1: SUBROUTINE ZSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU,
2: $ M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK,
3: $ IWORK, LIWORK, INFO )
4: IMPLICIT NONE
5: *
6: * -- LAPACK computational routine (version 3.2.1) --
7: *
8: * -- April 2009 --
9: *
10: * -- LAPACK is a software package provided by Univ. of Tennessee, --
11: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
12: *
13: * .. Scalar Arguments ..
14: CHARACTER JOBZ, RANGE
15: LOGICAL TRYRAC
16: INTEGER IL, INFO, IU, LDZ, NZC, LIWORK, LWORK, M, N
17: DOUBLE PRECISION VL, VU
18: * ..
19: * .. Array Arguments ..
20: INTEGER ISUPPZ( * ), IWORK( * )
21: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * )
22: COMPLEX*16 Z( LDZ, * )
23: * ..
24: *
25: * Purpose
26: * =======
27: *
28: * ZSTEMR computes selected eigenvalues and, optionally, eigenvectors
29: * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has
30: * a well defined set of pairwise different real eigenvalues, the corresponding
31: * real eigenvectors are pairwise orthogonal.
32: *
33: * The spectrum may be computed either completely or partially by specifying
34: * either an interval (VL,VU] or a range of indices IL:IU for the desired
35: * eigenvalues.
36: *
37: * Depending on the number of desired eigenvalues, these are computed either
38: * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are
39: * computed by the use of various suitable L D L^T factorizations near clusters
40: * of close eigenvalues (referred to as RRRs, Relatively Robust
41: * Representations). An informal sketch of the algorithm follows.
42: *
43: * For each unreduced block (submatrix) of T,
44: * (a) Compute T - sigma I = L D L^T, so that L and D
45: * define all the wanted eigenvalues to high relative accuracy.
46: * This means that small relative changes in the entries of D and L
47: * cause only small relative changes in the eigenvalues and
48: * eigenvectors. The standard (unfactored) representation of the
49: * tridiagonal matrix T does not have this property in general.
50: * (b) Compute the eigenvalues to suitable accuracy.
51: * If the eigenvectors are desired, the algorithm attains full
52: * accuracy of the computed eigenvalues only right before
53: * the corresponding vectors have to be computed, see steps c) and d).
54: * (c) For each cluster of close eigenvalues, select a new
55: * shift close to the cluster, find a new factorization, and refine
56: * the shifted eigenvalues to suitable accuracy.
57: * (d) For each eigenvalue with a large enough relative separation compute
58: * the corresponding eigenvector by forming a rank revealing twisted
59: * factorization. Go back to (c) for any clusters that remain.
60: *
61: * For more details, see:
62: * - Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
63: * to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
64: * Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
65: * - Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
66: * Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
67: * 2004. Also LAPACK Working Note 154.
68: * - Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
69: * tridiagonal eigenvalue/eigenvector problem",
70: * Computer Science Division Technical Report No. UCB/CSD-97-971,
71: * UC Berkeley, May 1997.
72: *
73: * Further Details
74: * 1.ZSTEMR works only on machines which follow IEEE-754
75: * floating-point standard in their handling of infinities and NaNs.
76: * This permits the use of efficient inner loops avoiding a check for
77: * zero divisors.
78: *
79: * 2. LAPACK routines can be used to reduce a complex Hermitean matrix to
80: * real symmetric tridiagonal form.
81: *
82: * (Any complex Hermitean tridiagonal matrix has real values on its diagonal
83: * and potentially complex numbers on its off-diagonals. By applying a
84: * similarity transform with an appropriate diagonal matrix
85: * diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean
86: * matrix can be transformed into a real symmetric matrix and complex
87: * arithmetic can be entirely avoided.)
88: *
89: * While the eigenvectors of the real symmetric tridiagonal matrix are real,
90: * the eigenvectors of original complex Hermitean matrix have complex entries
91: * in general.
92: * Since LAPACK drivers overwrite the matrix data with the eigenvectors,
93: * ZSTEMR accepts complex workspace to facilitate interoperability
94: * with ZUNMTR or ZUPMTR.
95: *
96: * Arguments
97: * =========
98: *
99: * JOBZ (input) CHARACTER*1
100: * = 'N': Compute eigenvalues only;
101: * = 'V': Compute eigenvalues and eigenvectors.
102: *
103: * RANGE (input) CHARACTER*1
104: * = 'A': all eigenvalues will be found.
105: * = 'V': all eigenvalues in the half-open interval (VL,VU]
106: * will be found.
107: * = 'I': the IL-th through IU-th eigenvalues will be found.
108: *
109: * N (input) INTEGER
110: * The order of the matrix. N >= 0.
111: *
112: * D (input/output) DOUBLE PRECISION array, dimension (N)
113: * On entry, the N diagonal elements of the tridiagonal matrix
114: * T. On exit, D is overwritten.
115: *
116: * E (input/output) DOUBLE PRECISION array, dimension (N)
117: * On entry, the (N-1) subdiagonal elements of the tridiagonal
118: * matrix T in elements 1 to N-1 of E. E(N) need not be set on
119: * input, but is used internally as workspace.
120: * On exit, E is overwritten.
121: *
122: * VL (input) DOUBLE PRECISION
123: * VU (input) DOUBLE PRECISION
124: * If RANGE='V', the lower and upper bounds of the interval to
125: * be searched for eigenvalues. VL < VU.
126: * Not referenced if RANGE = 'A' or 'I'.
127: *
128: * IL (input) INTEGER
129: * IU (input) INTEGER
130: * If RANGE='I', the indices (in ascending order) of the
131: * smallest and largest eigenvalues to be returned.
132: * 1 <= IL <= IU <= N, if N > 0.
133: * Not referenced if RANGE = 'A' or 'V'.
134: *
135: * M (output) INTEGER
136: * The total number of eigenvalues found. 0 <= M <= N.
137: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
138: *
139: * W (output) DOUBLE PRECISION array, dimension (N)
140: * The first M elements contain the selected eigenvalues in
141: * ascending order.
142: *
143: * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M) )
144: * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z
145: * contain the orthonormal eigenvectors of the matrix T
146: * corresponding to the selected eigenvalues, with the i-th
147: * column of Z holding the eigenvector associated with W(i).
148: * If JOBZ = 'N', then Z is not referenced.
149: * Note: the user must ensure that at least max(1,M) columns are
150: * supplied in the array Z; if RANGE = 'V', the exact value of M
151: * is not known in advance and can be computed with a workspace
152: * query by setting NZC = -1, see below.
153: *
154: * LDZ (input) INTEGER
155: * The leading dimension of the array Z. LDZ >= 1, and if
156: * JOBZ = 'V', then LDZ >= max(1,N).
157: *
158: * NZC (input) INTEGER
159: * The number of eigenvectors to be held in the array Z.
160: * If RANGE = 'A', then NZC >= max(1,N).
161: * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU].
162: * If RANGE = 'I', then NZC >= IU-IL+1.
163: * If NZC = -1, then a workspace query is assumed; the
164: * routine calculates the number of columns of the array Z that
165: * are needed to hold the eigenvectors.
166: * This value is returned as the first entry of the Z array, and
167: * no error message related to NZC is issued by XERBLA.
168: *
169: * ISUPPZ (output) INTEGER ARRAY, dimension ( 2*max(1,M) )
170: * The support of the eigenvectors in Z, i.e., the indices
171: * indicating the nonzero elements in Z. The i-th computed eigenvector
172: * is nonzero only in elements ISUPPZ( 2*i-1 ) through
173: * ISUPPZ( 2*i ). This is relevant in the case when the matrix
174: * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0.
175: *
176: * TRYRAC (input/output) LOGICAL
177: * If TRYRAC.EQ..TRUE., indicates that the code should check whether
178: * the tridiagonal matrix defines its eigenvalues to high relative
179: * accuracy. If so, the code uses relative-accuracy preserving
180: * algorithms that might be (a bit) slower depending on the matrix.
181: * If the matrix does not define its eigenvalues to high relative
182: * accuracy, the code can uses possibly faster algorithms.
183: * If TRYRAC.EQ..FALSE., the code is not required to guarantee
184: * relatively accurate eigenvalues and can use the fastest possible
185: * techniques.
186: * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix
187: * does not define its eigenvalues to high relative accuracy.
188: *
189: * WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
190: * On exit, if INFO = 0, WORK(1) returns the optimal
191: * (and minimal) LWORK.
192: *
193: * LWORK (input) INTEGER
194: * The dimension of the array WORK. LWORK >= max(1,18*N)
195: * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'.
196: * If LWORK = -1, then a workspace query is assumed; the routine
197: * only calculates the optimal size of the WORK array, returns
198: * this value as the first entry of the WORK array, and no error
199: * message related to LWORK is issued by XERBLA.
200: *
201: * IWORK (workspace/output) INTEGER array, dimension (LIWORK)
202: * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
203: *
204: * LIWORK (input) INTEGER
205: * The dimension of the array IWORK. LIWORK >= max(1,10*N)
206: * if the eigenvectors are desired, and LIWORK >= max(1,8*N)
207: * if only the eigenvalues are to be computed.
208: * If LIWORK = -1, then a workspace query is assumed; the
209: * routine only calculates the optimal size of the IWORK array,
210: * returns this value as the first entry of the IWORK array, and
211: * no error message related to LIWORK is issued by XERBLA.
212: *
213: * INFO (output) INTEGER
214: * On exit, INFO
215: * = 0: successful exit
216: * < 0: if INFO = -i, the i-th argument had an illegal value
217: * > 0: if INFO = 1X, internal error in DLARRE,
218: * if INFO = 2X, internal error in ZLARRV.
219: * Here, the digit X = ABS( IINFO ) < 10, where IINFO is
220: * the nonzero error code returned by DLARRE or
221: * ZLARRV, respectively.
222: *
223: *
224: * Further Details
225: * ===============
226: *
227: * Based on contributions by
228: * Beresford Parlett, University of California, Berkeley, USA
229: * Jim Demmel, University of California, Berkeley, USA
230: * Inderjit Dhillon, University of Texas, Austin, USA
231: * Osni Marques, LBNL/NERSC, USA
232: * Christof Voemel, University of California, Berkeley, USA
233: *
234: * =====================================================================
235: *
236: * .. Parameters ..
237: DOUBLE PRECISION ZERO, ONE, FOUR, MINRGP
238: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0,
239: $ FOUR = 4.0D0,
240: $ MINRGP = 1.0D-3 )
241: * ..
242: * .. Local Scalars ..
243: LOGICAL ALLEIG, INDEIG, LQUERY, VALEIG, WANTZ, ZQUERY
244: INTEGER I, IBEGIN, IEND, IFIRST, IIL, IINDBL, IINDW,
245: $ IINDWK, IINFO, IINSPL, IIU, ILAST, IN, INDD,
246: $ INDE2, INDERR, INDGP, INDGRS, INDWRK, ITMP,
247: $ ITMP2, J, JBLK, JJ, LIWMIN, LWMIN, NSPLIT,
248: $ NZCMIN, OFFSET, WBEGIN, WEND
249: DOUBLE PRECISION BIGNUM, CS, EPS, PIVMIN, R1, R2, RMAX, RMIN,
250: $ RTOL1, RTOL2, SAFMIN, SCALE, SMLNUM, SN,
251: $ THRESH, TMP, TNRM, WL, WU
252: * ..
253: * ..
254: * .. External Functions ..
255: LOGICAL LSAME
256: DOUBLE PRECISION DLAMCH, DLANST
257: EXTERNAL LSAME, DLAMCH, DLANST
258: * ..
259: * .. External Subroutines ..
260: EXTERNAL DCOPY, DLAE2, DLAEV2, DLARRC, DLARRE, DLARRJ,
261: $ DLARRR, DLASRT, DSCAL, XERBLA, ZLARRV, ZSWAP
262: * ..
263: * .. Intrinsic Functions ..
264: INTRINSIC MAX, MIN, SQRT
265:
266:
267: * ..
268: * .. Executable Statements ..
269: *
270: * Test the input parameters.
271: *
272: WANTZ = LSAME( JOBZ, 'V' )
273: ALLEIG = LSAME( RANGE, 'A' )
274: VALEIG = LSAME( RANGE, 'V' )
275: INDEIG = LSAME( RANGE, 'I' )
276: *
277: LQUERY = ( ( LWORK.EQ.-1 ).OR.( LIWORK.EQ.-1 ) )
278: ZQUERY = ( NZC.EQ.-1 )
279:
280: * DSTEMR needs WORK of size 6*N, IWORK of size 3*N.
281: * In addition, DLARRE needs WORK of size 6*N, IWORK of size 5*N.
282: * Furthermore, ZLARRV needs WORK of size 12*N, IWORK of size 7*N.
283: IF( WANTZ ) THEN
284: LWMIN = 18*N
285: LIWMIN = 10*N
286: ELSE
287: * need less workspace if only the eigenvalues are wanted
288: LWMIN = 12*N
289: LIWMIN = 8*N
290: ENDIF
291:
292: WL = ZERO
293: WU = ZERO
294: IIL = 0
295: IIU = 0
296:
297: IF( VALEIG ) THEN
298: * We do not reference VL, VU in the cases RANGE = 'I','A'
299: * The interval (WL, WU] contains all the wanted eigenvalues.
300: * It is either given by the user or computed in DLARRE.
301: WL = VL
302: WU = VU
303: ELSEIF( INDEIG ) THEN
304: * We do not reference IL, IU in the cases RANGE = 'V','A'
305: IIL = IL
306: IIU = IU
307: ENDIF
308: *
309: INFO = 0
310: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
311: INFO = -1
312: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
313: INFO = -2
314: ELSE IF( N.LT.0 ) THEN
315: INFO = -3
316: ELSE IF( VALEIG .AND. N.GT.0 .AND. WU.LE.WL ) THEN
317: INFO = -7
318: ELSE IF( INDEIG .AND. ( IIL.LT.1 .OR. IIL.GT.N ) ) THEN
319: INFO = -8
320: ELSE IF( INDEIG .AND. ( IIU.LT.IIL .OR. IIU.GT.N ) ) THEN
321: INFO = -9
322: ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
323: INFO = -13
324: ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
325: INFO = -17
326: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
327: INFO = -19
328: END IF
329: *
330: * Get machine constants.
331: *
332: SAFMIN = DLAMCH( 'Safe minimum' )
333: EPS = DLAMCH( 'Precision' )
334: SMLNUM = SAFMIN / EPS
335: BIGNUM = ONE / SMLNUM
336: RMIN = SQRT( SMLNUM )
337: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
338: *
339: IF( INFO.EQ.0 ) THEN
340: WORK( 1 ) = LWMIN
341: IWORK( 1 ) = LIWMIN
342: *
343: IF( WANTZ .AND. ALLEIG ) THEN
344: NZCMIN = N
345: ELSE IF( WANTZ .AND. VALEIG ) THEN
346: CALL DLARRC( 'T', N, VL, VU, D, E, SAFMIN,
347: $ NZCMIN, ITMP, ITMP2, INFO )
348: ELSE IF( WANTZ .AND. INDEIG ) THEN
349: NZCMIN = IIU-IIL+1
350: ELSE
351: * WANTZ .EQ. FALSE.
352: NZCMIN = 0
353: ENDIF
354: IF( ZQUERY .AND. INFO.EQ.0 ) THEN
355: Z( 1,1 ) = NZCMIN
356: ELSE IF( NZC.LT.NZCMIN .AND. .NOT.ZQUERY ) THEN
357: INFO = -14
358: END IF
359: END IF
360:
361: IF( INFO.NE.0 ) THEN
362: *
363: CALL XERBLA( 'ZSTEMR', -INFO )
364: *
365: RETURN
366: ELSE IF( LQUERY .OR. ZQUERY ) THEN
367: RETURN
368: END IF
369: *
370: * Handle N = 0, 1, and 2 cases immediately
371: *
372: M = 0
373: IF( N.EQ.0 )
374: $ RETURN
375: *
376: IF( N.EQ.1 ) THEN
377: IF( ALLEIG .OR. INDEIG ) THEN
378: M = 1
379: W( 1 ) = D( 1 )
380: ELSE
381: IF( WL.LT.D( 1 ) .AND. WU.GE.D( 1 ) ) THEN
382: M = 1
383: W( 1 ) = D( 1 )
384: END IF
385: END IF
386: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
387: Z( 1, 1 ) = ONE
388: ISUPPZ(1) = 1
389: ISUPPZ(2) = 1
390: END IF
391: RETURN
392: END IF
393: *
394: IF( N.EQ.2 ) THEN
395: IF( .NOT.WANTZ ) THEN
396: CALL DLAE2( D(1), E(1), D(2), R1, R2 )
397: ELSE IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
398: CALL DLAEV2( D(1), E(1), D(2), R1, R2, CS, SN )
399: END IF
400: IF( ALLEIG.OR.
401: $ (VALEIG.AND.(R2.GT.WL).AND.
402: $ (R2.LE.WU)).OR.
403: $ (INDEIG.AND.(IIL.EQ.1)) ) THEN
404: M = M+1
405: W( M ) = R2
406: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
407: Z( 1, M ) = -SN
408: Z( 2, M ) = CS
409: * Note: At most one of SN and CS can be zero.
410: IF (SN.NE.ZERO) THEN
411: IF (CS.NE.ZERO) THEN
412: ISUPPZ(2*M-1) = 1
413: ISUPPZ(2*M-1) = 2
414: ELSE
415: ISUPPZ(2*M-1) = 1
416: ISUPPZ(2*M-1) = 1
417: END IF
418: ELSE
419: ISUPPZ(2*M-1) = 2
420: ISUPPZ(2*M) = 2
421: END IF
422: ENDIF
423: ENDIF
424: IF( ALLEIG.OR.
425: $ (VALEIG.AND.(R1.GT.WL).AND.
426: $ (R1.LE.WU)).OR.
427: $ (INDEIG.AND.(IIU.EQ.2)) ) THEN
428: M = M+1
429: W( M ) = R1
430: IF( WANTZ.AND.(.NOT.ZQUERY) ) THEN
431: Z( 1, M ) = CS
432: Z( 2, M ) = SN
433: * Note: At most one of SN and CS can be zero.
434: IF (SN.NE.ZERO) THEN
435: IF (CS.NE.ZERO) THEN
436: ISUPPZ(2*M-1) = 1
437: ISUPPZ(2*M-1) = 2
438: ELSE
439: ISUPPZ(2*M-1) = 1
440: ISUPPZ(2*M-1) = 1
441: END IF
442: ELSE
443: ISUPPZ(2*M-1) = 2
444: ISUPPZ(2*M) = 2
445: END IF
446: ENDIF
447: ENDIF
448: RETURN
449: END IF
450:
451: * Continue with general N
452:
453: INDGRS = 1
454: INDERR = 2*N + 1
455: INDGP = 3*N + 1
456: INDD = 4*N + 1
457: INDE2 = 5*N + 1
458: INDWRK = 6*N + 1
459: *
460: IINSPL = 1
461: IINDBL = N + 1
462: IINDW = 2*N + 1
463: IINDWK = 3*N + 1
464: *
465: * Scale matrix to allowable range, if necessary.
466: * The allowable range is related to the PIVMIN parameter; see the
467: * comments in DLARRD. The preference for scaling small values
468: * up is heuristic; we expect users' matrices not to be close to the
469: * RMAX threshold.
470: *
471: SCALE = ONE
472: TNRM = DLANST( 'M', N, D, E )
473: IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
474: SCALE = RMIN / TNRM
475: ELSE IF( TNRM.GT.RMAX ) THEN
476: SCALE = RMAX / TNRM
477: END IF
478: IF( SCALE.NE.ONE ) THEN
479: CALL DSCAL( N, SCALE, D, 1 )
480: CALL DSCAL( N-1, SCALE, E, 1 )
481: TNRM = TNRM*SCALE
482: IF( VALEIG ) THEN
483: * If eigenvalues in interval have to be found,
484: * scale (WL, WU] accordingly
485: WL = WL*SCALE
486: WU = WU*SCALE
487: ENDIF
488: END IF
489: *
490: * Compute the desired eigenvalues of the tridiagonal after splitting
491: * into smaller subblocks if the corresponding off-diagonal elements
492: * are small
493: * THRESH is the splitting parameter for DLARRE
494: * A negative THRESH forces the old splitting criterion based on the
495: * size of the off-diagonal. A positive THRESH switches to splitting
496: * which preserves relative accuracy.
497: *
498: IF( TRYRAC ) THEN
499: * Test whether the matrix warrants the more expensive relative approach.
500: CALL DLARRR( N, D, E, IINFO )
501: ELSE
502: * The user does not care about relative accurately eigenvalues
503: IINFO = -1
504: ENDIF
505: * Set the splitting criterion
506: IF (IINFO.EQ.0) THEN
507: THRESH = EPS
508: ELSE
509: THRESH = -EPS
510: * relative accuracy is desired but T does not guarantee it
511: TRYRAC = .FALSE.
512: ENDIF
513: *
514: IF( TRYRAC ) THEN
515: * Copy original diagonal, needed to guarantee relative accuracy
516: CALL DCOPY(N,D,1,WORK(INDD),1)
517: ENDIF
518: * Store the squares of the offdiagonal values of T
519: DO 5 J = 1, N-1
520: WORK( INDE2+J-1 ) = E(J)**2
521: 5 CONTINUE
522:
523: * Set the tolerance parameters for bisection
524: IF( .NOT.WANTZ ) THEN
525: * DLARRE computes the eigenvalues to full precision.
526: RTOL1 = FOUR * EPS
527: RTOL2 = FOUR * EPS
528: ELSE
529: * DLARRE computes the eigenvalues to less than full precision.
530: * ZLARRV will refine the eigenvalue approximations, and we only
531: * need less accurate initial bisection in DLARRE.
532: * Note: these settings do only affect the subset case and DLARRE
533: RTOL1 = SQRT(EPS)
534: RTOL2 = MAX( SQRT(EPS)*5.0D-3, FOUR * EPS )
535: ENDIF
536: CALL DLARRE( RANGE, N, WL, WU, IIL, IIU, D, E,
537: $ WORK(INDE2), RTOL1, RTOL2, THRESH, NSPLIT,
538: $ IWORK( IINSPL ), M, W, WORK( INDERR ),
539: $ WORK( INDGP ), IWORK( IINDBL ),
540: $ IWORK( IINDW ), WORK( INDGRS ), PIVMIN,
541: $ WORK( INDWRK ), IWORK( IINDWK ), IINFO )
542: IF( IINFO.NE.0 ) THEN
543: INFO = 10 + ABS( IINFO )
544: RETURN
545: END IF
546: * Note that if RANGE .NE. 'V', DLARRE computes bounds on the desired
547: * part of the spectrum. All desired eigenvalues are contained in
548: * (WL,WU]
549:
550:
551: IF( WANTZ ) THEN
552: *
553: * Compute the desired eigenvectors corresponding to the computed
554: * eigenvalues
555: *
556: CALL ZLARRV( N, WL, WU, D, E,
557: $ PIVMIN, IWORK( IINSPL ), M,
558: $ 1, M, MINRGP, RTOL1, RTOL2,
559: $ W, WORK( INDERR ), WORK( INDGP ), IWORK( IINDBL ),
560: $ IWORK( IINDW ), WORK( INDGRS ), Z, LDZ,
561: $ ISUPPZ, WORK( INDWRK ), IWORK( IINDWK ), IINFO )
562: IF( IINFO.NE.0 ) THEN
563: INFO = 20 + ABS( IINFO )
564: RETURN
565: END IF
566: ELSE
567: * DLARRE computes eigenvalues of the (shifted) root representation
568: * ZLARRV returns the eigenvalues of the unshifted matrix.
569: * However, if the eigenvectors are not desired by the user, we need
570: * to apply the corresponding shifts from DLARRE to obtain the
571: * eigenvalues of the original matrix.
572: DO 20 J = 1, M
573: ITMP = IWORK( IINDBL+J-1 )
574: W( J ) = W( J ) + E( IWORK( IINSPL+ITMP-1 ) )
575: 20 CONTINUE
576: END IF
577: *
578:
579: IF ( TRYRAC ) THEN
580: * Refine computed eigenvalues so that they are relatively accurate
581: * with respect to the original matrix T.
582: IBEGIN = 1
583: WBEGIN = 1
584: DO 39 JBLK = 1, IWORK( IINDBL+M-1 )
585: IEND = IWORK( IINSPL+JBLK-1 )
586: IN = IEND - IBEGIN + 1
587: WEND = WBEGIN - 1
588: * check if any eigenvalues have to be refined in this block
589: 36 CONTINUE
590: IF( WEND.LT.M ) THEN
591: IF( IWORK( IINDBL+WEND ).EQ.JBLK ) THEN
592: WEND = WEND + 1
593: GO TO 36
594: END IF
595: END IF
596: IF( WEND.LT.WBEGIN ) THEN
597: IBEGIN = IEND + 1
598: GO TO 39
599: END IF
600:
601: OFFSET = IWORK(IINDW+WBEGIN-1)-1
602: IFIRST = IWORK(IINDW+WBEGIN-1)
603: ILAST = IWORK(IINDW+WEND-1)
604: RTOL2 = FOUR * EPS
605: CALL DLARRJ( IN,
606: $ WORK(INDD+IBEGIN-1), WORK(INDE2+IBEGIN-1),
607: $ IFIRST, ILAST, RTOL2, OFFSET, W(WBEGIN),
608: $ WORK( INDERR+WBEGIN-1 ),
609: $ WORK( INDWRK ), IWORK( IINDWK ), PIVMIN,
610: $ TNRM, IINFO )
611: IBEGIN = IEND + 1
612: WBEGIN = WEND + 1
613: 39 CONTINUE
614: ENDIF
615: *
616: * If matrix was scaled, then rescale eigenvalues appropriately.
617: *
618: IF( SCALE.NE.ONE ) THEN
619: CALL DSCAL( M, ONE / SCALE, W, 1 )
620: END IF
621: *
622: * If eigenvalues are not in increasing order, then sort them,
623: * possibly along with eigenvectors.
624: *
625: IF( NSPLIT.GT.1 ) THEN
626: IF( .NOT. WANTZ ) THEN
627: CALL DLASRT( 'I', M, W, IINFO )
628: IF( IINFO.NE.0 ) THEN
629: INFO = 3
630: RETURN
631: END IF
632: ELSE
633: DO 60 J = 1, M - 1
634: I = 0
635: TMP = W( J )
636: DO 50 JJ = J + 1, M
637: IF( W( JJ ).LT.TMP ) THEN
638: I = JJ
639: TMP = W( JJ )
640: END IF
641: 50 CONTINUE
642: IF( I.NE.0 ) THEN
643: W( I ) = W( J )
644: W( J ) = TMP
645: IF( WANTZ ) THEN
646: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
647: ITMP = ISUPPZ( 2*I-1 )
648: ISUPPZ( 2*I-1 ) = ISUPPZ( 2*J-1 )
649: ISUPPZ( 2*J-1 ) = ITMP
650: ITMP = ISUPPZ( 2*I )
651: ISUPPZ( 2*I ) = ISUPPZ( 2*J )
652: ISUPPZ( 2*J ) = ITMP
653: END IF
654: END IF
655: 60 CONTINUE
656: END IF
657: ENDIF
658: *
659: *
660: WORK( 1 ) = LWMIN
661: IWORK( 1 ) = LIWMIN
662: RETURN
663: *
664: * End of ZSTEMR
665: *
666: END
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