1: *> \brief \b DLARRE given the tridiagonal matrix T, sets small off-diagonal elements to zero and for each unreduced block Ti, finds base representations and eigenvalues.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLARRE + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarre.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarre.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarre.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2,
22: * RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M,
23: * W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN,
24: * WORK, IWORK, INFO )
25: *
26: * .. Scalar Arguments ..
27: * CHARACTER RANGE
28: * INTEGER IL, INFO, IU, M, N, NSPLIT
29: * DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
30: * ..
31: * .. Array Arguments ..
32: * INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ),
33: * $ INDEXW( * )
34: * DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ),
35: * $ W( * ),WERR( * ), WGAP( * ), WORK( * )
36: * ..
37: *
38: *
39: *> \par Purpose:
40: * =============
41: *>
42: *> \verbatim
43: *>
44: *> To find the desired eigenvalues of a given real symmetric
45: *> tridiagonal matrix T, DLARRE sets any "small" off-diagonal
46: *> elements to zero, and for each unreduced block T_i, it finds
47: *> (a) a suitable shift at one end of the block's spectrum,
48: *> (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
49: *> (c) eigenvalues of each L_i D_i L_i^T.
50: *> The representations and eigenvalues found are then used by
51: *> DSTEMR to compute the eigenvectors of T.
52: *> The accuracy varies depending on whether bisection is used to
53: *> find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
54: *> conpute all and then discard any unwanted one.
55: *> As an added benefit, DLARRE also outputs the n
56: *> Gerschgorin intervals for the matrices L_i D_i L_i^T.
57: *> \endverbatim
58: *
59: * Arguments:
60: * ==========
61: *
62: *> \param[in] RANGE
63: *> \verbatim
64: *> RANGE is CHARACTER*1
65: *> = 'A': ("All") all eigenvalues will be found.
66: *> = 'V': ("Value") all eigenvalues in the half-open interval
67: *> (VL, VU] will be found.
68: *> = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
69: *> entire matrix) will be found.
70: *> \endverbatim
71: *>
72: *> \param[in] N
73: *> \verbatim
74: *> N is INTEGER
75: *> The order of the matrix. N > 0.
76: *> \endverbatim
77: *>
78: *> \param[in,out] VL
79: *> \verbatim
80: *> VL is DOUBLE PRECISION
81: *> If RANGE='V', the lower bound for the eigenvalues.
82: *> Eigenvalues less than or equal to VL, or greater than VU,
83: *> will not be returned. VL < VU.
84: *> If RANGE='I' or ='A', DLARRE computes bounds on the desired
85: *> part of the spectrum.
86: *> \endverbatim
87: *>
88: *> \param[in,out] VU
89: *> \verbatim
90: *> VU is DOUBLE PRECISION
91: *> If RANGE='V', the upper bound for the eigenvalues.
92: *> Eigenvalues less than or equal to VL, or greater than VU,
93: *> will not be returned. VL < VU.
94: *> If RANGE='I' or ='A', DLARRE computes bounds on the desired
95: *> part of the spectrum.
96: *> \endverbatim
97: *>
98: *> \param[in] IL
99: *> \verbatim
100: *> IL is INTEGER
101: *> If RANGE='I', the index of the
102: *> smallest eigenvalue to be returned.
103: *> 1 <= IL <= IU <= N.
104: *> \endverbatim
105: *>
106: *> \param[in] IU
107: *> \verbatim
108: *> IU is INTEGER
109: *> If RANGE='I', the index of the
110: *> largest eigenvalue to be returned.
111: *> 1 <= IL <= IU <= N.
112: *> \endverbatim
113: *>
114: *> \param[in,out] D
115: *> \verbatim
116: *> D is DOUBLE PRECISION array, dimension (N)
117: *> On entry, the N diagonal elements of the tridiagonal
118: *> matrix T.
119: *> On exit, the N diagonal elements of the diagonal
120: *> matrices D_i.
121: *> \endverbatim
122: *>
123: *> \param[in,out] E
124: *> \verbatim
125: *> E is DOUBLE PRECISION array, dimension (N)
126: *> On entry, the first (N-1) entries contain the subdiagonal
127: *> elements of the tridiagonal matrix T; E(N) need not be set.
128: *> On exit, E contains the subdiagonal elements of the unit
129: *> bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
130: *> 1 <= I <= NSPLIT, contain the base points sigma_i on output.
131: *> \endverbatim
132: *>
133: *> \param[in,out] E2
134: *> \verbatim
135: *> E2 is DOUBLE PRECISION array, dimension (N)
136: *> On entry, the first (N-1) entries contain the SQUARES of the
137: *> subdiagonal elements of the tridiagonal matrix T;
138: *> E2(N) need not be set.
139: *> On exit, the entries E2( ISPLIT( I ) ),
140: *> 1 <= I <= NSPLIT, have been set to zero
141: *> \endverbatim
142: *>
143: *> \param[in] RTOL1
144: *> \verbatim
145: *> RTOL1 is DOUBLE PRECISION
146: *> \endverbatim
147: *>
148: *> \param[in] RTOL2
149: *> \verbatim
150: *> RTOL2 is DOUBLE PRECISION
151: *> Parameters for bisection.
152: *> An interval [LEFT,RIGHT] has converged if
153: *> RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
154: *> \endverbatim
155: *>
156: *> \param[in] SPLTOL
157: *> \verbatim
158: *> SPLTOL is DOUBLE PRECISION
159: *> The threshold for splitting.
160: *> \endverbatim
161: *>
162: *> \param[out] NSPLIT
163: *> \verbatim
164: *> NSPLIT is INTEGER
165: *> The number of blocks T splits into. 1 <= NSPLIT <= N.
166: *> \endverbatim
167: *>
168: *> \param[out] ISPLIT
169: *> \verbatim
170: *> ISPLIT is INTEGER array, dimension (N)
171: *> The splitting points, at which T breaks up into blocks.
172: *> The first block consists of rows/columns 1 to ISPLIT(1),
173: *> the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
174: *> etc., and the NSPLIT-th consists of rows/columns
175: *> ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
176: *> \endverbatim
177: *>
178: *> \param[out] M
179: *> \verbatim
180: *> M is INTEGER
181: *> The total number of eigenvalues (of all L_i D_i L_i^T)
182: *> found.
183: *> \endverbatim
184: *>
185: *> \param[out] W
186: *> \verbatim
187: *> W is DOUBLE PRECISION array, dimension (N)
188: *> The first M elements contain the eigenvalues. The
189: *> eigenvalues of each of the blocks, L_i D_i L_i^T, are
190: *> sorted in ascending order ( DLARRE may use the
191: *> remaining N-M elements as workspace).
192: *> \endverbatim
193: *>
194: *> \param[out] WERR
195: *> \verbatim
196: *> WERR is DOUBLE PRECISION array, dimension (N)
197: *> The error bound on the corresponding eigenvalue in W.
198: *> \endverbatim
199: *>
200: *> \param[out] WGAP
201: *> \verbatim
202: *> WGAP is DOUBLE PRECISION array, dimension (N)
203: *> The separation from the right neighbor eigenvalue in W.
204: *> The gap is only with respect to the eigenvalues of the same block
205: *> as each block has its own representation tree.
206: *> Exception: at the right end of a block we store the left gap
207: *> \endverbatim
208: *>
209: *> \param[out] IBLOCK
210: *> \verbatim
211: *> IBLOCK is INTEGER array, dimension (N)
212: *> The indices of the blocks (submatrices) associated with the
213: *> corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
214: *> W(i) belongs to the first block from the top, =2 if W(i)
215: *> belongs to the second block, etc.
216: *> \endverbatim
217: *>
218: *> \param[out] INDEXW
219: *> \verbatim
220: *> INDEXW is INTEGER array, dimension (N)
221: *> The indices of the eigenvalues within each block (submatrix);
222: *> for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
223: *> i-th eigenvalue W(i) is the 10-th eigenvalue in block 2
224: *> \endverbatim
225: *>
226: *> \param[out] GERS
227: *> \verbatim
228: *> GERS is DOUBLE PRECISION array, dimension (2*N)
229: *> The N Gerschgorin intervals (the i-th Gerschgorin interval
230: *> is (GERS(2*i-1), GERS(2*i)).
231: *> \endverbatim
232: *>
233: *> \param[out] PIVMIN
234: *> \verbatim
235: *> PIVMIN is DOUBLE PRECISION
236: *> The minimum pivot in the Sturm sequence for T.
237: *> \endverbatim
238: *>
239: *> \param[out] WORK
240: *> \verbatim
241: *> WORK is DOUBLE PRECISION array, dimension (6*N)
242: *> Workspace.
243: *> \endverbatim
244: *>
245: *> \param[out] IWORK
246: *> \verbatim
247: *> IWORK is INTEGER array, dimension (5*N)
248: *> Workspace.
249: *> \endverbatim
250: *>
251: *> \param[out] INFO
252: *> \verbatim
253: *> INFO is INTEGER
254: *> = 0: successful exit
255: *> > 0: A problem occurred in DLARRE.
256: *> < 0: One of the called subroutines signaled an internal problem.
257: *> Needs inspection of the corresponding parameter IINFO
258: *> for further information.
259: *>
260: *> =-1: Problem in DLARRD.
261: *> = 2: No base representation could be found in MAXTRY iterations.
262: *> Increasing MAXTRY and recompilation might be a remedy.
263: *> =-3: Problem in DLARRB when computing the refined root
264: *> representation for DLASQ2.
265: *> =-4: Problem in DLARRB when preforming bisection on the
266: *> desired part of the spectrum.
267: *> =-5: Problem in DLASQ2.
268: *> =-6: Problem in DLASQ2.
269: *> \endverbatim
270: *
271: * Authors:
272: * ========
273: *
274: *> \author Univ. of Tennessee
275: *> \author Univ. of California Berkeley
276: *> \author Univ. of Colorado Denver
277: *> \author NAG Ltd.
278: *
279: *> \date June 2016
280: *
281: *> \ingroup OTHERauxiliary
282: *
283: *> \par Further Details:
284: * =====================
285: *>
286: *> \verbatim
287: *>
288: *> The base representations are required to suffer very little
289: *> element growth and consequently define all their eigenvalues to
290: *> high relative accuracy.
291: *> \endverbatim
292: *
293: *> \par Contributors:
294: * ==================
295: *>
296: *> Beresford Parlett, University of California, Berkeley, USA \n
297: *> Jim Demmel, University of California, Berkeley, USA \n
298: *> Inderjit Dhillon, University of Texas, Austin, USA \n
299: *> Osni Marques, LBNL/NERSC, USA \n
300: *> Christof Voemel, University of California, Berkeley, USA \n
301: *>
302: * =====================================================================
303: SUBROUTINE DLARRE( RANGE, N, VL, VU, IL, IU, D, E, E2,
304: $ RTOL1, RTOL2, SPLTOL, NSPLIT, ISPLIT, M,
305: $ W, WERR, WGAP, IBLOCK, INDEXW, GERS, PIVMIN,
306: $ WORK, IWORK, INFO )
307: *
308: * -- LAPACK auxiliary routine (version 3.8.0) --
309: * -- LAPACK is a software package provided by Univ. of Tennessee, --
310: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
311: * June 2016
312: *
313: * .. Scalar Arguments ..
314: CHARACTER RANGE
315: INTEGER IL, INFO, IU, M, N, NSPLIT
316: DOUBLE PRECISION PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU
317: * ..
318: * .. Array Arguments ..
319: INTEGER IBLOCK( * ), ISPLIT( * ), IWORK( * ),
320: $ INDEXW( * )
321: DOUBLE PRECISION D( * ), E( * ), E2( * ), GERS( * ),
322: $ W( * ),WERR( * ), WGAP( * ), WORK( * )
323: * ..
324: *
325: * =====================================================================
326: *
327: * .. Parameters ..
328: DOUBLE PRECISION FAC, FOUR, FOURTH, FUDGE, HALF, HNDRD,
329: $ MAXGROWTH, ONE, PERT, TWO, ZERO
330: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
331: $ TWO = 2.0D0, FOUR=4.0D0,
332: $ HNDRD = 100.0D0,
333: $ PERT = 8.0D0,
334: $ HALF = ONE/TWO, FOURTH = ONE/FOUR, FAC= HALF,
335: $ MAXGROWTH = 64.0D0, FUDGE = 2.0D0 )
336: INTEGER MAXTRY, ALLRNG, INDRNG, VALRNG
337: PARAMETER ( MAXTRY = 6, ALLRNG = 1, INDRNG = 2,
338: $ VALRNG = 3 )
339: * ..
340: * .. Local Scalars ..
341: LOGICAL FORCEB, NOREP, USEDQD
342: INTEGER CNT, CNT1, CNT2, I, IBEGIN, IDUM, IEND, IINFO,
343: $ IN, INDL, INDU, IRANGE, J, JBLK, MB, MM,
344: $ WBEGIN, WEND
345: DOUBLE PRECISION AVGAP, BSRTOL, CLWDTH, DMAX, DPIVOT, EABS,
346: $ EMAX, EOLD, EPS, GL, GU, ISLEFT, ISRGHT, RTL,
347: $ RTOL, S1, S2, SAFMIN, SGNDEF, SIGMA, SPDIAM,
348: $ TAU, TMP, TMP1
349:
350:
351: * ..
352: * .. Local Arrays ..
353: INTEGER ISEED( 4 )
354: * ..
355: * .. External Functions ..
356: LOGICAL LSAME
357: DOUBLE PRECISION DLAMCH
358: EXTERNAL DLAMCH, LSAME
359:
360: * ..
361: * .. External Subroutines ..
362: EXTERNAL DCOPY, DLARNV, DLARRA, DLARRB, DLARRC, DLARRD,
363: $ DLASQ2, DLARRK
364: * ..
365: * .. Intrinsic Functions ..
366: INTRINSIC ABS, MAX, MIN
367:
368: * ..
369: * .. Executable Statements ..
370: *
371:
372: INFO = 0
373: *
374: * Quick return if possible
375: *
376: IF( N.LE.0 ) THEN
377: RETURN
378: END IF
379: *
380: * Decode RANGE
381: *
382: IF( LSAME( RANGE, 'A' ) ) THEN
383: IRANGE = ALLRNG
384: ELSE IF( LSAME( RANGE, 'V' ) ) THEN
385: IRANGE = VALRNG
386: ELSE IF( LSAME( RANGE, 'I' ) ) THEN
387: IRANGE = INDRNG
388: END IF
389:
390: M = 0
391:
392: * Get machine constants
393: SAFMIN = DLAMCH( 'S' )
394: EPS = DLAMCH( 'P' )
395:
396: * Set parameters
397: RTL = SQRT(EPS)
398: BSRTOL = SQRT(EPS)
399:
400: * Treat case of 1x1 matrix for quick return
401: IF( N.EQ.1 ) THEN
402: IF( (IRANGE.EQ.ALLRNG).OR.
403: $ ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR.
404: $ ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN
405: M = 1
406: W(1) = D(1)
407: * The computation error of the eigenvalue is zero
408: WERR(1) = ZERO
409: WGAP(1) = ZERO
410: IBLOCK( 1 ) = 1
411: INDEXW( 1 ) = 1
412: GERS(1) = D( 1 )
413: GERS(2) = D( 1 )
414: ENDIF
415: * store the shift for the initial RRR, which is zero in this case
416: E(1) = ZERO
417: RETURN
418: END IF
419:
420: * General case: tridiagonal matrix of order > 1
421: *
422: * Init WERR, WGAP. Compute Gerschgorin intervals and spectral diameter.
423: * Compute maximum off-diagonal entry and pivmin.
424: GL = D(1)
425: GU = D(1)
426: EOLD = ZERO
427: EMAX = ZERO
428: E(N) = ZERO
429: DO 5 I = 1,N
430: WERR(I) = ZERO
431: WGAP(I) = ZERO
432: EABS = ABS( E(I) )
433: IF( EABS .GE. EMAX ) THEN
434: EMAX = EABS
435: END IF
436: TMP1 = EABS + EOLD
437: GERS( 2*I-1) = D(I) - TMP1
438: GL = MIN( GL, GERS( 2*I - 1))
439: GERS( 2*I ) = D(I) + TMP1
440: GU = MAX( GU, GERS(2*I) )
441: EOLD = EABS
442: 5 CONTINUE
443: * The minimum pivot allowed in the Sturm sequence for T
444: PIVMIN = SAFMIN * MAX( ONE, EMAX**2 )
445: * Compute spectral diameter. The Gerschgorin bounds give an
446: * estimate that is wrong by at most a factor of SQRT(2)
447: SPDIAM = GU - GL
448:
449: * Compute splitting points
450: CALL DLARRA( N, D, E, E2, SPLTOL, SPDIAM,
451: $ NSPLIT, ISPLIT, IINFO )
452:
453: * Can force use of bisection instead of faster DQDS.
454: * Option left in the code for future multisection work.
455: FORCEB = .FALSE.
456:
457: * Initialize USEDQD, DQDS should be used for ALLRNG unless someone
458: * explicitly wants bisection.
459: USEDQD = (( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB))
460:
461: IF( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ) THEN
462: * Set interval [VL,VU] that contains all eigenvalues
463: VL = GL
464: VU = GU
465: ELSE
466: * We call DLARRD to find crude approximations to the eigenvalues
467: * in the desired range. In case IRANGE = INDRNG, we also obtain the
468: * interval (VL,VU] that contains all the wanted eigenvalues.
469: * An interval [LEFT,RIGHT] has converged if
470: * RIGHT-LEFT.LT.RTOL*MAX(ABS(LEFT),ABS(RIGHT))
471: * DLARRD needs a WORK of size 4*N, IWORK of size 3*N
472: CALL DLARRD( RANGE, 'B', N, VL, VU, IL, IU, GERS,
473: $ BSRTOL, D, E, E2, PIVMIN, NSPLIT, ISPLIT,
474: $ MM, W, WERR, VL, VU, IBLOCK, INDEXW,
475: $ WORK, IWORK, IINFO )
476: IF( IINFO.NE.0 ) THEN
477: INFO = -1
478: RETURN
479: ENDIF
480: * Make sure that the entries M+1 to N in W, WERR, IBLOCK, INDEXW are 0
481: DO 14 I = MM+1,N
482: W( I ) = ZERO
483: WERR( I ) = ZERO
484: IBLOCK( I ) = 0
485: INDEXW( I ) = 0
486: 14 CONTINUE
487: END IF
488:
489:
490: ***
491: * Loop over unreduced blocks
492: IBEGIN = 1
493: WBEGIN = 1
494: DO 170 JBLK = 1, NSPLIT
495: IEND = ISPLIT( JBLK )
496: IN = IEND - IBEGIN + 1
497:
498: * 1 X 1 block
499: IF( IN.EQ.1 ) THEN
500: IF( (IRANGE.EQ.ALLRNG).OR.( (IRANGE.EQ.VALRNG).AND.
501: $ ( D( IBEGIN ).GT.VL ).AND.( D( IBEGIN ).LE.VU ) )
502: $ .OR. ( (IRANGE.EQ.INDRNG).AND.(IBLOCK(WBEGIN).EQ.JBLK))
503: $ ) THEN
504: M = M + 1
505: W( M ) = D( IBEGIN )
506: WERR(M) = ZERO
507: * The gap for a single block doesn't matter for the later
508: * algorithm and is assigned an arbitrary large value
509: WGAP(M) = ZERO
510: IBLOCK( M ) = JBLK
511: INDEXW( M ) = 1
512: WBEGIN = WBEGIN + 1
513: ENDIF
514: * E( IEND ) holds the shift for the initial RRR
515: E( IEND ) = ZERO
516: IBEGIN = IEND + 1
517: GO TO 170
518: END IF
519: *
520: * Blocks of size larger than 1x1
521: *
522: * E( IEND ) will hold the shift for the initial RRR, for now set it =0
523: E( IEND ) = ZERO
524: *
525: * Find local outer bounds GL,GU for the block
526: GL = D(IBEGIN)
527: GU = D(IBEGIN)
528: DO 15 I = IBEGIN , IEND
529: GL = MIN( GERS( 2*I-1 ), GL )
530: GU = MAX( GERS( 2*I ), GU )
531: 15 CONTINUE
532: SPDIAM = GU - GL
533:
534: IF(.NOT. ((IRANGE.EQ.ALLRNG).AND.(.NOT.FORCEB)) ) THEN
535: * Count the number of eigenvalues in the current block.
536: MB = 0
537: DO 20 I = WBEGIN,MM
538: IF( IBLOCK(I).EQ.JBLK ) THEN
539: MB = MB+1
540: ELSE
541: GOTO 21
542: ENDIF
543: 20 CONTINUE
544: 21 CONTINUE
545:
546: IF( MB.EQ.0) THEN
547: * No eigenvalue in the current block lies in the desired range
548: * E( IEND ) holds the shift for the initial RRR
549: E( IEND ) = ZERO
550: IBEGIN = IEND + 1
551: GO TO 170
552: ELSE
553:
554: * Decide whether dqds or bisection is more efficient
555: USEDQD = ( (MB .GT. FAC*IN) .AND. (.NOT.FORCEB) )
556: WEND = WBEGIN + MB - 1
557: * Calculate gaps for the current block
558: * In later stages, when representations for individual
559: * eigenvalues are different, we use SIGMA = E( IEND ).
560: SIGMA = ZERO
561: DO 30 I = WBEGIN, WEND - 1
562: WGAP( I ) = MAX( ZERO,
563: $ W(I+1)-WERR(I+1) - (W(I)+WERR(I)) )
564: 30 CONTINUE
565: WGAP( WEND ) = MAX( ZERO,
566: $ VU - SIGMA - (W( WEND )+WERR( WEND )))
567: * Find local index of the first and last desired evalue.
568: INDL = INDEXW(WBEGIN)
569: INDU = INDEXW( WEND )
570: ENDIF
571: ENDIF
572: IF(( (IRANGE.EQ.ALLRNG) .AND. (.NOT. FORCEB) ).OR.USEDQD) THEN
573: * Case of DQDS
574: * Find approximations to the extremal eigenvalues of the block
575: CALL DLARRK( IN, 1, GL, GU, D(IBEGIN),
576: $ E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO )
577: IF( IINFO.NE.0 ) THEN
578: INFO = -1
579: RETURN
580: ENDIF
581: ISLEFT = MAX(GL, TMP - TMP1
582: $ - HNDRD * EPS* ABS(TMP - TMP1))
583:
584: CALL DLARRK( IN, IN, GL, GU, D(IBEGIN),
585: $ E2(IBEGIN), PIVMIN, RTL, TMP, TMP1, IINFO )
586: IF( IINFO.NE.0 ) THEN
587: INFO = -1
588: RETURN
589: ENDIF
590: ISRGHT = MIN(GU, TMP + TMP1
591: $ + HNDRD * EPS * ABS(TMP + TMP1))
592: * Improve the estimate of the spectral diameter
593: SPDIAM = ISRGHT - ISLEFT
594: ELSE
595: * Case of bisection
596: * Find approximations to the wanted extremal eigenvalues
597: ISLEFT = MAX(GL, W(WBEGIN) - WERR(WBEGIN)
598: $ - HNDRD * EPS*ABS(W(WBEGIN)- WERR(WBEGIN) ))
599: ISRGHT = MIN(GU,W(WEND) + WERR(WEND)
600: $ + HNDRD * EPS * ABS(W(WEND)+ WERR(WEND)))
601: ENDIF
602:
603:
604: * Decide whether the base representation for the current block
605: * L_JBLK D_JBLK L_JBLK^T = T_JBLK - sigma_JBLK I
606: * should be on the left or the right end of the current block.
607: * The strategy is to shift to the end which is "more populated"
608: * Furthermore, decide whether to use DQDS for the computation of
609: * the eigenvalue approximations at the end of DLARRE or bisection.
610: * dqds is chosen if all eigenvalues are desired or the number of
611: * eigenvalues to be computed is large compared to the blocksize.
612: IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
613: * If all the eigenvalues have to be computed, we use dqd
614: USEDQD = .TRUE.
615: * INDL is the local index of the first eigenvalue to compute
616: INDL = 1
617: INDU = IN
618: * MB = number of eigenvalues to compute
619: MB = IN
620: WEND = WBEGIN + MB - 1
621: * Define 1/4 and 3/4 points of the spectrum
622: S1 = ISLEFT + FOURTH * SPDIAM
623: S2 = ISRGHT - FOURTH * SPDIAM
624: ELSE
625: * DLARRD has computed IBLOCK and INDEXW for each eigenvalue
626: * approximation.
627: * choose sigma
628: IF( USEDQD ) THEN
629: S1 = ISLEFT + FOURTH * SPDIAM
630: S2 = ISRGHT - FOURTH * SPDIAM
631: ELSE
632: TMP = MIN(ISRGHT,VU) - MAX(ISLEFT,VL)
633: S1 = MAX(ISLEFT,VL) + FOURTH * TMP
634: S2 = MIN(ISRGHT,VU) - FOURTH * TMP
635: ENDIF
636: ENDIF
637:
638: * Compute the negcount at the 1/4 and 3/4 points
639: IF(MB.GT.1) THEN
640: CALL DLARRC( 'T', IN, S1, S2, D(IBEGIN),
641: $ E(IBEGIN), PIVMIN, CNT, CNT1, CNT2, IINFO)
642: ENDIF
643:
644: IF(MB.EQ.1) THEN
645: SIGMA = GL
646: SGNDEF = ONE
647: ELSEIF( CNT1 - INDL .GE. INDU - CNT2 ) THEN
648: IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
649: SIGMA = MAX(ISLEFT,GL)
650: ELSEIF( USEDQD ) THEN
651: * use Gerschgorin bound as shift to get pos def matrix
652: * for dqds
653: SIGMA = ISLEFT
654: ELSE
655: * use approximation of the first desired eigenvalue of the
656: * block as shift
657: SIGMA = MAX(ISLEFT,VL)
658: ENDIF
659: SGNDEF = ONE
660: ELSE
661: IF( ( IRANGE.EQ.ALLRNG ) .AND. (.NOT.FORCEB) ) THEN
662: SIGMA = MIN(ISRGHT,GU)
663: ELSEIF( USEDQD ) THEN
664: * use Gerschgorin bound as shift to get neg def matrix
665: * for dqds
666: SIGMA = ISRGHT
667: ELSE
668: * use approximation of the first desired eigenvalue of the
669: * block as shift
670: SIGMA = MIN(ISRGHT,VU)
671: ENDIF
672: SGNDEF = -ONE
673: ENDIF
674:
675:
676: * An initial SIGMA has been chosen that will be used for computing
677: * T - SIGMA I = L D L^T
678: * Define the increment TAU of the shift in case the initial shift
679: * needs to be refined to obtain a factorization with not too much
680: * element growth.
681: IF( USEDQD ) THEN
682: * The initial SIGMA was to the outer end of the spectrum
683: * the matrix is definite and we need not retreat.
684: TAU = SPDIAM*EPS*N + TWO*PIVMIN
685: TAU = MAX( TAU,TWO*EPS*ABS(SIGMA) )
686: ELSE
687: IF(MB.GT.1) THEN
688: CLWDTH = W(WEND) + WERR(WEND) - W(WBEGIN) - WERR(WBEGIN)
689: AVGAP = ABS(CLWDTH / DBLE(WEND-WBEGIN))
690: IF( SGNDEF.EQ.ONE ) THEN
691: TAU = HALF*MAX(WGAP(WBEGIN),AVGAP)
692: TAU = MAX(TAU,WERR(WBEGIN))
693: ELSE
694: TAU = HALF*MAX(WGAP(WEND-1),AVGAP)
695: TAU = MAX(TAU,WERR(WEND))
696: ENDIF
697: ELSE
698: TAU = WERR(WBEGIN)
699: ENDIF
700: ENDIF
701: *
702: DO 80 IDUM = 1, MAXTRY
703: * Compute L D L^T factorization of tridiagonal matrix T - sigma I.
704: * Store D in WORK(1:IN), L in WORK(IN+1:2*IN), and reciprocals of
705: * pivots in WORK(2*IN+1:3*IN)
706: DPIVOT = D( IBEGIN ) - SIGMA
707: WORK( 1 ) = DPIVOT
708: DMAX = ABS( WORK(1) )
709: J = IBEGIN
710: DO 70 I = 1, IN - 1
711: WORK( 2*IN+I ) = ONE / WORK( I )
712: TMP = E( J )*WORK( 2*IN+I )
713: WORK( IN+I ) = TMP
714: DPIVOT = ( D( J+1 )-SIGMA ) - TMP*E( J )
715: WORK( I+1 ) = DPIVOT
716: DMAX = MAX( DMAX, ABS(DPIVOT) )
717: J = J + 1
718: 70 CONTINUE
719: * check for element growth
720: IF( DMAX .GT. MAXGROWTH*SPDIAM ) THEN
721: NOREP = .TRUE.
722: ELSE
723: NOREP = .FALSE.
724: ENDIF
725: IF( USEDQD .AND. .NOT.NOREP ) THEN
726: * Ensure the definiteness of the representation
727: * All entries of D (of L D L^T) must have the same sign
728: DO 71 I = 1, IN
729: TMP = SGNDEF*WORK( I )
730: IF( TMP.LT.ZERO ) NOREP = .TRUE.
731: 71 CONTINUE
732: ENDIF
733: IF(NOREP) THEN
734: * Note that in the case of IRANGE=ALLRNG, we use the Gerschgorin
735: * shift which makes the matrix definite. So we should end up
736: * here really only in the case of IRANGE = VALRNG or INDRNG.
737: IF( IDUM.EQ.MAXTRY-1 ) THEN
738: IF( SGNDEF.EQ.ONE ) THEN
739: * The fudged Gerschgorin shift should succeed
740: SIGMA =
741: $ GL - FUDGE*SPDIAM*EPS*N - FUDGE*TWO*PIVMIN
742: ELSE
743: SIGMA =
744: $ GU + FUDGE*SPDIAM*EPS*N + FUDGE*TWO*PIVMIN
745: END IF
746: ELSE
747: SIGMA = SIGMA - SGNDEF * TAU
748: TAU = TWO * TAU
749: END IF
750: ELSE
751: * an initial RRR is found
752: GO TO 83
753: END IF
754: 80 CONTINUE
755: * if the program reaches this point, no base representation could be
756: * found in MAXTRY iterations.
757: INFO = 2
758: RETURN
759:
760: 83 CONTINUE
761: * At this point, we have found an initial base representation
762: * T - SIGMA I = L D L^T with not too much element growth.
763: * Store the shift.
764: E( IEND ) = SIGMA
765: * Store D and L.
766: CALL DCOPY( IN, WORK, 1, D( IBEGIN ), 1 )
767: CALL DCOPY( IN-1, WORK( IN+1 ), 1, E( IBEGIN ), 1 )
768:
769:
770: IF(MB.GT.1 ) THEN
771: *
772: * Perturb each entry of the base representation by a small
773: * (but random) relative amount to overcome difficulties with
774: * glued matrices.
775: *
776: DO 122 I = 1, 4
777: ISEED( I ) = 1
778: 122 CONTINUE
779:
780: CALL DLARNV(2, ISEED, 2*IN-1, WORK(1))
781: DO 125 I = 1,IN-1
782: D(IBEGIN+I-1) = D(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(I))
783: E(IBEGIN+I-1) = E(IBEGIN+I-1)*(ONE+EPS*PERT*WORK(IN+I))
784: 125 CONTINUE
785: D(IEND) = D(IEND)*(ONE+EPS*FOUR*WORK(IN))
786: *
787: ENDIF
788: *
789: * Don't update the Gerschgorin intervals because keeping track
790: * of the updates would be too much work in DLARRV.
791: * We update W instead and use it to locate the proper Gerschgorin
792: * intervals.
793:
794: * Compute the required eigenvalues of L D L' by bisection or dqds
795: IF ( .NOT.USEDQD ) THEN
796: * If DLARRD has been used, shift the eigenvalue approximations
797: * according to their representation. This is necessary for
798: * a uniform DLARRV since dqds computes eigenvalues of the
799: * shifted representation. In DLARRV, W will always hold the
800: * UNshifted eigenvalue approximation.
801: DO 134 J=WBEGIN,WEND
802: W(J) = W(J) - SIGMA
803: WERR(J) = WERR(J) + ABS(W(J)) * EPS
804: 134 CONTINUE
805: * call DLARRB to reduce eigenvalue error of the approximations
806: * from DLARRD
807: DO 135 I = IBEGIN, IEND-1
808: WORK( I ) = D( I ) * E( I )**2
809: 135 CONTINUE
810: * use bisection to find EV from INDL to INDU
811: CALL DLARRB(IN, D(IBEGIN), WORK(IBEGIN),
812: $ INDL, INDU, RTOL1, RTOL2, INDL-1,
813: $ W(WBEGIN), WGAP(WBEGIN), WERR(WBEGIN),
814: $ WORK( 2*N+1 ), IWORK, PIVMIN, SPDIAM,
815: $ IN, IINFO )
816: IF( IINFO .NE. 0 ) THEN
817: INFO = -4
818: RETURN
819: END IF
820: * DLARRB computes all gaps correctly except for the last one
821: * Record distance to VU/GU
822: WGAP( WEND ) = MAX( ZERO,
823: $ ( VU-SIGMA ) - ( W( WEND ) + WERR( WEND ) ) )
824: DO 138 I = INDL, INDU
825: M = M + 1
826: IBLOCK(M) = JBLK
827: INDEXW(M) = I
828: 138 CONTINUE
829: ELSE
830: * Call dqds to get all eigs (and then possibly delete unwanted
831: * eigenvalues).
832: * Note that dqds finds the eigenvalues of the L D L^T representation
833: * of T to high relative accuracy. High relative accuracy
834: * might be lost when the shift of the RRR is subtracted to obtain
835: * the eigenvalues of T. However, T is not guaranteed to define its
836: * eigenvalues to high relative accuracy anyway.
837: * Set RTOL to the order of the tolerance used in DLASQ2
838: * This is an ESTIMATED error, the worst case bound is 4*N*EPS
839: * which is usually too large and requires unnecessary work to be
840: * done by bisection when computing the eigenvectors
841: RTOL = LOG(DBLE(IN)) * FOUR * EPS
842: J = IBEGIN
843: DO 140 I = 1, IN - 1
844: WORK( 2*I-1 ) = ABS( D( J ) )
845: WORK( 2*I ) = E( J )*E( J )*WORK( 2*I-1 )
846: J = J + 1
847: 140 CONTINUE
848: WORK( 2*IN-1 ) = ABS( D( IEND ) )
849: WORK( 2*IN ) = ZERO
850: CALL DLASQ2( IN, WORK, IINFO )
851: IF( IINFO .NE. 0 ) THEN
852: * If IINFO = -5 then an index is part of a tight cluster
853: * and should be changed. The index is in IWORK(1) and the
854: * gap is in WORK(N+1)
855: INFO = -5
856: RETURN
857: ELSE
858: * Test that all eigenvalues are positive as expected
859: DO 149 I = 1, IN
860: IF( WORK( I ).LT.ZERO ) THEN
861: INFO = -6
862: RETURN
863: ENDIF
864: 149 CONTINUE
865: END IF
866: IF( SGNDEF.GT.ZERO ) THEN
867: DO 150 I = INDL, INDU
868: M = M + 1
869: W( M ) = WORK( IN-I+1 )
870: IBLOCK( M ) = JBLK
871: INDEXW( M ) = I
872: 150 CONTINUE
873: ELSE
874: DO 160 I = INDL, INDU
875: M = M + 1
876: W( M ) = -WORK( I )
877: IBLOCK( M ) = JBLK
878: INDEXW( M ) = I
879: 160 CONTINUE
880: END IF
881:
882: DO 165 I = M - MB + 1, M
883: * the value of RTOL below should be the tolerance in DLASQ2
884: WERR( I ) = RTOL * ABS( W(I) )
885: 165 CONTINUE
886: DO 166 I = M - MB + 1, M - 1
887: * compute the right gap between the intervals
888: WGAP( I ) = MAX( ZERO,
889: $ W(I+1)-WERR(I+1) - (W(I)+WERR(I)) )
890: 166 CONTINUE
891: WGAP( M ) = MAX( ZERO,
892: $ ( VU-SIGMA ) - ( W( M ) + WERR( M ) ) )
893: END IF
894: * proceed with next block
895: IBEGIN = IEND + 1
896: WBEGIN = WEND + 1
897: 170 CONTINUE
898: *
899:
900: RETURN
901: *
902: * end of DLARRE
903: *
904: END
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