1: *> \brief <b> DSTEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSTEVR + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevr.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevr.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevr.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
22: * M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
23: * LIWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE
27: * INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
28: * DOUBLE PRECISION ABSTOL, VL, VU
29: * ..
30: * .. Array Arguments ..
31: * INTEGER ISUPPZ( * ), IWORK( * )
32: * DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DSTEVR computes selected eigenvalues and, optionally, eigenvectors
42: *> of a real symmetric tridiagonal matrix T. Eigenvalues and
43: *> eigenvectors can be selected by specifying either a range of values
44: *> or a range of indices for the desired eigenvalues.
45: *>
46: *> Whenever possible, DSTEVR calls DSTEMR to compute the
47: *> eigenspectrum using Relatively Robust Representations. DSTEMR
48: *> computes eigenvalues by the dqds algorithm, while orthogonal
49: *> eigenvectors are computed from various "good" L D L^T representations
50: *> (also known as Relatively Robust Representations). Gram-Schmidt
51: *> orthogonalization is avoided as far as possible. More specifically,
52: *> the various steps of the algorithm are as follows. For the i-th
53: *> unreduced block of T,
54: *> (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
55: *> is a relatively robust representation,
56: *> (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
57: *> relative accuracy by the dqds algorithm,
58: *> (c) If there is a cluster of close eigenvalues, "choose" sigma_i
59: *> close to the cluster, and go to step (a),
60: *> (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
61: *> compute the corresponding eigenvector by forming a
62: *> rank-revealing twisted factorization.
63: *> The desired accuracy of the output can be specified by the input
64: *> parameter ABSTOL.
65: *>
66: *> For more details, see "A new O(n^2) algorithm for the symmetric
67: *> tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon,
68: *> Computer Science Division Technical Report No. UCB//CSD-97-971,
69: *> UC Berkeley, May 1997.
70: *>
71: *>
72: *> Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested
73: *> on machines which conform to the ieee-754 floating point standard.
74: *> DSTEVR calls DSTEBZ and DSTEIN on non-ieee machines and
75: *> when partial spectrum requests are made.
76: *>
77: *> Normal execution of DSTEMR may create NaNs and infinities and
78: *> hence may abort due to a floating point exception in environments
79: *> which do not handle NaNs and infinities in the ieee standard default
80: *> manner.
81: *> \endverbatim
82: *
83: * Arguments:
84: * ==========
85: *
86: *> \param[in] JOBZ
87: *> \verbatim
88: *> JOBZ is CHARACTER*1
89: *> = 'N': Compute eigenvalues only;
90: *> = 'V': Compute eigenvalues and eigenvectors.
91: *> \endverbatim
92: *>
93: *> \param[in] RANGE
94: *> \verbatim
95: *> RANGE is CHARACTER*1
96: *> = 'A': all eigenvalues will be found.
97: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
98: *> will be found.
99: *> = 'I': the IL-th through IU-th eigenvalues will be found.
100: *> For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and
101: *> DSTEIN are called
102: *> \endverbatim
103: *>
104: *> \param[in] N
105: *> \verbatim
106: *> N is INTEGER
107: *> The order of the matrix. N >= 0.
108: *> \endverbatim
109: *>
110: *> \param[in,out] D
111: *> \verbatim
112: *> D is DOUBLE PRECISION array, dimension (N)
113: *> On entry, the n diagonal elements of the tridiagonal matrix
114: *> A.
115: *> On exit, D may be multiplied by a constant factor chosen
116: *> to avoid over/underflow in computing the eigenvalues.
117: *> \endverbatim
118: *>
119: *> \param[in,out] E
120: *> \verbatim
121: *> E is DOUBLE PRECISION array, dimension (max(1,N-1))
122: *> On entry, the (n-1) subdiagonal elements of the tridiagonal
123: *> matrix A in elements 1 to N-1 of E.
124: *> On exit, E may be multiplied by a constant factor chosen
125: *> to avoid over/underflow in computing the eigenvalues.
126: *> \endverbatim
127: *>
128: *> \param[in] VL
129: *> \verbatim
130: *> VL is DOUBLE PRECISION
131: *> If RANGE='V', the lower bound of the interval to
132: *> be searched for eigenvalues. VL < VU.
133: *> Not referenced if RANGE = 'A' or 'I'.
134: *> \endverbatim
135: *>
136: *> \param[in] VU
137: *> \verbatim
138: *> VU is DOUBLE PRECISION
139: *> If RANGE='V', the upper bound of the interval to
140: *> be searched for eigenvalues. VL < VU.
141: *> Not referenced if RANGE = 'A' or 'I'.
142: *> \endverbatim
143: *>
144: *> \param[in] IL
145: *> \verbatim
146: *> IL is INTEGER
147: *> If RANGE='I', the index of the
148: *> smallest eigenvalue to be returned.
149: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
150: *> Not referenced if RANGE = 'A' or 'V'.
151: *> \endverbatim
152: *>
153: *> \param[in] IU
154: *> \verbatim
155: *> IU is INTEGER
156: *> If RANGE='I', the index of the
157: *> largest eigenvalue to be returned.
158: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
159: *> Not referenced if RANGE = 'A' or 'V'.
160: *> \endverbatim
161: *>
162: *> \param[in] ABSTOL
163: *> \verbatim
164: *> ABSTOL is DOUBLE PRECISION
165: *> The absolute error tolerance for the eigenvalues.
166: *> An approximate eigenvalue is accepted as converged
167: *> when it is determined to lie in an interval [a,b]
168: *> of width less than or equal to
169: *>
170: *> ABSTOL + EPS * max( |a|,|b| ) ,
171: *>
172: *> where EPS is the machine precision. If ABSTOL is less than
173: *> or equal to zero, then EPS*|T| will be used in its place,
174: *> where |T| is the 1-norm of the tridiagonal matrix obtained
175: *> by reducing A to tridiagonal form.
176: *>
177: *> See "Computing Small Singular Values of Bidiagonal Matrices
178: *> with Guaranteed High Relative Accuracy," by Demmel and
179: *> Kahan, LAPACK Working Note #3.
180: *>
181: *> If high relative accuracy is important, set ABSTOL to
182: *> DLAMCH( 'Safe minimum' ). Doing so will guarantee that
183: *> eigenvalues are computed to high relative accuracy when
184: *> possible in future releases. The current code does not
185: *> make any guarantees about high relative accuracy, but
186: *> future releases will. See J. Barlow and J. Demmel,
187: *> "Computing Accurate Eigensystems of Scaled Diagonally
188: *> Dominant Matrices", LAPACK Working Note #7, for a discussion
189: *> of which matrices define their eigenvalues to high relative
190: *> accuracy.
191: *> \endverbatim
192: *>
193: *> \param[out] M
194: *> \verbatim
195: *> M is INTEGER
196: *> The total number of eigenvalues found. 0 <= M <= N.
197: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
198: *> \endverbatim
199: *>
200: *> \param[out] W
201: *> \verbatim
202: *> W is DOUBLE PRECISION array, dimension (N)
203: *> The first M elements contain the selected eigenvalues in
204: *> ascending order.
205: *> \endverbatim
206: *>
207: *> \param[out] Z
208: *> \verbatim
209: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
210: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
211: *> contain the orthonormal eigenvectors of the matrix A
212: *> corresponding to the selected eigenvalues, with the i-th
213: *> column of Z holding the eigenvector associated with W(i).
214: *> Note: the user must ensure that at least max(1,M) columns are
215: *> supplied in the array Z; if RANGE = 'V', the exact value of M
216: *> is not known in advance and an upper bound must be used.
217: *> \endverbatim
218: *>
219: *> \param[in] LDZ
220: *> \verbatim
221: *> LDZ is INTEGER
222: *> The leading dimension of the array Z. LDZ >= 1, and if
223: *> JOBZ = 'V', LDZ >= max(1,N).
224: *> \endverbatim
225: *>
226: *> \param[out] ISUPPZ
227: *> \verbatim
228: *> ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
229: *> The support of the eigenvectors in Z, i.e., the indices
230: *> indicating the nonzero elements in Z. The i-th eigenvector
231: *> is nonzero only in elements ISUPPZ( 2*i-1 ) through
232: *> ISUPPZ( 2*i ).
233: *> Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1
234: *> \endverbatim
235: *>
236: *> \param[out] WORK
237: *> \verbatim
238: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
239: *> On exit, if INFO = 0, WORK(1) returns the optimal (and
240: *> minimal) LWORK.
241: *> \endverbatim
242: *>
243: *> \param[in] LWORK
244: *> \verbatim
245: *> LWORK is INTEGER
246: *> The dimension of the array WORK. LWORK >= max(1,20*N).
247: *>
248: *> If LWORK = -1, then a workspace query is assumed; the routine
249: *> only calculates the optimal sizes of the WORK and IWORK
250: *> arrays, returns these values as the first entries of the WORK
251: *> and IWORK arrays, and no error message related to LWORK or
252: *> LIWORK is issued by XERBLA.
253: *> \endverbatim
254: *>
255: *> \param[out] IWORK
256: *> \verbatim
257: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
258: *> On exit, if INFO = 0, IWORK(1) returns the optimal (and
259: *> minimal) LIWORK.
260: *> \endverbatim
261: *>
262: *> \param[in] LIWORK
263: *> \verbatim
264: *> LIWORK is INTEGER
265: *> The dimension of the array IWORK. LIWORK >= max(1,10*N).
266: *>
267: *> If LIWORK = -1, then a workspace query is assumed; the
268: *> routine only calculates the optimal sizes of the WORK and
269: *> IWORK arrays, returns these values as the first entries of
270: *> the WORK and IWORK arrays, and no error message related to
271: *> LWORK or LIWORK is issued by XERBLA.
272: *> \endverbatim
273: *>
274: *> \param[out] INFO
275: *> \verbatim
276: *> INFO is INTEGER
277: *> = 0: successful exit
278: *> < 0: if INFO = -i, the i-th argument had an illegal value
279: *> > 0: Internal error
280: *> \endverbatim
281: *
282: * Authors:
283: * ========
284: *
285: *> \author Univ. of Tennessee
286: *> \author Univ. of California Berkeley
287: *> \author Univ. of Colorado Denver
288: *> \author NAG Ltd.
289: *
290: *> \ingroup doubleOTHEReigen
291: *
292: *> \par Contributors:
293: * ==================
294: *>
295: *> Inderjit Dhillon, IBM Almaden, USA \n
296: *> Osni Marques, LBNL/NERSC, USA \n
297: *> Ken Stanley, Computer Science Division, University of
298: *> California at Berkeley, USA \n
299: *>
300: * =====================================================================
301: SUBROUTINE DSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
302: $ M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK,
303: $ LIWORK, INFO )
304: *
305: * -- LAPACK driver routine --
306: * -- LAPACK is a software package provided by Univ. of Tennessee, --
307: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
308: *
309: * .. Scalar Arguments ..
310: CHARACTER JOBZ, RANGE
311: INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
312: DOUBLE PRECISION ABSTOL, VL, VU
313: * ..
314: * .. Array Arguments ..
315: INTEGER ISUPPZ( * ), IWORK( * )
316: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
317: * ..
318: *
319: * =====================================================================
320: *
321: * .. Parameters ..
322: DOUBLE PRECISION ZERO, ONE, TWO
323: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
324: * ..
325: * .. Local Scalars ..
326: LOGICAL ALLEIG, INDEIG, TEST, LQUERY, VALEIG, WANTZ,
327: $ TRYRAC
328: CHARACTER ORDER
329: INTEGER I, IEEEOK, IMAX, INDIBL, INDIFL, INDISP,
330: $ INDIWO, ISCALE, ITMP1, J, JJ, LIWMIN, LWMIN,
331: $ NSPLIT
332: DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
333: $ TMP1, TNRM, VLL, VUU
334: * ..
335: * .. External Functions ..
336: LOGICAL LSAME
337: INTEGER ILAENV
338: DOUBLE PRECISION DLAMCH, DLANST
339: EXTERNAL LSAME, ILAENV, DLAMCH, DLANST
340: * ..
341: * .. External Subroutines ..
342: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEMR, DSTEIN, DSTERF,
343: $ DSWAP, XERBLA
344: * ..
345: * .. Intrinsic Functions ..
346: INTRINSIC MAX, MIN, SQRT
347: * ..
348: * .. Executable Statements ..
349: *
350: *
351: * Test the input parameters.
352: *
353: IEEEOK = ILAENV( 10, 'DSTEVR', 'N', 1, 2, 3, 4 )
354: *
355: WANTZ = LSAME( JOBZ, 'V' )
356: ALLEIG = LSAME( RANGE, 'A' )
357: VALEIG = LSAME( RANGE, 'V' )
358: INDEIG = LSAME( RANGE, 'I' )
359: *
360: LQUERY = ( ( LWORK.EQ.-1 ) .OR. ( LIWORK.EQ.-1 ) )
361: LWMIN = MAX( 1, 20*N )
362: LIWMIN = MAX( 1, 10*N )
363: *
364: *
365: INFO = 0
366: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
367: INFO = -1
368: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
369: INFO = -2
370: ELSE IF( N.LT.0 ) THEN
371: INFO = -3
372: ELSE
373: IF( VALEIG ) THEN
374: IF( N.GT.0 .AND. VU.LE.VL )
375: $ INFO = -7
376: ELSE IF( INDEIG ) THEN
377: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
378: INFO = -8
379: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
380: INFO = -9
381: END IF
382: END IF
383: END IF
384: IF( INFO.EQ.0 ) THEN
385: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
386: INFO = -14
387: END IF
388: END IF
389: *
390: IF( INFO.EQ.0 ) THEN
391: WORK( 1 ) = LWMIN
392: IWORK( 1 ) = LIWMIN
393: *
394: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
395: INFO = -17
396: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
397: INFO = -19
398: END IF
399: END IF
400: *
401: IF( INFO.NE.0 ) THEN
402: CALL XERBLA( 'DSTEVR', -INFO )
403: RETURN
404: ELSE IF( LQUERY ) THEN
405: RETURN
406: END IF
407: *
408: * Quick return if possible
409: *
410: M = 0
411: IF( N.EQ.0 )
412: $ RETURN
413: *
414: IF( N.EQ.1 ) THEN
415: IF( ALLEIG .OR. INDEIG ) THEN
416: M = 1
417: W( 1 ) = D( 1 )
418: ELSE
419: IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
420: M = 1
421: W( 1 ) = D( 1 )
422: END IF
423: END IF
424: IF( WANTZ )
425: $ Z( 1, 1 ) = ONE
426: RETURN
427: END IF
428: *
429: * Get machine constants.
430: *
431: SAFMIN = DLAMCH( 'Safe minimum' )
432: EPS = DLAMCH( 'Precision' )
433: SMLNUM = SAFMIN / EPS
434: BIGNUM = ONE / SMLNUM
435: RMIN = SQRT( SMLNUM )
436: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
437: *
438: *
439: * Scale matrix to allowable range, if necessary.
440: *
441: ISCALE = 0
442: IF( VALEIG ) THEN
443: VLL = VL
444: VUU = VU
445: END IF
446: *
447: TNRM = DLANST( 'M', N, D, E )
448: IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
449: ISCALE = 1
450: SIGMA = RMIN / TNRM
451: ELSE IF( TNRM.GT.RMAX ) THEN
452: ISCALE = 1
453: SIGMA = RMAX / TNRM
454: END IF
455: IF( ISCALE.EQ.1 ) THEN
456: CALL DSCAL( N, SIGMA, D, 1 )
457: CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
458: IF( VALEIG ) THEN
459: VLL = VL*SIGMA
460: VUU = VU*SIGMA
461: END IF
462: END IF
463:
464: * Initialize indices into workspaces. Note: These indices are used only
465: * if DSTERF or DSTEMR fail.
466:
467: * IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in DSTEBZ and
468: * stores the block indices of each of the M<=N eigenvalues.
469: INDIBL = 1
470: * IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in DSTEBZ and
471: * stores the starting and finishing indices of each block.
472: INDISP = INDIBL + N
473: * IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
474: * that corresponding to eigenvectors that fail to converge in
475: * DSTEIN. This information is discarded; if any fail, the driver
476: * returns INFO > 0.
477: INDIFL = INDISP + N
478: * INDIWO is the offset of the remaining integer workspace.
479: INDIWO = INDISP + N
480: *
481: * If all eigenvalues are desired, then
482: * call DSTERF or DSTEMR. If this fails for some eigenvalue, then
483: * try DSTEBZ.
484: *
485: *
486: TEST = .FALSE.
487: IF( INDEIG ) THEN
488: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
489: TEST = .TRUE.
490: END IF
491: END IF
492: IF( ( ALLEIG .OR. TEST ) .AND. IEEEOK.EQ.1 ) THEN
493: CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
494: IF( .NOT.WANTZ ) THEN
495: CALL DCOPY( N, D, 1, W, 1 )
496: CALL DSTERF( N, W, WORK, INFO )
497: ELSE
498: CALL DCOPY( N, D, 1, WORK( N+1 ), 1 )
499: IF (ABSTOL .LE. TWO*N*EPS) THEN
500: TRYRAC = .TRUE.
501: ELSE
502: TRYRAC = .FALSE.
503: END IF
504: CALL DSTEMR( JOBZ, 'A', N, WORK( N+1 ), WORK, VL, VU, IL,
505: $ IU, M, W, Z, LDZ, N, ISUPPZ, TRYRAC,
506: $ WORK( 2*N+1 ), LWORK-2*N, IWORK, LIWORK, INFO )
507: *
508: END IF
509: IF( INFO.EQ.0 ) THEN
510: M = N
511: GO TO 10
512: END IF
513: INFO = 0
514: END IF
515: *
516: * Otherwise, call DSTEBZ and, if eigenvectors are desired, DSTEIN.
517: *
518: IF( WANTZ ) THEN
519: ORDER = 'B'
520: ELSE
521: ORDER = 'E'
522: END IF
523:
524: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
525: $ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ), WORK,
526: $ IWORK( INDIWO ), INFO )
527: *
528: IF( WANTZ ) THEN
529: CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
530: $ Z, LDZ, WORK, IWORK( INDIWO ), IWORK( INDIFL ),
531: $ INFO )
532: END IF
533: *
534: * If matrix was scaled, then rescale eigenvalues appropriately.
535: *
536: 10 CONTINUE
537: IF( ISCALE.EQ.1 ) THEN
538: IF( INFO.EQ.0 ) THEN
539: IMAX = M
540: ELSE
541: IMAX = INFO - 1
542: END IF
543: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
544: END IF
545: *
546: * If eigenvalues are not in order, then sort them, along with
547: * eigenvectors.
548: *
549: IF( WANTZ ) THEN
550: DO 30 J = 1, M - 1
551: I = 0
552: TMP1 = W( J )
553: DO 20 JJ = J + 1, M
554: IF( W( JJ ).LT.TMP1 ) THEN
555: I = JJ
556: TMP1 = W( JJ )
557: END IF
558: 20 CONTINUE
559: *
560: IF( I.NE.0 ) THEN
561: ITMP1 = IWORK( I )
562: W( I ) = W( J )
563: IWORK( I ) = IWORK( J )
564: W( J ) = TMP1
565: IWORK( J ) = ITMP1
566: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
567: END IF
568: 30 CONTINUE
569: END IF
570: *
571: * Causes problems with tests 19 & 20:
572: * IF (wantz .and. INDEIG ) Z( 1,1) = Z(1,1) / 1.002 + .002
573: *
574: *
575: WORK( 1 ) = LWMIN
576: IWORK( 1 ) = LIWMIN
577: RETURN
578: *
579: * End of DSTEVR
580: *
581: END
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