1: *> \brief <b> DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY 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 DSYEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyevx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
22: * ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
23: * IFAIL, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE, UPLO
27: * INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
28: * DOUBLE PRECISION ABSTOL, VL, VU
29: * ..
30: * .. Array Arguments ..
31: * INTEGER IFAIL( * ), IWORK( * )
32: * DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DSYEVX computes selected eigenvalues and, optionally, eigenvectors
42: *> of a real symmetric matrix A. Eigenvalues and eigenvectors can be
43: *> selected by specifying either a range of values or a range of indices
44: *> for the desired eigenvalues.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] JOBZ
51: *> \verbatim
52: *> JOBZ is CHARACTER*1
53: *> = 'N': Compute eigenvalues only;
54: *> = 'V': Compute eigenvalues and eigenvectors.
55: *> \endverbatim
56: *>
57: *> \param[in] RANGE
58: *> \verbatim
59: *> RANGE is CHARACTER*1
60: *> = 'A': all eigenvalues will be found.
61: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
62: *> will be found.
63: *> = 'I': the IL-th through IU-th eigenvalues will be found.
64: *> \endverbatim
65: *>
66: *> \param[in] UPLO
67: *> \verbatim
68: *> UPLO is CHARACTER*1
69: *> = 'U': Upper triangle of A is stored;
70: *> = 'L': Lower triangle of A is stored.
71: *> \endverbatim
72: *>
73: *> \param[in] N
74: *> \verbatim
75: *> N is INTEGER
76: *> The order of the matrix A. N >= 0.
77: *> \endverbatim
78: *>
79: *> \param[in,out] A
80: *> \verbatim
81: *> A is DOUBLE PRECISION array, dimension (LDA, N)
82: *> On entry, the symmetric matrix A. If UPLO = 'U', the
83: *> leading N-by-N upper triangular part of A contains the
84: *> upper triangular part of the matrix A. If UPLO = 'L',
85: *> the leading N-by-N lower triangular part of A contains
86: *> the lower triangular part of the matrix A.
87: *> On exit, the lower triangle (if UPLO='L') or the upper
88: *> triangle (if UPLO='U') of A, including the diagonal, is
89: *> destroyed.
90: *> \endverbatim
91: *>
92: *> \param[in] LDA
93: *> \verbatim
94: *> LDA is INTEGER
95: *> The leading dimension of the array A. LDA >= max(1,N).
96: *> \endverbatim
97: *>
98: *> \param[in] VL
99: *> \verbatim
100: *> VL is DOUBLE PRECISION
101: *> If RANGE='V', the lower bound of the interval to
102: *> be searched for eigenvalues. VL < VU.
103: *> Not referenced if RANGE = 'A' or 'I'.
104: *> \endverbatim
105: *>
106: *> \param[in] VU
107: *> \verbatim
108: *> VU is DOUBLE PRECISION
109: *> If RANGE='V', the upper bound of the interval to
110: *> be searched for eigenvalues. VL < VU.
111: *> Not referenced if RANGE = 'A' or 'I'.
112: *> \endverbatim
113: *>
114: *> \param[in] IL
115: *> \verbatim
116: *> IL is INTEGER
117: *> If RANGE='I', the index of the
118: *> smallest eigenvalue to be returned.
119: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
120: *> Not referenced if RANGE = 'A' or 'V'.
121: *> \endverbatim
122: *>
123: *> \param[in] IU
124: *> \verbatim
125: *> IU is INTEGER
126: *> If RANGE='I', the index of the
127: *> largest eigenvalue to be returned.
128: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
129: *> Not referenced if RANGE = 'A' or 'V'.
130: *> \endverbatim
131: *>
132: *> \param[in] ABSTOL
133: *> \verbatim
134: *> ABSTOL is DOUBLE PRECISION
135: *> The absolute error tolerance for the eigenvalues.
136: *> An approximate eigenvalue is accepted as converged
137: *> when it is determined to lie in an interval [a,b]
138: *> of width less than or equal to
139: *>
140: *> ABSTOL + EPS * max( |a|,|b| ) ,
141: *>
142: *> where EPS is the machine precision. If ABSTOL is less than
143: *> or equal to zero, then EPS*|T| will be used in its place,
144: *> where |T| is the 1-norm of the tridiagonal matrix obtained
145: *> by reducing A to tridiagonal form.
146: *>
147: *> Eigenvalues will be computed most accurately when ABSTOL is
148: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
149: *> If this routine returns with INFO>0, indicating that some
150: *> eigenvectors did not converge, try setting ABSTOL to
151: *> 2*DLAMCH('S').
152: *>
153: *> See "Computing Small Singular Values of Bidiagonal Matrices
154: *> with Guaranteed High Relative Accuracy," by Demmel and
155: *> Kahan, LAPACK Working Note #3.
156: *> \endverbatim
157: *>
158: *> \param[out] M
159: *> \verbatim
160: *> M is INTEGER
161: *> The total number of eigenvalues found. 0 <= M <= N.
162: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
163: *> \endverbatim
164: *>
165: *> \param[out] W
166: *> \verbatim
167: *> W is DOUBLE PRECISION array, dimension (N)
168: *> On normal exit, the first M elements contain the selected
169: *> eigenvalues in ascending order.
170: *> \endverbatim
171: *>
172: *> \param[out] Z
173: *> \verbatim
174: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
175: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
176: *> contain the orthonormal eigenvectors of the matrix A
177: *> corresponding to the selected eigenvalues, with the i-th
178: *> column of Z holding the eigenvector associated with W(i).
179: *> If an eigenvector fails to converge, then that column of Z
180: *> contains the latest approximation to the eigenvector, and the
181: *> index of the eigenvector is returned in IFAIL.
182: *> If JOBZ = 'N', then Z is not referenced.
183: *> Note: the user must ensure that at least max(1,M) columns are
184: *> supplied in the array Z; if RANGE = 'V', the exact value of M
185: *> is not known in advance and an upper bound must be used.
186: *> \endverbatim
187: *>
188: *> \param[in] LDZ
189: *> \verbatim
190: *> LDZ is INTEGER
191: *> The leading dimension of the array Z. LDZ >= 1, and if
192: *> JOBZ = 'V', LDZ >= max(1,N).
193: *> \endverbatim
194: *>
195: *> \param[out] WORK
196: *> \verbatim
197: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
198: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
199: *> \endverbatim
200: *>
201: *> \param[in] LWORK
202: *> \verbatim
203: *> LWORK is INTEGER
204: *> The length of the array WORK. LWORK >= 1, when N <= 1;
205: *> otherwise 8*N.
206: *> For optimal efficiency, LWORK >= (NB+3)*N,
207: *> where NB is the max of the blocksize for DSYTRD and DORMTR
208: *> returned by ILAENV.
209: *>
210: *> If LWORK = -1, then a workspace query is assumed; the routine
211: *> only calculates the optimal size of the WORK array, returns
212: *> this value as the first entry of the WORK array, and no error
213: *> message related to LWORK is issued by XERBLA.
214: *> \endverbatim
215: *>
216: *> \param[out] IWORK
217: *> \verbatim
218: *> IWORK is INTEGER array, dimension (5*N)
219: *> \endverbatim
220: *>
221: *> \param[out] IFAIL
222: *> \verbatim
223: *> IFAIL is INTEGER array, dimension (N)
224: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
225: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
226: *> indices of the eigenvectors that failed to converge.
227: *> If JOBZ = 'N', then IFAIL is not referenced.
228: *> \endverbatim
229: *>
230: *> \param[out] INFO
231: *> \verbatim
232: *> INFO is INTEGER
233: *> = 0: successful exit
234: *> < 0: if INFO = -i, the i-th argument had an illegal value
235: *> > 0: if INFO = i, then i eigenvectors failed to converge.
236: *> Their indices are stored in array IFAIL.
237: *> \endverbatim
238: *
239: * Authors:
240: * ========
241: *
242: *> \author Univ. of Tennessee
243: *> \author Univ. of California Berkeley
244: *> \author Univ. of Colorado Denver
245: *> \author NAG Ltd.
246: *
247: *> \date June 2016
248: *
249: *> \ingroup doubleSYeigen
250: *
251: * =====================================================================
252: SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
253: $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
254: $ IFAIL, INFO )
255: *
256: * -- LAPACK driver routine (version 3.6.1) --
257: * -- LAPACK is a software package provided by Univ. of Tennessee, --
258: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
259: * June 2016
260: *
261: * .. Scalar Arguments ..
262: CHARACTER JOBZ, RANGE, UPLO
263: INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
264: DOUBLE PRECISION ABSTOL, VL, VU
265: * ..
266: * .. Array Arguments ..
267: INTEGER IFAIL( * ), IWORK( * )
268: DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
269: * ..
270: *
271: * =====================================================================
272: *
273: * .. Parameters ..
274: DOUBLE PRECISION ZERO, ONE
275: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
276: * ..
277: * .. Local Scalars ..
278: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
279: $ WANTZ
280: CHARACTER ORDER
281: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
282: $ INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
283: $ ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN,
284: $ LWKOPT, NB, NSPLIT
285: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
286: $ SIGMA, SMLNUM, TMP1, VLL, VUU
287: * ..
288: * .. External Functions ..
289: LOGICAL LSAME
290: INTEGER ILAENV
291: DOUBLE PRECISION DLAMCH, DLANSY
292: EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
293: * ..
294: * .. External Subroutines ..
295: EXTERNAL DCOPY, DLACPY, DORGTR, DORMTR, DSCAL, DSTEBZ,
296: $ DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA
297: * ..
298: * .. Intrinsic Functions ..
299: INTRINSIC MAX, MIN, SQRT
300: * ..
301: * .. Executable Statements ..
302: *
303: * Test the input parameters.
304: *
305: LOWER = LSAME( UPLO, 'L' )
306: WANTZ = LSAME( JOBZ, 'V' )
307: ALLEIG = LSAME( RANGE, 'A' )
308: VALEIG = LSAME( RANGE, 'V' )
309: INDEIG = LSAME( RANGE, 'I' )
310: LQUERY = ( LWORK.EQ.-1 )
311: *
312: INFO = 0
313: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
314: INFO = -1
315: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
316: INFO = -2
317: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
318: INFO = -3
319: ELSE IF( N.LT.0 ) THEN
320: INFO = -4
321: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
322: INFO = -6
323: ELSE
324: IF( VALEIG ) THEN
325: IF( N.GT.0 .AND. VU.LE.VL )
326: $ INFO = -8
327: ELSE IF( INDEIG ) THEN
328: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
329: INFO = -9
330: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
331: INFO = -10
332: END IF
333: END IF
334: END IF
335: IF( INFO.EQ.0 ) THEN
336: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
337: INFO = -15
338: END IF
339: END IF
340: *
341: IF( INFO.EQ.0 ) THEN
342: IF( N.LE.1 ) THEN
343: LWKMIN = 1
344: WORK( 1 ) = LWKMIN
345: ELSE
346: LWKMIN = 8*N
347: NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
348: NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
349: LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
350: WORK( 1 ) = LWKOPT
351: END IF
352: *
353: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
354: $ INFO = -17
355: END IF
356: *
357: IF( INFO.NE.0 ) THEN
358: CALL XERBLA( 'DSYEVX', -INFO )
359: RETURN
360: ELSE IF( LQUERY ) THEN
361: RETURN
362: END IF
363: *
364: * Quick return if possible
365: *
366: M = 0
367: IF( N.EQ.0 ) THEN
368: RETURN
369: END IF
370: *
371: IF( N.EQ.1 ) THEN
372: IF( ALLEIG .OR. INDEIG ) THEN
373: M = 1
374: W( 1 ) = A( 1, 1 )
375: ELSE
376: IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
377: M = 1
378: W( 1 ) = A( 1, 1 )
379: END IF
380: END IF
381: IF( WANTZ )
382: $ Z( 1, 1 ) = ONE
383: RETURN
384: END IF
385: *
386: * Get machine constants.
387: *
388: SAFMIN = DLAMCH( 'Safe minimum' )
389: EPS = DLAMCH( 'Precision' )
390: SMLNUM = SAFMIN / EPS
391: BIGNUM = ONE / SMLNUM
392: RMIN = SQRT( SMLNUM )
393: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
394: *
395: * Scale matrix to allowable range, if necessary.
396: *
397: ISCALE = 0
398: ABSTLL = ABSTOL
399: IF( VALEIG ) THEN
400: VLL = VL
401: VUU = VU
402: END IF
403: ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
404: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
405: ISCALE = 1
406: SIGMA = RMIN / ANRM
407: ELSE IF( ANRM.GT.RMAX ) THEN
408: ISCALE = 1
409: SIGMA = RMAX / ANRM
410: END IF
411: IF( ISCALE.EQ.1 ) THEN
412: IF( LOWER ) THEN
413: DO 10 J = 1, N
414: CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
415: 10 CONTINUE
416: ELSE
417: DO 20 J = 1, N
418: CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
419: 20 CONTINUE
420: END IF
421: IF( ABSTOL.GT.0 )
422: $ ABSTLL = ABSTOL*SIGMA
423: IF( VALEIG ) THEN
424: VLL = VL*SIGMA
425: VUU = VU*SIGMA
426: END IF
427: END IF
428: *
429: * Call DSYTRD to reduce symmetric matrix to tridiagonal form.
430: *
431: INDTAU = 1
432: INDE = INDTAU + N
433: INDD = INDE + N
434: INDWRK = INDD + N
435: LLWORK = LWORK - INDWRK + 1
436: CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
437: $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
438: *
439: * If all eigenvalues are desired and ABSTOL is less than or equal to
440: * zero, then call DSTERF or DORGTR and SSTEQR. If this fails for
441: * some eigenvalue, then try DSTEBZ.
442: *
443: TEST = .FALSE.
444: IF( INDEIG ) THEN
445: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
446: TEST = .TRUE.
447: END IF
448: END IF
449: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
450: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
451: INDEE = INDWRK + 2*N
452: IF( .NOT.WANTZ ) THEN
453: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
454: CALL DSTERF( N, W, WORK( INDEE ), INFO )
455: ELSE
456: CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ )
457: CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
458: $ WORK( INDWRK ), LLWORK, IINFO )
459: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
460: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
461: $ WORK( INDWRK ), INFO )
462: IF( INFO.EQ.0 ) THEN
463: DO 30 I = 1, N
464: IFAIL( I ) = 0
465: 30 CONTINUE
466: END IF
467: END IF
468: IF( INFO.EQ.0 ) THEN
469: M = N
470: GO TO 40
471: END IF
472: INFO = 0
473: END IF
474: *
475: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
476: *
477: IF( WANTZ ) THEN
478: ORDER = 'B'
479: ELSE
480: ORDER = 'E'
481: END IF
482: INDIBL = 1
483: INDISP = INDIBL + N
484: INDIWO = INDISP + N
485: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
486: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
487: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
488: $ IWORK( INDIWO ), INFO )
489: *
490: IF( WANTZ ) THEN
491: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
492: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
493: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
494: *
495: * Apply orthogonal matrix used in reduction to tridiagonal
496: * form to eigenvectors returned by DSTEIN.
497: *
498: INDWKN = INDE
499: LLWRKN = LWORK - INDWKN + 1
500: CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
501: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
502: END IF
503: *
504: * If matrix was scaled, then rescale eigenvalues appropriately.
505: *
506: 40 CONTINUE
507: IF( ISCALE.EQ.1 ) THEN
508: IF( INFO.EQ.0 ) THEN
509: IMAX = M
510: ELSE
511: IMAX = INFO - 1
512: END IF
513: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
514: END IF
515: *
516: * If eigenvalues are not in order, then sort them, along with
517: * eigenvectors.
518: *
519: IF( WANTZ ) THEN
520: DO 60 J = 1, M - 1
521: I = 0
522: TMP1 = W( J )
523: DO 50 JJ = J + 1, M
524: IF( W( JJ ).LT.TMP1 ) THEN
525: I = JJ
526: TMP1 = W( JJ )
527: END IF
528: 50 CONTINUE
529: *
530: IF( I.NE.0 ) THEN
531: ITMP1 = IWORK( INDIBL+I-1 )
532: W( I ) = W( J )
533: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
534: W( J ) = TMP1
535: IWORK( INDIBL+J-1 ) = ITMP1
536: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
537: IF( INFO.NE.0 ) THEN
538: ITMP1 = IFAIL( I )
539: IFAIL( I ) = IFAIL( J )
540: IFAIL( J ) = ITMP1
541: END IF
542: END IF
543: 60 CONTINUE
544: END IF
545: *
546: * Set WORK(1) to optimal workspace size.
547: *
548: WORK( 1 ) = LWKOPT
549: *
550: RETURN
551: *
552: * End of DSYEVX
553: *
554: END
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