1: *> \brief <b> ZHBEVX 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 ZHBEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhbevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhbevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhbevx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
22: * VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
23: * IWORK, IFAIL, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE, UPLO
27: * INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
28: * DOUBLE PRECISION ABSTOL, VL, VU
29: * ..
30: * .. Array Arguments ..
31: * INTEGER IFAIL( * ), IWORK( * )
32: * DOUBLE PRECISION RWORK( * ), W( * )
33: * COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
34: * $ Z( LDZ, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> ZHBEVX computes selected eigenvalues and, optionally, eigenvectors
44: *> of a complex Hermitian band matrix A. Eigenvalues and eigenvectors
45: *> can be selected by specifying either a range of values or a range of
46: *> indices for the desired eigenvalues.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] JOBZ
53: *> \verbatim
54: *> JOBZ is CHARACTER*1
55: *> = 'N': Compute eigenvalues only;
56: *> = 'V': Compute eigenvalues and eigenvectors.
57: *> \endverbatim
58: *>
59: *> \param[in] RANGE
60: *> \verbatim
61: *> RANGE is CHARACTER*1
62: *> = 'A': all eigenvalues will be found;
63: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
64: *> will be found;
65: *> = 'I': the IL-th through IU-th eigenvalues will be found.
66: *> \endverbatim
67: *>
68: *> \param[in] UPLO
69: *> \verbatim
70: *> UPLO is CHARACTER*1
71: *> = 'U': Upper triangle of A is stored;
72: *> = 'L': Lower triangle of A is stored.
73: *> \endverbatim
74: *>
75: *> \param[in] N
76: *> \verbatim
77: *> N is INTEGER
78: *> The order of the matrix A. N >= 0.
79: *> \endverbatim
80: *>
81: *> \param[in] KD
82: *> \verbatim
83: *> KD is INTEGER
84: *> The number of superdiagonals of the matrix A if UPLO = 'U',
85: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
86: *> \endverbatim
87: *>
88: *> \param[in,out] AB
89: *> \verbatim
90: *> AB is COMPLEX*16 array, dimension (LDAB, N)
91: *> On entry, the upper or lower triangle of the Hermitian band
92: *> matrix A, stored in the first KD+1 rows of the array. The
93: *> j-th column of A is stored in the j-th column of the array AB
94: *> as follows:
95: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
96: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
97: *>
98: *> On exit, AB is overwritten by values generated during the
99: *> reduction to tridiagonal form.
100: *> \endverbatim
101: *>
102: *> \param[in] LDAB
103: *> \verbatim
104: *> LDAB is INTEGER
105: *> The leading dimension of the array AB. LDAB >= KD + 1.
106: *> \endverbatim
107: *>
108: *> \param[out] Q
109: *> \verbatim
110: *> Q is COMPLEX*16 array, dimension (LDQ, N)
111: *> If JOBZ = 'V', the N-by-N unitary matrix used in the
112: *> reduction to tridiagonal form.
113: *> If JOBZ = 'N', the array Q is not referenced.
114: *> \endverbatim
115: *>
116: *> \param[in] LDQ
117: *> \verbatim
118: *> LDQ is INTEGER
119: *> The leading dimension of the array Q. If JOBZ = 'V', then
120: *> LDQ >= max(1,N).
121: *> \endverbatim
122: *>
123: *> \param[in] VL
124: *> \verbatim
125: *> VL is DOUBLE PRECISION
126: *> If RANGE='V', the lower bound of the interval to
127: *> be searched for eigenvalues. VL < VU.
128: *> Not referenced if RANGE = 'A' or 'I'.
129: *> \endverbatim
130: *>
131: *> \param[in] VU
132: *> \verbatim
133: *> VU is DOUBLE PRECISION
134: *> If RANGE='V', the upper bound of the interval to
135: *> be searched for eigenvalues. VL < VU.
136: *> Not referenced if RANGE = 'A' or 'I'.
137: *> \endverbatim
138: *>
139: *> \param[in] IL
140: *> \verbatim
141: *> IL is INTEGER
142: *> If RANGE='I', the index of the
143: *> smallest eigenvalue to be returned.
144: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
145: *> Not referenced if RANGE = 'A' or 'V'.
146: *> \endverbatim
147: *>
148: *> \param[in] IU
149: *> \verbatim
150: *> IU is INTEGER
151: *> If RANGE='I', the index of the
152: *> largest eigenvalue to be returned.
153: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
154: *> Not referenced if RANGE = 'A' or 'V'.
155: *> \endverbatim
156: *>
157: *> \param[in] ABSTOL
158: *> \verbatim
159: *> ABSTOL is DOUBLE PRECISION
160: *> The absolute error tolerance for the eigenvalues.
161: *> An approximate eigenvalue is accepted as converged
162: *> when it is determined to lie in an interval [a,b]
163: *> of width less than or equal to
164: *>
165: *> ABSTOL + EPS * max( |a|,|b| ) ,
166: *>
167: *> where EPS is the machine precision. If ABSTOL is less than
168: *> or equal to zero, then EPS*|T| will be used in its place,
169: *> where |T| is the 1-norm of the tridiagonal matrix obtained
170: *> by reducing AB to tridiagonal form.
171: *>
172: *> Eigenvalues will be computed most accurately when ABSTOL is
173: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
174: *> If this routine returns with INFO>0, indicating that some
175: *> eigenvectors did not converge, try setting ABSTOL to
176: *> 2*DLAMCH('S').
177: *>
178: *> See "Computing Small Singular Values of Bidiagonal Matrices
179: *> with Guaranteed High Relative Accuracy," by Demmel and
180: *> Kahan, LAPACK Working Note #3.
181: *> \endverbatim
182: *>
183: *> \param[out] M
184: *> \verbatim
185: *> M is INTEGER
186: *> The total number of eigenvalues found. 0 <= M <= N.
187: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
188: *> \endverbatim
189: *>
190: *> \param[out] W
191: *> \verbatim
192: *> W is DOUBLE PRECISION array, dimension (N)
193: *> The first M elements contain the selected eigenvalues in
194: *> ascending order.
195: *> \endverbatim
196: *>
197: *> \param[out] Z
198: *> \verbatim
199: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
200: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
201: *> contain the orthonormal eigenvectors of the matrix A
202: *> corresponding to the selected eigenvalues, with the i-th
203: *> column of Z holding the eigenvector associated with W(i).
204: *> If an eigenvector fails to converge, then that column of Z
205: *> contains the latest approximation to the eigenvector, and the
206: *> index of the eigenvector is returned in IFAIL.
207: *> If JOBZ = 'N', then Z is not referenced.
208: *> Note: the user must ensure that at least max(1,M) columns are
209: *> supplied in the array Z; if RANGE = 'V', the exact value of M
210: *> is not known in advance and an upper bound must be used.
211: *> \endverbatim
212: *>
213: *> \param[in] LDZ
214: *> \verbatim
215: *> LDZ is INTEGER
216: *> The leading dimension of the array Z. LDZ >= 1, and if
217: *> JOBZ = 'V', LDZ >= max(1,N).
218: *> \endverbatim
219: *>
220: *> \param[out] WORK
221: *> \verbatim
222: *> WORK is COMPLEX*16 array, dimension (N)
223: *> \endverbatim
224: *>
225: *> \param[out] RWORK
226: *> \verbatim
227: *> RWORK is DOUBLE PRECISION array, dimension (7*N)
228: *> \endverbatim
229: *>
230: *> \param[out] IWORK
231: *> \verbatim
232: *> IWORK is INTEGER array, dimension (5*N)
233: *> \endverbatim
234: *>
235: *> \param[out] IFAIL
236: *> \verbatim
237: *> IFAIL is INTEGER array, dimension (N)
238: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
239: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
240: *> indices of the eigenvectors that failed to converge.
241: *> If JOBZ = 'N', then IFAIL is not referenced.
242: *> \endverbatim
243: *>
244: *> \param[out] INFO
245: *> \verbatim
246: *> INFO is INTEGER
247: *> = 0: successful exit
248: *> < 0: if INFO = -i, the i-th argument had an illegal value
249: *> > 0: if INFO = i, then i eigenvectors failed to converge.
250: *> Their indices are stored in array IFAIL.
251: *> \endverbatim
252: *
253: * Authors:
254: * ========
255: *
256: *> \author Univ. of Tennessee
257: *> \author Univ. of California Berkeley
258: *> \author Univ. of Colorado Denver
259: *> \author NAG Ltd.
260: *
261: *> \date June 2016
262: *
263: *> \ingroup complex16OTHEReigen
264: *
265: * =====================================================================
266: SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
267: $ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
268: $ IWORK, IFAIL, INFO )
269: *
270: * -- LAPACK driver routine (version 3.7.0) --
271: * -- LAPACK is a software package provided by Univ. of Tennessee, --
272: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
273: * June 2016
274: *
275: * .. Scalar Arguments ..
276: CHARACTER JOBZ, RANGE, UPLO
277: INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
278: DOUBLE PRECISION ABSTOL, VL, VU
279: * ..
280: * .. Array Arguments ..
281: INTEGER IFAIL( * ), IWORK( * )
282: DOUBLE PRECISION RWORK( * ), W( * )
283: COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
284: $ Z( LDZ, * )
285: * ..
286: *
287: * =====================================================================
288: *
289: * .. Parameters ..
290: DOUBLE PRECISION ZERO, ONE
291: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
292: COMPLEX*16 CZERO, CONE
293: PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
294: $ CONE = ( 1.0D0, 0.0D0 ) )
295: * ..
296: * .. Local Scalars ..
297: LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
298: CHARACTER ORDER
299: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
300: $ INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1,
301: $ J, JJ, NSPLIT
302: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
303: $ SIGMA, SMLNUM, TMP1, VLL, VUU
304: COMPLEX*16 CTMP1
305: * ..
306: * .. External Functions ..
307: LOGICAL LSAME
308: DOUBLE PRECISION DLAMCH, ZLANHB
309: EXTERNAL LSAME, DLAMCH, ZLANHB
310: * ..
311: * .. External Subroutines ..
312: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZCOPY,
313: $ ZGEMV, ZHBTRD, ZLACPY, ZLASCL, ZSTEIN, ZSTEQR,
314: $ ZSWAP
315: * ..
316: * .. Intrinsic Functions ..
317: INTRINSIC DBLE, MAX, MIN, SQRT
318: * ..
319: * .. Executable Statements ..
320: *
321: * Test the input parameters.
322: *
323: WANTZ = LSAME( JOBZ, 'V' )
324: ALLEIG = LSAME( RANGE, 'A' )
325: VALEIG = LSAME( RANGE, 'V' )
326: INDEIG = LSAME( RANGE, 'I' )
327: LOWER = LSAME( UPLO, 'L' )
328: *
329: INFO = 0
330: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
331: INFO = -1
332: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
333: INFO = -2
334: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
335: INFO = -3
336: ELSE IF( N.LT.0 ) THEN
337: INFO = -4
338: ELSE IF( KD.LT.0 ) THEN
339: INFO = -5
340: ELSE IF( LDAB.LT.KD+1 ) THEN
341: INFO = -7
342: ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
343: INFO = -9
344: ELSE
345: IF( VALEIG ) THEN
346: IF( N.GT.0 .AND. VU.LE.VL )
347: $ INFO = -11
348: ELSE IF( INDEIG ) THEN
349: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
350: INFO = -12
351: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
352: INFO = -13
353: END IF
354: END IF
355: END IF
356: IF( INFO.EQ.0 ) THEN
357: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
358: $ INFO = -18
359: END IF
360: *
361: IF( INFO.NE.0 ) THEN
362: CALL XERBLA( 'ZHBEVX', -INFO )
363: RETURN
364: END IF
365: *
366: * Quick return if possible
367: *
368: M = 0
369: IF( N.EQ.0 )
370: $ RETURN
371: *
372: IF( N.EQ.1 ) THEN
373: M = 1
374: IF( LOWER ) THEN
375: CTMP1 = AB( 1, 1 )
376: ELSE
377: CTMP1 = AB( KD+1, 1 )
378: END IF
379: TMP1 = DBLE( CTMP1 )
380: IF( VALEIG ) THEN
381: IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
382: $ M = 0
383: END IF
384: IF( M.EQ.1 ) THEN
385: W( 1 ) = CTMP1
386: IF( WANTZ )
387: $ Z( 1, 1 ) = CONE
388: END IF
389: RETURN
390: END IF
391: *
392: * Get machine constants.
393: *
394: SAFMIN = DLAMCH( 'Safe minimum' )
395: EPS = DLAMCH( 'Precision' )
396: SMLNUM = SAFMIN / EPS
397: BIGNUM = ONE / SMLNUM
398: RMIN = SQRT( SMLNUM )
399: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
400: *
401: * Scale matrix to allowable range, if necessary.
402: *
403: ISCALE = 0
404: ABSTLL = ABSTOL
405: IF( VALEIG ) THEN
406: VLL = VL
407: VUU = VU
408: ELSE
409: VLL = ZERO
410: VUU = ZERO
411: END IF
412: ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
413: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
414: ISCALE = 1
415: SIGMA = RMIN / ANRM
416: ELSE IF( ANRM.GT.RMAX ) THEN
417: ISCALE = 1
418: SIGMA = RMAX / ANRM
419: END IF
420: IF( ISCALE.EQ.1 ) THEN
421: IF( LOWER ) THEN
422: CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
423: ELSE
424: CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
425: END IF
426: IF( ABSTOL.GT.0 )
427: $ ABSTLL = ABSTOL*SIGMA
428: IF( VALEIG ) THEN
429: VLL = VL*SIGMA
430: VUU = VU*SIGMA
431: END IF
432: END IF
433: *
434: * Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
435: *
436: INDD = 1
437: INDE = INDD + N
438: INDRWK = INDE + N
439: INDWRK = 1
440: CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, RWORK( INDD ),
441: $ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
442: *
443: * If all eigenvalues are desired and ABSTOL is less than or equal
444: * to zero, then call DSTERF or ZSTEQR. If this fails for some
445: * eigenvalue, then try DSTEBZ.
446: *
447: TEST = .FALSE.
448: IF (INDEIG) THEN
449: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
450: TEST = .TRUE.
451: END IF
452: END IF
453: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
454: CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
455: INDEE = INDRWK + 2*N
456: IF( .NOT.WANTZ ) THEN
457: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
458: CALL DSTERF( N, W, RWORK( INDEE ), INFO )
459: ELSE
460: CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
461: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
462: CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
463: $ RWORK( INDRWK ), INFO )
464: IF( INFO.EQ.0 ) THEN
465: DO 10 I = 1, N
466: IFAIL( I ) = 0
467: 10 CONTINUE
468: END IF
469: END IF
470: IF( INFO.EQ.0 ) THEN
471: M = N
472: GO TO 30
473: END IF
474: INFO = 0
475: END IF
476: *
477: * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
478: *
479: IF( WANTZ ) THEN
480: ORDER = 'B'
481: ELSE
482: ORDER = 'E'
483: END IF
484: INDIBL = 1
485: INDISP = INDIBL + N
486: INDIWK = INDISP + N
487: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
488: $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
489: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
490: $ IWORK( INDIWK ), INFO )
491: *
492: IF( WANTZ ) THEN
493: CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
494: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
495: $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
496: *
497: * Apply unitary matrix used in reduction to tridiagonal
498: * form to eigenvectors returned by ZSTEIN.
499: *
500: DO 20 J = 1, M
501: CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
502: CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
503: $ Z( 1, J ), 1 )
504: 20 CONTINUE
505: END IF
506: *
507: * If matrix was scaled, then rescale eigenvalues appropriately.
508: *
509: 30 CONTINUE
510: IF( ISCALE.EQ.1 ) THEN
511: IF( INFO.EQ.0 ) THEN
512: IMAX = M
513: ELSE
514: IMAX = INFO - 1
515: END IF
516: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
517: END IF
518: *
519: * If eigenvalues are not in order, then sort them, along with
520: * eigenvectors.
521: *
522: IF( WANTZ ) THEN
523: DO 50 J = 1, M - 1
524: I = 0
525: TMP1 = W( J )
526: DO 40 JJ = J + 1, M
527: IF( W( JJ ).LT.TMP1 ) THEN
528: I = JJ
529: TMP1 = W( JJ )
530: END IF
531: 40 CONTINUE
532: *
533: IF( I.NE.0 ) THEN
534: ITMP1 = IWORK( INDIBL+I-1 )
535: W( I ) = W( J )
536: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
537: W( J ) = TMP1
538: IWORK( INDIBL+J-1 ) = ITMP1
539: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
540: IF( INFO.NE.0 ) THEN
541: ITMP1 = IFAIL( I )
542: IFAIL( I ) = IFAIL( J )
543: IFAIL( J ) = ITMP1
544: END IF
545: END IF
546: 50 CONTINUE
547: END IF
548: *
549: RETURN
550: *
551: * End of ZHBEVX
552: *
553: END
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