1: *> \brief <b> ZHEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE 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 ZHEEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zheevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
22: * ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
23: * IWORK, 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 RWORK( * ), W( * )
33: * COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> ZHEEVX computes selected eigenvalues and, optionally, eigenvectors
43: *> of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
44: *> be selected by specifying either a range of values or a range of
45: *> indices for the desired eigenvalues.
46: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] JOBZ
52: *> \verbatim
53: *> JOBZ is CHARACTER*1
54: *> = 'N': Compute eigenvalues only;
55: *> = 'V': Compute eigenvalues and eigenvectors.
56: *> \endverbatim
57: *>
58: *> \param[in] RANGE
59: *> \verbatim
60: *> RANGE is CHARACTER*1
61: *> = 'A': all eigenvalues will be found.
62: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
63: *> will be found.
64: *> = 'I': the IL-th through IU-th eigenvalues will be found.
65: *> \endverbatim
66: *>
67: *> \param[in] UPLO
68: *> \verbatim
69: *> UPLO is CHARACTER*1
70: *> = 'U': Upper triangle of A is stored;
71: *> = 'L': Lower triangle of A is stored.
72: *> \endverbatim
73: *>
74: *> \param[in] N
75: *> \verbatim
76: *> N is INTEGER
77: *> The order of the matrix A. N >= 0.
78: *> \endverbatim
79: *>
80: *> \param[in,out] A
81: *> \verbatim
82: *> A is COMPLEX*16 array, dimension (LDA, N)
83: *> On entry, the Hermitian matrix A. If UPLO = 'U', the
84: *> leading N-by-N upper triangular part of A contains the
85: *> upper triangular part of the matrix A. If UPLO = 'L',
86: *> the leading N-by-N lower triangular part of A contains
87: *> the lower triangular part of the matrix A.
88: *> On exit, the lower triangle (if UPLO='L') or the upper
89: *> triangle (if UPLO='U') of A, including the diagonal, is
90: *> destroyed.
91: *> \endverbatim
92: *>
93: *> \param[in] LDA
94: *> \verbatim
95: *> LDA is INTEGER
96: *> The leading dimension of the array A. LDA >= max(1,N).
97: *> \endverbatim
98: *>
99: *> \param[in] VL
100: *> \verbatim
101: *> VL is DOUBLE PRECISION
102: *> If RANGE='V', the lower bound of the interval to
103: *> be searched for eigenvalues. VL < VU.
104: *> Not referenced if RANGE = 'A' or 'I'.
105: *> \endverbatim
106: *>
107: *> \param[in] VU
108: *> \verbatim
109: *> VU is DOUBLE PRECISION
110: *> If RANGE='V', the upper bound of the interval to
111: *> be searched for eigenvalues. VL < VU.
112: *> Not referenced if RANGE = 'A' or 'I'.
113: *> \endverbatim
114: *>
115: *> \param[in] IL
116: *> \verbatim
117: *> IL is INTEGER
118: *> If RANGE='I', the index of the
119: *> smallest eigenvalue to be returned.
120: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
121: *> Not referenced if RANGE = 'A' or 'V'.
122: *> \endverbatim
123: *>
124: *> \param[in] IU
125: *> \verbatim
126: *> IU is INTEGER
127: *> If RANGE='I', the index of the
128: *> largest eigenvalue to be returned.
129: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
130: *> Not referenced if RANGE = 'A' or 'V'.
131: *> \endverbatim
132: *>
133: *> \param[in] ABSTOL
134: *> \verbatim
135: *> ABSTOL is DOUBLE PRECISION
136: *> The absolute error tolerance for the eigenvalues.
137: *> An approximate eigenvalue is accepted as converged
138: *> when it is determined to lie in an interval [a,b]
139: *> of width less than or equal to
140: *>
141: *> ABSTOL + EPS * max( |a|,|b| ) ,
142: *>
143: *> where EPS is the machine precision. If ABSTOL is less than
144: *> or equal to zero, then EPS*|T| will be used in its place,
145: *> where |T| is the 1-norm of the tridiagonal matrix obtained
146: *> by reducing A to tridiagonal form.
147: *>
148: *> Eigenvalues will be computed most accurately when ABSTOL is
149: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
150: *> If this routine returns with INFO>0, indicating that some
151: *> eigenvectors did not converge, try setting ABSTOL to
152: *> 2*DLAMCH('S').
153: *>
154: *> See "Computing Small Singular Values of Bidiagonal Matrices
155: *> with Guaranteed High Relative Accuracy," by Demmel and
156: *> Kahan, LAPACK Working Note #3.
157: *> \endverbatim
158: *>
159: *> \param[out] M
160: *> \verbatim
161: *> M is INTEGER
162: *> The total number of eigenvalues found. 0 <= M <= N.
163: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
164: *> \endverbatim
165: *>
166: *> \param[out] W
167: *> \verbatim
168: *> W is DOUBLE PRECISION array, dimension (N)
169: *> On normal exit, the first M elements contain the selected
170: *> eigenvalues in ascending order.
171: *> \endverbatim
172: *>
173: *> \param[out] Z
174: *> \verbatim
175: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
176: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
177: *> contain the orthonormal eigenvectors of the matrix A
178: *> corresponding to the selected eigenvalues, with the i-th
179: *> column of Z holding the eigenvector associated with W(i).
180: *> If an eigenvector fails to converge, then that column of Z
181: *> contains the latest approximation to the eigenvector, and the
182: *> index of the eigenvector is returned in IFAIL.
183: *> If JOBZ = 'N', then Z is not referenced.
184: *> Note: the user must ensure that at least max(1,M) columns are
185: *> supplied in the array Z; if RANGE = 'V', the exact value of M
186: *> is not known in advance and an upper bound must be used.
187: *> \endverbatim
188: *>
189: *> \param[in] LDZ
190: *> \verbatim
191: *> LDZ is INTEGER
192: *> The leading dimension of the array Z. LDZ >= 1, and if
193: *> JOBZ = 'V', LDZ >= max(1,N).
194: *> \endverbatim
195: *>
196: *> \param[out] WORK
197: *> \verbatim
198: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
199: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
200: *> \endverbatim
201: *>
202: *> \param[in] LWORK
203: *> \verbatim
204: *> LWORK is INTEGER
205: *> The length of the array WORK. LWORK >= 1, when N <= 1;
206: *> otherwise 2*N.
207: *> For optimal efficiency, LWORK >= (NB+1)*N,
208: *> where NB is the max of the blocksize for ZHETRD and for
209: *> ZUNMTR as returned by ILAENV.
210: *>
211: *> If LWORK = -1, then a workspace query is assumed; the routine
212: *> only calculates the optimal size of the WORK array, returns
213: *> this value as the first entry of the WORK array, and no error
214: *> message related to LWORK is issued by XERBLA.
215: *> \endverbatim
216: *>
217: *> \param[out] RWORK
218: *> \verbatim
219: *> RWORK is DOUBLE PRECISION array, dimension (7*N)
220: *> \endverbatim
221: *>
222: *> \param[out] IWORK
223: *> \verbatim
224: *> IWORK is INTEGER array, dimension (5*N)
225: *> \endverbatim
226: *>
227: *> \param[out] IFAIL
228: *> \verbatim
229: *> IFAIL is INTEGER array, dimension (N)
230: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
231: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
232: *> indices of the eigenvectors that failed to converge.
233: *> If JOBZ = 'N', then IFAIL is not referenced.
234: *> \endverbatim
235: *>
236: *> \param[out] INFO
237: *> \verbatim
238: *> INFO is INTEGER
239: *> = 0: successful exit
240: *> < 0: if INFO = -i, the i-th argument had an illegal value
241: *> > 0: if INFO = i, then i eigenvectors failed to converge.
242: *> Their indices are stored in array IFAIL.
243: *> \endverbatim
244: *
245: * Authors:
246: * ========
247: *
248: *> \author Univ. of Tennessee
249: *> \author Univ. of California Berkeley
250: *> \author Univ. of Colorado Denver
251: *> \author NAG Ltd.
252: *
253: *> \ingroup complex16HEeigen
254: *
255: * =====================================================================
256: SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
257: $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
258: $ IWORK, IFAIL, INFO )
259: *
260: * -- LAPACK driver routine --
261: * -- LAPACK is a software package provided by Univ. of Tennessee, --
262: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
263: *
264: * .. Scalar Arguments ..
265: CHARACTER JOBZ, RANGE, UPLO
266: INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
267: DOUBLE PRECISION ABSTOL, VL, VU
268: * ..
269: * .. Array Arguments ..
270: INTEGER IFAIL( * ), IWORK( * )
271: DOUBLE PRECISION RWORK( * ), W( * )
272: COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
273: * ..
274: *
275: * =====================================================================
276: *
277: * .. Parameters ..
278: DOUBLE PRECISION ZERO, ONE
279: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
280: COMPLEX*16 CONE
281: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
282: * ..
283: * .. Local Scalars ..
284: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
285: $ WANTZ
286: CHARACTER ORDER
287: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
288: $ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
289: $ ITMP1, J, JJ, LLWORK, LWKMIN, LWKOPT, NB,
290: $ NSPLIT
291: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
292: $ SIGMA, SMLNUM, TMP1, VLL, VUU
293: * ..
294: * .. External Functions ..
295: LOGICAL LSAME
296: INTEGER ILAENV
297: DOUBLE PRECISION DLAMCH, ZLANHE
298: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE
299: * ..
300: * .. External Subroutines ..
301: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
302: $ ZHETRD, ZLACPY, ZSTEIN, ZSTEQR, ZSWAP, ZUNGTR,
303: $ ZUNMTR
304: * ..
305: * .. Intrinsic Functions ..
306: INTRINSIC DBLE, MAX, MIN, SQRT
307: * ..
308: * .. Executable Statements ..
309: *
310: * Test the input parameters.
311: *
312: LOWER = LSAME( UPLO, 'L' )
313: WANTZ = LSAME( JOBZ, 'V' )
314: ALLEIG = LSAME( RANGE, 'A' )
315: VALEIG = LSAME( RANGE, 'V' )
316: INDEIG = LSAME( RANGE, 'I' )
317: LQUERY = ( LWORK.EQ.-1 )
318: *
319: INFO = 0
320: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
321: INFO = -1
322: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
323: INFO = -2
324: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
325: INFO = -3
326: ELSE IF( N.LT.0 ) THEN
327: INFO = -4
328: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
329: INFO = -6
330: ELSE
331: IF( VALEIG ) THEN
332: IF( N.GT.0 .AND. VU.LE.VL )
333: $ INFO = -8
334: ELSE IF( INDEIG ) THEN
335: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
336: INFO = -9
337: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
338: INFO = -10
339: END IF
340: END IF
341: END IF
342: IF( INFO.EQ.0 ) THEN
343: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
344: INFO = -15
345: END IF
346: END IF
347: *
348: IF( INFO.EQ.0 ) THEN
349: IF( N.LE.1 ) THEN
350: LWKMIN = 1
351: WORK( 1 ) = LWKMIN
352: ELSE
353: LWKMIN = 2*N
354: NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
355: NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
356: LWKOPT = MAX( 1, ( NB + 1 )*N )
357: WORK( 1 ) = LWKOPT
358: END IF
359: *
360: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
361: $ INFO = -17
362: END IF
363: *
364: IF( INFO.NE.0 ) THEN
365: CALL XERBLA( 'ZHEEVX', -INFO )
366: RETURN
367: ELSE IF( LQUERY ) THEN
368: RETURN
369: END IF
370: *
371: * Quick return if possible
372: *
373: M = 0
374: IF( N.EQ.0 ) THEN
375: RETURN
376: END IF
377: *
378: IF( N.EQ.1 ) THEN
379: IF( ALLEIG .OR. INDEIG ) THEN
380: M = 1
381: W( 1 ) = DBLE( A( 1, 1 ) )
382: ELSE IF( VALEIG ) THEN
383: IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
384: $ THEN
385: M = 1
386: W( 1 ) = DBLE( A( 1, 1 ) )
387: END IF
388: END IF
389: IF( WANTZ )
390: $ Z( 1, 1 ) = CONE
391: RETURN
392: END IF
393: *
394: * Get machine constants.
395: *
396: SAFMIN = DLAMCH( 'Safe minimum' )
397: EPS = DLAMCH( 'Precision' )
398: SMLNUM = SAFMIN / EPS
399: BIGNUM = ONE / SMLNUM
400: RMIN = SQRT( SMLNUM )
401: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
402: *
403: * Scale matrix to allowable range, if necessary.
404: *
405: ISCALE = 0
406: ABSTLL = ABSTOL
407: IF( VALEIG ) THEN
408: VLL = VL
409: VUU = VU
410: END IF
411: ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK )
412: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
413: ISCALE = 1
414: SIGMA = RMIN / ANRM
415: ELSE IF( ANRM.GT.RMAX ) THEN
416: ISCALE = 1
417: SIGMA = RMAX / ANRM
418: END IF
419: IF( ISCALE.EQ.1 ) THEN
420: IF( LOWER ) THEN
421: DO 10 J = 1, N
422: CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
423: 10 CONTINUE
424: ELSE
425: DO 20 J = 1, N
426: CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
427: 20 CONTINUE
428: END IF
429: IF( ABSTOL.GT.0 )
430: $ ABSTLL = ABSTOL*SIGMA
431: IF( VALEIG ) THEN
432: VLL = VL*SIGMA
433: VUU = VU*SIGMA
434: END IF
435: END IF
436: *
437: * Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
438: *
439: INDD = 1
440: INDE = INDD + N
441: INDRWK = INDE + N
442: INDTAU = 1
443: INDWRK = INDTAU + N
444: LLWORK = LWORK - INDWRK + 1
445: CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDD ), RWORK( INDE ),
446: $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
447: *
448: * If all eigenvalues are desired and ABSTOL is less than or equal to
449: * zero, then call DSTERF or ZUNGTR and ZSTEQR. If this fails for
450: * some eigenvalue, then try DSTEBZ.
451: *
452: TEST = .FALSE.
453: IF( INDEIG ) THEN
454: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
455: TEST = .TRUE.
456: END IF
457: END IF
458: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
459: CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
460: INDEE = INDRWK + 2*N
461: IF( .NOT.WANTZ ) THEN
462: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
463: CALL DSTERF( N, W, RWORK( INDEE ), INFO )
464: ELSE
465: CALL ZLACPY( 'A', N, N, A, LDA, Z, LDZ )
466: CALL ZUNGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
467: $ WORK( INDWRK ), LLWORK, IINFO )
468: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
469: CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
470: $ RWORK( INDRWK ), INFO )
471: IF( INFO.EQ.0 ) THEN
472: DO 30 I = 1, N
473: IFAIL( I ) = 0
474: 30 CONTINUE
475: END IF
476: END IF
477: IF( INFO.EQ.0 ) THEN
478: M = N
479: GO TO 40
480: END IF
481: INFO = 0
482: END IF
483: *
484: * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
485: *
486: IF( WANTZ ) THEN
487: ORDER = 'B'
488: ELSE
489: ORDER = 'E'
490: END IF
491: INDIBL = 1
492: INDISP = INDIBL + N
493: INDIWK = INDISP + N
494: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
495: $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
496: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
497: $ IWORK( INDIWK ), INFO )
498: *
499: IF( WANTZ ) THEN
500: CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
501: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
502: $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
503: *
504: * Apply unitary matrix used in reduction to tridiagonal
505: * form to eigenvectors returned by ZSTEIN.
506: *
507: CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
508: $ LDZ, WORK( INDWRK ), LLWORK, IINFO )
509: END IF
510: *
511: * If matrix was scaled, then rescale eigenvalues appropriately.
512: *
513: 40 CONTINUE
514: IF( ISCALE.EQ.1 ) THEN
515: IF( INFO.EQ.0 ) THEN
516: IMAX = M
517: ELSE
518: IMAX = INFO - 1
519: END IF
520: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
521: END IF
522: *
523: * If eigenvalues are not in order, then sort them, along with
524: * eigenvectors.
525: *
526: IF( WANTZ ) THEN
527: DO 60 J = 1, M - 1
528: I = 0
529: TMP1 = W( J )
530: DO 50 JJ = J + 1, M
531: IF( W( JJ ).LT.TMP1 ) THEN
532: I = JJ
533: TMP1 = W( JJ )
534: END IF
535: 50 CONTINUE
536: *
537: IF( I.NE.0 ) THEN
538: ITMP1 = IWORK( INDIBL+I-1 )
539: W( I ) = W( J )
540: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
541: W( J ) = TMP1
542: IWORK( INDIBL+J-1 ) = ITMP1
543: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
544: IF( INFO.NE.0 ) THEN
545: ITMP1 = IFAIL( I )
546: IFAIL( I ) = IFAIL( J )
547: IFAIL( J ) = ITMP1
548: END IF
549: END IF
550: 60 CONTINUE
551: END IF
552: *
553: * Set WORK(1) to optimal complex workspace size.
554: *
555: WORK( 1 ) = LWKOPT
556: *
557: RETURN
558: *
559: * End of ZHEEVX
560: *
561: END
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