1: *> \brief <b> ZHPEVX 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 ZHPEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhpevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhpevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhpevx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
22: * ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,
23: * IFAIL, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE, UPLO
27: * INTEGER IL, INFO, IU, 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 AP( * ), WORK( * ), Z( LDZ, * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> ZHPEVX computes selected eigenvalues and, optionally, eigenvectors
43: *> of a complex Hermitian matrix A in packed storage.
44: *> Eigenvalues/vectors can be selected by specifying either a range of
45: *> values or a range of 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] AP
81: *> \verbatim
82: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
83: *> On entry, the upper or lower triangle of the Hermitian matrix
84: *> A, packed columnwise in a linear array. The j-th column of A
85: *> is stored in the array AP as follows:
86: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
87: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
88: *>
89: *> On exit, AP is overwritten by values generated during the
90: *> reduction to tridiagonal form. If UPLO = 'U', the diagonal
91: *> and first superdiagonal of the tridiagonal matrix T overwrite
92: *> the corresponding elements of A, and if UPLO = 'L', the
93: *> diagonal and first subdiagonal of T overwrite the
94: *> corresponding elements of A.
95: *> \endverbatim
96: *>
97: *> \param[in] VL
98: *> \verbatim
99: *> VL is DOUBLE PRECISION
100: *> If RANGE='V', the lower bound of the interval to
101: *> be searched for eigenvalues. VL < VU.
102: *> Not referenced if RANGE = 'A' or 'I'.
103: *> \endverbatim
104: *>
105: *> \param[in] VU
106: *> \verbatim
107: *> VU is DOUBLE PRECISION
108: *> If RANGE='V', the upper bound of the interval to
109: *> be searched for eigenvalues. VL < VU.
110: *> Not referenced if RANGE = 'A' or 'I'.
111: *> \endverbatim
112: *>
113: *> \param[in] IL
114: *> \verbatim
115: *> IL is INTEGER
116: *> If RANGE='I', the index of the
117: *> smallest eigenvalue to be returned.
118: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
119: *> Not referenced if RANGE = 'A' or 'V'.
120: *> \endverbatim
121: *>
122: *> \param[in] IU
123: *> \verbatim
124: *> IU is INTEGER
125: *> If RANGE='I', the index of the
126: *> largest eigenvalue to be returned.
127: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
128: *> Not referenced if RANGE = 'A' or 'V'.
129: *> \endverbatim
130: *>
131: *> \param[in] ABSTOL
132: *> \verbatim
133: *> ABSTOL is DOUBLE PRECISION
134: *> The absolute error tolerance for the eigenvalues.
135: *> An approximate eigenvalue is accepted as converged
136: *> when it is determined to lie in an interval [a,b]
137: *> of width less than or equal to
138: *>
139: *> ABSTOL + EPS * max( |a|,|b| ) ,
140: *>
141: *> where EPS is the machine precision. If ABSTOL is less than
142: *> or equal to zero, then EPS*|T| will be used in its place,
143: *> where |T| is the 1-norm of the tridiagonal matrix obtained
144: *> by reducing AP to tridiagonal form.
145: *>
146: *> Eigenvalues will be computed most accurately when ABSTOL is
147: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
148: *> If this routine returns with INFO>0, indicating that some
149: *> eigenvectors did not converge, try setting ABSTOL to
150: *> 2*DLAMCH('S').
151: *>
152: *> See "Computing Small Singular Values of Bidiagonal Matrices
153: *> with Guaranteed High Relative Accuracy," by Demmel and
154: *> Kahan, LAPACK Working Note #3.
155: *> \endverbatim
156: *>
157: *> \param[out] M
158: *> \verbatim
159: *> M is INTEGER
160: *> The total number of eigenvalues found. 0 <= M <= N.
161: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
162: *> \endverbatim
163: *>
164: *> \param[out] W
165: *> \verbatim
166: *> W is DOUBLE PRECISION array, dimension (N)
167: *> If INFO = 0, the selected eigenvalues in ascending order.
168: *> \endverbatim
169: *>
170: *> \param[out] Z
171: *> \verbatim
172: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
173: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
174: *> contain the orthonormal eigenvectors of the matrix A
175: *> corresponding to the selected eigenvalues, with the i-th
176: *> column of Z holding the eigenvector associated with W(i).
177: *> If an eigenvector fails to converge, then that column of Z
178: *> contains the latest approximation to the eigenvector, and
179: *> the index of the eigenvector is returned in IFAIL.
180: *> If JOBZ = 'N', then Z is not referenced.
181: *> Note: the user must ensure that at least max(1,M) columns are
182: *> supplied in the array Z; if RANGE = 'V', the exact value of M
183: *> is not known in advance and an upper bound must be used.
184: *> \endverbatim
185: *>
186: *> \param[in] LDZ
187: *> \verbatim
188: *> LDZ is INTEGER
189: *> The leading dimension of the array Z. LDZ >= 1, and if
190: *> JOBZ = 'V', LDZ >= max(1,N).
191: *> \endverbatim
192: *>
193: *> \param[out] WORK
194: *> \verbatim
195: *> WORK is COMPLEX*16 array, dimension (2*N)
196: *> \endverbatim
197: *>
198: *> \param[out] RWORK
199: *> \verbatim
200: *> RWORK is DOUBLE PRECISION array, dimension (7*N)
201: *> \endverbatim
202: *>
203: *> \param[out] IWORK
204: *> \verbatim
205: *> IWORK is INTEGER array, dimension (5*N)
206: *> \endverbatim
207: *>
208: *> \param[out] IFAIL
209: *> \verbatim
210: *> IFAIL is INTEGER array, dimension (N)
211: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
212: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
213: *> indices of the eigenvectors that failed to converge.
214: *> If JOBZ = 'N', then IFAIL is not referenced.
215: *> \endverbatim
216: *>
217: *> \param[out] INFO
218: *> \verbatim
219: *> INFO is INTEGER
220: *> = 0: successful exit
221: *> < 0: if INFO = -i, the i-th argument had an illegal value
222: *> > 0: if INFO = i, then i eigenvectors failed to converge.
223: *> Their indices are stored in array IFAIL.
224: *> \endverbatim
225: *
226: * Authors:
227: * ========
228: *
229: *> \author Univ. of Tennessee
230: *> \author Univ. of California Berkeley
231: *> \author Univ. of Colorado Denver
232: *> \author NAG Ltd.
233: *
234: *> \date June 2016
235: *
236: *> \ingroup complex16OTHEReigen
237: *
238: * =====================================================================
239: SUBROUTINE ZHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
240: $ ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,
241: $ IFAIL, INFO )
242: *
243: * -- LAPACK driver routine (version 3.7.0) --
244: * -- LAPACK is a software package provided by Univ. of Tennessee, --
245: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
246: * June 2016
247: *
248: * .. Scalar Arguments ..
249: CHARACTER JOBZ, RANGE, UPLO
250: INTEGER IL, INFO, IU, LDZ, M, N
251: DOUBLE PRECISION ABSTOL, VL, VU
252: * ..
253: * .. Array Arguments ..
254: INTEGER IFAIL( * ), IWORK( * )
255: DOUBLE PRECISION RWORK( * ), W( * )
256: COMPLEX*16 AP( * ), WORK( * ), Z( LDZ, * )
257: * ..
258: *
259: * =====================================================================
260: *
261: * .. Parameters ..
262: DOUBLE PRECISION ZERO, ONE
263: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
264: COMPLEX*16 CONE
265: PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) )
266: * ..
267: * .. Local Scalars ..
268: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
269: CHARACTER ORDER
270: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
271: $ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
272: $ ITMP1, J, JJ, NSPLIT
273: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
274: $ SIGMA, SMLNUM, TMP1, VLL, VUU
275: * ..
276: * .. External Functions ..
277: LOGICAL LSAME
278: DOUBLE PRECISION DLAMCH, ZLANHP
279: EXTERNAL LSAME, DLAMCH, ZLANHP
280: * ..
281: * .. External Subroutines ..
282: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
283: $ ZHPTRD, ZSTEIN, ZSTEQR, ZSWAP, ZUPGTR, ZUPMTR
284: * ..
285: * .. Intrinsic Functions ..
286: INTRINSIC DBLE, MAX, MIN, SQRT
287: * ..
288: * .. Executable Statements ..
289: *
290: * Test the input parameters.
291: *
292: WANTZ = LSAME( JOBZ, 'V' )
293: ALLEIG = LSAME( RANGE, 'A' )
294: VALEIG = LSAME( RANGE, 'V' )
295: INDEIG = LSAME( RANGE, 'I' )
296: *
297: INFO = 0
298: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
299: INFO = -1
300: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
301: INFO = -2
302: ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
303: $ THEN
304: INFO = -3
305: ELSE IF( N.LT.0 ) THEN
306: INFO = -4
307: ELSE
308: IF( VALEIG ) THEN
309: IF( N.GT.0 .AND. VU.LE.VL )
310: $ INFO = -7
311: ELSE IF( INDEIG ) THEN
312: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
313: INFO = -8
314: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
315: INFO = -9
316: END IF
317: END IF
318: END IF
319: IF( INFO.EQ.0 ) THEN
320: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
321: $ INFO = -14
322: END IF
323: *
324: IF( INFO.NE.0 ) THEN
325: CALL XERBLA( 'ZHPEVX', -INFO )
326: RETURN
327: END IF
328: *
329: * Quick return if possible
330: *
331: M = 0
332: IF( N.EQ.0 )
333: $ RETURN
334: *
335: IF( N.EQ.1 ) THEN
336: IF( ALLEIG .OR. INDEIG ) THEN
337: M = 1
338: W( 1 ) = AP( 1 )
339: ELSE
340: IF( VL.LT.DBLE( AP( 1 ) ) .AND. VU.GE.DBLE( AP( 1 ) ) ) THEN
341: M = 1
342: W( 1 ) = AP( 1 )
343: END IF
344: END IF
345: IF( WANTZ )
346: $ Z( 1, 1 ) = CONE
347: RETURN
348: END IF
349: *
350: * Get machine constants.
351: *
352: SAFMIN = DLAMCH( 'Safe minimum' )
353: EPS = DLAMCH( 'Precision' )
354: SMLNUM = SAFMIN / EPS
355: BIGNUM = ONE / SMLNUM
356: RMIN = SQRT( SMLNUM )
357: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
358: *
359: * Scale matrix to allowable range, if necessary.
360: *
361: ISCALE = 0
362: ABSTLL = ABSTOL
363: IF( VALEIG ) THEN
364: VLL = VL
365: VUU = VU
366: ELSE
367: VLL = ZERO
368: VUU = ZERO
369: END IF
370: ANRM = ZLANHP( 'M', UPLO, N, AP, RWORK )
371: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
372: ISCALE = 1
373: SIGMA = RMIN / ANRM
374: ELSE IF( ANRM.GT.RMAX ) THEN
375: ISCALE = 1
376: SIGMA = RMAX / ANRM
377: END IF
378: IF( ISCALE.EQ.1 ) THEN
379: CALL ZDSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
380: IF( ABSTOL.GT.0 )
381: $ ABSTLL = ABSTOL*SIGMA
382: IF( VALEIG ) THEN
383: VLL = VL*SIGMA
384: VUU = VU*SIGMA
385: END IF
386: END IF
387: *
388: * Call ZHPTRD to reduce Hermitian packed matrix to tridiagonal form.
389: *
390: INDD = 1
391: INDE = INDD + N
392: INDRWK = INDE + N
393: INDTAU = 1
394: INDWRK = INDTAU + N
395: CALL ZHPTRD( UPLO, N, AP, RWORK( INDD ), RWORK( INDE ),
396: $ WORK( INDTAU ), IINFO )
397: *
398: * If all eigenvalues are desired and ABSTOL is less than or equal
399: * to zero, then call DSTERF or ZUPGTR and ZSTEQR. If this fails
400: * for some eigenvalue, then try DSTEBZ.
401: *
402: TEST = .FALSE.
403: IF (INDEIG) THEN
404: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
405: TEST = .TRUE.
406: END IF
407: END IF
408: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
409: CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
410: INDEE = INDRWK + 2*N
411: IF( .NOT.WANTZ ) THEN
412: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
413: CALL DSTERF( N, W, RWORK( INDEE ), INFO )
414: ELSE
415: CALL ZUPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
416: $ WORK( INDWRK ), IINFO )
417: CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
418: CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
419: $ RWORK( INDRWK ), INFO )
420: IF( INFO.EQ.0 ) THEN
421: DO 10 I = 1, N
422: IFAIL( I ) = 0
423: 10 CONTINUE
424: END IF
425: END IF
426: IF( INFO.EQ.0 ) THEN
427: M = N
428: GO TO 20
429: END IF
430: INFO = 0
431: END IF
432: *
433: * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
434: *
435: IF( WANTZ ) THEN
436: ORDER = 'B'
437: ELSE
438: ORDER = 'E'
439: END IF
440: INDIBL = 1
441: INDISP = INDIBL + N
442: INDIWK = INDISP + N
443: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
444: $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
445: $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
446: $ IWORK( INDIWK ), INFO )
447: *
448: IF( WANTZ ) THEN
449: CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
450: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
451: $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
452: *
453: * Apply unitary matrix used in reduction to tridiagonal
454: * form to eigenvectors returned by ZSTEIN.
455: *
456: INDWRK = INDTAU + N
457: CALL ZUPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
458: $ WORK( INDWRK ), IINFO )
459: END IF
460: *
461: * If matrix was scaled, then rescale eigenvalues appropriately.
462: *
463: 20 CONTINUE
464: IF( ISCALE.EQ.1 ) THEN
465: IF( INFO.EQ.0 ) THEN
466: IMAX = M
467: ELSE
468: IMAX = INFO - 1
469: END IF
470: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
471: END IF
472: *
473: * If eigenvalues are not in order, then sort them, along with
474: * eigenvectors.
475: *
476: IF( WANTZ ) THEN
477: DO 40 J = 1, M - 1
478: I = 0
479: TMP1 = W( J )
480: DO 30 JJ = J + 1, M
481: IF( W( JJ ).LT.TMP1 ) THEN
482: I = JJ
483: TMP1 = W( JJ )
484: END IF
485: 30 CONTINUE
486: *
487: IF( I.NE.0 ) THEN
488: ITMP1 = IWORK( INDIBL+I-1 )
489: W( I ) = W( J )
490: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
491: W( J ) = TMP1
492: IWORK( INDIBL+J-1 ) = ITMP1
493: CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
494: IF( INFO.NE.0 ) THEN
495: ITMP1 = IFAIL( I )
496: IFAIL( I ) = IFAIL( J )
497: IFAIL( J ) = ITMP1
498: END IF
499: END IF
500: 40 CONTINUE
501: END IF
502: *
503: RETURN
504: *
505: * End of ZHPEVX
506: *
507: END
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