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>
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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 < 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: *> \ingroup complex16OTHERauxiliary
270: *
271: *> \par Contributors:
272: * ==================
273: *>
274: *> Beresford Parlett, University of California, Berkeley, USA \n
275: *> Jim Demmel, University of California, Berkeley, USA \n
276: *> Inderjit Dhillon, University of Texas, Austin, USA \n
277: *> Osni Marques, LBNL/NERSC, USA \n
278: *> Christof Voemel, University of California, Berkeley, USA
279: *
280: * =====================================================================
281: SUBROUTINE ZLARRV( N, VL, VU, D, L, PIVMIN,
282: $ ISPLIT, M, DOL, DOU, MINRGP,
283: $ RTOL1, RTOL2, W, WERR, WGAP,
284: $ IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
285: $ WORK, IWORK, INFO )
286: *
287: * -- LAPACK auxiliary routine --
288: * -- LAPACK is a software package provided by Univ. of Tennessee, --
289: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
290: *
291: * .. Scalar Arguments ..
292: INTEGER DOL, DOU, INFO, LDZ, M, N
293: DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
294: * ..
295: * .. Array Arguments ..
296: INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
297: $ ISUPPZ( * ), IWORK( * )
298: DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
299: $ WGAP( * ), WORK( * )
300: COMPLEX*16 Z( LDZ, * )
301: * ..
302: *
303: * =====================================================================
304: *
305: * .. Parameters ..
306: INTEGER MAXITR
307: PARAMETER ( MAXITR = 10 )
308: COMPLEX*16 CZERO
309: PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ) )
310: DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, HALF
311: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
312: $ TWO = 2.0D0, THREE = 3.0D0,
313: $ FOUR = 4.0D0, HALF = 0.5D0)
314: * ..
315: * .. Local Scalars ..
316: LOGICAL ESKIP, NEEDBS, STP2II, TRYRQC, USEDBS, USEDRQ
317: INTEGER DONE, I, IBEGIN, IDONE, IEND, II, IINDC1,
318: $ IINDC2, IINDR, IINDWK, IINFO, IM, IN, INDEIG,
319: $ INDLD, INDLLD, INDWRK, ISUPMN, ISUPMX, ITER,
320: $ ITMP1, J, JBLK, K, MINIWSIZE, MINWSIZE, NCLUS,
321: $ NDEPTH, NEGCNT, NEWCLS, NEWFST, NEWFTT, NEWLST,
322: $ NEWSIZ, OFFSET, OLDCLS, OLDFST, OLDIEN, OLDLST,
323: $ OLDNCL, P, PARITY, Q, WBEGIN, WEND, WINDEX,
324: $ WINDMN, WINDPL, ZFROM, ZTO, ZUSEDL, ZUSEDU,
325: $ ZUSEDW
326: INTEGER INDIN1, INDIN2
327: DOUBLE PRECISION BSTRES, BSTW, EPS, FUDGE, GAP, GAPTOL, GL, GU,
328: $ LAMBDA, LEFT, LGAP, MINGMA, NRMINV, RESID,
329: $ RGAP, RIGHT, RQCORR, RQTOL, SAVGAP, SGNDEF,
330: $ SIGMA, SPDIAM, SSIGMA, TAU, TMP, TOL, ZTZ
331: * ..
332: * .. External Functions ..
333: DOUBLE PRECISION DLAMCH
334: EXTERNAL DLAMCH
335: * ..
336: * .. External Subroutines ..
337: EXTERNAL DCOPY, DLARRB, DLARRF, ZDSCAL, ZLAR1V,
338: $ ZLASET
339: * ..
340: * .. Intrinsic Functions ..
341: INTRINSIC ABS, DBLE, MAX, MIN
342: INTRINSIC DCMPLX
343: * ..
344: * .. Executable Statements ..
345: * ..
346:
347: INFO = 0
348: *
349: * Quick return if possible
350: *
351: IF( (N.LE.0).OR.(M.LE.0) ) THEN
352: RETURN
353: END IF
354: *
355: * The first N entries of WORK are reserved for the eigenvalues
356: INDLD = N+1
357: INDLLD= 2*N+1
358: INDIN1 = 3*N + 1
359: INDIN2 = 4*N + 1
360: INDWRK = 5*N + 1
361: MINWSIZE = 12 * N
362:
363: DO 5 I= 1,MINWSIZE
364: WORK( I ) = ZERO
365: 5 CONTINUE
366:
367: * IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
368: * factorization used to compute the FP vector
369: IINDR = 0
370: * IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
371: * layer and the one above.
372: IINDC1 = N
373: IINDC2 = 2*N
374: IINDWK = 3*N + 1
375:
376: MINIWSIZE = 7 * N
377: DO 10 I= 1,MINIWSIZE
378: IWORK( I ) = 0
379: 10 CONTINUE
380:
381: ZUSEDL = 1
382: IF(DOL.GT.1) THEN
383: * Set lower bound for use of Z
384: ZUSEDL = DOL-1
385: ENDIF
386: ZUSEDU = M
387: IF(DOU.LT.M) THEN
388: * Set lower bound for use of Z
389: ZUSEDU = DOU+1
390: ENDIF
391: * The width of the part of Z that is used
392: ZUSEDW = ZUSEDU - ZUSEDL + 1
393:
394:
395: CALL ZLASET( 'Full', N, ZUSEDW, CZERO, CZERO,
396: $ Z(1,ZUSEDL), LDZ )
397:
398: EPS = DLAMCH( 'Precision' )
399: RQTOL = TWO * EPS
400: *
401: * Set expert flags for standard code.
402: TRYRQC = .TRUE.
403:
404: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
405: ELSE
406: * Only selected eigenpairs are computed. Since the other evalues
407: * are not refined by RQ iteration, bisection has to compute to full
408: * accuracy.
409: RTOL1 = FOUR * EPS
410: RTOL2 = FOUR * EPS
411: ENDIF
412:
413: * The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
414: * desired eigenvalues. The support of the nonzero eigenvector
415: * entries is contained in the interval IBEGIN:IEND.
416: * Remark that if k eigenpairs are desired, then the eigenvectors
417: * are stored in k contiguous columns of Z.
418:
419: * DONE is the number of eigenvectors already computed
420: DONE = 0
421: IBEGIN = 1
422: WBEGIN = 1
423: DO 170 JBLK = 1, IBLOCK( M )
424: IEND = ISPLIT( JBLK )
425: SIGMA = L( IEND )
426: * Find the eigenvectors of the submatrix indexed IBEGIN
427: * through IEND.
428: WEND = WBEGIN - 1
429: 15 CONTINUE
430: IF( WEND.LT.M ) THEN
431: IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN
432: WEND = WEND + 1
433: GO TO 15
434: END IF
435: END IF
436: IF( WEND.LT.WBEGIN ) THEN
437: IBEGIN = IEND + 1
438: GO TO 170
439: ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN
440: IBEGIN = IEND + 1
441: WBEGIN = WEND + 1
442: GO TO 170
443: END IF
444:
445: * Find local spectral diameter of the block
446: GL = GERS( 2*IBEGIN-1 )
447: GU = GERS( 2*IBEGIN )
448: DO 20 I = IBEGIN+1 , IEND
449: GL = MIN( GERS( 2*I-1 ), GL )
450: GU = MAX( GERS( 2*I ), GU )
451: 20 CONTINUE
452: SPDIAM = GU - GL
453:
454: * OLDIEN is the last index of the previous block
455: OLDIEN = IBEGIN - 1
456: * Calculate the size of the current block
457: IN = IEND - IBEGIN + 1
458: * The number of eigenvalues in the current block
459: IM = WEND - WBEGIN + 1
460:
461: * This is for a 1x1 block
462: IF( IBEGIN.EQ.IEND ) THEN
463: DONE = DONE+1
464: Z( IBEGIN, WBEGIN ) = DCMPLX( ONE, ZERO )
465: ISUPPZ( 2*WBEGIN-1 ) = IBEGIN
466: ISUPPZ( 2*WBEGIN ) = IBEGIN
467: W( WBEGIN ) = W( WBEGIN ) + SIGMA
468: WORK( WBEGIN ) = W( WBEGIN )
469: IBEGIN = IEND + 1
470: WBEGIN = WBEGIN + 1
471: GO TO 170
472: END IF
473:
474: * The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
475: * Note that these can be approximations, in this case, the corresp.
476: * entries of WERR give the size of the uncertainty interval.
477: * The eigenvalue approximations will be refined when necessary as
478: * high relative accuracy is required for the computation of the
479: * corresponding eigenvectors.
480: CALL DCOPY( IM, W( WBEGIN ), 1,
481: $ WORK( WBEGIN ), 1 )
482:
483: * We store in W the eigenvalue approximations w.r.t. the original
484: * matrix T.
485: DO 30 I=1,IM
486: W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA
487: 30 CONTINUE
488:
489:
490: * NDEPTH is the current depth of the representation tree
491: NDEPTH = 0
492: * PARITY is either 1 or 0
493: PARITY = 1
494: * NCLUS is the number of clusters for the next level of the
495: * representation tree, we start with NCLUS = 1 for the root
496: NCLUS = 1
497: IWORK( IINDC1+1 ) = 1
498: IWORK( IINDC1+2 ) = IM
499:
500: * IDONE is the number of eigenvectors already computed in the current
501: * block
502: IDONE = 0
503: * loop while( IDONE.LT.IM )
504: * generate the representation tree for the current block and
505: * compute the eigenvectors
506: 40 CONTINUE
507: IF( IDONE.LT.IM ) THEN
508: * This is a crude protection against infinitely deep trees
509: IF( NDEPTH.GT.M ) THEN
510: INFO = -2
511: RETURN
512: ENDIF
513: * breadth first processing of the current level of the representation
514: * tree: OLDNCL = number of clusters on current level
515: OLDNCL = NCLUS
516: * reset NCLUS to count the number of child clusters
517: NCLUS = 0
518: *
519: PARITY = 1 - PARITY
520: IF( PARITY.EQ.0 ) THEN
521: OLDCLS = IINDC1
522: NEWCLS = IINDC2
523: ELSE
524: OLDCLS = IINDC2
525: NEWCLS = IINDC1
526: END IF
527: * Process the clusters on the current level
528: DO 150 I = 1, OLDNCL
529: J = OLDCLS + 2*I
530: * OLDFST, OLDLST = first, last index of current cluster.
531: * cluster indices start with 1 and are relative
532: * to WBEGIN when accessing W, WGAP, WERR, Z
533: OLDFST = IWORK( J-1 )
534: OLDLST = IWORK( J )
535: IF( NDEPTH.GT.0 ) THEN
536: * Retrieve relatively robust representation (RRR) of cluster
537: * that has been computed at the previous level
538: * The RRR is stored in Z and overwritten once the eigenvectors
539: * have been computed or when the cluster is refined
540:
541: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
542: * Get representation from location of the leftmost evalue
543: * of the cluster
544: J = WBEGIN + OLDFST - 1
545: ELSE
546: IF(WBEGIN+OLDFST-1.LT.DOL) THEN
547: * Get representation from the left end of Z array
548: J = DOL - 1
549: ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN
550: * Get representation from the right end of Z array
551: J = DOU
552: ELSE
553: J = WBEGIN + OLDFST - 1
554: ENDIF
555: ENDIF
556: DO 45 K = 1, IN - 1
557: D( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
558: $ J ) )
559: L( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
560: $ J+1 ) )
561: 45 CONTINUE
562: D( IEND ) = DBLE( Z( IEND, J ) )
563: SIGMA = DBLE( Z( IEND, J+1 ) )
564:
565: * Set the corresponding entries in Z to zero
566: CALL ZLASET( 'Full', IN, 2, CZERO, CZERO,
567: $ Z( IBEGIN, J), LDZ )
568: END IF
569:
570: * Compute DL and DLL of current RRR
571: DO 50 J = IBEGIN, IEND-1
572: TMP = D( J )*L( J )
573: WORK( INDLD-1+J ) = TMP
574: WORK( INDLLD-1+J ) = TMP*L( J )
575: 50 CONTINUE
576:
577: IF( NDEPTH.GT.0 ) THEN
578: * P and Q are index of the first and last eigenvalue to compute
579: * within the current block
580: P = INDEXW( WBEGIN-1+OLDFST )
581: Q = INDEXW( WBEGIN-1+OLDLST )
582: * Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET
583: * through the Q-OFFSET elements of these arrays are to be used.
584: * OFFSET = P-OLDFST
585: OFFSET = INDEXW( WBEGIN ) - 1
586: * perform limited bisection (if necessary) to get approximate
587: * eigenvalues to the precision needed.
588: CALL DLARRB( IN, D( IBEGIN ),
589: $ WORK(INDLLD+IBEGIN-1),
590: $ P, Q, RTOL1, RTOL2, OFFSET,
591: $ WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN),
592: $ WORK( INDWRK ), IWORK( IINDWK ),
593: $ PIVMIN, SPDIAM, IN, IINFO )
594: IF( IINFO.NE.0 ) THEN
595: INFO = -1
596: RETURN
597: ENDIF
598: * We also recompute the extremal gaps. W holds all eigenvalues
599: * of the unshifted matrix and must be used for computation
600: * of WGAP, the entries of WORK might stem from RRRs with
601: * different shifts. The gaps from WBEGIN-1+OLDFST to
602: * WBEGIN-1+OLDLST are correctly computed in DLARRB.
603: * However, we only allow the gaps to become greater since
604: * this is what should happen when we decrease WERR
605: IF( OLDFST.GT.1) THEN
606: WGAP( WBEGIN+OLDFST-2 ) =
607: $ MAX(WGAP(WBEGIN+OLDFST-2),
608: $ W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1)
609: $ - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) )
610: ENDIF
611: IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN
612: WGAP( WBEGIN+OLDLST-1 ) =
613: $ MAX(WGAP(WBEGIN+OLDLST-1),
614: $ W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST)
615: $ - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) )
616: ENDIF
617: * Each time the eigenvalues in WORK get refined, we store
618: * the newly found approximation with all shifts applied in W
619: DO 53 J=OLDFST,OLDLST
620: W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA
621: 53 CONTINUE
622: END IF
623:
624: * Process the current node.
625: NEWFST = OLDFST
626: DO 140 J = OLDFST, OLDLST
627: IF( J.EQ.OLDLST ) THEN
628: * we are at the right end of the cluster, this is also the
629: * boundary of the child cluster
630: NEWLST = J
631: ELSE IF ( WGAP( WBEGIN + J -1).GE.
632: $ MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN
633: * the right relative gap is big enough, the child cluster
634: * (NEWFST,..,NEWLST) is well separated from the following
635: NEWLST = J
636: ELSE
637: * inside a child cluster, the relative gap is not
638: * big enough.
639: GOTO 140
640: END IF
641:
642: * Compute size of child cluster found
643: NEWSIZ = NEWLST - NEWFST + 1
644:
645: * NEWFTT is the place in Z where the new RRR or the computed
646: * eigenvector is to be stored
647: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
648: * Store representation at location of the leftmost evalue
649: * of the cluster
650: NEWFTT = WBEGIN + NEWFST - 1
651: ELSE
652: IF(WBEGIN+NEWFST-1.LT.DOL) THEN
653: * Store representation at the left end of Z array
654: NEWFTT = DOL - 1
655: ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN
656: * Store representation at the right end of Z array
657: NEWFTT = DOU
658: ELSE
659: NEWFTT = WBEGIN + NEWFST - 1
660: ENDIF
661: ENDIF
662:
663: IF( NEWSIZ.GT.1) THEN
664: *
665: * Current child is not a singleton but a cluster.
666: * Compute and store new representation of child.
667: *
668: *
669: * Compute left and right cluster gap.
670: *
671: * LGAP and RGAP are not computed from WORK because
672: * the eigenvalue approximations may stem from RRRs
673: * different shifts. However, W hold all eigenvalues
674: * of the unshifted matrix. Still, the entries in WGAP
675: * have to be computed from WORK since the entries
676: * in W might be of the same order so that gaps are not
677: * exhibited correctly for very close eigenvalues.
678: IF( NEWFST.EQ.1 ) THEN
679: LGAP = MAX( ZERO,
680: $ W(WBEGIN)-WERR(WBEGIN) - VL )
681: ELSE
682: LGAP = WGAP( WBEGIN+NEWFST-2 )
683: ENDIF
684: RGAP = WGAP( WBEGIN+NEWLST-1 )
685: *
686: * Compute left- and rightmost eigenvalue of child
687: * to high precision in order to shift as close
688: * as possible and obtain as large relative gaps
689: * as possible
690: *
691: DO 55 K =1,2
692: IF(K.EQ.1) THEN
693: P = INDEXW( WBEGIN-1+NEWFST )
694: ELSE
695: P = INDEXW( WBEGIN-1+NEWLST )
696: ENDIF
697: OFFSET = INDEXW( WBEGIN ) - 1
698: CALL DLARRB( IN, D(IBEGIN),
699: $ WORK( INDLLD+IBEGIN-1 ),P,P,
700: $ RQTOL, RQTOL, OFFSET,
701: $ WORK(WBEGIN),WGAP(WBEGIN),
702: $ WERR(WBEGIN),WORK( INDWRK ),
703: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
704: $ IN, IINFO )
705: 55 CONTINUE
706: *
707: IF((WBEGIN+NEWLST-1.LT.DOL).OR.
708: $ (WBEGIN+NEWFST-1.GT.DOU)) THEN
709: * if the cluster contains no desired eigenvalues
710: * skip the computation of that branch of the rep. tree
711: *
712: * We could skip before the refinement of the extremal
713: * eigenvalues of the child, but then the representation
714: * tree could be different from the one when nothing is
715: * skipped. For this reason we skip at this place.
716: IDONE = IDONE + NEWLST - NEWFST + 1
717: GOTO 139
718: ENDIF
719: *
720: * Compute RRR of child cluster.
721: * Note that the new RRR is stored in Z
722: *
723: * DLARRF needs LWORK = 2*N
724: CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
725: $ WORK(INDLD+IBEGIN-1),
726: $ NEWFST, NEWLST, WORK(WBEGIN),
727: $ WGAP(WBEGIN), WERR(WBEGIN),
728: $ SPDIAM, LGAP, RGAP, PIVMIN, TAU,
729: $ WORK( INDIN1 ), WORK( INDIN2 ),
730: $ WORK( INDWRK ), IINFO )
731: * In the complex case, DLARRF cannot write
732: * the new RRR directly into Z and needs an intermediate
733: * workspace
734: DO 56 K = 1, IN-1
735: Z( IBEGIN+K-1, NEWFTT ) =
736: $ DCMPLX( WORK( INDIN1+K-1 ), ZERO )
737: Z( IBEGIN+K-1, NEWFTT+1 ) =
738: $ DCMPLX( WORK( INDIN2+K-1 ), ZERO )
739: 56 CONTINUE
740: Z( IEND, NEWFTT ) =
741: $ DCMPLX( WORK( INDIN1+IN-1 ), ZERO )
742: IF( IINFO.EQ.0 ) THEN
743: * a new RRR for the cluster was found by DLARRF
744: * update shift and store it
745: SSIGMA = SIGMA + TAU
746: Z( IEND, NEWFTT+1 ) = DCMPLX( SSIGMA, ZERO )
747: * WORK() are the midpoints and WERR() the semi-width
748: * Note that the entries in W are unchanged.
749: DO 116 K = NEWFST, NEWLST
750: FUDGE =
751: $ THREE*EPS*ABS(WORK(WBEGIN+K-1))
752: WORK( WBEGIN + K - 1 ) =
753: $ WORK( WBEGIN + K - 1) - TAU
754: FUDGE = FUDGE +
755: $ FOUR*EPS*ABS(WORK(WBEGIN+K-1))
756: * Fudge errors
757: WERR( WBEGIN + K - 1 ) =
758: $ WERR( WBEGIN + K - 1 ) + FUDGE
759: * Gaps are not fudged. Provided that WERR is small
760: * when eigenvalues are close, a zero gap indicates
761: * that a new representation is needed for resolving
762: * the cluster. A fudge could lead to a wrong decision
763: * of judging eigenvalues 'separated' which in
764: * reality are not. This could have a negative impact
765: * on the orthogonality of the computed eigenvectors.
766: 116 CONTINUE
767:
768: NCLUS = NCLUS + 1
769: K = NEWCLS + 2*NCLUS
770: IWORK( K-1 ) = NEWFST
771: IWORK( K ) = NEWLST
772: ELSE
773: INFO = -2
774: RETURN
775: ENDIF
776: ELSE
777: *
778: * Compute eigenvector of singleton
779: *
780: ITER = 0
781: *
782: TOL = FOUR * LOG(DBLE(IN)) * EPS
783: *
784: K = NEWFST
785: WINDEX = WBEGIN + K - 1
786: WINDMN = MAX(WINDEX - 1,1)
787: WINDPL = MIN(WINDEX + 1,M)
788: LAMBDA = WORK( WINDEX )
789: DONE = DONE + 1
790: * Check if eigenvector computation is to be skipped
791: IF((WINDEX.LT.DOL).OR.
792: $ (WINDEX.GT.DOU)) THEN
793: ESKIP = .TRUE.
794: GOTO 125
795: ELSE
796: ESKIP = .FALSE.
797: ENDIF
798: LEFT = WORK( WINDEX ) - WERR( WINDEX )
799: RIGHT = WORK( WINDEX ) + WERR( WINDEX )
800: INDEIG = INDEXW( WINDEX )
801: * Note that since we compute the eigenpairs for a child,
802: * all eigenvalue approximations are w.r.t the same shift.
803: * In this case, the entries in WORK should be used for
804: * computing the gaps since they exhibit even very small
805: * differences in the eigenvalues, as opposed to the
806: * entries in W which might "look" the same.
807:
808: IF( K .EQ. 1) THEN
809: * In the case RANGE='I' and with not much initial
810: * accuracy in LAMBDA and VL, the formula
811: * LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
812: * can lead to an overestimation of the left gap and
813: * thus to inadequately early RQI 'convergence'.
814: * Prevent this by forcing a small left gap.
815: LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
816: ELSE
817: LGAP = WGAP(WINDMN)
818: ENDIF
819: IF( K .EQ. IM) THEN
820: * In the case RANGE='I' and with not much initial
821: * accuracy in LAMBDA and VU, the formula
822: * can lead to an overestimation of the right gap and
823: * thus to inadequately early RQI 'convergence'.
824: * Prevent this by forcing a small right gap.
825: RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
826: ELSE
827: RGAP = WGAP(WINDEX)
828: ENDIF
829: GAP = MIN( LGAP, RGAP )
830: IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN
831: * The eigenvector support can become wrong
832: * because significant entries could be cut off due to a
833: * large GAPTOL parameter in LAR1V. Prevent this.
834: GAPTOL = ZERO
835: ELSE
836: GAPTOL = GAP * EPS
837: ENDIF
838: ISUPMN = IN
839: ISUPMX = 1
840: * Update WGAP so that it holds the minimum gap
841: * to the left or the right. This is crucial in the
842: * case where bisection is used to ensure that the
843: * eigenvalue is refined up to the required precision.
844: * The correct value is restored afterwards.
845: SAVGAP = WGAP(WINDEX)
846: WGAP(WINDEX) = GAP
847: * We want to use the Rayleigh Quotient Correction
848: * as often as possible since it converges quadratically
849: * when we are close enough to the desired eigenvalue.
850: * However, the Rayleigh Quotient can have the wrong sign
851: * and lead us away from the desired eigenvalue. In this
852: * case, the best we can do is to use bisection.
853: USEDBS = .FALSE.
854: USEDRQ = .FALSE.
855: * Bisection is initially turned off unless it is forced
856: NEEDBS = .NOT.TRYRQC
857: 120 CONTINUE
858: * Check if bisection should be used to refine eigenvalue
859: IF(NEEDBS) THEN
860: * Take the bisection as new iterate
861: USEDBS = .TRUE.
862: ITMP1 = IWORK( IINDR+WINDEX )
863: OFFSET = INDEXW( WBEGIN ) - 1
864: CALL DLARRB( IN, D(IBEGIN),
865: $ WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG,
866: $ ZERO, TWO*EPS, OFFSET,
867: $ WORK(WBEGIN),WGAP(WBEGIN),
868: $ WERR(WBEGIN),WORK( INDWRK ),
869: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
870: $ ITMP1, IINFO )
871: IF( IINFO.NE.0 ) THEN
872: INFO = -3
873: RETURN
874: ENDIF
875: LAMBDA = WORK( WINDEX )
876: * Reset twist index from inaccurate LAMBDA to
877: * force computation of true MINGMA
878: IWORK( IINDR+WINDEX ) = 0
879: ENDIF
880: * Given LAMBDA, compute the eigenvector.
881: CALL ZLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
882: $ L( IBEGIN ), WORK(INDLD+IBEGIN-1),
883: $ WORK(INDLLD+IBEGIN-1),
884: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
885: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
886: $ IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ),
887: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
888: IF(ITER .EQ. 0) THEN
889: BSTRES = RESID
890: BSTW = LAMBDA
891: ELSEIF(RESID.LT.BSTRES) THEN
892: BSTRES = RESID
893: BSTW = LAMBDA
894: ENDIF
895: ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 ))
896: ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX ))
897: ITER = ITER + 1
898:
899: * sin alpha <= |resid|/gap
900: * Note that both the residual and the gap are
901: * proportional to the matrix, so ||T|| doesn't play
902: * a role in the quotient
903:
904: *
905: * Convergence test for Rayleigh-Quotient iteration
906: * (omitted when Bisection has been used)
907: *
908: IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
909: $ RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS)
910: $ THEN
911: * We need to check that the RQCORR update doesn't
912: * move the eigenvalue away from the desired one and
913: * towards a neighbor. -> protection with bisection
914: IF(INDEIG.LE.NEGCNT) THEN
915: * The wanted eigenvalue lies to the left
916: SGNDEF = -ONE
917: ELSE
918: * The wanted eigenvalue lies to the right
919: SGNDEF = ONE
920: ENDIF
921: * We only use the RQCORR if it improves the
922: * the iterate reasonably.
923: IF( ( RQCORR*SGNDEF.GE.ZERO )
924: $ .AND.( LAMBDA + RQCORR.LE. RIGHT)
925: $ .AND.( LAMBDA + RQCORR.GE. LEFT)
926: $ ) THEN
927: USEDRQ = .TRUE.
928: * Store new midpoint of bisection interval in WORK
929: IF(SGNDEF.EQ.ONE) THEN
930: * The current LAMBDA is on the left of the true
931: * eigenvalue
932: LEFT = LAMBDA
933: * We prefer to assume that the error estimate
934: * is correct. We could make the interval not
935: * as a bracket but to be modified if the RQCORR
936: * chooses to. In this case, the RIGHT side should
937: * be modified as follows:
938: * RIGHT = MAX(RIGHT, LAMBDA + RQCORR)
939: ELSE
940: * The current LAMBDA is on the right of the true
941: * eigenvalue
942: RIGHT = LAMBDA
943: * See comment about assuming the error estimate is
944: * correct above.
945: * LEFT = MIN(LEFT, LAMBDA + RQCORR)
946: ENDIF
947: WORK( WINDEX ) =
948: $ HALF * (RIGHT + LEFT)
949: * Take RQCORR since it has the correct sign and
950: * improves the iterate reasonably
951: LAMBDA = LAMBDA + RQCORR
952: * Update width of error interval
953: WERR( WINDEX ) =
954: $ HALF * (RIGHT-LEFT)
955: ELSE
956: NEEDBS = .TRUE.
957: ENDIF
958: IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN
959: * The eigenvalue is computed to bisection accuracy
960: * compute eigenvector and stop
961: USEDBS = .TRUE.
962: GOTO 120
963: ELSEIF( ITER.LT.MAXITR ) THEN
964: GOTO 120
965: ELSEIF( ITER.EQ.MAXITR ) THEN
966: NEEDBS = .TRUE.
967: GOTO 120
968: ELSE
969: INFO = 5
970: RETURN
971: END IF
972: ELSE
973: STP2II = .FALSE.
974: IF(USEDRQ .AND. USEDBS .AND.
975: $ BSTRES.LE.RESID) THEN
976: LAMBDA = BSTW
977: STP2II = .TRUE.
978: ENDIF
979: IF (STP2II) THEN
980: * improve error angle by second step
981: CALL ZLAR1V( IN, 1, IN, LAMBDA,
982: $ D( IBEGIN ), L( IBEGIN ),
983: $ WORK(INDLD+IBEGIN-1),
984: $ WORK(INDLLD+IBEGIN-1),
985: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
986: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
987: $ IWORK( IINDR+WINDEX ),
988: $ ISUPPZ( 2*WINDEX-1 ),
989: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
990: ENDIF
991: WORK( WINDEX ) = LAMBDA
992: END IF
993: *
994: * Compute FP-vector support w.r.t. whole matrix
995: *
996: ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN
997: ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN
998: ZFROM = ISUPPZ( 2*WINDEX-1 )
999: ZTO = ISUPPZ( 2*WINDEX )
1000: ISUPMN = ISUPMN + OLDIEN
1001: ISUPMX = ISUPMX + OLDIEN
1002: * Ensure vector is ok if support in the RQI has changed
1003: IF(ISUPMN.LT.ZFROM) THEN
1004: DO 122 II = ISUPMN,ZFROM-1
1005: Z( II, WINDEX ) = ZERO
1006: 122 CONTINUE
1007: ENDIF
1008: IF(ISUPMX.GT.ZTO) THEN
1009: DO 123 II = ZTO+1,ISUPMX
1010: Z( II, WINDEX ) = ZERO
1011: 123 CONTINUE
1012: ENDIF
1013: CALL ZDSCAL( ZTO-ZFROM+1, NRMINV,
1014: $ Z( ZFROM, WINDEX ), 1 )
1015: 125 CONTINUE
1016: * Update W
1017: W( WINDEX ) = LAMBDA+SIGMA
1018: * Recompute the gaps on the left and right
1019: * But only allow them to become larger and not
1020: * smaller (which can only happen through "bad"
1021: * cancellation and doesn't reflect the theory
1022: * where the initial gaps are underestimated due
1023: * to WERR being too crude.)
1024: IF(.NOT.ESKIP) THEN
1025: IF( K.GT.1) THEN
1026: WGAP( WINDMN ) = MAX( WGAP(WINDMN),
1027: $ W(WINDEX)-WERR(WINDEX)
1028: $ - W(WINDMN)-WERR(WINDMN) )
1029: ENDIF
1030: IF( WINDEX.LT.WEND ) THEN
1031: WGAP( WINDEX ) = MAX( SAVGAP,
1032: $ W( WINDPL )-WERR( WINDPL )
1033: $ - W( WINDEX )-WERR( WINDEX) )
1034: ENDIF
1035: ENDIF
1036: IDONE = IDONE + 1
1037: ENDIF
1038: * here ends the code for the current child
1039: *
1040: 139 CONTINUE
1041: * Proceed to any remaining child nodes
1042: NEWFST = J + 1
1043: 140 CONTINUE
1044: 150 CONTINUE
1045: NDEPTH = NDEPTH + 1
1046: GO TO 40
1047: END IF
1048: IBEGIN = IEND + 1
1049: WBEGIN = WEND + 1
1050: 170 CONTINUE
1051: *
1052:
1053: RETURN
1054: *
1055: * End of ZLARRV
1056: *
1057: END
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