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1.1 bertrand 1: SUBROUTINE DLARRV( N, VL, VU, D, L, PIVMIN,
2: $ ISPLIT, M, DOL, DOU, MINRGP,
3: $ RTOL1, RTOL2, W, WERR, WGAP,
4: $ IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ,
5: $ WORK, IWORK, INFO )
6: *
7: * -- LAPACK auxiliary routine (version 3.2) --
8: * -- LAPACK is a software package provided by Univ. of Tennessee, --
9: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
10: * November 2006
11: *
12: * .. Scalar Arguments ..
13: INTEGER DOL, DOU, INFO, LDZ, M, N
14: DOUBLE PRECISION MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
15: * ..
16: * .. Array Arguments ..
17: INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ),
18: $ ISUPPZ( * ), IWORK( * )
19: DOUBLE PRECISION D( * ), GERS( * ), L( * ), W( * ), WERR( * ),
20: $ WGAP( * ), WORK( * )
21: DOUBLE PRECISION Z( LDZ, * )
22: * ..
23: *
24: * Purpose
25: * =======
26: *
27: * DLARRV computes the eigenvectors of the tridiagonal matrix
28: * T = L D L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L^T.
29: * The input eigenvalues should have been computed by DLARRE.
30: *
31: * Arguments
32: * =========
33: *
34: * N (input) INTEGER
35: * The order of the matrix. N >= 0.
36: *
37: * VL (input) DOUBLE PRECISION
38: * VU (input) DOUBLE PRECISION
39: * Lower and upper bounds of the interval that contains the desired
40: * eigenvalues. VL < VU. Needed to compute gaps on the left or right
41: * end of the extremal eigenvalues in the desired RANGE.
42: *
43: * D (input/output) DOUBLE PRECISION array, dimension (N)
44: * On entry, the N diagonal elements of the diagonal matrix D.
45: * On exit, D may be overwritten.
46: *
47: * L (input/output) DOUBLE PRECISION array, dimension (N)
48: * On entry, the (N-1) subdiagonal elements of the unit
49: * bidiagonal matrix L are in elements 1 to N-1 of L
50: * (if the matrix is not splitted.) At the end of each block
51: * is stored the corresponding shift as given by DLARRE.
52: * On exit, L is overwritten.
53: *
54: * PIVMIN (in) DOUBLE PRECISION
55: * The minimum pivot allowed in the Sturm sequence.
56: *
57: * ISPLIT (input) INTEGER array, dimension (N)
58: * The splitting points, at which T breaks up into blocks.
59: * The first block consists of rows/columns 1 to
60: * ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
61: * through ISPLIT( 2 ), etc.
62: *
63: * M (input) INTEGER
64: * The total number of input eigenvalues. 0 <= M <= N.
65: *
66: * DOL (input) INTEGER
67: * DOU (input) INTEGER
68: * If the user wants to compute only selected eigenvectors from all
69: * the eigenvalues supplied, he can specify an index range DOL:DOU.
70: * Or else the setting DOL=1, DOU=M should be applied.
71: * Note that DOL and DOU refer to the order in which the eigenvalues
72: * are stored in W.
73: * If the user wants to compute only selected eigenpairs, then
74: * the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
75: * computed eigenvectors. All other columns of Z are set to zero.
76: *
77: * MINRGP (input) DOUBLE PRECISION
78: *
79: * RTOL1 (input) DOUBLE PRECISION
80: * RTOL2 (input) DOUBLE PRECISION
81: * Parameters for bisection.
82: * An interval [LEFT,RIGHT] has converged if
83: * RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
84: *
85: * W (input/output) DOUBLE PRECISION array, dimension (N)
86: * The first M elements of W contain the APPROXIMATE eigenvalues for
87: * which eigenvectors are to be computed. The eigenvalues
88: * should be grouped by split-off block and ordered from
89: * smallest to largest within the block ( The output array
90: * W from DLARRE is expected here ). Furthermore, they are with
91: * respect to the shift of the corresponding root representation
92: * for their block. On exit, W holds the eigenvalues of the
93: * UNshifted matrix.
94: *
95: * WERR (input/output) DOUBLE PRECISION array, dimension (N)
96: * The first M elements contain the semiwidth of the uncertainty
97: * interval of the corresponding eigenvalue in W
98: *
99: * WGAP (input/output) DOUBLE PRECISION array, dimension (N)
100: * The separation from the right neighbor eigenvalue in W.
101: *
102: * IBLOCK (input) INTEGER array, dimension (N)
103: * The indices of the blocks (submatrices) associated with the
104: * corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
105: * W(i) belongs to the first block from the top, =2 if W(i)
106: * belongs to the second block, etc.
107: *
108: * INDEXW (input) INTEGER array, dimension (N)
109: * The indices of the eigenvalues within each block (submatrix);
110: * for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
111: * i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.
112: *
113: * GERS (input) DOUBLE PRECISION array, dimension (2*N)
114: * The N Gerschgorin intervals (the i-th Gerschgorin interval
115: * is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
116: * be computed from the original UNshifted matrix.
117: *
118: * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
119: * If INFO = 0, the first M columns of Z contain the
120: * orthonormal eigenvectors of the matrix T
121: * corresponding to the input eigenvalues, with the i-th
122: * column of Z holding the eigenvector associated with W(i).
123: * Note: the user must ensure that at least max(1,M) columns are
124: * supplied in the array Z.
125: *
126: * LDZ (input) INTEGER
127: * The leading dimension of the array Z. LDZ >= 1, and if
128: * JOBZ = 'V', LDZ >= max(1,N).
129: *
130: * ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
131: * The support of the eigenvectors in Z, i.e., the indices
132: * indicating the nonzero elements in Z. The I-th eigenvector
133: * is nonzero only in elements ISUPPZ( 2*I-1 ) through
134: * ISUPPZ( 2*I ).
135: *
136: * WORK (workspace) DOUBLE PRECISION array, dimension (12*N)
137: *
138: * IWORK (workspace) INTEGER array, dimension (7*N)
139: *
140: * INFO (output) INTEGER
141: * = 0: successful exit
142: *
143: * > 0: A problem occured in DLARRV.
144: * < 0: One of the called subroutines signaled an internal problem.
145: * Needs inspection of the corresponding parameter IINFO
146: * for further information.
147: *
148: * =-1: Problem in DLARRB when refining a child's eigenvalues.
149: * =-2: Problem in DLARRF when computing the RRR of a child.
150: * When a child is inside a tight cluster, it can be difficult
151: * to find an RRR. A partial remedy from the user's point of
152: * view is to make the parameter MINRGP smaller and recompile.
153: * However, as the orthogonality of the computed vectors is
154: * proportional to 1/MINRGP, the user should be aware that
155: * he might be trading in precision when he decreases MINRGP.
156: * =-3: Problem in DLARRB when refining a single eigenvalue
157: * after the Rayleigh correction was rejected.
158: * = 5: The Rayleigh Quotient Iteration failed to converge to
159: * full accuracy in MAXITR steps.
160: *
161: * Further Details
162: * ===============
163: *
164: * Based on contributions by
165: * Beresford Parlett, University of California, Berkeley, USA
166: * Jim Demmel, University of California, Berkeley, USA
167: * Inderjit Dhillon, University of Texas, Austin, USA
168: * Osni Marques, LBNL/NERSC, USA
169: * Christof Voemel, University of California, Berkeley, USA
170: *
171: * =====================================================================
172: *
173: * .. Parameters ..
174: INTEGER MAXITR
175: PARAMETER ( MAXITR = 10 )
176: DOUBLE PRECISION ZERO, ONE, TWO, THREE, FOUR, HALF
177: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
178: $ TWO = 2.0D0, THREE = 3.0D0,
179: $ FOUR = 4.0D0, HALF = 0.5D0)
180: * ..
181: * .. Local Scalars ..
182: LOGICAL ESKIP, NEEDBS, STP2II, TRYRQC, USEDBS, USEDRQ
183: INTEGER DONE, I, IBEGIN, IDONE, IEND, II, IINDC1,
184: $ IINDC2, IINDR, IINDWK, IINFO, IM, IN, INDEIG,
185: $ INDLD, INDLLD, INDWRK, ISUPMN, ISUPMX, ITER,
186: $ ITMP1, J, JBLK, K, MINIWSIZE, MINWSIZE, NCLUS,
187: $ NDEPTH, NEGCNT, NEWCLS, NEWFST, NEWFTT, NEWLST,
188: $ NEWSIZ, OFFSET, OLDCLS, OLDFST, OLDIEN, OLDLST,
189: $ OLDNCL, P, PARITY, Q, WBEGIN, WEND, WINDEX,
190: $ WINDMN, WINDPL, ZFROM, ZTO, ZUSEDL, ZUSEDU,
191: $ ZUSEDW
192: DOUBLE PRECISION BSTRES, BSTW, EPS, FUDGE, GAP, GAPTOL, GL, GU,
193: $ LAMBDA, LEFT, LGAP, MINGMA, NRMINV, RESID,
194: $ RGAP, RIGHT, RQCORR, RQTOL, SAVGAP, SGNDEF,
195: $ SIGMA, SPDIAM, SSIGMA, TAU, TMP, TOL, ZTZ
196: * ..
197: * .. External Functions ..
198: DOUBLE PRECISION DLAMCH
199: EXTERNAL DLAMCH
200: * ..
201: * .. External Subroutines ..
202: EXTERNAL DCOPY, DLAR1V, DLARRB, DLARRF, DLASET,
203: $ DSCAL
204: * ..
205: * .. Intrinsic Functions ..
206: INTRINSIC ABS, DBLE, MAX, MIN
207: * ..
208: * .. Executable Statements ..
209: * ..
210:
211: * The first N entries of WORK are reserved for the eigenvalues
212: INDLD = N+1
213: INDLLD= 2*N+1
214: INDWRK= 3*N+1
215: MINWSIZE = 12 * N
216:
217: DO 5 I= 1,MINWSIZE
218: WORK( I ) = ZERO
219: 5 CONTINUE
220:
221: * IWORK(IINDR+1:IINDR+N) hold the twist indices R for the
222: * factorization used to compute the FP vector
223: IINDR = 0
224: * IWORK(IINDC1+1:IINC2+N) are used to store the clusters of the current
225: * layer and the one above.
226: IINDC1 = N
227: IINDC2 = 2*N
228: IINDWK = 3*N + 1
229:
230: MINIWSIZE = 7 * N
231: DO 10 I= 1,MINIWSIZE
232: IWORK( I ) = 0
233: 10 CONTINUE
234:
235: ZUSEDL = 1
236: IF(DOL.GT.1) THEN
237: * Set lower bound for use of Z
238: ZUSEDL = DOL-1
239: ENDIF
240: ZUSEDU = M
241: IF(DOU.LT.M) THEN
242: * Set lower bound for use of Z
243: ZUSEDU = DOU+1
244: ENDIF
245: * The width of the part of Z that is used
246: ZUSEDW = ZUSEDU - ZUSEDL + 1
247:
248:
249: CALL DLASET( 'Full', N, ZUSEDW, ZERO, ZERO,
250: $ Z(1,ZUSEDL), LDZ )
251:
252: EPS = DLAMCH( 'Precision' )
253: RQTOL = TWO * EPS
254: *
255: * Set expert flags for standard code.
256: TRYRQC = .TRUE.
257:
258: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
259: ELSE
260: * Only selected eigenpairs are computed. Since the other evalues
261: * are not refined by RQ iteration, bisection has to compute to full
262: * accuracy.
263: RTOL1 = FOUR * EPS
264: RTOL2 = FOUR * EPS
265: ENDIF
266:
267: * The entries WBEGIN:WEND in W, WERR, WGAP correspond to the
268: * desired eigenvalues. The support of the nonzero eigenvector
269: * entries is contained in the interval IBEGIN:IEND.
270: * Remark that if k eigenpairs are desired, then the eigenvectors
271: * are stored in k contiguous columns of Z.
272:
273: * DONE is the number of eigenvectors already computed
274: DONE = 0
275: IBEGIN = 1
276: WBEGIN = 1
277: DO 170 JBLK = 1, IBLOCK( M )
278: IEND = ISPLIT( JBLK )
279: SIGMA = L( IEND )
280: * Find the eigenvectors of the submatrix indexed IBEGIN
281: * through IEND.
282: WEND = WBEGIN - 1
283: 15 CONTINUE
284: IF( WEND.LT.M ) THEN
285: IF( IBLOCK( WEND+1 ).EQ.JBLK ) THEN
286: WEND = WEND + 1
287: GO TO 15
288: END IF
289: END IF
290: IF( WEND.LT.WBEGIN ) THEN
291: IBEGIN = IEND + 1
292: GO TO 170
293: ELSEIF( (WEND.LT.DOL).OR.(WBEGIN.GT.DOU) ) THEN
294: IBEGIN = IEND + 1
295: WBEGIN = WEND + 1
296: GO TO 170
297: END IF
298:
299: * Find local spectral diameter of the block
300: GL = GERS( 2*IBEGIN-1 )
301: GU = GERS( 2*IBEGIN )
302: DO 20 I = IBEGIN+1 , IEND
303: GL = MIN( GERS( 2*I-1 ), GL )
304: GU = MAX( GERS( 2*I ), GU )
305: 20 CONTINUE
306: SPDIAM = GU - GL
307:
308: * OLDIEN is the last index of the previous block
309: OLDIEN = IBEGIN - 1
310: * Calculate the size of the current block
311: IN = IEND - IBEGIN + 1
312: * The number of eigenvalues in the current block
313: IM = WEND - WBEGIN + 1
314:
315: * This is for a 1x1 block
316: IF( IBEGIN.EQ.IEND ) THEN
317: DONE = DONE+1
318: Z( IBEGIN, WBEGIN ) = ONE
319: ISUPPZ( 2*WBEGIN-1 ) = IBEGIN
320: ISUPPZ( 2*WBEGIN ) = IBEGIN
321: W( WBEGIN ) = W( WBEGIN ) + SIGMA
322: WORK( WBEGIN ) = W( WBEGIN )
323: IBEGIN = IEND + 1
324: WBEGIN = WBEGIN + 1
325: GO TO 170
326: END IF
327:
328: * The desired (shifted) eigenvalues are stored in W(WBEGIN:WEND)
329: * Note that these can be approximations, in this case, the corresp.
330: * entries of WERR give the size of the uncertainty interval.
331: * The eigenvalue approximations will be refined when necessary as
332: * high relative accuracy is required for the computation of the
333: * corresponding eigenvectors.
334: CALL DCOPY( IM, W( WBEGIN ), 1,
335: & WORK( WBEGIN ), 1 )
336:
337: * We store in W the eigenvalue approximations w.r.t. the original
338: * matrix T.
339: DO 30 I=1,IM
340: W(WBEGIN+I-1) = W(WBEGIN+I-1)+SIGMA
341: 30 CONTINUE
342:
343:
344: * NDEPTH is the current depth of the representation tree
345: NDEPTH = 0
346: * PARITY is either 1 or 0
347: PARITY = 1
348: * NCLUS is the number of clusters for the next level of the
349: * representation tree, we start with NCLUS = 1 for the root
350: NCLUS = 1
351: IWORK( IINDC1+1 ) = 1
352: IWORK( IINDC1+2 ) = IM
353:
354: * IDONE is the number of eigenvectors already computed in the current
355: * block
356: IDONE = 0
357: * loop while( IDONE.LT.IM )
358: * generate the representation tree for the current block and
359: * compute the eigenvectors
360: 40 CONTINUE
361: IF( IDONE.LT.IM ) THEN
362: * This is a crude protection against infinitely deep trees
363: IF( NDEPTH.GT.M ) THEN
364: INFO = -2
365: RETURN
366: ENDIF
367: * breadth first processing of the current level of the representation
368: * tree: OLDNCL = number of clusters on current level
369: OLDNCL = NCLUS
370: * reset NCLUS to count the number of child clusters
371: NCLUS = 0
372: *
373: PARITY = 1 - PARITY
374: IF( PARITY.EQ.0 ) THEN
375: OLDCLS = IINDC1
376: NEWCLS = IINDC2
377: ELSE
378: OLDCLS = IINDC2
379: NEWCLS = IINDC1
380: END IF
381: * Process the clusters on the current level
382: DO 150 I = 1, OLDNCL
383: J = OLDCLS + 2*I
384: * OLDFST, OLDLST = first, last index of current cluster.
385: * cluster indices start with 1 and are relative
386: * to WBEGIN when accessing W, WGAP, WERR, Z
387: OLDFST = IWORK( J-1 )
388: OLDLST = IWORK( J )
389: IF( NDEPTH.GT.0 ) THEN
390: * Retrieve relatively robust representation (RRR) of cluster
391: * that has been computed at the previous level
392: * The RRR is stored in Z and overwritten once the eigenvectors
393: * have been computed or when the cluster is refined
394:
395: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
396: * Get representation from location of the leftmost evalue
397: * of the cluster
398: J = WBEGIN + OLDFST - 1
399: ELSE
400: IF(WBEGIN+OLDFST-1.LT.DOL) THEN
401: * Get representation from the left end of Z array
402: J = DOL - 1
403: ELSEIF(WBEGIN+OLDFST-1.GT.DOU) THEN
404: * Get representation from the right end of Z array
405: J = DOU
406: ELSE
407: J = WBEGIN + OLDFST - 1
408: ENDIF
409: ENDIF
410: CALL DCOPY( IN, Z( IBEGIN, J ), 1, D( IBEGIN ), 1 )
411: CALL DCOPY( IN-1, Z( IBEGIN, J+1 ), 1, L( IBEGIN ),
412: $ 1 )
413: SIGMA = Z( IEND, J+1 )
414:
415: * Set the corresponding entries in Z to zero
416: CALL DLASET( 'Full', IN, 2, ZERO, ZERO,
417: $ Z( IBEGIN, J), LDZ )
418: END IF
419:
420: * Compute DL and DLL of current RRR
421: DO 50 J = IBEGIN, IEND-1
422: TMP = D( J )*L( J )
423: WORK( INDLD-1+J ) = TMP
424: WORK( INDLLD-1+J ) = TMP*L( J )
425: 50 CONTINUE
426:
427: IF( NDEPTH.GT.0 ) THEN
428: * P and Q are index of the first and last eigenvalue to compute
429: * within the current block
430: P = INDEXW( WBEGIN-1+OLDFST )
431: Q = INDEXW( WBEGIN-1+OLDLST )
432: * Offset for the arrays WORK, WGAP and WERR, i.e., th P-OFFSET
433: * thru' Q-OFFSET elements of these arrays are to be used.
434: C OFFSET = P-OLDFST
435: OFFSET = INDEXW( WBEGIN ) - 1
436: * perform limited bisection (if necessary) to get approximate
437: * eigenvalues to the precision needed.
438: CALL DLARRB( IN, D( IBEGIN ),
439: $ WORK(INDLLD+IBEGIN-1),
440: $ P, Q, RTOL1, RTOL2, OFFSET,
441: $ WORK(WBEGIN),WGAP(WBEGIN),WERR(WBEGIN),
442: $ WORK( INDWRK ), IWORK( IINDWK ),
443: $ PIVMIN, SPDIAM, IN, IINFO )
444: IF( IINFO.NE.0 ) THEN
445: INFO = -1
446: RETURN
447: ENDIF
448: * We also recompute the extremal gaps. W holds all eigenvalues
449: * of the unshifted matrix and must be used for computation
450: * of WGAP, the entries of WORK might stem from RRRs with
451: * different shifts. The gaps from WBEGIN-1+OLDFST to
452: * WBEGIN-1+OLDLST are correctly computed in DLARRB.
453: * However, we only allow the gaps to become greater since
454: * this is what should happen when we decrease WERR
455: IF( OLDFST.GT.1) THEN
456: WGAP( WBEGIN+OLDFST-2 ) =
457: $ MAX(WGAP(WBEGIN+OLDFST-2),
458: $ W(WBEGIN+OLDFST-1)-WERR(WBEGIN+OLDFST-1)
459: $ - W(WBEGIN+OLDFST-2)-WERR(WBEGIN+OLDFST-2) )
460: ENDIF
461: IF( WBEGIN + OLDLST -1 .LT. WEND ) THEN
462: WGAP( WBEGIN+OLDLST-1 ) =
463: $ MAX(WGAP(WBEGIN+OLDLST-1),
464: $ W(WBEGIN+OLDLST)-WERR(WBEGIN+OLDLST)
465: $ - W(WBEGIN+OLDLST-1)-WERR(WBEGIN+OLDLST-1) )
466: ENDIF
467: * Each time the eigenvalues in WORK get refined, we store
468: * the newly found approximation with all shifts applied in W
469: DO 53 J=OLDFST,OLDLST
470: W(WBEGIN+J-1) = WORK(WBEGIN+J-1)+SIGMA
471: 53 CONTINUE
472: END IF
473:
474: * Process the current node.
475: NEWFST = OLDFST
476: DO 140 J = OLDFST, OLDLST
477: IF( J.EQ.OLDLST ) THEN
478: * we are at the right end of the cluster, this is also the
479: * boundary of the child cluster
480: NEWLST = J
481: ELSE IF ( WGAP( WBEGIN + J -1).GE.
482: $ MINRGP* ABS( WORK(WBEGIN + J -1) ) ) THEN
483: * the right relative gap is big enough, the child cluster
484: * (NEWFST,..,NEWLST) is well separated from the following
485: NEWLST = J
486: ELSE
487: * inside a child cluster, the relative gap is not
488: * big enough.
489: GOTO 140
490: END IF
491:
492: * Compute size of child cluster found
493: NEWSIZ = NEWLST - NEWFST + 1
494:
495: * NEWFTT is the place in Z where the new RRR or the computed
496: * eigenvector is to be stored
497: IF((DOL.EQ.1).AND.(DOU.EQ.M)) THEN
498: * Store representation at location of the leftmost evalue
499: * of the cluster
500: NEWFTT = WBEGIN + NEWFST - 1
501: ELSE
502: IF(WBEGIN+NEWFST-1.LT.DOL) THEN
503: * Store representation at the left end of Z array
504: NEWFTT = DOL - 1
505: ELSEIF(WBEGIN+NEWFST-1.GT.DOU) THEN
506: * Store representation at the right end of Z array
507: NEWFTT = DOU
508: ELSE
509: NEWFTT = WBEGIN + NEWFST - 1
510: ENDIF
511: ENDIF
512:
513: IF( NEWSIZ.GT.1) THEN
514: *
515: * Current child is not a singleton but a cluster.
516: * Compute and store new representation of child.
517: *
518: *
519: * Compute left and right cluster gap.
520: *
521: * LGAP and RGAP are not computed from WORK because
522: * the eigenvalue approximations may stem from RRRs
523: * different shifts. However, W hold all eigenvalues
524: * of the unshifted matrix. Still, the entries in WGAP
525: * have to be computed from WORK since the entries
526: * in W might be of the same order so that gaps are not
527: * exhibited correctly for very close eigenvalues.
528: IF( NEWFST.EQ.1 ) THEN
529: LGAP = MAX( ZERO,
530: $ W(WBEGIN)-WERR(WBEGIN) - VL )
531: ELSE
532: LGAP = WGAP( WBEGIN+NEWFST-2 )
533: ENDIF
534: RGAP = WGAP( WBEGIN+NEWLST-1 )
535: *
536: * Compute left- and rightmost eigenvalue of child
537: * to high precision in order to shift as close
538: * as possible and obtain as large relative gaps
539: * as possible
540: *
541: DO 55 K =1,2
542: IF(K.EQ.1) THEN
543: P = INDEXW( WBEGIN-1+NEWFST )
544: ELSE
545: P = INDEXW( WBEGIN-1+NEWLST )
546: ENDIF
547: OFFSET = INDEXW( WBEGIN ) - 1
548: CALL DLARRB( IN, D(IBEGIN),
549: $ WORK( INDLLD+IBEGIN-1 ),P,P,
550: $ RQTOL, RQTOL, OFFSET,
551: $ WORK(WBEGIN),WGAP(WBEGIN),
552: $ WERR(WBEGIN),WORK( INDWRK ),
553: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
554: $ IN, IINFO )
555: 55 CONTINUE
556: *
557: IF((WBEGIN+NEWLST-1.LT.DOL).OR.
558: $ (WBEGIN+NEWFST-1.GT.DOU)) THEN
559: * if the cluster contains no desired eigenvalues
560: * skip the computation of that branch of the rep. tree
561: *
562: * We could skip before the refinement of the extremal
563: * eigenvalues of the child, but then the representation
564: * tree could be different from the one when nothing is
565: * skipped. For this reason we skip at this place.
566: IDONE = IDONE + NEWLST - NEWFST + 1
567: GOTO 139
568: ENDIF
569: *
570: * Compute RRR of child cluster.
571: * Note that the new RRR is stored in Z
572: *
573: C DLARRF needs LWORK = 2*N
574: CALL DLARRF( IN, D( IBEGIN ), L( IBEGIN ),
575: $ WORK(INDLD+IBEGIN-1),
576: $ NEWFST, NEWLST, WORK(WBEGIN),
577: $ WGAP(WBEGIN), WERR(WBEGIN),
578: $ SPDIAM, LGAP, RGAP, PIVMIN, TAU,
579: $ Z(IBEGIN, NEWFTT),Z(IBEGIN, NEWFTT+1),
580: $ WORK( INDWRK ), IINFO )
581: IF( IINFO.EQ.0 ) THEN
582: * a new RRR for the cluster was found by DLARRF
583: * update shift and store it
584: SSIGMA = SIGMA + TAU
585: Z( IEND, NEWFTT+1 ) = SSIGMA
586: * WORK() are the midpoints and WERR() the semi-width
587: * Note that the entries in W are unchanged.
588: DO 116 K = NEWFST, NEWLST
589: FUDGE =
590: $ THREE*EPS*ABS(WORK(WBEGIN+K-1))
591: WORK( WBEGIN + K - 1 ) =
592: $ WORK( WBEGIN + K - 1) - TAU
593: FUDGE = FUDGE +
594: $ FOUR*EPS*ABS(WORK(WBEGIN+K-1))
595: * Fudge errors
596: WERR( WBEGIN + K - 1 ) =
597: $ WERR( WBEGIN + K - 1 ) + FUDGE
598: * Gaps are not fudged. Provided that WERR is small
599: * when eigenvalues are close, a zero gap indicates
600: * that a new representation is needed for resolving
601: * the cluster. A fudge could lead to a wrong decision
602: * of judging eigenvalues 'separated' which in
603: * reality are not. This could have a negative impact
604: * on the orthogonality of the computed eigenvectors.
605: 116 CONTINUE
606:
607: NCLUS = NCLUS + 1
608: K = NEWCLS + 2*NCLUS
609: IWORK( K-1 ) = NEWFST
610: IWORK( K ) = NEWLST
611: ELSE
612: INFO = -2
613: RETURN
614: ENDIF
615: ELSE
616: *
617: * Compute eigenvector of singleton
618: *
619: ITER = 0
620: *
621: TOL = FOUR * LOG(DBLE(IN)) * EPS
622: *
623: K = NEWFST
624: WINDEX = WBEGIN + K - 1
625: WINDMN = MAX(WINDEX - 1,1)
626: WINDPL = MIN(WINDEX + 1,M)
627: LAMBDA = WORK( WINDEX )
628: DONE = DONE + 1
629: * Check if eigenvector computation is to be skipped
630: IF((WINDEX.LT.DOL).OR.
631: $ (WINDEX.GT.DOU)) THEN
632: ESKIP = .TRUE.
633: GOTO 125
634: ELSE
635: ESKIP = .FALSE.
636: ENDIF
637: LEFT = WORK( WINDEX ) - WERR( WINDEX )
638: RIGHT = WORK( WINDEX ) + WERR( WINDEX )
639: INDEIG = INDEXW( WINDEX )
640: * Note that since we compute the eigenpairs for a child,
641: * all eigenvalue approximations are w.r.t the same shift.
642: * In this case, the entries in WORK should be used for
643: * computing the gaps since they exhibit even very small
644: * differences in the eigenvalues, as opposed to the
645: * entries in W which might "look" the same.
646:
647: IF( K .EQ. 1) THEN
648: * In the case RANGE='I' and with not much initial
649: * accuracy in LAMBDA and VL, the formula
650: * LGAP = MAX( ZERO, (SIGMA - VL) + LAMBDA )
651: * can lead to an overestimation of the left gap and
652: * thus to inadequately early RQI 'convergence'.
653: * Prevent this by forcing a small left gap.
654: LGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
655: ELSE
656: LGAP = WGAP(WINDMN)
657: ENDIF
658: IF( K .EQ. IM) THEN
659: * In the case RANGE='I' and with not much initial
660: * accuracy in LAMBDA and VU, the formula
661: * can lead to an overestimation of the right gap and
662: * thus to inadequately early RQI 'convergence'.
663: * Prevent this by forcing a small right gap.
664: RGAP = EPS*MAX(ABS(LEFT),ABS(RIGHT))
665: ELSE
666: RGAP = WGAP(WINDEX)
667: ENDIF
668: GAP = MIN( LGAP, RGAP )
669: IF(( K .EQ. 1).OR.(K .EQ. IM)) THEN
670: * The eigenvector support can become wrong
671: * because significant entries could be cut off due to a
672: * large GAPTOL parameter in LAR1V. Prevent this.
673: GAPTOL = ZERO
674: ELSE
675: GAPTOL = GAP * EPS
676: ENDIF
677: ISUPMN = IN
678: ISUPMX = 1
679: * Update WGAP so that it holds the minimum gap
680: * to the left or the right. This is crucial in the
681: * case where bisection is used to ensure that the
682: * eigenvalue is refined up to the required precision.
683: * The correct value is restored afterwards.
684: SAVGAP = WGAP(WINDEX)
685: WGAP(WINDEX) = GAP
686: * We want to use the Rayleigh Quotient Correction
687: * as often as possible since it converges quadratically
688: * when we are close enough to the desired eigenvalue.
689: * However, the Rayleigh Quotient can have the wrong sign
690: * and lead us away from the desired eigenvalue. In this
691: * case, the best we can do is to use bisection.
692: USEDBS = .FALSE.
693: USEDRQ = .FALSE.
694: * Bisection is initially turned off unless it is forced
695: NEEDBS = .NOT.TRYRQC
696: 120 CONTINUE
697: * Check if bisection should be used to refine eigenvalue
698: IF(NEEDBS) THEN
699: * Take the bisection as new iterate
700: USEDBS = .TRUE.
701: ITMP1 = IWORK( IINDR+WINDEX )
702: OFFSET = INDEXW( WBEGIN ) - 1
703: CALL DLARRB( IN, D(IBEGIN),
704: $ WORK(INDLLD+IBEGIN-1),INDEIG,INDEIG,
705: $ ZERO, TWO*EPS, OFFSET,
706: $ WORK(WBEGIN),WGAP(WBEGIN),
707: $ WERR(WBEGIN),WORK( INDWRK ),
708: $ IWORK( IINDWK ), PIVMIN, SPDIAM,
709: $ ITMP1, IINFO )
710: IF( IINFO.NE.0 ) THEN
711: INFO = -3
712: RETURN
713: ENDIF
714: LAMBDA = WORK( WINDEX )
715: * Reset twist index from inaccurate LAMBDA to
716: * force computation of true MINGMA
717: IWORK( IINDR+WINDEX ) = 0
718: ENDIF
719: * Given LAMBDA, compute the eigenvector.
720: CALL DLAR1V( IN, 1, IN, LAMBDA, D( IBEGIN ),
721: $ L( IBEGIN ), WORK(INDLD+IBEGIN-1),
722: $ WORK(INDLLD+IBEGIN-1),
723: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
724: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
725: $ IWORK( IINDR+WINDEX ), ISUPPZ( 2*WINDEX-1 ),
726: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
727: IF(ITER .EQ. 0) THEN
728: BSTRES = RESID
729: BSTW = LAMBDA
730: ELSEIF(RESID.LT.BSTRES) THEN
731: BSTRES = RESID
732: BSTW = LAMBDA
733: ENDIF
734: ISUPMN = MIN(ISUPMN,ISUPPZ( 2*WINDEX-1 ))
735: ISUPMX = MAX(ISUPMX,ISUPPZ( 2*WINDEX ))
736: ITER = ITER + 1
737:
738: * sin alpha <= |resid|/gap
739: * Note that both the residual and the gap are
740: * proportional to the matrix, so ||T|| doesn't play
741: * a role in the quotient
742:
743: *
744: * Convergence test for Rayleigh-Quotient iteration
745: * (omitted when Bisection has been used)
746: *
747: IF( RESID.GT.TOL*GAP .AND. ABS( RQCORR ).GT.
748: $ RQTOL*ABS( LAMBDA ) .AND. .NOT. USEDBS)
749: $ THEN
750: * We need to check that the RQCORR update doesn't
751: * move the eigenvalue away from the desired one and
752: * towards a neighbor. -> protection with bisection
753: IF(INDEIG.LE.NEGCNT) THEN
754: * The wanted eigenvalue lies to the left
755: SGNDEF = -ONE
756: ELSE
757: * The wanted eigenvalue lies to the right
758: SGNDEF = ONE
759: ENDIF
760: * We only use the RQCORR if it improves the
761: * the iterate reasonably.
762: IF( ( RQCORR*SGNDEF.GE.ZERO )
763: $ .AND.( LAMBDA + RQCORR.LE. RIGHT)
764: $ .AND.( LAMBDA + RQCORR.GE. LEFT)
765: $ ) THEN
766: USEDRQ = .TRUE.
767: * Store new midpoint of bisection interval in WORK
768: IF(SGNDEF.EQ.ONE) THEN
769: * The current LAMBDA is on the left of the true
770: * eigenvalue
771: LEFT = LAMBDA
772: * We prefer to assume that the error estimate
773: * is correct. We could make the interval not
774: * as a bracket but to be modified if the RQCORR
775: * chooses to. In this case, the RIGHT side should
776: * be modified as follows:
777: * RIGHT = MAX(RIGHT, LAMBDA + RQCORR)
778: ELSE
779: * The current LAMBDA is on the right of the true
780: * eigenvalue
781: RIGHT = LAMBDA
782: * See comment about assuming the error estimate is
783: * correct above.
784: * LEFT = MIN(LEFT, LAMBDA + RQCORR)
785: ENDIF
786: WORK( WINDEX ) =
787: $ HALF * (RIGHT + LEFT)
788: * Take RQCORR since it has the correct sign and
789: * improves the iterate reasonably
790: LAMBDA = LAMBDA + RQCORR
791: * Update width of error interval
792: WERR( WINDEX ) =
793: $ HALF * (RIGHT-LEFT)
794: ELSE
795: NEEDBS = .TRUE.
796: ENDIF
797: IF(RIGHT-LEFT.LT.RQTOL*ABS(LAMBDA)) THEN
798: * The eigenvalue is computed to bisection accuracy
799: * compute eigenvector and stop
800: USEDBS = .TRUE.
801: GOTO 120
802: ELSEIF( ITER.LT.MAXITR ) THEN
803: GOTO 120
804: ELSEIF( ITER.EQ.MAXITR ) THEN
805: NEEDBS = .TRUE.
806: GOTO 120
807: ELSE
808: INFO = 5
809: RETURN
810: END IF
811: ELSE
812: STP2II = .FALSE.
813: IF(USEDRQ .AND. USEDBS .AND.
814: $ BSTRES.LE.RESID) THEN
815: LAMBDA = BSTW
816: STP2II = .TRUE.
817: ENDIF
818: IF (STP2II) THEN
819: * improve error angle by second step
820: CALL DLAR1V( IN, 1, IN, LAMBDA,
821: $ D( IBEGIN ), L( IBEGIN ),
822: $ WORK(INDLD+IBEGIN-1),
823: $ WORK(INDLLD+IBEGIN-1),
824: $ PIVMIN, GAPTOL, Z( IBEGIN, WINDEX ),
825: $ .NOT.USEDBS, NEGCNT, ZTZ, MINGMA,
826: $ IWORK( IINDR+WINDEX ),
827: $ ISUPPZ( 2*WINDEX-1 ),
828: $ NRMINV, RESID, RQCORR, WORK( INDWRK ) )
829: ENDIF
830: WORK( WINDEX ) = LAMBDA
831: END IF
832: *
833: * Compute FP-vector support w.r.t. whole matrix
834: *
835: ISUPPZ( 2*WINDEX-1 ) = ISUPPZ( 2*WINDEX-1 )+OLDIEN
836: ISUPPZ( 2*WINDEX ) = ISUPPZ( 2*WINDEX )+OLDIEN
837: ZFROM = ISUPPZ( 2*WINDEX-1 )
838: ZTO = ISUPPZ( 2*WINDEX )
839: ISUPMN = ISUPMN + OLDIEN
840: ISUPMX = ISUPMX + OLDIEN
841: * Ensure vector is ok if support in the RQI has changed
842: IF(ISUPMN.LT.ZFROM) THEN
843: DO 122 II = ISUPMN,ZFROM-1
844: Z( II, WINDEX ) = ZERO
845: 122 CONTINUE
846: ENDIF
847: IF(ISUPMX.GT.ZTO) THEN
848: DO 123 II = ZTO+1,ISUPMX
849: Z( II, WINDEX ) = ZERO
850: 123 CONTINUE
851: ENDIF
852: CALL DSCAL( ZTO-ZFROM+1, NRMINV,
853: $ Z( ZFROM, WINDEX ), 1 )
854: 125 CONTINUE
855: * Update W
856: W( WINDEX ) = LAMBDA+SIGMA
857: * Recompute the gaps on the left and right
858: * But only allow them to become larger and not
859: * smaller (which can only happen through "bad"
860: * cancellation and doesn't reflect the theory
861: * where the initial gaps are underestimated due
862: * to WERR being too crude.)
863: IF(.NOT.ESKIP) THEN
864: IF( K.GT.1) THEN
865: WGAP( WINDMN ) = MAX( WGAP(WINDMN),
866: $ W(WINDEX)-WERR(WINDEX)
867: $ - W(WINDMN)-WERR(WINDMN) )
868: ENDIF
869: IF( WINDEX.LT.WEND ) THEN
870: WGAP( WINDEX ) = MAX( SAVGAP,
871: $ W( WINDPL )-WERR( WINDPL )
872: $ - W( WINDEX )-WERR( WINDEX) )
873: ENDIF
874: ENDIF
875: IDONE = IDONE + 1
876: ENDIF
877: * here ends the code for the current child
878: *
879: 139 CONTINUE
880: * Proceed to any remaining child nodes
881: NEWFST = J + 1
882: 140 CONTINUE
883: 150 CONTINUE
884: NDEPTH = NDEPTH + 1
885: GO TO 40
886: END IF
887: IBEGIN = IEND + 1
888: WBEGIN = WEND + 1
889: 170 CONTINUE
890: *
891:
892: RETURN
893: *
894: * End of DLARRV
895: *
896: END