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>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlarrv.f">
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 3.7.1) --
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: *
352: * Quick return if possible
353: *
354: IF( N.LE.0 ) THEN
355: RETURN
356: END IF
357: *
358: * The first N entries of WORK are reserved for the eigenvalues
359: INDLD = N+1
360: INDLLD= 2*N+1
361: INDIN1 = 3*N + 1
362: INDIN2 = 4*N + 1
363: INDWRK = 5*N + 1
364: MINWSIZE = 12 * N
365:
366: DO 5 I= 1,MINWSIZE
367: WORK( I ) = ZERO
368: 5 CONTINUE
369:
370: * IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
371: * factorization used to compute the FP vector
372: IINDR = 0
373: * IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
374: * layer and the one above.
375: IINDC1 = N
376: IINDC2 = 2*N
377: IINDWK = 3*N + 1
378:
379: MINIWSIZE = 7 * N
380: DO 10 I= 1,MINIWSIZE
381: IWORK( I ) = 0
382: 10 CONTINUE
383:
384: ZUSEDL = 1
385: IF(DOL.GT.1) THEN
386: * Set lower bound for use of Z
387: ZUSEDL = DOL-1
388: ENDIF
389: ZUSEDU = M
390: IF(DOU.LT.M) THEN
391: * Set lower bound for use of Z
392: ZUSEDU = DOU+1
393: ENDIF
394: * The width of the part of Z that is used
395: ZUSEDW = ZUSEDU - ZUSEDL + 1
396:
397:
398: CALL ZLASET( 'Full', N, ZUSEDW, CZERO, CZERO,
399: $ Z(1,ZUSEDL), LDZ )
400:
401: EPS = DLAMCH( 'Precision' )
402: RQTOL = TWO * EPS
403: *
404: * Set expert flags for standard code.
405: TRYRQC = .TRUE.
406:
407: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
408: ELSE
409: * Only selected eigenpairs are computed. Since the other evalues
410: * are not refined by RQ iteration, bisection has to compute to full
411: * accuracy.
412: RTOL1 = FOUR * EPS
413: RTOL2 = FOUR * EPS
414: ENDIF
415:
416: * The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
417: * desired eigenvalues. The support of the nonzero eigenvector
418: * entries is contained in the interval IBEGIN:IEND.
419: * Remark that if k eigenpairs are desired, then the eigenvectors
420: * are stored in k contiguous columns of Z.
421:
422: * DONE is the number of eigenvectors already computed
423: DONE = 0
424: IBEGIN = 1
425: WBEGIN = 1
426: DO 170 JBLK = 1, IBLOCK( M )
427: IEND = ISPLIT( JBLK )
428: SIGMA = L( IEND )
429: * Find the eigenvectors of the submatrix indexed IBEGIN
430: * through IEND.
431: WEND = WBEGIN - 1
432: 15 CONTINUE
433: IF( WEND.LT.M ) THEN
434: IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN
435: WEND = WEND + 1
436: GO TO 15
437: END IF
438: END IF
439: IF( WEND.LT.WBEGIN ) THEN
440: IBEGIN = IEND + 1
441: GO TO 170
442: ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN
443: IBEGIN = IEND + 1
444: WBEGIN = WEND + 1
445: GO TO 170
446: END IF
447:
448: * Find local spectral diameter of the block
449: GL = GERS( 2*IBEGIN-1 )
450: GU = GERS( 2*IBEGIN )
451: DO 20 I = IBEGIN+1 , IEND
452: GL = MIN( GERS( 2*I-1 ), GL )
453: GU = MAX( GERS( 2*I ), GU )
454: 20 CONTINUE
455: SPDIAM = GU - GL
456:
457: * OLDIEN is the last index of the previous block
458: OLDIEN = IBEGIN - 1
459: * Calculate the size of the current block
460: IN = IEND - IBEGIN + 1
461: * The number of eigenvalues in the current block
462: IM = WEND - WBEGIN + 1
463:
464: * This is for a 1x1 block
465: IF( IBEGIN.EQ.IEND ) THEN
466: DONE = DONE+1
467: Z( IBEGIN, WBEGIN ) = DCMPLX( ONE, ZERO )
468: ISUPPZ( 2*WBEGIN-1 ) = IBEGIN
469: ISUPPZ( 2*WBEGIN ) = IBEGIN
470: W( WBEGIN ) = W( WBEGIN ) + SIGMA
471: WORK( WBEGIN ) = W( WBEGIN )
472: IBEGIN = IEND + 1
473: WBEGIN = WBEGIN + 1
474: GO TO 170
475: END IF
476:
477: * The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
478: * Note that these can be approximations, in this case, the corresp.
479: * entries of WERR give the size of the uncertainty interval.
480: * The eigenvalue approximations will be refined when necessary as
481: * high relative accuracy is required for the computation of the
482: * corresponding eigenvectors.
483: CALL DCOPY( IM, W( WBEGIN ), 1,
484: $ WORK( WBEGIN ), 1 )
485:
486: * We store in W the eigenvalue approximations w.r.t. the original
487: * matrix T.
488: DO 30 I=1,IM
489: W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA
490: 30 CONTINUE
491:
492:
493: * NDEPTH is the current depth of the representation tree
494: NDEPTH = 0
495: * PARITY is either 1 or 0
496: PARITY = 1
497: * NCLUS is the number of clusters for the next level of the
498: * representation tree, we start with NCLUS = 1 for the root
499: NCLUS = 1
500: IWORK( IINDC1+1 ) = 1
501: IWORK( IINDC1+2 ) = IM
502:
503: * IDONE is the number of eigenvectors already computed in the current
504: * block
505: IDONE = 0
506: * loop while( IDONE.LT.IM )
507: * generate the representation tree for the current block and
508: * compute the eigenvectors
509: 40 CONTINUE
510: IF( IDONE.LT.IM ) THEN
511: * This is a crude protection against infinitely deep trees
512: IF( NDEPTH.GT.M ) THEN
513: INFO = -2
514: RETURN
515: ENDIF
516: * breadth first processing of the current level of the representation
517: * tree: OLDNCL = number of clusters on current level
518: OLDNCL = NCLUS
519: * reset NCLUS to count the number of child clusters
520: NCLUS = 0
521: *
522: PARITY = 1 - PARITY
523: IF( PARITY.EQ.0 ) THEN
524: OLDCLS = IINDC1
525: NEWCLS = IINDC2
526: ELSE
527: OLDCLS = IINDC2
528: NEWCLS = IINDC1
529: END IF
530: * Process the clusters on the current level
531: DO 150 I = 1, OLDNCL
532: J = OLDCLS + 2*I
533: * OLDFST, OLDLST = first, last index of current cluster.
534: * cluster indices start with 1 and are relative
535: * to WBEGIN when accessing W, WGAP, WERR, Z
536: OLDFST = IWORK( J-1 )
537: OLDLST = IWORK( J )
538: IF( NDEPTH.GT.0 ) THEN
539: * Retrieve relatively robust representation (RRR) of cluster
540: * that has been computed at the previous level
541: * The RRR is stored in Z and overwritten once the eigenvectors
542: * have been computed or when the cluster is refined
543:
544: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
545: * Get representation from location of the leftmost evalue
546: * of the cluster
547: J = WBEGIN + OLDFST - 1
548: ELSE
549: IF(WBEGIN+OLDFST-1.LT.DOL) THEN
550: * Get representation from the left end of Z array
551: J = DOL - 1
552: ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN
553: * Get representation from the right end of Z array
554: J = DOU
555: ELSE
556: J = WBEGIN + OLDFST - 1
557: ENDIF
558: ENDIF
559: DO 45 K = 1, IN - 1
560: D( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
561: $ J ) )
562: L( IBEGIN+K-1 ) = DBLE( Z( IBEGIN+K-1,
563: $ J+1 ) )
564: 45 CONTINUE
565: D( IEND ) = DBLE( Z( IEND, J ) )
566: SIGMA = DBLE( Z( IEND, J+1 ) )
567:
568: * Set the corresponding entries in Z to zero
569: CALL ZLASET( 'Full', IN, 2, CZERO, CZERO,
570: $ Z( IBEGIN, J), LDZ )
571: END IF
572:
573: * Compute DL and DLL of current RRR
574: DO 50 J = IBEGIN, IEND-1
575: TMP = D( J )*L( J )
576: WORK( INDLD-1+J ) = TMP
577: WORK( INDLLD-1+J ) = TMP*L( J )
578: 50 CONTINUE
579:
580: IF( NDEPTH.GT.0 ) THEN
581: * P and Q are index of the first and last eigenvalue to compute
582: * within the current block
583: P = INDEXW( WBEGIN-1+OLDFST )
584: Q = INDEXW( WBEGIN-1+OLDLST )
585: * Offset for the arrays WORK, WGAP and WERR, i.e., the P-OFFSET
586: * through the Q-OFFSET elements of these arrays are to be used.
587: * OFFSET = P-OLDFST
588: OFFSET = INDEXW( WBEGIN ) - 1
589: * perform limited bisection (if necessary) to get approximate
590: * eigenvalues to the precision needed.
591: CALL DLARRB( IN, D( IBEGIN ),
592: $ WORK(INDLLD+IBEGIN-1),
593: $ P, Q, RTOL1, RTOL2, OFFSET,
594: $ WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN),
595: $ WORK( INDWRK ), IWORK( IINDWK ),
596: $ PIVMIN, SPDIAM, IN, IINFO )
597: IF( IINFO.NE.0 ) THEN
598: INFO = -1
599: RETURN
600: ENDIF
601: * We also recompute the extremal gaps. W holds all eigenvalues
602: * of the unshifted matrix and must be used for computation
603: * of WGAP, the entries of WORK might stem from RRRs with
604: * different shifts. The gaps from WBEGIN-1+OLDFST to
605: * WBEGIN-1+OLDLST are correctly computed in DLARRB.
606: * However, we only allow the gaps to become greater since
607: * this is what should happen when we decrease WERR
608: IF( OLDFST.GT.1) THEN
609: WGAP( WBEGIN+OLDFST-2 ) =
610: $ MAX(WGAP(WBEGIN+OLDFST-2),
611: $ W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1)
612: $ - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) )
613: ENDIF
614: IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN
615: WGAP( WBEGIN+OLDLST-1 ) =
616: $ MAX(WGAP(WBEGIN+OLDLST-1),
617: $ W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST)
618: $ - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) )
619: ENDIF
620: * Each time the eigenvalues in WORK get refined, we store
621: * the newly found approximation with all shifts applied in W
622: DO 53 J=OLDFST,OLDLST
623: W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA
624: 53 CONTINUE
625: END IF
626:
627: * Process the current node.
628: NEWFST = OLDFST
629: DO 140 J = OLDFST, OLDLST
630: IF( J.EQ.OLDLST ) THEN
631: * we are at the right end of the cluster, this is also the
632: * boundary of the child cluster
633: NEWLST = J
634: ELSE IF ( WGAP( WBEGIN + J -1).GE.
635: $ MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN
636: * the right relative gap is big enough, the child cluster
637: * (NEWFST,..,NEWLST) is well separated from the following
638: NEWLST = J
639: ELSE
640: * inside a child cluster, the relative gap is not
641: * big enough.
642: GOTO 140
643: END IF
644:
645: * Compute size of child cluster found
646: NEWSIZ = NEWLST - NEWFST + 1
647:
648: * NEWFTT is the place in Z where the new RRR or the computed
649: * eigenvector is to be stored
650: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
651: * Store representation at location of the leftmost evalue
652: * of the cluster
653: NEWFTT = WBEGIN + NEWFST - 1
654: ELSE
655: IF(WBEGIN+NEWFST-1.LT.DOL) THEN
656: * Store representation at the left end of Z array
657: NEWFTT = DOL - 1
658: ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN
659: * Store representation at the right end of Z array
660: NEWFTT = DOU
661: ELSE
662: NEWFTT = WBEGIN + NEWFST - 1
663: ENDIF
664: ENDIF
665:
666: IF( NEWSIZ.GT.1) THEN
667: *
668: * Current child is not a singleton but a cluster.
669: * Compute and store new representation of child.
670: *
671: *
672: * Compute left and right cluster gap.
673: *
674: * LGAP and RGAP are not computed from WORK because
675: * the eigenvalue approximations may stem from RRRs
676: * different shifts. However, W hold all eigenvalues
677: * of the unshifted matrix. Still, the entries in WGAP
678: * have to be computed from WORK since the entries
679: * in W might be of the same order so that gaps are not
680: * exhibited correctly for very close eigenvalues.
681: IF( NEWFST.EQ.1 ) THEN
682: LGAP = MAX( ZERO,
683: $ W(WBEGIN)-WERR(WBEGIN) - VL )
684: ELSE
685: LGAP = WGAP( WBEGIN+NEWFST-2 )
686: ENDIF
687: RGAP = WGAP( WBEGIN+NEWLST-1 )
688: *
689: * Compute left- and rightmost eigenvalue of child
690: * to high precision in order to shift as close
691: * as possible and obtain as large relative gaps
692: * as possible
693: *
694: DO 55 K =1,2
695: IF(K.EQ.1) THEN
696: P = INDEXW( WBEGIN-1+NEWFST )
697: ELSE
698: P = INDEXW( WBEGIN-1+NEWLST )
699: ENDIF
700: OFFSET = INDEXW( WBEGIN ) - 1
701: CALL DLARRB( IN, D(IBEGIN),
702: $ WORK( INDLLD+IBEGIN-1 ),P,P,
703: $ RQTOL, RQTOL, OFFSET,
704: $ WORK(WBEGIN),WGAP(WBEGIN),
705: $ WERR(WBEGIN),WORK( INDWRK ),
706: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
707: $ IN, IINFO )
708: 55 CONTINUE
709: *
710: IF((WBEGIN+NEWLST-1.LT.DOL).OR.
711: $ (WBEGIN+NEWFST-1.GT.DOU)) THEN
712: * if the cluster contains no desired eigenvalues
713: * skip the computation of that branch of the rep. tree
714: *
715: * We could skip before the refinement of the extremal
716: * eigenvalues of the child, but then the representation
717: * tree could be different from the one when nothing is
718: * skipped. For this reason we skip at this place.
719: IDONE = IDONE + NEWLST - NEWFST + 1
720: GOTO 139
721: ENDIF
722: *
723: * Compute RRR of child cluster.
724: * Note that the new RRR is stored in Z
725: *
726: * DLARRF needs LWORK = 2*N
727: CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
728: $ WORK(INDLD+IBEGIN-1),
729: $ NEWFST, NEWLST, WORK(WBEGIN),
730: $ WGAP(WBEGIN), WERR(WBEGIN),
731: $ SPDIAM, LGAP, RGAP, PIVMIN, TAU,
732: $ WORK( INDIN1 ), WORK( INDIN2 ),
733: $ WORK( INDWRK ), IINFO )
734: * In the complex case, DLARRF cannot write
735: * the new RRR directly into Z and needs an intermediate
736: * workspace
737: DO 56 K = 1, IN-1
738: Z( IBEGIN+K-1, NEWFTT ) =
739: $ DCMPLX( WORK( INDIN1+K-1 ), ZERO )
740: Z( IBEGIN+K-1, NEWFTT+1 ) =
741: $ DCMPLX( WORK( INDIN2+K-1 ), ZERO )
742: 56 CONTINUE
743: Z( IEND, NEWFTT ) =
744: $ DCMPLX( WORK( INDIN1+IN-1 ), ZERO )
745: IF( IINFO.EQ.0 ) THEN
746: * a new RRR for the cluster was found by DLARRF
747: * update shift and store it
748: SSIGMA = SIGMA + TAU
749: Z( IEND, NEWFTT+1 ) = DCMPLX( SSIGMA, ZERO )
750: * WORK() are the midpoints and WERR() the semi-width
751: * Note that the entries in W are unchanged.
752: DO 116 K = NEWFST, NEWLST
753: FUDGE =
754: $ THREE*EPS*ABS(WORK(WBEGIN+K-1))
755: WORK( WBEGIN + K - 1 ) =
756: $ WORK( WBEGIN + K - 1) - TAU
757: FUDGE = FUDGE +
758: $ FOUR*EPS*ABS(WORK(WBEGIN+K-1))
759: * Fudge errors
760: WERR( WBEGIN + K - 1 ) =
761: $ WERR( WBEGIN + K - 1 ) + FUDGE
762: * Gaps are not fudged. Provided that WERR is small
763: * when eigenvalues are close, a zero gap indicates
764: * that a new representation is needed for resolving
765: * the cluster. A fudge could lead to a wrong decision
766: * of judging eigenvalues 'separated' which in
767: * reality are not. This could have a negative impact
768: * on the orthogonality of the computed eigenvectors.
769: 116 CONTINUE
770:
771: NCLUS = NCLUS + 1
772: K = NEWCLS + 2*NCLUS
773: IWORK( K-1 ) = NEWFST
774: IWORK( K ) = NEWLST
775: ELSE
776: INFO = -2
777: RETURN
778: ENDIF
779: ELSE
780: *
781: * Compute eigenvector of singleton
782: *
783: ITER = 0
784: *
785: TOL = FOUR * LOG(DBLE(IN)) * EPS
786: *
787: K = NEWFST
788: WINDEX = WBEGIN + K - 1
789: WINDMN = MAX(WINDEX - 1,1)
790: WINDPL = MIN(WINDEX + 1,M)
791: LAMBDA = WORK( WINDEX )
792: DONE = DONE + 1
793: * Check if eigenvector computation is to be skipped
794: IF((WINDEX.LT.DOL).OR.
795: $ (WINDEX.GT.DOU)) THEN
796: ESKIP = .TRUE.
797: GOTO 125
798: ELSE
799: ESKIP = .FALSE.
800: ENDIF
801: LEFT = WORK( WINDEX ) - WERR( WINDEX )
802: RIGHT = WORK( WINDEX ) + WERR( WINDEX )
803: INDEIG = INDEXW( WINDEX )
804: * Note that since we compute the eigenpairs for a child,
805: * all eigenvalue approximations are w.r.t the same shift.
806: * In this case, the entries in WORK should be used for
807: * computing the gaps since they exhibit even very small
808: * differences in the eigenvalues, as opposed to the
809: * entries in W which might "look" the same.
810:
811: IF( K .EQ. 1) THEN
812: * In the case RANGE='I' and with not much initial
813: * accuracy in LAMBDA and VL, the formula
814: * LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
815: * can lead to an overestimation of the left gap and
816: * thus to inadequately early RQI 'convergence'.
817: * Prevent this by forcing a small left gap.
818: LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
819: ELSE
820: LGAP = WGAP(WINDMN)
821: ENDIF
822: IF( K .EQ. IM) THEN
823: * In the case RANGE='I' and with not much initial
824: * accuracy in LAMBDA and VU, the formula
825: * can lead to an overestimation of the right gap and
826: * thus to inadequately early RQI 'convergence'.
827: * Prevent this by forcing a small right gap.
828: RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
829: ELSE
830: RGAP = WGAP(WINDEX)
831: ENDIF
832: GAP = MIN( LGAP, RGAP )
833: IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN
834: * The eigenvector support can become wrong
835: * because significant entries could be cut off due to a
836: * large GAPTOL parameter in LAR1V. Prevent this.
837: GAPTOL = ZERO
838: ELSE
839: GAPTOL = GAP * EPS
840: ENDIF
841: ISUPMN = IN
842: ISUPMX = 1
843: * Update WGAP so that it holds the minimum gap
844: * to the left or the right. This is crucial in the
845: * case where bisection is used to ensure that the
846: * eigenvalue is refined up to the required precision.
847: * The correct value is restored afterwards.
848: SAVGAP = WGAP(WINDEX)
849: WGAP(WINDEX) = GAP
850: * We want to use the Rayleigh Quotient Correction
851: * as often as possible since it converges quadratically
852: * when we are close enough to the desired eigenvalue.
853: * However, the Rayleigh Quotient can have the wrong sign
854: * and lead us away from the desired eigenvalue. In this
855: * case, the best we can do is to use bisection.
856: USEDBS = .FALSE.
857: USEDRQ = .FALSE.
858: * Bisection is initially turned off unless it is forced
859: NEEDBS = .NOT.TRYRQC
860: 120 CONTINUE
861: * Check if bisection should be used to refine eigenvalue
862: IF(NEEDBS) THEN
863: * Take the bisection as new iterate
864: USEDBS = .TRUE.
865: ITMP1 = IWORK( IINDR+WINDEX )
866: OFFSET = INDEXW( WBEGIN ) - 1
867: CALL DLARRB( IN, D(IBEGIN),
868: $ WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG,
869: $ ZERO, TWO*EPS, OFFSET,
870: $ WORK(WBEGIN),WGAP(WBEGIN),
871: $ WERR(WBEGIN),WORK( INDWRK ),
872: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
873: $ ITMP1, IINFO )
874: IF( IINFO.NE.0 ) THEN
875: INFO = -3
876: RETURN
877: ENDIF
878: LAMBDA = WORK( WINDEX )
879: * Reset twist index from inaccurate LAMBDA to
880: * force computation of true MINGMA
881: IWORK( IINDR+WINDEX ) = 0
882: ENDIF
883: * Given LAMBDA, compute the eigenvector.
884: CALL ZLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
885: $ L( IBEGIN ), WORK(INDLD+IBEGIN-1),
886: $ WORK(INDLLD+IBEGIN-1),
887: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
888: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
889: $ IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ),
890: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
891: IF(ITER .EQ. 0) THEN
892: BSTRES = RESID
893: BSTW = LAMBDA
894: ELSEIF(RESID.LT.BSTRES) THEN
895: BSTRES = RESID
896: BSTW = LAMBDA
897: ENDIF
898: ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 ))
899: ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX ))
900: ITER = ITER + 1
901:
902: * sin alpha <= |resid|/gap
903: * Note that both the residual and the gap are
904: * proportional to the matrix, so ||T|| doesn't play
905: * a role in the quotient
906:
907: *
908: * Convergence test for Rayleigh-Quotient iteration
909: * (omitted when Bisection has been used)
910: *
911: IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
912: $ RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS)
913: $ THEN
914: * We need to check that the RQCORR update doesn't
915: * move the eigenvalue away from the desired one and
916: * towards a neighbor. -> protection with bisection
917: IF(INDEIG.LE.NEGCNT) THEN
918: * The wanted eigenvalue lies to the left
919: SGNDEF = -ONE
920: ELSE
921: * The wanted eigenvalue lies to the right
922: SGNDEF = ONE
923: ENDIF
924: * We only use the RQCORR if it improves the
925: * the iterate reasonably.
926: IF( ( RQCORR*SGNDEF.GE.ZERO )
927: $ .AND.( LAMBDA + RQCORR.LE. RIGHT)
928: $ .AND.( LAMBDA + RQCORR.GE. LEFT)
929: $ ) THEN
930: USEDRQ = .TRUE.
931: * Store new midpoint of bisection interval in WORK
932: IF(SGNDEF.EQ.ONE) THEN
933: * The current LAMBDA is on the left of the true
934: * eigenvalue
935: LEFT = LAMBDA
936: * We prefer to assume that the error estimate
937: * is correct. We could make the interval not
938: * as a bracket but to be modified if the RQCORR
939: * chooses to. In this case, the RIGHT side should
940: * be modified as follows:
941: * RIGHT = MAX(RIGHT, LAMBDA + RQCORR)
942: ELSE
943: * The current LAMBDA is on the right of the true
944: * eigenvalue
945: RIGHT = LAMBDA
946: * See comment about assuming the error estimate is
947: * correct above.
948: * LEFT = MIN(LEFT, LAMBDA + RQCORR)
949: ENDIF
950: WORK( WINDEX ) =
951: $ HALF * (RIGHT + LEFT)
952: * Take RQCORR since it has the correct sign and
953: * improves the iterate reasonably
954: LAMBDA = LAMBDA + RQCORR
955: * Update width of error interval
956: WERR( WINDEX ) =
957: $ HALF * (RIGHT-LEFT)
958: ELSE
959: NEEDBS = .TRUE.
960: ENDIF
961: IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN
962: * The eigenvalue is computed to bisection accuracy
963: * compute eigenvector and stop
964: USEDBS = .TRUE.
965: GOTO 120
966: ELSEIF( ITER.LT.MAXITR ) THEN
967: GOTO 120
968: ELSEIF( ITER.EQ.MAXITR ) THEN
969: NEEDBS = .TRUE.
970: GOTO 120
971: ELSE
972: INFO = 5
973: RETURN
974: END IF
975: ELSE
976: STP2II = .FALSE.
977: IF(USEDRQ .AND. USEDBS .AND.
978: $ BSTRES.LE.RESID) THEN
979: LAMBDA = BSTW
980: STP2II = .TRUE.
981: ENDIF
982: IF (STP2II) THEN
983: * improve error angle by second step
984: CALL ZLAR1V( IN, 1, IN, LAMBDA,
985: $ D( IBEGIN ), L( IBEGIN ),
986: $ WORK(INDLD+IBEGIN-1),
987: $ WORK(INDLLD+IBEGIN-1),
988: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
989: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
990: $ IWORK( IINDR+WINDEX ),
991: $ ISUPPZ( 2*WINDEX-1 ),
992: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
993: ENDIF
994: WORK( WINDEX ) = LAMBDA
995: END IF
996: *
997: * Compute FP-vector support w.r.t. whole matrix
998: *
999: ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN
1000: ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN
1001: ZFROM = ISUPPZ( 2*WINDEX-1 )
1002: ZTO = ISUPPZ( 2*WINDEX )
1003: ISUPMN = ISUPMN + OLDIEN
1004: ISUPMX = ISUPMX + OLDIEN
1005: * Ensure vector is ok if support in the RQI has changed
1006: IF(ISUPMN.LT.ZFROM) THEN
1007: DO 122 II = ISUPMN,ZFROM-1
1008: Z( II, WINDEX ) = ZERO
1009: 122 CONTINUE
1010: ENDIF
1011: IF(ISUPMX.GT.ZTO) THEN
1012: DO 123 II = ZTO+1,ISUPMX
1013: Z( II, WINDEX ) = ZERO
1014: 123 CONTINUE
1015: ENDIF
1016: CALL ZDSCAL( ZTO-ZFROM+1, NRMINV,
1017: $ Z( ZFROM, WINDEX ), 1 )
1018: 125 CONTINUE
1019: * Update W
1020: W( WINDEX ) = LAMBDA+SIGMA
1021: * Recompute the gaps on the left and right
1022: * But only allow them to become larger and not
1023: * smaller (which can only happen through "bad"
1024: * cancellation and doesn't reflect the theory
1025: * where the initial gaps are underestimated due
1026: * to WERR being too crude.)
1027: IF(.NOT.ESKIP) THEN
1028: IF( K.GT.1) THEN
1029: WGAP( WINDMN ) = MAX( WGAP(WINDMN),
1030: $ W(WINDEX)-WERR(WINDEX)
1031: $ - W(WINDMN)-WERR(WINDMN) )
1032: ENDIF
1033: IF( WINDEX.LT.WEND ) THEN
1034: WGAP( WINDEX ) = MAX( SAVGAP,
1035: $ W( WINDPL )-WERR( WINDPL )
1036: $ - W( WINDEX )-WERR( WINDEX) )
1037: ENDIF
1038: ENDIF
1039: IDONE = IDONE + 1
1040: ENDIF
1041: * here ends the code for the current child
1042: *
1043: 139 CONTINUE
1044: * Proceed to any remaining child nodes
1045: NEWFST = J + 1
1046: 140 CONTINUE
1047: 150 CONTINUE
1048: NDEPTH = NDEPTH + 1
1049: GO TO 40
1050: END IF
1051: IBEGIN = IEND + 1
1052: WBEGIN = WEND + 1
1053: 170 CONTINUE
1054: *
1055:
1056: RETURN
1057: *
1058: * End of ZLARRV
1059: *
1060: END
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