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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16: *> \endhtmlonly
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: *> \endverbatim
136: *>
137: *> \param[in] VU
138: *> \verbatim
139: *> VU is DOUBLE PRECISION
140: *>
141: *> If RANGE='V', the lower and upper bounds of the interval to
142: *> be searched for eigenvalues. VL < VU.
143: *> Not referenced if RANGE = 'A' or 'I'.
144: *> \endverbatim
145: *>
146: *> \param[in] IL
147: *> \verbatim
148: *> IL is INTEGER
149: *> \endverbatim
150: *>
151: *> \param[in] IU
152: *> \verbatim
153: *> IU is INTEGER
154: *>
155: *> If RANGE='I', the indices (in ascending order) of the
156: *> smallest and largest eigenvalues to be returned.
157: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
158: *> Not referenced if RANGE = 'A' or 'V'.
159: *> \endverbatim
160: *>
161: *> \param[in] ABSTOL
162: *> \verbatim
163: *> ABSTOL is DOUBLE PRECISION
164: *> The absolute error tolerance for the eigenvalues.
165: *> An approximate eigenvalue is accepted as converged
166: *> when it is determined to lie in an interval [a,b]
167: *> of width less than or equal to
168: *>
169: *> ABSTOL + EPS * max( |a|,|b| ) ,
170: *>
171: *> where EPS is the machine precision. If ABSTOL is less than
172: *> or equal to zero, then EPS*|T| will be used in its place,
173: *> where |T| is the 1-norm of the tridiagonal matrix obtained
174: *> by reducing C to tridiagonal form, where C is the symmetric
175: *> matrix of the standard symmetric problem to which the
176: *> generalized problem is transformed.
177: *>
178: *> Eigenvalues will be computed most accurately when ABSTOL is
179: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
180: *> If this routine returns with INFO>0, indicating that some
181: *> eigenvectors did not converge, try setting ABSTOL to
182: *> 2*DLAMCH('S').
183: *> \endverbatim
184: *>
185: *> \param[out] M
186: *> \verbatim
187: *> M is INTEGER
188: *> The total number of eigenvalues found. 0 <= M <= N.
189: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
190: *> \endverbatim
191: *>
192: *> \param[out] W
193: *> \verbatim
194: *> W is DOUBLE PRECISION array, dimension (N)
195: *> The first M elements contain the selected
196: *> eigenvalues in ascending order.
197: *> \endverbatim
198: *>
199: *> \param[out] Z
200: *> \verbatim
201: *> Z is COMPLEX*16 array, dimension (LDZ, max(1,M))
202: *> If JOBZ = 'N', then Z is not referenced.
203: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
204: *> contain the orthonormal eigenvectors of the matrix A
205: *> corresponding to the selected eigenvalues, with the i-th
206: *> column of Z holding the eigenvector associated with W(i).
207: *> The eigenvectors are normalized as follows:
208: *> if ITYPE = 1 or 2, Z**T*B*Z = I;
209: *> if ITYPE = 3, Z**T*inv(B)*Z = I.
210: *>
211: *> If an eigenvector fails to converge, then that column of Z
212: *> contains the latest approximation to the eigenvector, and the
213: *> index of the eigenvector is returned in IFAIL.
214: *> Note: the user must ensure that at least max(1,M) columns are
215: *> supplied in the array Z; if RANGE = 'V', the exact value of M
216: *> is not known in advance and an upper bound must be used.
217: *> \endverbatim
218: *>
219: *> \param[in] LDZ
220: *> \verbatim
221: *> LDZ is INTEGER
222: *> The leading dimension of the array Z. LDZ >= 1, and if
223: *> JOBZ = 'V', LDZ >= max(1,N).
224: *> \endverbatim
225: *>
226: *> \param[out] WORK
227: *> \verbatim
228: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
229: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
230: *> \endverbatim
231: *>
232: *> \param[in] LWORK
233: *> \verbatim
234: *> LWORK is INTEGER
235: *> The length of the array WORK. LWORK >= max(1,2*N).
236: *> For optimal efficiency, LWORK >= (NB+1)*N,
237: *> where NB is the blocksize for ZHETRD returned by ILAENV.
238: *>
239: *> If LWORK = -1, then a workspace query is assumed; the routine
240: *> only calculates the optimal size of the WORK array, returns
241: *> this value as the first entry of the WORK array, and no error
242: *> message related to LWORK is issued by XERBLA.
243: *> \endverbatim
244: *>
245: *> \param[out] RWORK
246: *> \verbatim
247: *> RWORK is DOUBLE PRECISION array, dimension (7*N)
248: *> \endverbatim
249: *>
250: *> \param[out] IWORK
251: *> \verbatim
252: *> IWORK is INTEGER array, dimension (5*N)
253: *> \endverbatim
254: *>
255: *> \param[out] IFAIL
256: *> \verbatim
257: *> IFAIL is INTEGER array, dimension (N)
258: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
259: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
260: *> indices of the eigenvectors that failed to converge.
261: *> If JOBZ = 'N', then IFAIL is not referenced.
262: *> \endverbatim
263: *>
264: *> \param[out] INFO
265: *> \verbatim
266: *> INFO is INTEGER
267: *> = 0: successful exit
268: *> < 0: if INFO = -i, the i-th argument had an illegal value
269: *> > 0: ZPOTRF or ZHEEVX returned an error code:
270: *> <= N: if INFO = i, ZHEEVX failed to converge;
271: *> i eigenvectors failed to converge. Their indices
272: *> are stored in array IFAIL.
273: *> > N: if INFO = N + i, for 1 <= i <= N, then the leading
274: *> minor of order i of B is not positive definite.
275: *> The factorization of B could not be completed and
276: *> no eigenvalues or eigenvectors were computed.
277: *> \endverbatim
278: *
279: * Authors:
280: * ========
281: *
282: *> \author Univ. of Tennessee
283: *> \author Univ. of California Berkeley
284: *> \author Univ. of Colorado Denver
285: *> \author NAG Ltd.
286: *
287: *> \date November 2015
288: *
289: *> \ingroup complex16HEeigen
290: *
291: *> \par Contributors:
292: * ==================
293: *>
294: *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
295: *
296: * =====================================================================
297: SUBROUTINE ZHEGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB,
298: $ VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK,
299: $ LWORK, RWORK, IWORK, IFAIL, INFO )
300: *
301: * -- LAPACK driver routine (version 3.6.0) --
302: * -- LAPACK is a software package provided by Univ. of Tennessee, --
303: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
304: * November 2015
305: *
306: * .. Scalar Arguments ..
307: CHARACTER JOBZ, RANGE, UPLO
308: INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
309: DOUBLE PRECISION ABSTOL, VL, VU
310: * ..
311: * .. Array Arguments ..
312: INTEGER IFAIL( * ), IWORK( * )
313: DOUBLE PRECISION RWORK( * ), W( * )
314: COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ),
315: $ Z( LDZ, * )
316: * ..
317: *
318: * =====================================================================
319: *
320: * .. Parameters ..
321: COMPLEX*16 CONE
322: PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
323: * ..
324: * .. Local Scalars ..
325: LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
326: CHARACTER TRANS
327: INTEGER LWKOPT, NB
328: * ..
329: * .. External Functions ..
330: LOGICAL LSAME
331: INTEGER ILAENV
332: EXTERNAL LSAME, ILAENV
333: * ..
334: * .. External Subroutines ..
335: EXTERNAL XERBLA, ZHEEVX, ZHEGST, ZPOTRF, ZTRMM, ZTRSM
336: * ..
337: * .. Intrinsic Functions ..
338: INTRINSIC MAX, MIN
339: * ..
340: * .. Executable Statements ..
341: *
342: * Test the input parameters.
343: *
344: WANTZ = LSAME( JOBZ, 'V' )
345: UPPER = LSAME( UPLO, 'U' )
346: ALLEIG = LSAME( RANGE, 'A' )
347: VALEIG = LSAME( RANGE, 'V' )
348: INDEIG = LSAME( RANGE, 'I' )
349: LQUERY = ( LWORK.EQ.-1 )
350: *
351: INFO = 0
352: IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
353: INFO = -1
354: ELSE IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
355: INFO = -2
356: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
357: INFO = -3
358: ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
359: INFO = -4
360: ELSE IF( N.LT.0 ) THEN
361: INFO = -5
362: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
363: INFO = -7
364: ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
365: INFO = -9
366: ELSE
367: IF( VALEIG ) THEN
368: IF( N.GT.0 .AND. VU.LE.VL )
369: $ INFO = -11
370: ELSE IF( INDEIG ) THEN
371: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
372: INFO = -12
373: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
374: INFO = -13
375: END IF
376: END IF
377: END IF
378: IF (INFO.EQ.0) THEN
379: IF (LDZ.LT.1 .OR. (WANTZ .AND. LDZ.LT.N)) THEN
380: INFO = -18
381: END IF
382: END IF
383: *
384: IF( INFO.EQ.0 ) THEN
385: NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
386: LWKOPT = MAX( 1, ( NB + 1 )*N )
387: WORK( 1 ) = LWKOPT
388: *
389: IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN
390: INFO = -20
391: END IF
392: END IF
393: *
394: IF( INFO.NE.0 ) THEN
395: CALL XERBLA( 'ZHEGVX', -INFO )
396: RETURN
397: ELSE IF( LQUERY ) THEN
398: RETURN
399: END IF
400: *
401: * Quick return if possible
402: *
403: M = 0
404: IF( N.EQ.0 ) THEN
405: RETURN
406: END IF
407: *
408: * Form a Cholesky factorization of B.
409: *
410: CALL ZPOTRF( UPLO, N, B, LDB, INFO )
411: IF( INFO.NE.0 ) THEN
412: INFO = N + INFO
413: RETURN
414: END IF
415: *
416: * Transform problem to standard eigenvalue problem and solve.
417: *
418: CALL ZHEGST( ITYPE, UPLO, N, A, LDA, B, LDB, INFO )
419: CALL ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL,
420: $ M, W, Z, LDZ, WORK, LWORK, RWORK, IWORK, IFAIL,
421: $ INFO )
422: *
423: IF( WANTZ ) THEN
424: *
425: * Backtransform eigenvectors to the original problem.
426: *
427: IF( INFO.GT.0 )
428: $ M = INFO - 1
429: IF( ITYPE.EQ.1 .OR. ITYPE.EQ.2 ) THEN
430: *
431: * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
432: * backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
433: *
434: IF( UPPER ) THEN
435: TRANS = 'N'
436: ELSE
437: TRANS = 'C'
438: END IF
439: *
440: CALL ZTRSM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
441: $ LDB, Z, LDZ )
442: *
443: ELSE IF( ITYPE.EQ.3 ) THEN
444: *
445: * For B*A*x=(lambda)*x;
446: * backtransform eigenvectors: x = L*y or U**H *y
447: *
448: IF( UPPER ) THEN
449: TRANS = 'C'
450: ELSE
451: TRANS = 'N'
452: END IF
453: *
454: CALL ZTRMM( 'Left', UPLO, TRANS, 'Non-unit', N, M, CONE, B,
455: $ LDB, Z, LDZ )
456: END IF
457: END IF
458: *
459: * Set WORK(1) to optimal complex workspace size.
460: *
461: WORK( 1 ) = LWKOPT
462: *
463: RETURN
464: *
465: * End of ZHEGVX
466: *
467: END
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