sigma = lsigma
429  shift = sleft
430  GOTO 100
431  ENDIF
432  ELSE IF(indx.EQ.2) THEN
433  tmp = abs( work( n ) )
434  znm2 = one
435  prod = one
436  oldp = one
437  DO 16 i = n-1, 1, -1
438  IF( prod .LE. eps ) THEN
439  prod = ((work(i+1)*lplus(i+1))/(work(i)*lplus(i)))*oldp
440  ELSE
441  prod = prod*abs(lplus(i))
442  END IF
443  oldp = prod
444  znm2 = znm2 + prod**2
445  tmp = max( tmp, abs( work( i ) * prod ))
446  16 CONTINUE
447  rrr2 = tmp/( spdiam * sqrt( znm2 ) )
448  IF (rrr2.LE.maxgrowth2) THEN
449  sigma = rsigma
450  shift = sright
451  GOTO 100
452  ENDIF
453  END IF
454  ENDIF
455 
456  50 CONTINUE
457 
458  IF (ktry.LT.ktrymax) THEN
459 * If we are here, both shifts failed also the RRR test.
460 * Back off to the outside
461  lsigma = max( lsigma - ldelta,
462  $ lsigma - ldmax)
463  rsigma = min( rsigma + rdelta,
464  $ rsigma + rdmax )
465  ldelta = two * ldelta
466  rdelta = two * rdelta
467  ktry = ktry + 1
468  GOTO 5
469  ELSE
470 * None of the representations investigated satisfied our
471 * criteria. Take the best one we found.
472  IF((smlgrowth.LT.fail).OR.nofail) THEN
473  lsigma = bestshift
474  rsigma = bestshift
475  forcer = .true.
476  GOTO 5
477  ELSE
478  info = 1
479  RETURN
480  ENDIF
481  END IF
482 
483  100 CONTINUE
484  IF (shift.EQ.sleft) THEN
485  ELSEIF (shift.EQ.sright) THEN
486 * store new L and D back into DPLUS, LPLUS
487  CALL dcopy( n, work, 1, dplus, 1 )
488  CALL dcopy( n-1, work(n+1), 1, lplus, 1 )
489  ENDIF
490 
491  RETURN
492 *
493 * End of DLARRF
494 *
495  END
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84
subroutine dlarrf(N, D, L, LD, CLSTRT, CLEND, W, WGAP, WERR, SPDIAM, CLGAPL, CLGAPR, PIVMIN, SIGMA, DPLUS, LPLUS, WORK, INFO)
DLARRF finds a new relatively robust representation such that at least one of the eigenvalues is rela...
Definition: dlarrf.f:195