1: *> \brief \b ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZLARRV + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlarrv.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZLARRV( N, VL, VU, D, L, PIVMIN,
22: * ISPLIT, M, DOL, DOU, MINRGP,
23: * RTOL1, RTOL2, W, WERR, WGAP,
24: * IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
25: * WORK, IWORK, INFO )
26: *
27: * .. Scalar Arguments ..
28: * INTEGER DOL, DOU, INFO, LDZ, M, N
29: * DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
30: * ..
31: * .. Array Arguments ..
32: * INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
33: * $ ISUPPZ( * ), IWORK( * )
34: * DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
35: * $ WGAP( * ), WORK( * )
36: * COMPLEX*16 Z( LDZ, * )
37: * ..
38: *
39: *
40: *> \par Purpose:
41: * =============
42: *>
43: *> \verbatim
44: *>
45: *> ZLARRV computes the eigenvectors of the tridiagonal matrix
46: *> T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
47: *> The input eigenvalues should have been computed by DLARRE.
48: *> \endverbatim
49: *
50: * Arguments:
51: * ==========
52: *
53: *> \param[in] N
54: *> \verbatim
55: *> N is INTEGER
56: *> The order of the matrix. N >= 0.
57: *> \endverbatim
58: *>
59: *> \param[in] VL
60: *> \verbatim
61: *> VL is DOUBLE PRECISION
62: *> Lower bound of the interval that contains the desired
63: *> eigenvalues. VL < VU. Needed to compute gaps on the left or right
64: *> end of the extremal eigenvalues in the desired RANGE.
65: *> \endverbatim
66: *>
67: *> \param[in] VU
68: *> \verbatim
69: *> VU is DOUBLE PRECISION
70: *> Upper bound of the interval that contains the desired
71: *> eigenvalues. VL < VU. Needed to compute gaps on the left or right
72: *> end of the extremal eigenvalues in the desired RANGE.
73: *> \endverbatim
74: *>
75: *> \param[in,out] D
76: *> \verbatim
77: *> D is DOUBLE PRECISION array, dimension (N)
78: *> On entry, the N diagonal elements of the diagonal matrix D.
79: *> On exit, D may be overwritten.
80: *> \endverbatim
81: *>
82: *> \param[in,out] L
83: *> \verbatim
84: *> L is DOUBLE PRECISION array, dimension (N)
85: *> On entry, the (N-1) subdiagonal elements of the unit
86: *> bidiagonal matrix L are in elements 1 to N-1 of L
87: *> (if the matrix is not split.) At the end of each block
88: *> is stored the corresponding shift as given by DLARRE.
89: *> On exit, L is overwritten.
90: *> \endverbatim
91: *>
92: *> \param[in] PIVMIN
93: *> \verbatim
94: *> PIVMIN is DOUBLE PRECISION
95: *> The minimum pivot allowed in the Sturm sequence.
96: *> \endverbatim
97: *>
98: *> \param[in] ISPLIT
99: *> \verbatim
100: *> ISPLIT is INTEGER array, dimension (N)
101: *> The splitting points, at which T breaks up into blocks.
102: *> The first block consists of rows/columns 1 to
103: *> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
104: *> through ISPLIT( 2 ), etc.
105: *> \endverbatim
106: *>
107: *> \param[in] M
108: *> \verbatim
109: *> M is INTEGER
110: *> The total number of input eigenvalues. 0 <= M <= N.
111: *> \endverbatim
112: *>
113: *> \param[in] DOL
114: *> \verbatim
115: *> DOL is INTEGER
116: *> \endverbatim
117: *>
118: *> \param[in] DOU
119: *> \verbatim
120: *> DOU is INTEGER
121: *> If the user wants to compute only selected eigenvectors from all
122: *> the eigenvalues supplied, he can specify an index range DOL:DOU.
123: *> Or else the setting DOL=1, DOU=M should be applied.
124: *> Note that DOL and DOU refer to the order in which the eigenvalues
125: *> are stored in W.
126: *> If the user wants to compute only selected eigenpairs, then
127: *> the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
128: *> computed eigenvectors. All other columns of Z are set to zero.
129: *> \endverbatim
130: *>
131: *> \param[in] MINRGP
132: *> \verbatim
133: *> MINRGP is DOUBLE PRECISION
134: *> \endverbatim
135: *>
136: *> \param[in] RTOL1
137: *> \verbatim
138: *> RTOL1 is DOUBLE PRECISION
139: *> \endverbatim
140: *>
141: *> \param[in] RTOL2
142: *> \verbatim
143: *> RTOL2 is DOUBLE PRECISION
144: *> Parameters for bisection.
145: *> An interval [LEFT,RIGHT] has converged if
146: *> RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
147: *> \endverbatim
148: *>
149: *> \param[in,out] W
150: *> \verbatim
151: *> W is DOUBLE PRECISION array, dimension (N)
152: *> The first M elements of W contain the APPROXIMATE eigenvalues for
153: *> which eigenvectors are to be computed. The eigenvalues
154: *> should be grouped by split-off block and ordered from
155: *> smallest to largest within the block ( The output array
156: *> W from DLARRE is expected here ). Furthermore, they are with
157: *> respect to the shift of the corresponding root representation
158: *> for their block. On exit, W holds the eigenvalues of the
159: *> UNshifted matrix.
160: *> \endverbatim
161: *>
162: *> \param[in,out] WERR
163: *> \verbatim
164: *> WERR is DOUBLE PRECISION array, dimension (N)
165: *> The first M elements contain the semiwidth of the uncertainty
166: *> interval of the corresponding eigenvalue in W
167: *> \endverbatim
168: *>
169: *> \param[in,out] WGAP
170: *> \verbatim
171: *> WGAP is DOUBLE PRECISION array, dimension (N)
172: *> The separation from the right neighbor eigenvalue in W.
173: *> \endverbatim
174: *>
175: *> \param[in] IBLOCK
176: *> \verbatim
177: *> IBLOCK is INTEGER array, dimension (N)
178: *> The indices of the blocks (submatrices) associated with the
179: *> corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
180: *> W(i) belongs to the first block from the top, =2 if W(i)
181: *> belongs to the second block, etc.
182: *> \endverbatim
183: *>
184: *> \param[in] INDEXW
185: *> \verbatim
186: *> INDEXW is INTEGER array, dimension (N)
187: *> The indices of the eigenvalues within each block (submatrix);
188: *> for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
189: *> i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.
190: *> \endverbatim
191: *>
192: *> \param[in] GERS
193: *> \verbatim
194: *> GERS is DOUBLE PRECISION array, dimension (2*N)
195: *> The N Gerschgorin intervals (the i-th Gerschgorin interval
196: *> is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
197: *> be computed from the original UNshifted matrix.
198: *> \endverbatim
199: *>
200: *> \param[out] Z
201: *> \verbatim
202: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M) )
203: *> If INFO = 0, the first M columns of Z contain the
204: *> orthonormal eigenvectors of the matrix T
205: *> corresponding to the input eigenvalues, with the i-th
206: *> column of Z holding the eigenvector associated with W(i).
207: *> Note: the user must ensure that at least max(1,M) columns are
208: *> supplied in the array Z.
209: *> \endverbatim
210: *>
211: *> \param[in] LDZ
212: *> \verbatim
213: *> LDZ is INTEGER
214: *> The leading dimension of the array Z. LDZ >= 1, and if
215: *> JOBZ = 'V', LDZ >= max(1,N).
216: *> \endverbatim
217: *>
218: *> \param[out] ISUPPZ
219: *> \verbatim
220: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
221: *> The support of the eigenvectors in Z, i.e., the indices
222: *> indicating the nonzero elements in Z. The I-th eigenvector
223: *> is nonzero only in elements ISUPPZ( 2*I-1 ) through
224: *> ISUPPZ( 2*I ).
225: *> \endverbatim
226: *>
227: *> \param[out] WORK
228: *> \verbatim
229: *> WORK is DOUBLE PRECISION array, dimension (12*N)
230: *> \endverbatim
231: *>
232: *> \param[out] IWORK
233: *> \verbatim
234: *> IWORK is INTEGER array, dimension (7*N)
235: *> \endverbatim
236: *>
237: *> \param[out] INFO
238: *> \verbatim
239: *> INFO is INTEGER
240: *> = 0: successful exit
241: *>
242: *> > 0: A problem occurred in ZLARRV.
243: *> < 0: One of the called subroutines signaled an internal problem.
244: *> Needs inspection of the corresponding parameter IINFO
245: *> for further information.
246: *>
247: *> =-1: Problem in DLARRB when refining a child's eigenvalues.
248: *> =-2: Problem in DLARRF when computing the RRR of a child.
249: *> When a child is inside a tight cluster, it can be difficult
250: *> to find an RRR. A partial remedy from the user's point of
251: *> view is to make the parameter MINRGP smaller and recompile.
252: *> However, as the orthogonality of the computed vectors is
253: *> proportional to 1/MINRGP, the user should be aware that
254: *> he might be trading in precision when he decreases MINRGP.
255: *> =-3: Problem in DLARRB when refining a single eigenvalue
256: *> after the Rayleigh correction was rejected.
257: *> = 5: The Rayleigh Quotient Iteration failed to converge to
258: *> full accuracy in MAXITR steps.
259: *> \endverbatim
260: *
261: * Authors:
262: * ========
263: *
264: *> \author Univ. of Tennessee
265: *> \author Univ. of California Berkeley
266: *> \author Univ. of Colorado Denver
267: *> \author NAG Ltd.
268: *
269: *> \date June 2016
270: *
271: *> \ingroup complex16OTHERauxiliary
272: *
273: *> \par Contributors:
274: * ==================
275: *>
276: *> Beresford Parlett, University of California, Berkeley, USA \n
277: *> Jim Demmel, University of California, Berkeley, USA \n
278: *> Inderjit Dhillon, University of Texas, Austin, USA \n
279: *> Osni Marques, LBNL/NERSC, USA \n
280: *> Christof Voemel, University of California, Berkeley, USA
281: *
282: * =====================================================================
283: SUBROUTINE ZLARRV( N, VL, VU, D, L, PIVMIN,
284: $ ISPLIT, M, DOL, DOU, MINRGP,
285: $ RTOL1, RTOL2, W, WERR, WGAP,
286: $ IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
287: $ WORK, IWORK, INFO )
288: *
289: * -- LAPACK auxiliary routine (version 3.6.1) --
290: * -- LAPACK is a software package provided by Univ. of Tennessee, --
291: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
292: * June 2016
293: *
294: * .. Scalar Arguments ..
295: INTEGER DOL, DOU, INFO, LDZ, M, N
296: DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
297: * ..
298: * .. Array Arguments ..
299: INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
300: $ ISUPPZ( * ), IWORK( * )
301: DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
302: $ WGAP( * ), WORK( * )
303: COMPLEX*16 Z( LDZ, * )
304: * ..
305: *
306: * =====================================================================
307: *
308: * .. Parameters ..
309: INTEGER MAXITR
310: PARAMETER ( MAXITR = 10 )
311: COMPLEX*16 CZERO
312: PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ) )
313: DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, HALF
314: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
315: $ TWO = 2.0D0, THREE = 3.0D0,
316: $ FOUR = 4.0D0, HALF = 0.5D0)
317: * ..
318: * .. Local Scalars ..
319: LOGICAL ESKIP, NEEDBS, STP2II, TRYRQC, USEDBS, USEDRQ
320: INTEGER DONE, I, IBEGIN, IDONE, IEND, II, IINDC1,
321: $ IINDC2, IINDR, IINDWK, IINFO, IM, IN, INDEIG,
322: $ INDLD, INDLLD, INDWRK, ISUPMN, ISUPMX, ITER,
323: $ ITMP1, J, JBLK, K, MINIWSIZE, MINWSIZE, NCLUS,
324: $ NDEPTH, NEGCNT, NEWCLS, NEWFST, NEWFTT, NEWLST,
325: $ NEWSIZ, OFFSET, OLDCLS, OLDFST, OLDIEN, OLDLST,
326: $ OLDNCL, P, PARITY, Q, WBEGIN, WEND, WINDEX,
327: $ WINDMN, WINDPL, ZFROM, ZTO, ZUSEDL, ZUSEDU,
328: $ ZUSEDW
329: INTEGER INDIN1, INDIN2
330: DOUBLE PRECISION BSTRES, BSTW, EPS, FUDGE, GAP, GAPTOL, GL, GU,
331: $ LAMBDA, LEFT, LGAP, MINGMA, NRMINV, RESID,
332: $ RGAP, RIGHT, RQCORR, RQTOL, SAVGAP, SGNDEF,
333: $ SIGMA, SPDIAM, SSIGMA, TAU, TMP, TOL, ZTZ
334: * ..
335: * .. External Functions ..
336: DOUBLE PRECISION DLAMCH
337: EXTERNAL DLAMCH
338: * ..
339: * .. External Subroutines ..
340: EXTERNAL DCOPY, DLARRB, DLARRF, ZDSCAL, ZLAR1V,
341: $ ZLASET
342: * ..
343: * .. Intrinsic Functions ..
344: INTRINSIC ABS, DBLE, MAX, MIN
345: INTRINSIC DCMPLX
346: * ..
347: * .. Executable Statements ..
348: * ..
349:
350: INFO = 0
351: * The first N entries of WORK are reserved for the eigenvalues
352: INDLD = N+1
353: INDLLD= 2*N+1
354: INDIN1 = 3*N + 1
355: INDIN2 = 4*N + 1
356: INDWRK = 5*N + 1
357: MINWSIZE = 12 * N
358:
359: DO 5 I= 1,MINWSIZE
360: WORK( I ) = ZERO
361: 5 CONTINUE
362:
363: * IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
364: * factorization used to compute the FP vector
365: IINDR = 0
366: * IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
367: * layer and the one above.
368: IINDC1 = N
369: IINDC2 = 2*N
370: IINDWK = 3*N + 1
371:
372: MINIWSIZE = 7 * N
373: DO 10 I= 1,MINIWSIZE
374: IWORK( I ) = 0
375: 10 CONTINUE
376:
377: ZUSEDL = 1
378: IF(DOL.GT.1) THEN
379: * Set lower bound for use of Z
380: ZUSEDL = DOL-1
381: ENDIF
382: ZUSEDU = M
383: IF(DOU.LT.M) THEN
384: * Set lower bound for use of Z
385: ZUSEDU = DOU+1
386: ENDIF
387: * The width of the part of Z that is used
388: ZUSEDW = ZUSEDU - ZUSEDL + 1
389:
390:
391: CALL ZLASET( 'Full', N, ZUSEDW, CZERO, CZERO,
392: $ Z(1,ZUSEDL), LDZ )
393:
394: EPS = DLAMCH( 'Precision' )
395: RQTOL = TWO * EPS
396: *
397: * Set expert flags for standard code.
398: TRYRQC = .TRUE.
399:
400: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
401: ELSE
402: * Only selected eigenpairs are computed. Since the other evalues
403: * are not refined by RQ iteration, bisection has to compute to full
404: * accuracy.
405: RTOL1 = FOUR * EPS
406: RTOL2 = FOUR * EPS
407: ENDIF
408:
409: * The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
410: * desired eigenvalues. The support of the nonzero eigenvector
411: * entries is contained in the interval IBEGIN:IEND.
412: * Remark that if k eigenpairs are desired, then the eigenvectors
413: * are stored in k contiguous columns of Z.
414:
415: * DONE is the number of eigenvectors already computed
416: DONE = 0
417: IBEGIN = 1
418: WBEGIN = 1
419: DO 170 JBLK = 1, IBLOCK( M )
420: IEND = ISPLIT( JBLK )
421: SIGMA = L( IEND )
422: * Find the eigenvectors of the submatrix indexed IBEGIN
423: * through IEND.
424: WEND = WBEGIN - 1
425: 15 CONTINUE
426: IF( WEND.LT.M ) THEN
427: IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN
428: WEND = WEND + 1
429: GO TO 15
430: END IF
431: END IF
432: IF( WEND.LT.WBEGIN ) THEN
433: IBEGIN = IEND + 1
434: GO TO 170
435: ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN
436: IBEGIN = IEND + 1
437: WBEGIN = WEND + 1
438: GO TO 170
439: END IF
440:
441: * Find local spectral diameter of the block
442: GL = GERS( 2*IBEGIN-1 )
443: GU = GERS( 2*IBEGIN )
444: DO 20 I = IBEGIN+1 , IEND
445: GL = MIN( GERS( 2*I-1 ), GL )
446: GU = MAX( GERS( 2*I ), GU )
447: 20 CONTINUE
448: SPDIAM = GU - GL
449:
450: * OLDIEN is the last index of the previous block
451: OLDIEN = IBEGIN - 1
452: * Calculate the size of the current block
453: IN = IEND - IBEGIN + 1
454: * The number of eigenvalues in the current block
455: IM = WEND - WBEGIN + 1
456:
457: * This is for a 1x1 block
458: IF( IBEGIN.EQ.IEND ) THEN
459: DONE = DONE+1
460: Z( IBEGIN, WBEGIN ) = DCMPLX( ONE, ZERO )
461: ISUPPZ( 2*WBEGIN-1 ) = IBEGIN
462: ISUPPZ( 2*WBEGIN ) = IBEGIN
463: W( WBEGIN ) = W( WBEGIN ) + SIGMA
464: WORK( WBEGIN ) = W( WBEGIN )
465: IBEGIN = IEND + 1
466: WBEGIN = WBEGIN + 1
467: GO TO 170
468: END IF
469:
470: * The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
471: * Note that these can be approximations, in this case, the corresp.
472: * entries of WERR give the size of the uncertainty interval.
473: * The eigenvalue approximations will be refined when necessary as
474: * high relative accuracy is required for the computation of the
475: * corresponding eigenvectors.
476: CALL DCOPY( IM, W( WBEGIN ), 1,
477: $ WORK( WBEGIN ), 1 )
478:
479: * We store in W the eigenvalue approximations w.r.t. the original
480: * matrix T.
481: DO 30 I=1,IM
482: W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA
483: 30 CONTINUE
484:
485:
486: * NDEPTH is the current depth of the representation tree
487: NDEPTH = 0
488: * PARITY is either 1 or 0
489: PARITY = 1
490: * NCLUS is the number of clusters for the next level of the
491: * representation tree, we start with NCLUS = 1 for the root
492: NCLUS = 1
493: IWORK( IINDC1+1 ) = 1
494: IWORK( IINDC1+2 ) = IM
495:
496: * IDONE is the number of eigenvectors already computed in the current
497: * block
498: IDONE = 0
499: * loop while( IDONE.LT.IM )
500: * generate the representation tree for the current block and
501: * compute the eigenvectors
502: 40 CONTINUE
503: IF( IDONE.LT.IM ) THEN
504: * This is a crude protection against infinitely deep trees
505: IF( NDEPTH.GT.M ) THEN
506: INFO = -2
507: RETURN
508: ENDIF
509: * breadth first processing of the current level of the representation
510: * tree: OLDNCL = number of clusters on current level
511: OLDNCL = NCLUS
512: * reset NCLUS to count the number of child clusters
513: NCLUS = 0
514: *
515: PARITY = 1 - PARITY
516: IF( PARITY.EQ.0 ) THEN
517: OLDCLS = IINDC1
518: NEWCLS = IINDC2
519: ELSE
520: OLDCLS = IINDC2
521: NEWCLS = IINDC1
522: END IF
523: * Process the clusters on the current level
524: DO 150 I = 1, OLDNCL
525: J = OLDCLS + 2*I
526: * OLDFST, OLDLST = first, last index of current cluster.
527: * cluster indices start with 1 and are relative
528: * to WBEGIN when accessing W, WGAP, WERR, Z
529: OLDFST = IWORK( J-1 )
530: OLDLST = IWORK( J )
531: IF( NDEPTH.GT.0 ) THEN
532: * Retrieve relatively robust representation (RRR) of cluster
533: * that has been computed at the previous level
534: * The RRR is stored in Z and overwritten once the eigenvectors
535: * have been computed or when the cluster is refined
536:
537: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
538: * Get representation from location of the leftmost evalue
539: * of the cluster
540: J = WBEGIN + OLDFST - 1
541: ELSE
542: IF(WBEGIN+OLDFST-1.LT.DOL) THEN
543: * Get representation from the left end of Z array
544: J = DOL - 1
545: ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN
546: * Get representation from the right end of Z array
547: J = DOU
548: ELSE
549: J = WBEGIN + OLDFST - 1
550: ENDIF
551: ENDIF
552: DO 45 K = 1, IN - 1
553: D( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
554: $ J ) )
555: L( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
556: $ J+1 ) )
557: 45 CONTINUE
558: D( IEND ) = DBLE( Z( IEND, J ) )
559: SIGMA = DBLE( Z( IEND, J+1 ) )
560:
561: * Set the corresponding entries in Z to zero
562: CALL ZLASET( 'Full', IN, 2, CZERO, CZERO,
563: $ Z( IBEGIN, J), LDZ )
564: END IF
565:
566: * Compute DL and DLL of current RRR
567: DO 50 J = IBEGIN, IEND-1
568: TMP = D( J )*L( J )
569: WORK( INDLD-1+J ) = TMP
570: WORK( INDLLD-1+J ) = TMP*L( J )
571: 50 CONTINUE
572:
573: IF( NDEPTH.GT.0 ) THEN
574: * P and Q are index of the first and last eigenvalue to compute
575: * within the current block
576: P = INDEXW( WBEGIN-1+OLDFST )
577: Q = INDEXW( WBEGIN-1+OLDLST )
578: * Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET
579: * through the Q-OFFSET elements of these arrays are to be used.
580: * OFFSET = P-OLDFST
581: OFFSET = INDEXW( WBEGIN ) - 1
582: * perform limited bisection (if necessary) to get approximate
583: * eigenvalues to the precision needed.
584: CALL DLARRB( IN, D( IBEGIN ),
585: $ WORK(INDLLD+IBEGIN-1),
586: $ P, Q, RTOL1, RTOL2, OFFSET,
587: $ WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN),
588: $ WORK( INDWRK ), IWORK( IINDWK ),
589: $ PIVMIN, SPDIAM, IN, IINFO )
590: IF( IINFO.NE.0 ) THEN
591: INFO = -1
592: RETURN
593: ENDIF
594: * We also recompute the extremal gaps. W holds all eigenvalues
595: * of the unshifted matrix and must be used for computation
596: * of WGAP, the entries of WORK might stem from RRRs with
597: * different shifts. The gaps from WBEGIN-1+OLDFST to
598: * WBEGIN-1+OLDLST are correctly computed in DLARRB.
599: * However, we only allow the gaps to become greater since
600: * this is what should happen when we decrease WERR
601: IF( OLDFST.GT.1) THEN
602: WGAP( WBEGIN+OLDFST-2 ) =
603: $ MAX(WGAP(WBEGIN+OLDFST-2),
604: $ W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1)
605: $ - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) )
606: ENDIF
607: IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN
608: WGAP( WBEGIN+OLDLST-1 ) =
609: $ MAX(WGAP(WBEGIN+OLDLST-1),
610: $ W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST)
611: $ - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) )
612: ENDIF
613: * Each time the eigenvalues in WORK get refined, we store
614: * the newly found approximation with all shifts applied in W
615: DO 53 J=OLDFST,OLDLST
616: W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA
617: 53 CONTINUE
618: END IF
619:
620: * Process the current node.
621: NEWFST = OLDFST
622: DO 140 J = OLDFST, OLDLST
623: IF( J.EQ.OLDLST ) THEN
624: * we are at the right end of the cluster, this is also the
625: * boundary of the child cluster
626: NEWLST = J
627: ELSE IF ( WGAP( WBEGIN + J -1).GE.
628: $ MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN
629: * the right relative gap is big enough, the child cluster
630: * (NEWFST,..,NEWLST) is well separated from the following
631: NEWLST = J
632: ELSE
633: * inside a child cluster, the relative gap is not
634: * big enough.
635: GOTO 140
636: END IF
637:
638: * Compute size of child cluster found
639: NEWSIZ = NEWLST - NEWFST + 1
640:
641: * NEWFTT is the place in Z where the new RRR or the computed
642: * eigenvector is to be stored
643: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
644: * Store representation at location of the leftmost evalue
645: * of the cluster
646: NEWFTT = WBEGIN + NEWFST - 1
647: ELSE
648: IF(WBEGIN+NEWFST-1.LT.DOL) THEN
649: * Store representation at the left end of Z array
650: NEWFTT = DOL - 1
651: ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN
652: * Store representation at the right end of Z array
653: NEWFTT = DOU
654: ELSE
655: NEWFTT = WBEGIN + NEWFST - 1
656: ENDIF
657: ENDIF
658:
659: IF( NEWSIZ.GT.1) THEN
660: *
661: * Current child is not a singleton but a cluster.
662: * Compute and store new representation of child.
663: *
664: *
665: * Compute left and right cluster gap.
666: *
667: * LGAP and RGAP are not computed from WORK because
668: * the eigenvalue approximations may stem from RRRs
669: * different shifts. However, W hold all eigenvalues
670: * of the unshifted matrix. Still, the entries in WGAP
671: * have to be computed from WORK since the entries
672: * in W might be of the same order so that gaps are not
673: * exhibited correctly for very close eigenvalues.
674: IF( NEWFST.EQ.1 ) THEN
675: LGAP = MAX( ZERO,
676: $ W(WBEGIN)-WERR(WBEGIN) - VL )
677: ELSE
678: LGAP = WGAP( WBEGIN+NEWFST-2 )
679: ENDIF
680: RGAP = WGAP( WBEGIN+NEWLST-1 )
681: *
682: * Compute left- and rightmost eigenvalue of child
683: * to high precision in order to shift as close
684: * as possible and obtain as large relative gaps
685: * as possible
686: *
687: DO 55 K =1,2
688: IF(K.EQ.1) THEN
689: P = INDEXW( WBEGIN-1+NEWFST )
690: ELSE
691: P = INDEXW( WBEGIN-1+NEWLST )
692: ENDIF
693: OFFSET = INDEXW( WBEGIN ) - 1
694: CALL DLARRB( IN, D(IBEGIN),
695: $ WORK( INDLLD+IBEGIN-1 ),P,P,
696: $ RQTOL, RQTOL, OFFSET,
697: $ WORK(WBEGIN),WGAP(WBEGIN),
698: $ WERR(WBEGIN),WORK( INDWRK ),
699: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
700: $ IN, IINFO )
701: 55 CONTINUE
702: *
703: IF((WBEGIN+NEWLST-1.LT.DOL).OR.
704: $ (WBEGIN+NEWFST-1.GT.DOU)) THEN
705: * if the cluster contains no desired eigenvalues
706: * skip the computation of that branch of the rep. tree
707: *
708: * We could skip before the refinement of the extremal
709: * eigenvalues of the child, but then the representation
710: * tree could be different from the one when nothing is
711: * skipped. For this reason we skip at this place.
712: IDONE = IDONE + NEWLST - NEWFST + 1
713: GOTO 139
714: ENDIF
715: *
716: * Compute RRR of child cluster.
717: * Note that the new RRR is stored in Z
718: *
719: * DLARRF needs LWORK = 2*N
720: CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
721: $ WORK(INDLD+IBEGIN-1),
722: $ NEWFST, NEWLST, WORK(WBEGIN),
723: $ WGAP(WBEGIN), WERR(WBEGIN),
724: $ SPDIAM, LGAP, RGAP, PIVMIN, TAU,
725: $ WORK( INDIN1 ), WORK( INDIN2 ),
726: $ WORK( INDWRK ), IINFO )
727: * In the complex case, DLARRF cannot write
728: * the new RRR directly into Z and needs an intermediate
729: * workspace
730: DO 56 K = 1, IN-1
731: Z( IBEGIN+K-1, NEWFTT ) =
732: $ DCMPLX( WORK( INDIN1+K-1 ), ZERO )
733: Z( IBEGIN+K-1, NEWFTT+1 ) =
734: $ DCMPLX( WORK( INDIN2+K-1 ), ZERO )
735: 56 CONTINUE
736: Z( IEND, NEWFTT ) =
737: $ DCMPLX( WORK( INDIN1+IN-1 ), ZERO )
738: IF( IINFO.EQ.0 ) THEN
739: * a new RRR for the cluster was found by DLARRF
740: * update shift and store it
741: SSIGMA = SIGMA + TAU
742: Z( IEND, NEWFTT+1 ) = DCMPLX( SSIGMA, ZERO )
743: * WORK() are the midpoints and WERR() the semi-width
744: * Note that the entries in W are unchanged.
745: DO 116 K = NEWFST, NEWLST
746: FUDGE =
747: $ THREE*EPS*ABS(WORK(WBEGIN+K-1))
748: WORK( WBEGIN + K - 1 ) =
749: $ WORK( WBEGIN + K - 1) - TAU
750: FUDGE = FUDGE +
751: $ FOUR*EPS*ABS(WORK(WBEGIN+K-1))
752: * Fudge errors
753: WERR( WBEGIN + K - 1 ) =
754: $ WERR( WBEGIN + K - 1 ) + FUDGE
755: * Gaps are not fudged. Provided that WERR is small
756: * when eigenvalues are close, a zero gap indicates
757: * that a new representation is needed for resolving
758: * the cluster. A fudge could lead to a wrong decision
759: * of judging eigenvalues 'separated' which in
760: * reality are not. This could have a negative impact
761: * on the orthogonality of the computed eigenvectors.
762: 116 CONTINUE
763:
764: NCLUS = NCLUS + 1
765: K = NEWCLS + 2*NCLUS
766: IWORK( K-1 ) = NEWFST
767: IWORK( K ) = NEWLST
768: ELSE
769: INFO = -2
770: RETURN
771: ENDIF
772: ELSE
773: *
774: * Compute eigenvector of singleton
775: *
776: ITER = 0
777: *
778: TOL = FOUR * LOG(DBLE(IN)) * EPS
779: *
780: K = NEWFST
781: WINDEX = WBEGIN + K - 1
782: WINDMN = MAX(WINDEX - 1,1)
783: WINDPL = MIN(WINDEX + 1,M)
784: LAMBDA = WORK( WINDEX )
785: DONE = DONE + 1
786: * Check if eigenvector computation is to be skipped
787: IF((WINDEX.LT.DOL).OR.
788: $ (WINDEX.GT.DOU)) THEN
789: ESKIP = .TRUE.
790: GOTO 125
791: ELSE
792: ESKIP = .FALSE.
793: ENDIF
794: LEFT = WORK( WINDEX ) - WERR( WINDEX )
795: RIGHT = WORK( WINDEX ) + WERR( WINDEX )
796: INDEIG = INDEXW( WINDEX )
797: * Note that since we compute the eigenpairs for a child,
798: * all eigenvalue approximations are w.r.t the same shift.
799: * In this case, the entries in WORK should be used for
800: * computing the gaps since they exhibit even very small
801: * differences in the eigenvalues, as opposed to the
802: * entries in W which might "look" the same.
803:
804: IF( K .EQ. 1) THEN
805: * In the case RANGE='I' and with not much initial
806: * accuracy in LAMBDA and VL, the formula
807: * LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
808: * can lead to an overestimation of the left gap and
809: * thus to inadequately early RQI 'convergence'.
810: * Prevent this by forcing a small left gap.
811: LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
812: ELSE
813: LGAP = WGAP(WINDMN)
814: ENDIF
815: IF( K .EQ. IM) THEN
816: * In the case RANGE='I' and with not much initial
817: * accuracy in LAMBDA and VU, the formula
818: * can lead to an overestimation of the right gap and
819: * thus to inadequately early RQI 'convergence'.
820: * Prevent this by forcing a small right gap.
821: RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
822: ELSE
823: RGAP = WGAP(WINDEX)
824: ENDIF
825: GAP = MIN( LGAP, RGAP )
826: IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN
827: * The eigenvector support can become wrong
828: * because significant entries could be cut off due to a
829: * large GAPTOL parameter in LAR1V. Prevent this.
830: GAPTOL = ZERO
831: ELSE
832: GAPTOL = GAP * EPS
833: ENDIF
834: ISUPMN = IN
835: ISUPMX = 1
836: * Update WGAP so that it holds the minimum gap
837: * to the left or the right. This is crucial in the
838: * case where bisection is used to ensure that the
839: * eigenvalue is refined up to the required precision.
840: * The correct value is restored afterwards.
841: SAVGAP = WGAP(WINDEX)
842: WGAP(WINDEX) = GAP
843: * We want to use the Rayleigh Quotient Correction
844: * as often as possible since it converges quadratically
845: * when we are close enough to the desired eigenvalue.
846: * However, the Rayleigh Quotient can have the wrong sign
847: * and lead us away from the desired eigenvalue. In this
848: * case, the best we can do is to use bisection.
849: USEDBS = .FALSE.
850: USEDRQ = .FALSE.
851: * Bisection is initially turned off unless it is forced
852: NEEDBS = .NOT.TRYRQC
853: 120 CONTINUE
854: * Check if bisection should be used to refine eigenvalue
855: IF(NEEDBS) THEN
856: * Take the bisection as new iterate
857: USEDBS = .TRUE.
858: ITMP1 = IWORK( IINDR+WINDEX )
859: OFFSET = INDEXW( WBEGIN ) - 1
860: CALL DLARRB( IN, D(IBEGIN),
861: $ WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG,
862: $ ZERO, TWO*EPS, OFFSET,
863: $ WORK(WBEGIN),WGAP(WBEGIN),
864: $ WERR(WBEGIN),WORK( INDWRK ),
865: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
866: $ ITMP1, IINFO )
867: IF( IINFO.NE.0 ) THEN
868: INFO = -3
869: RETURN
870: ENDIF
871: LAMBDA = WORK( WINDEX )
872: * Reset twist index from inaccurate LAMBDA to
873: * force computation of true MINGMA
874: IWORK( IINDR+WINDEX ) = 0
875: ENDIF
876: * Given LAMBDA, compute the eigenvector.
877: CALL ZLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
878: $ L( IBEGIN ), WORK(INDLD+IBEGIN-1),
879: $ WORK(INDLLD+IBEGIN-1),
880: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
881: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
882: $ IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ),
883: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
884: IF(ITER .EQ. 0) THEN
885: BSTRES = RESID
886: BSTW = LAMBDA
887: ELSEIF(RESID.LT.BSTRES) THEN
888: BSTRES = RESID
889: BSTW = LAMBDA
890: ENDIF
891: ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 ))
892: ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX ))
893: ITER = ITER + 1
894:
895: * sin alpha <= |resid|/gap
896: * Note that both the residual and the gap are
897: * proportional to the matrix, so ||T|| doesn't play
898: * a role in the quotient
899:
900: *
901: * Convergence test for Rayleigh-Quotient iteration
902: * (omitted when Bisection has been used)
903: *
904: IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
905: $ RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS)
906: $ THEN
907: * We need to check that the RQCORR update doesn't
908: * move the eigenvalue away from the desired one and
909: * towards a neighbor. -> protection with bisection
910: IF(INDEIG.LE.NEGCNT) THEN
911: * The wanted eigenvalue lies to the left
912: SGNDEF = -ONE
913: ELSE
914: * The wanted eigenvalue lies to the right
915: SGNDEF = ONE
916: ENDIF
917: * We only use the RQCORR if it improves the
918: * the iterate reasonably.
919: IF( ( RQCORR*SGNDEF.GE.ZERO )
920: $ .AND.( LAMBDA + RQCORR.LE. RIGHT)
921: $ .AND.( LAMBDA + RQCORR.GE. LEFT)
922: $ ) THEN
923: USEDRQ = .TRUE.
924: * Store new midpoint of bisection interval in WORK
925: IF(SGNDEF.EQ.ONE) THEN
926: * The current LAMBDA is on the left of the true
927: * eigenvalue
928: LEFT = LAMBDA
929: * We prefer to assume that the error estimate
930: * is correct. We could make the interval not
931: * as a bracket but to be modified if the RQCORR
932: * chooses to. In this case, the RIGHT side should
933: * be modified as follows:
934: * RIGHT = MAX(RIGHT, LAMBDA + RQCORR)
935: ELSE
936: * The current LAMBDA is on the right of the true
937: * eigenvalue
938: RIGHT = LAMBDA
939: * See comment about assuming the error estimate is
940: * correct above.
941: * LEFT = MIN(LEFT, LAMBDA + RQCORR)
942: ENDIF
943: WORK( WINDEX ) =
944: $ HALF * (RIGHT + LEFT)
945: * Take RQCORR since it has the correct sign and
946: * improves the iterate reasonably
947: LAMBDA = LAMBDA + RQCORR
948: * Update width of error interval
949: WERR( WINDEX ) =
950: $ HALF * (RIGHT-LEFT)
951: ELSE
952: NEEDBS = .TRUE.
953: ENDIF
954: IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN
955: * The eigenvalue is computed to bisection accuracy
956: * compute eigenvector and stop
957: USEDBS = .TRUE.
958: GOTO 120
959: ELSEIF( ITER.LT.MAXITR ) THEN
960: GOTO 120
961: ELSEIF( ITER.EQ.MAXITR ) THEN
962: NEEDBS = .TRUE.
963: GOTO 120
964: ELSE
965: INFO = 5
966: RETURN
967: END IF
968: ELSE
969: STP2II = .FALSE.
970: IF(USEDRQ .AND. USEDBS .AND.
971: $ BSTRES.LE.RESID) THEN
972: LAMBDA = BSTW
973: STP2II = .TRUE.
974: ENDIF
975: IF (STP2II) THEN
976: * improve error angle by second step
977: CALL ZLAR1V( IN, 1, IN, LAMBDA,
978: $ D( IBEGIN ), L( IBEGIN ),
979: $ WORK(INDLD+IBEGIN-1),
980: $ WORK(INDLLD+IBEGIN-1),
981: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
982: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
983: $ IWORK( IINDR+WINDEX ),
984: $ ISUPPZ( 2*WINDEX-1 ),
985: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
986: ENDIF
987: WORK( WINDEX ) = LAMBDA
988: END IF
989: *
990: * Compute FP-vector support w.r.t. whole matrix
991: *
992: ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN
993: ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN
994: ZFROM = ISUPPZ( 2*WINDEX-1 )
995: ZTO = ISUPPZ( 2*WINDEX )
996: ISUPMN = ISUPMN + OLDIEN
997: ISUPMX = ISUPMX + OLDIEN
998: * Ensure vector is ok if support in the RQI has changed
999: IF(ISUPMN.LT.ZFROM) THEN
1000: DO 122 II = ISUPMN,ZFROM-1
1001: Z( II, WINDEX ) = ZERO
1002: 122 CONTINUE
1003: ENDIF
1004: IF(ISUPMX.GT.ZTO) THEN
1005: DO 123 II = ZTO+1,ISUPMX
1006: Z( II, WINDEX ) = ZERO
1007: 123 CONTINUE
1008: ENDIF
1009: CALL ZDSCAL( ZTO-ZFROM+1, NRMINV,
1010: $ Z( ZFROM, WINDEX ), 1 )
1011: 125 CONTINUE
1012: * Update W
1013: W( WINDEX ) = LAMBDA+SIGMA
1014: * Recompute the gaps on the left and right
1015: * But only allow them to become larger and not
1016: * smaller (which can only happen through "bad"
1017: * cancellation and doesn't reflect the theory
1018: * where the initial gaps are underestimated due
1019: * to WERR being too crude.)
1020: IF(.NOT.ESKIP) THEN
1021: IF( K.GT.1) THEN
1022: WGAP( WINDMN ) = MAX( WGAP(WINDMN),
1023: $ W(WINDEX)-WERR(WINDEX)
1024: $ - W(WINDMN)-WERR(WINDMN) )
1025: ENDIF
1026: IF( WINDEX.LT.WEND ) THEN
1027: WGAP( WINDEX ) = MAX( SAVGAP,
1028: $ W( WINDPL )-WERR( WINDPL )
1029: $ - W( WINDEX )-WERR( WINDEX) )
1030: ENDIF
1031: ENDIF
1032: IDONE = IDONE + 1
1033: ENDIF
1034: * here ends the code for the current child
1035: *
1036: 139 CONTINUE
1037: * Proceed to any remaining child nodes
1038: NEWFST = J + 1
1039: 140 CONTINUE
1040: 150 CONTINUE
1041: NDEPTH = NDEPTH + 1
1042: GO TO 40
1043: END IF
1044: IBEGIN = IEND + 1
1045: WBEGIN = WEND + 1
1046: 170 CONTINUE
1047: *
1048:
1049: RETURN
1050: *
1051: * End of ZLARRV
1052: *
1053: END
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