1: SUBROUTINE DLAEBZ( IJOB, NITMAX, N, MMAX, MINP, NBMIN, ABSTOL,
2: $ RELTOL, PIVMIN, D, E, E2, NVAL, AB, C, MOUT,
3: $ NAB, WORK, IWORK, INFO )
4: *
5: * -- LAPACK auxiliary routine (version 3.3.1) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * -- April 2011 --
9: *
10: * .. Scalar Arguments ..
11: INTEGER IJOB, INFO, MINP, MMAX, MOUT, N, NBMIN, NITMAX
12: DOUBLE PRECISION ABSTOL, PIVMIN, RELTOL
13: * ..
14: * .. Array Arguments ..
15: INTEGER IWORK( * ), NAB( MMAX, * ), NVAL( * )
16: DOUBLE PRECISION AB( MMAX, * ), C( * ), D( * ), E( * ), E2( * ),
17: $ WORK( * )
18: * ..
19: *
20: * Purpose
21: * =======
22: *
23: * DLAEBZ contains the iteration loops which compute and use the
24: * function N(w), which is the count of eigenvalues of a symmetric
25: * tridiagonal matrix T less than or equal to its argument w. It
26: * performs a choice of two types of loops:
27: *
28: * IJOB=1, followed by
29: * IJOB=2: It takes as input a list of intervals and returns a list of
30: * sufficiently small intervals whose union contains the same
31: * eigenvalues as the union of the original intervals.
32: * The input intervals are (AB(j,1),AB(j,2)], j=1,...,MINP.
33: * The output interval (AB(j,1),AB(j,2)] will contain
34: * eigenvalues NAB(j,1)+1,...,NAB(j,2), where 1 <= j <= MOUT.
35: *
36: * IJOB=3: It performs a binary search in each input interval
37: * (AB(j,1),AB(j,2)] for a point w(j) such that
38: * N(w(j))=NVAL(j), and uses C(j) as the starting point of
39: * the search. If such a w(j) is found, then on output
40: * AB(j,1)=AB(j,2)=w. If no such w(j) is found, then on output
41: * (AB(j,1),AB(j,2)] will be a small interval containing the
42: * point where N(w) jumps through NVAL(j), unless that point
43: * lies outside the initial interval.
44: *
45: * Note that the intervals are in all cases half-open intervals,
46: * i.e., of the form (a,b] , which includes b but not a .
47: *
48: * To avoid underflow, the matrix should be scaled so that its largest
49: * element is no greater than overflow**(1/2) * underflow**(1/4)
50: * in absolute value. To assure the most accurate computation
51: * of small eigenvalues, the matrix should be scaled to be
52: * not much smaller than that, either.
53: *
54: * See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
55: * Matrix", Report CS41, Computer Science Dept., Stanford
56: * University, July 21, 1966
57: *
58: * Note: the arguments are, in general, *not* checked for unreasonable
59: * values.
60: *
61: * Arguments
62: * =========
63: *
64: * IJOB (input) INTEGER
65: * Specifies what is to be done:
66: * = 1: Compute NAB for the initial intervals.
67: * = 2: Perform bisection iteration to find eigenvalues of T.
68: * = 3: Perform bisection iteration to invert N(w), i.e.,
69: * to find a point which has a specified number of
70: * eigenvalues of T to its left.
71: * Other values will cause DLAEBZ to return with INFO=-1.
72: *
73: * NITMAX (input) INTEGER
74: * The maximum number of "levels" of bisection to be
75: * performed, i.e., an interval of width W will not be made
76: * smaller than 2^(-NITMAX) * W. If not all intervals
77: * have converged after NITMAX iterations, then INFO is set
78: * to the number of non-converged intervals.
79: *
80: * N (input) INTEGER
81: * The dimension n of the tridiagonal matrix T. It must be at
82: * least 1.
83: *
84: * MMAX (input) INTEGER
85: * The maximum number of intervals. If more than MMAX intervals
86: * are generated, then DLAEBZ will quit with INFO=MMAX+1.
87: *
88: * MINP (input) INTEGER
89: * The initial number of intervals. It may not be greater than
90: * MMAX.
91: *
92: * NBMIN (input) INTEGER
93: * The smallest number of intervals that should be processed
94: * using a vector loop. If zero, then only the scalar loop
95: * will be used.
96: *
97: * ABSTOL (input) DOUBLE PRECISION
98: * The minimum (absolute) width of an interval. When an
99: * interval is narrower than ABSTOL, or than RELTOL times the
100: * larger (in magnitude) endpoint, then it is considered to be
101: * sufficiently small, i.e., converged. This must be at least
102: * zero.
103: *
104: * RELTOL (input) DOUBLE PRECISION
105: * The minimum relative width of an interval. When an interval
106: * is narrower than ABSTOL, or than RELTOL times the larger (in
107: * magnitude) endpoint, then it is considered to be
108: * sufficiently small, i.e., converged. Note: this should
109: * always be at least radix*machine epsilon.
110: *
111: * PIVMIN (input) DOUBLE PRECISION
112: * The minimum absolute value of a "pivot" in the Sturm
113: * sequence loop. This *must* be at least max |e(j)**2| *
114: * safe_min and at least safe_min, where safe_min is at least
115: * the smallest number that can divide one without overflow.
116: *
117: * D (input) DOUBLE PRECISION array, dimension (N)
118: * The diagonal elements of the tridiagonal matrix T.
119: *
120: * E (input) DOUBLE PRECISION array, dimension (N)
121: * The offdiagonal elements of the tridiagonal matrix T in
122: * positions 1 through N-1. E(N) is arbitrary.
123: *
124: * E2 (input) DOUBLE PRECISION array, dimension (N)
125: * The squares of the offdiagonal elements of the tridiagonal
126: * matrix T. E2(N) is ignored.
127: *
128: * NVAL (input/output) INTEGER array, dimension (MINP)
129: * If IJOB=1 or 2, not referenced.
130: * If IJOB=3, the desired values of N(w). The elements of NVAL
131: * will be reordered to correspond with the intervals in AB.
132: * Thus, NVAL(j) on output will not, in general be the same as
133: * NVAL(j) on input, but it will correspond with the interval
134: * (AB(j,1),AB(j,2)] on output.
135: *
136: * AB (input/output) DOUBLE PRECISION array, dimension (MMAX,2)
137: * The endpoints of the intervals. AB(j,1) is a(j), the left
138: * endpoint of the j-th interval, and AB(j,2) is b(j), the
139: * right endpoint of the j-th interval. The input intervals
140: * will, in general, be modified, split, and reordered by the
141: * calculation.
142: *
143: * C (input/output) DOUBLE PRECISION array, dimension (MMAX)
144: * If IJOB=1, ignored.
145: * If IJOB=2, workspace.
146: * If IJOB=3, then on input C(j) should be initialized to the
147: * first search point in the binary search.
148: *
149: * MOUT (output) INTEGER
150: * If IJOB=1, the number of eigenvalues in the intervals.
151: * If IJOB=2 or 3, the number of intervals output.
152: * If IJOB=3, MOUT will equal MINP.
153: *
154: * NAB (input/output) INTEGER array, dimension (MMAX,2)
155: * If IJOB=1, then on output NAB(i,j) will be set to N(AB(i,j)).
156: * If IJOB=2, then on input, NAB(i,j) should be set. It must
157: * satisfy the condition:
158: * N(AB(i,1)) <= NAB(i,1) <= NAB(i,2) <= N(AB(i,2)),
159: * which means that in interval i only eigenvalues
160: * NAB(i,1)+1,...,NAB(i,2) will be considered. Usually,
161: * NAB(i,j)=N(AB(i,j)), from a previous call to DLAEBZ with
162: * IJOB=1.
163: * On output, NAB(i,j) will contain
164: * max(na(k),min(nb(k),N(AB(i,j)))), where k is the index of
165: * the input interval that the output interval
166: * (AB(j,1),AB(j,2)] came from, and na(k) and nb(k) are the
167: * the input values of NAB(k,1) and NAB(k,2).
168: * If IJOB=3, then on output, NAB(i,j) contains N(AB(i,j)),
169: * unless N(w) > NVAL(i) for all search points w , in which
170: * case NAB(i,1) will not be modified, i.e., the output
171: * value will be the same as the input value (modulo
172: * reorderings -- see NVAL and AB), or unless N(w) < NVAL(i)
173: * for all search points w , in which case NAB(i,2) will
174: * not be modified. Normally, NAB should be set to some
175: * distinctive value(s) before DLAEBZ is called.
176: *
177: * WORK (workspace) DOUBLE PRECISION array, dimension (MMAX)
178: * Workspace.
179: *
180: * IWORK (workspace) INTEGER array, dimension (MMAX)
181: * Workspace.
182: *
183: * INFO (output) INTEGER
184: * = 0: All intervals converged.
185: * = 1--MMAX: The last INFO intervals did not converge.
186: * = MMAX+1: More than MMAX intervals were generated.
187: *
188: * Further Details
189: * ===============
190: *
191: * This routine is intended to be called only by other LAPACK
192: * routines, thus the interface is less user-friendly. It is intended
193: * for two purposes:
194: *
195: * (a) finding eigenvalues. In this case, DLAEBZ should have one or
196: * more initial intervals set up in AB, and DLAEBZ should be called
197: * with IJOB=1. This sets up NAB, and also counts the eigenvalues.
198: * Intervals with no eigenvalues would usually be thrown out at
199: * this point. Also, if not all the eigenvalues in an interval i
200: * are desired, NAB(i,1) can be increased or NAB(i,2) decreased.
201: * For example, set NAB(i,1)=NAB(i,2)-1 to get the largest
202: * eigenvalue. DLAEBZ is then called with IJOB=2 and MMAX
203: * no smaller than the value of MOUT returned by the call with
204: * IJOB=1. After this (IJOB=2) call, eigenvalues NAB(i,1)+1
205: * through NAB(i,2) are approximately AB(i,1) (or AB(i,2)) to the
206: * tolerance specified by ABSTOL and RELTOL.
207: *
208: * (b) finding an interval (a',b'] containing eigenvalues w(f),...,w(l).
209: * In this case, start with a Gershgorin interval (a,b). Set up
210: * AB to contain 2 search intervals, both initially (a,b). One
211: * NVAL element should contain f-1 and the other should contain l
212: * , while C should contain a and b, resp. NAB(i,1) should be -1
213: * and NAB(i,2) should be N+1, to flag an error if the desired
214: * interval does not lie in (a,b). DLAEBZ is then called with
215: * IJOB=3. On exit, if w(f-1) < w(f), then one of the intervals --
216: * j -- will have AB(j,1)=AB(j,2) and NAB(j,1)=NAB(j,2)=f-1, while
217: * if, to the specified tolerance, w(f-k)=...=w(f+r), k > 0 and r
218: * >= 0, then the interval will have N(AB(j,1))=NAB(j,1)=f-k and
219: * N(AB(j,2))=NAB(j,2)=f+r. The cases w(l) < w(l+1) and
220: * w(l-r)=...=w(l+k) are handled similarly.
221: *
222: * =====================================================================
223: *
224: * .. Parameters ..
225: DOUBLE PRECISION ZERO, TWO, HALF
226: PARAMETER ( ZERO = 0.0D0, TWO = 2.0D0,
227: $ HALF = 1.0D0 / TWO )
228: * ..
229: * .. Local Scalars ..
230: INTEGER ITMP1, ITMP2, J, JI, JIT, JP, KF, KFNEW, KL,
231: $ KLNEW
232: DOUBLE PRECISION TMP1, TMP2
233: * ..
234: * .. Intrinsic Functions ..
235: INTRINSIC ABS, MAX, MIN
236: * ..
237: * .. Executable Statements ..
238: *
239: * Check for Errors
240: *
241: INFO = 0
242: IF( IJOB.LT.1 .OR. IJOB.GT.3 ) THEN
243: INFO = -1
244: RETURN
245: END IF
246: *
247: * Initialize NAB
248: *
249: IF( IJOB.EQ.1 ) THEN
250: *
251: * Compute the number of eigenvalues in the initial intervals.
252: *
253: MOUT = 0
254: DO 30 JI = 1, MINP
255: DO 20 JP = 1, 2
256: TMP1 = D( 1 ) - AB( JI, JP )
257: IF( ABS( TMP1 ).LT.PIVMIN )
258: $ TMP1 = -PIVMIN
259: NAB( JI, JP ) = 0
260: IF( TMP1.LE.ZERO )
261: $ NAB( JI, JP ) = 1
262: *
263: DO 10 J = 2, N
264: TMP1 = D( J ) - E2( J-1 ) / TMP1 - AB( JI, JP )
265: IF( ABS( TMP1 ).LT.PIVMIN )
266: $ TMP1 = -PIVMIN
267: IF( TMP1.LE.ZERO )
268: $ NAB( JI, JP ) = NAB( JI, JP ) + 1
269: 10 CONTINUE
270: 20 CONTINUE
271: MOUT = MOUT + NAB( JI, 2 ) - NAB( JI, 1 )
272: 30 CONTINUE
273: RETURN
274: END IF
275: *
276: * Initialize for loop
277: *
278: * KF and KL have the following meaning:
279: * Intervals 1,...,KF-1 have converged.
280: * Intervals KF,...,KL still need to be refined.
281: *
282: KF = 1
283: KL = MINP
284: *
285: * If IJOB=2, initialize C.
286: * If IJOB=3, use the user-supplied starting point.
287: *
288: IF( IJOB.EQ.2 ) THEN
289: DO 40 JI = 1, MINP
290: C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
291: 40 CONTINUE
292: END IF
293: *
294: * Iteration loop
295: *
296: DO 130 JIT = 1, NITMAX
297: *
298: * Loop over intervals
299: *
300: IF( KL-KF+1.GE.NBMIN .AND. NBMIN.GT.0 ) THEN
301: *
302: * Begin of Parallel Version of the loop
303: *
304: DO 60 JI = KF, KL
305: *
306: * Compute N(c), the number of eigenvalues less than c
307: *
308: WORK( JI ) = D( 1 ) - C( JI )
309: IWORK( JI ) = 0
310: IF( WORK( JI ).LE.PIVMIN ) THEN
311: IWORK( JI ) = 1
312: WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
313: END IF
314: *
315: DO 50 J = 2, N
316: WORK( JI ) = D( J ) - E2( J-1 ) / WORK( JI ) - C( JI )
317: IF( WORK( JI ).LE.PIVMIN ) THEN
318: IWORK( JI ) = IWORK( JI ) + 1
319: WORK( JI ) = MIN( WORK( JI ), -PIVMIN )
320: END IF
321: 50 CONTINUE
322: 60 CONTINUE
323: *
324: IF( IJOB.LE.2 ) THEN
325: *
326: * IJOB=2: Choose all intervals containing eigenvalues.
327: *
328: KLNEW = KL
329: DO 70 JI = KF, KL
330: *
331: * Insure that N(w) is monotone
332: *
333: IWORK( JI ) = MIN( NAB( JI, 2 ),
334: $ MAX( NAB( JI, 1 ), IWORK( JI ) ) )
335: *
336: * Update the Queue -- add intervals if both halves
337: * contain eigenvalues.
338: *
339: IF( IWORK( JI ).EQ.NAB( JI, 2 ) ) THEN
340: *
341: * No eigenvalue in the upper interval:
342: * just use the lower interval.
343: *
344: AB( JI, 2 ) = C( JI )
345: *
346: ELSE IF( IWORK( JI ).EQ.NAB( JI, 1 ) ) THEN
347: *
348: * No eigenvalue in the lower interval:
349: * just use the upper interval.
350: *
351: AB( JI, 1 ) = C( JI )
352: ELSE
353: KLNEW = KLNEW + 1
354: IF( KLNEW.LE.MMAX ) THEN
355: *
356: * Eigenvalue in both intervals -- add upper to
357: * queue.
358: *
359: AB( KLNEW, 2 ) = AB( JI, 2 )
360: NAB( KLNEW, 2 ) = NAB( JI, 2 )
361: AB( KLNEW, 1 ) = C( JI )
362: NAB( KLNEW, 1 ) = IWORK( JI )
363: AB( JI, 2 ) = C( JI )
364: NAB( JI, 2 ) = IWORK( JI )
365: ELSE
366: INFO = MMAX + 1
367: END IF
368: END IF
369: 70 CONTINUE
370: IF( INFO.NE.0 )
371: $ RETURN
372: KL = KLNEW
373: ELSE
374: *
375: * IJOB=3: Binary search. Keep only the interval containing
376: * w s.t. N(w) = NVAL
377: *
378: DO 80 JI = KF, KL
379: IF( IWORK( JI ).LE.NVAL( JI ) ) THEN
380: AB( JI, 1 ) = C( JI )
381: NAB( JI, 1 ) = IWORK( JI )
382: END IF
383: IF( IWORK( JI ).GE.NVAL( JI ) ) THEN
384: AB( JI, 2 ) = C( JI )
385: NAB( JI, 2 ) = IWORK( JI )
386: END IF
387: 80 CONTINUE
388: END IF
389: *
390: ELSE
391: *
392: * End of Parallel Version of the loop
393: *
394: * Begin of Serial Version of the loop
395: *
396: KLNEW = KL
397: DO 100 JI = KF, KL
398: *
399: * Compute N(w), the number of eigenvalues less than w
400: *
401: TMP1 = C( JI )
402: TMP2 = D( 1 ) - TMP1
403: ITMP1 = 0
404: IF( TMP2.LE.PIVMIN ) THEN
405: ITMP1 = 1
406: TMP2 = MIN( TMP2, -PIVMIN )
407: END IF
408: *
409: DO 90 J = 2, N
410: TMP2 = D( J ) - E2( J-1 ) / TMP2 - TMP1
411: IF( TMP2.LE.PIVMIN ) THEN
412: ITMP1 = ITMP1 + 1
413: TMP2 = MIN( TMP2, -PIVMIN )
414: END IF
415: 90 CONTINUE
416: *
417: IF( IJOB.LE.2 ) THEN
418: *
419: * IJOB=2: Choose all intervals containing eigenvalues.
420: *
421: * Insure that N(w) is monotone
422: *
423: ITMP1 = MIN( NAB( JI, 2 ),
424: $ MAX( NAB( JI, 1 ), ITMP1 ) )
425: *
426: * Update the Queue -- add intervals if both halves
427: * contain eigenvalues.
428: *
429: IF( ITMP1.EQ.NAB( JI, 2 ) ) THEN
430: *
431: * No eigenvalue in the upper interval:
432: * just use the lower interval.
433: *
434: AB( JI, 2 ) = TMP1
435: *
436: ELSE IF( ITMP1.EQ.NAB( JI, 1 ) ) THEN
437: *
438: * No eigenvalue in the lower interval:
439: * just use the upper interval.
440: *
441: AB( JI, 1 ) = TMP1
442: ELSE IF( KLNEW.LT.MMAX ) THEN
443: *
444: * Eigenvalue in both intervals -- add upper to queue.
445: *
446: KLNEW = KLNEW + 1
447: AB( KLNEW, 2 ) = AB( JI, 2 )
448: NAB( KLNEW, 2 ) = NAB( JI, 2 )
449: AB( KLNEW, 1 ) = TMP1
450: NAB( KLNEW, 1 ) = ITMP1
451: AB( JI, 2 ) = TMP1
452: NAB( JI, 2 ) = ITMP1
453: ELSE
454: INFO = MMAX + 1
455: RETURN
456: END IF
457: ELSE
458: *
459: * IJOB=3: Binary search. Keep only the interval
460: * containing w s.t. N(w) = NVAL
461: *
462: IF( ITMP1.LE.NVAL( JI ) ) THEN
463: AB( JI, 1 ) = TMP1
464: NAB( JI, 1 ) = ITMP1
465: END IF
466: IF( ITMP1.GE.NVAL( JI ) ) THEN
467: AB( JI, 2 ) = TMP1
468: NAB( JI, 2 ) = ITMP1
469: END IF
470: END IF
471: 100 CONTINUE
472: KL = KLNEW
473: *
474: END IF
475: *
476: * Check for convergence
477: *
478: KFNEW = KF
479: DO 110 JI = KF, KL
480: TMP1 = ABS( AB( JI, 2 )-AB( JI, 1 ) )
481: TMP2 = MAX( ABS( AB( JI, 2 ) ), ABS( AB( JI, 1 ) ) )
482: IF( TMP1.LT.MAX( ABSTOL, PIVMIN, RELTOL*TMP2 ) .OR.
483: $ NAB( JI, 1 ).GE.NAB( JI, 2 ) ) THEN
484: *
485: * Converged -- Swap with position KFNEW,
486: * then increment KFNEW
487: *
488: IF( JI.GT.KFNEW ) THEN
489: TMP1 = AB( JI, 1 )
490: TMP2 = AB( JI, 2 )
491: ITMP1 = NAB( JI, 1 )
492: ITMP2 = NAB( JI, 2 )
493: AB( JI, 1 ) = AB( KFNEW, 1 )
494: AB( JI, 2 ) = AB( KFNEW, 2 )
495: NAB( JI, 1 ) = NAB( KFNEW, 1 )
496: NAB( JI, 2 ) = NAB( KFNEW, 2 )
497: AB( KFNEW, 1 ) = TMP1
498: AB( KFNEW, 2 ) = TMP2
499: NAB( KFNEW, 1 ) = ITMP1
500: NAB( KFNEW, 2 ) = ITMP2
501: IF( IJOB.EQ.3 ) THEN
502: ITMP1 = NVAL( JI )
503: NVAL( JI ) = NVAL( KFNEW )
504: NVAL( KFNEW ) = ITMP1
505: END IF
506: END IF
507: KFNEW = KFNEW + 1
508: END IF
509: 110 CONTINUE
510: KF = KFNEW
511: *
512: * Choose Midpoints
513: *
514: DO 120 JI = KF, KL
515: C( JI ) = HALF*( AB( JI, 1 )+AB( JI, 2 ) )
516: 120 CONTINUE
517: *
518: * If no more intervals to refine, quit.
519: *
520: IF( KF.GT.KL )
521: $ GO TO 140
522: 130 CONTINUE
523: *
524: * Converged
525: *
526: 140 CONTINUE
527: INFO = MAX( KL+1-KF, 0 )
528: MOUT = KL
529: *
530: RETURN
531: *
532: * End of DLAEBZ
533: *
534: END
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