1: *> \brief \b ZHEGVX
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
22: * VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
23: * LWORK, RWORK, IWORK, IFAIL, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE, UPLO
27: * INTEGER IL, INFO, ITYPE, IU, LDA, LDB, 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, * ), B( LDB, * ), WORK( * ),
34: * $ Z( LDZ, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> ZHEGVX computes selected eigenvalues, and optionally, eigenvectors
44: *> of a complex generalized Hermitian-definite eigenproblem, of the form
45: *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
46: *> B are assumed to be Hermitian and B is also positive definite.
47: *> Eigenvalues and eigenvectors can be selected by specifying either a
48: *> range of values or a range of indices for the desired eigenvalues.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] ITYPE
55: *> \verbatim
56: *> ITYPE is INTEGER
57: *> Specifies the problem type to be solved:
58: *> = 1: A*x = (lambda)*B*x
59: *> = 2: A*B*x = (lambda)*x
60: *> = 3: B*A*x = (lambda)*x
61: *> \endverbatim
62: *>
63: *> \param[in] JOBZ
64: *> \verbatim
65: *> JOBZ is CHARACTER*1
66: *> = 'N': Compute eigenvalues only;
67: *> = 'V': Compute eigenvalues and eigenvectors.
68: *> \endverbatim
69: *>
70: *> \param[in] RANGE
71: *> \verbatim
72: *> RANGE is CHARACTER*1
73: *> = 'A': all eigenvalues will be found.
74: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
75: *> will be found.
76: *> = 'I': the IL-th through IU-th eigenvalues will be found.
77: *> \endverbatim
78: *>
79: *> \param[in] UPLO
80: *> \verbatim
81: *> UPLO is CHARACTER*1
82: *> = 'U': Upper triangles of A and B are stored;
83: *> = 'L': Lower triangles of A and B are stored.
84: *> \endverbatim
85: *>
86: *> \param[in] N
87: *> \verbatim
88: *> N is INTEGER
89: *> The order of the matrices A and B. N >= 0.
90: *> \endverbatim
91: *>
92: *> \param[in,out] A
93: *> \verbatim
94: *> A is COMPLEX*16 array, dimension (LDA, N)
95: *> On entry, the Hermitian matrix A. If UPLO = 'U', the
96: *> leading N-by-N upper triangular part of A contains the
97: *> upper triangular part of the matrix A. If UPLO = 'L',
98: *> the leading N-by-N lower triangular part of A contains
99: *> the lower triangular part of the matrix A.
100: *>
101: *> On exit, the lower triangle (if UPLO='L') or the upper
102: *> triangle (if UPLO='U') of A, including the diagonal, is
103: *> destroyed.
104: *> \endverbatim
105: *>
106: *> \param[in] LDA
107: *> \verbatim
108: *> LDA is INTEGER
109: *> The leading dimension of the array A. LDA >= max(1,N).
110: *> \endverbatim
111: *>
112: *> \param[in,out] B
113: *> \verbatim
114: *> B is COMPLEX*16 array, dimension (LDB, N)
115: *> On entry, the Hermitian matrix B. If UPLO = 'U', the
116: *> leading N-by-N upper triangular part of B contains the
117: *> upper triangular part of the matrix B. If UPLO = 'L',
118: *> the leading N-by-N lower triangular part of B contains
119: *> the lower triangular part of the matrix B.
120: *>
121: *> On exit, if INFO <= N, the part of B containing the matrix is
122: *> overwritten by the triangular factor U or L from the Cholesky
123: *> factorization B = U**H*U or B = L*L**H.
124: *> \endverbatim
125: *>
126: *> \param[in] LDB
127: *> \verbatim
128: *> LDB is INTEGER
129: *> The leading dimension of the array B. LDB >= max(1,N).
130: *> \endverbatim
131: *>
132: *> \param[in] VL
133: *> \verbatim
134: *> VL is DOUBLE PRECISION
135: *>
136: *> If RANGE='V', the lower bound of the interval to
137: *> be searched for eigenvalues. VL < VU.
138: *> Not referenced if RANGE = 'A' or 'I'.
139: *> \endverbatim
140: *>
141: *> \param[in] VU
142: *> \verbatim
143: *> VU is DOUBLE PRECISION
144: *>
145: *> If RANGE='V', the upper bound of the interval to
146: *> be searched for eigenvalues. VL < VU.
147: *> Not referenced if RANGE = 'A' or 'I'.
148: *> \endverbatim
149: *>
150: *> \param[in] IL
151: *> \verbatim
152: *> IL is INTEGER
153: *>
154: *> If RANGE='I', the index of the
155: *> smallest eigenvalue to be returned.
156: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
157: *> Not referenced if RANGE = 'A' or 'V'.
158: *> \endverbatim
159: *>
160: *> \param[in] IU
161: *> \verbatim
162: *> IU is INTEGER
163: *>
164: *> If RANGE='I', the index of the
165: *> largest eigenvalue to be returned.
166: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
167: *> Not referenced if RANGE = 'A' or 'V'.
168: *> \endverbatim
169: *>
170: *> \param[in] ABSTOL
171: *> \verbatim
172: *> ABSTOL is DOUBLE PRECISION
173: *> The absolute error tolerance for the eigenvalues.
174: *> An approximate eigenvalue is accepted as converged
175: *> when it is determined to lie in an interval [a,b]
176: *> of width less than or equal to
177: *>
178: *> ABSTOL + EPS * max( |a|,|b| ) ,
179: *>
180: *> where EPS is the machine precision. If ABSTOL is less than
181: *> or equal to zero, then EPS*|T| will be used in its place,
182: *> where |T| is the 1-norm of the tridiagonal matrix obtained
183: *> by reducing C to tridiagonal form, where C is the symmetric
184: *> matrix of the standard symmetric problem to which the
185: *> generalized problem is transformed.
186: *>
187: *> Eigenvalues will be computed most accurately when ABSTOL is
188: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
189: *> If this routine returns with INFO>0, indicating that some
190: *> eigenvectors did not converge, try setting ABSTOL to
191: *> 2*DLAMCH('S').
192: *> \endverbatim
193: *>
194: *> \param[out] M
195: *> \verbatim
196: *> M is INTEGER
197: *> The total number of eigenvalues found. 0 <= M <= N.
198: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
199: *> \endverbatim
200: *>
201: *> \param[out] W
202: *> \verbatim
203: *> W is DOUBLE PRECISION array, dimension (N)
204: *> The first M elements contain the selected
205: *> eigenvalues in ascending order.
206: *> \endverbatim
207: *>
208: *> \param[out] Z
209: *> \verbatim
210: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
211: *> If JOBZ = 'N', then Z is not referenced.
212: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
213: *> contain the orthonormal eigenvectors of the matrix A
214: *> corresponding to the selected eigenvalues, with the i-th
215: *> column of Z holding the eigenvector associated with W(i).
216: *> The eigenvectors are normalized as follows:
217: *> if ITYPE = 1 or 2, Z**T*B*Z = I;
218: *> if ITYPE = 3, Z**T*inv(B)*Z = I.
219: *>
220: *> If an eigenvector fails to converge, then that column of Z
221: *> contains the latest approximation to the eigenvector, and the
222: *> index of the eigenvector is returned in IFAIL.
223: *> Note: the user must ensure that at least max(1,M) columns are
224: *> supplied in the array Z; if RANGE = 'V', the exact value of M
225: *> is not known in advance and an upper bound must be used.
226: *> \endverbatim
227: *>
228: *> \param[in] LDZ
229: *> \verbatim
230: *> LDZ is INTEGER
231: *> The leading dimension of the array Z. LDZ >= 1, and if
232: *> JOBZ = 'V', LDZ >= max(1,N).
233: *> \endverbatim
234: *>
235: *> \param[out] WORK
236: *> \verbatim
237: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
238: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
239: *> \endverbatim
240: *>
241: *> \param[in] LWORK
242: *> \verbatim
243: *> LWORK is INTEGER
244: *> The length of the array WORK. LWORK >= max(1,2*N).
245: *> For optimal efficiency, LWORK >= (NB+1)*N,
246: *> where NB is the blocksize for ZHETRD returned by ILAENV.
247: *>
248: *> If LWORK = -1, then a workspace query is assumed; the routine
249: *> only calculates the optimal size of the WORK array, returns
250: *> this value as the first entry of the WORK array, and no error
251: *> message related to LWORK is issued by XERBLA.
252: *> \endverbatim
253: *>
254: *> \param[out] RWORK
255: *> \verbatim
256: *> RWORK is DOUBLE PRECISION array, dimension (7*N)
257: *> \endverbatim
258: *>
259: *> \param[out] IWORK
260: *> \verbatim
261: *> IWORK is INTEGER array, dimension (5*N)
262: *> \endverbatim
263: *>
264: *> \param[out] IFAIL
265: *> \verbatim
266: *> IFAIL is INTEGER array, dimension (N)
267: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
268: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
269: *> indices of the eigenvectors that failed to converge.
270: *> If JOBZ = 'N', then IFAIL is not referenced.
271: *> \endverbatim
272: *>
273: *> \param[out] INFO
274: *> \verbatim
275: *> INFO is INTEGER
276: *> = 0: successful exit
277: *> < 0: if INFO = -i, the i-th argument had an illegal value
278: *> > 0: ZPOTRF or ZHEEVX returned an error code:
279: *> <= N: if INFO = i, ZHEEVX failed to converge;
280: *> i eigenvectors failed to converge. Their indices
281: *> are stored in array IFAIL.
282: *> > N: if INFO = N + i, for 1 <= i <= N, then the leading
283: *> minor of order i of B is not positive definite.
284: *> The factorization of B could not be completed and
285: *> no eigenvalues or eigenvectors were computed.
286: *> \endverbatim
287: *
288: * Authors:
289: * ========
290: *
291: *> \author Univ. of Tennessee
292: *> \author Univ. of California Berkeley
293: *> \author Univ. of Colorado Denver
294: *> \author NAG Ltd.
295: *
296: *> \ingroup complex16HEeigen
297: *
298: *> \par Contributors:
299: * ==================
300: *>
301: *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
302: *
303: * =====================================================================
304: SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
305: $ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
306: $ LWORK, RWORK, IWORK, IFAIL, INFO )
307: *
308: * -- LAPACK driver routine --
309: * -- LAPACK is a software package provided by Univ. of Tennessee, --
310: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
311: *
312: * .. Scalar Arguments ..
313: CHARACTER JOBZ, RANGE, UPLO
314: INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
315: DOUBLE PRECISION ABSTOL, VL, VU
316: * ..
317: * .. Array Arguments ..
318: INTEGER IFAIL( * ), IWORK( * )
319: DOUBLE PRECISION RWORK( * ), W( * )
320: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
321: $ Z( LDZ, * )
322: * ..
323: *
324: * =====================================================================
325: *
326: * .. Parameters ..
327: COMPLEX*16 CONE
328: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
329: * ..
330: * .. Local Scalars ..
331: LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
332: CHARACTER TRANS
333: INTEGER LWKOPT, NB
334: * ..
335: * .. External Functions ..
336: LOGICAL LSAME
337: INTEGER ILAENV
338: EXTERNAL LSAME, ILAENV
339: * ..
340: * .. External Subroutines ..
341: EXTERNAL XERBLA, ZHEEVX, ZHEGST, ZPOTRF, ZTRMM, ZTRSM
342: * ..
343: * .. Intrinsic Functions ..
344: INTRINSIC MAX, MIN
345: * ..
346: * .. Executable Statements ..
347: *
348: * Test the input parameters.
349: *
350: WANTZ = LSAME( JOBZ, 'V' )
351: UPPER = LSAME( UPLO, 'U' )
352: ALLEIG = LSAME( RANGE, 'A' )
353: VALEIG = LSAME( RANGE, 'V' )
354: INDEIG = LSAME( RANGE, 'I' )
355: LQUERY = ( LWORK.EQ.-1 )
356: *
357: INFO = 0
358: IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
359: INFO = -1
360: ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
361: INFO = -2
362: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
363: INFO = -3
364: ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
365: INFO = -4
366: ELSE IF( N.LT.0 ) THEN
367: INFO = -5
368: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
369: INFO = -7
370: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
371: INFO = -9
372: ELSE
373: IF( VALEIG ) THEN
374: IF( N.GT.0 .AND. VU.LE.VL )
375: $ INFO = -11
376: ELSE IF( INDEIG ) THEN
377: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
378: INFO = -12
379: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
380: INFO = -13
381: END IF
382: END IF
383: END IF
384: IF (INFO.EQ.0) THEN
385: IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
386: INFO = -18
387: END IF
388: END IF
389: *
390: IF( INFO.EQ.0 ) THEN
391: NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
392: LWKOPT = MAX( 1, ( NB + 1 )*N )
393: WORK( 1 ) = LWKOPT
394: *
395: IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN
396: INFO = -20
397: END IF
398: END IF
399: *
400: IF( INFO.NE.0 ) THEN
401: CALL XERBLA( 'ZHEGVX', -INFO )
402: RETURN
403: ELSE IF( LQUERY ) THEN
404: RETURN
405: END IF
406: *
407: * Quick return if possible
408: *
409: M = 0
410: IF( N.EQ.0 ) THEN
411: RETURN
412: END IF
413: *
414: * Form a Cholesky factorization of B.
415: *
416: CALL ZPOTRF( UPLO, N, B, LDB, INFO )
417: IF( INFO.NE.0 ) THEN
418: INFO = N + INFO
419: RETURN
420: END IF
421: *
422: * Transform problem to standard eigenvalue problem and solve.
423: *
424: CALL ZHEGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
425: CALL ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
426: $ M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL,
427: $ INFO )
428: *
429: IF( WANTZ ) THEN
430: *
431: * Backtransform eigenvectors to the original problem.
432: *
433: IF( INFO.GT.0 )
434: $ M = INFO - 1
435: IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
436: *
437: * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
438: * backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
439: *
440: IF( UPPER ) THEN
441: TRANS = 'N'
442: ELSE
443: TRANS = 'C'
444: END IF
445: *
446: CALL ZTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
447: $ LDB, Z, LDZ )
448: *
449: ELSE IF( ITYPE.EQ.3 ) THEN
450: *
451: * For B*A*x=(lambda)*x;
452: * backtransform eigenvectors: x = L*y or U**H *y
453: *
454: IF( UPPER ) THEN
455: TRANS = 'C'
456: ELSE
457: TRANS = 'N'
458: END IF
459: *
460: CALL ZTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
461: $ LDB, Z, LDZ )
462: END IF
463: END IF
464: *
465: * Set WORK(1) to optimal complex workspace size.
466: *
467: WORK( 1 ) = LWKOPT
468: *
469: RETURN
470: *
471: * End of ZHEGVX
472: *
473: END
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