LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
zhpgvx.f
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1 *> \brief \b ZHPGVX
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZHPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
22 * 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, ITYPE, 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( * ), BP( * ), WORK( * ), Z( LDZ, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> ZHPGVX computes selected eigenvalues and, optionally, eigenvectors
43 *> of a complex generalized Hermitian-definite eigenproblem, of the form
44 *> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and
45 *> B are assumed to be Hermitian, stored in packed format, and B is also
46 *> positive definite. Eigenvalues and eigenvectors can be selected by
47 *> specifying either a range of values or a range of indices for the
48 *> 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] AP
93 *> \verbatim
94 *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
95 *> On entry, the upper or lower triangle of the Hermitian matrix
96 *> A, packed columnwise in a linear array. The j-th column of A
97 *> is stored in the array AP as follows:
98 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
99 *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
100 *>
101 *> On exit, the contents of AP are destroyed.
102 *> \endverbatim
103 *>
104 *> \param[in,out] BP
105 *> \verbatim
106 *> BP is COMPLEX*16 array, dimension (N*(N+1)/2)
107 *> On entry, the upper or lower triangle of the Hermitian matrix
108 *> B, packed columnwise in a linear array. The j-th column of B
109 *> is stored in the array BP as follows:
110 *> if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
111 *> if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
112 *>
113 *> On exit, the triangular factor U or L from the Cholesky
114 *> factorization B = U**H*U or B = L*L**H, in the same storage
115 *> format as B.
116 *> \endverbatim
117 *>
118 *> \param[in] VL
119 *> \verbatim
120 *> VL is DOUBLE PRECISION
121 *>
122 *> If RANGE='V', the lower bound of the interval to
123 *> be searched for eigenvalues. VL < VU.
124 *> Not referenced if RANGE = 'A' or 'I'.
125 *> \endverbatim
126 *>
127 *> \param[in] VU
128 *> \verbatim
129 *> VU is DOUBLE PRECISION
130 *>
131 *> If RANGE='V', the upper bound of the interval to
132 *> be searched for eigenvalues. VL < VU.
133 *> Not referenced if RANGE = 'A' or 'I'.
134 *> \endverbatim
135 *>
136 *> \param[in] IL
137 *> \verbatim
138 *> IL is INTEGER
139 *>
140 *> If RANGE='I', the index of the
141 *> smallest eigenvalue to be returned.
142 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
143 *> Not referenced if RANGE = 'A' or 'V'.
144 *> \endverbatim
145 *>
146 *> \param[in] IU
147 *> \verbatim
148 *> IU is INTEGER
149 *>
150 *> If RANGE='I', the index of the
151 *> largest eigenvalue to be returned.
152 *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
153 *> Not referenced if RANGE = 'A' or 'V'.
154 *> \endverbatim
155 *>
156 *> \param[in] ABSTOL
157 *> \verbatim
158 *> ABSTOL is DOUBLE PRECISION
159 *> The absolute error tolerance for the eigenvalues.
160 *> An approximate eigenvalue is accepted as converged
161 *> when it is determined to lie in an interval [a,b]
162 *> of width less than or equal to
163 *>
164 *> ABSTOL + EPS * max( |a|,|b| ) ,
165 *>
166 *> where EPS is the machine precision. If ABSTOL is less than
167 *> or equal to zero, then EPS*|T| will be used in its place,
168 *> where |T| is the 1-norm of the tridiagonal matrix obtained
169 *> by reducing AP to tridiagonal form.
170 *>
171 *> Eigenvalues will be computed most accurately when ABSTOL is
172 *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
173 *> If this routine returns with INFO>0, indicating that some
174 *> eigenvectors did not converge, try setting ABSTOL to
175 *> 2*DLAMCH('S').
176 *> \endverbatim
177 *>
178 *> \param[out] M
179 *> \verbatim
180 *> M is INTEGER
181 *> The total number of eigenvalues found. 0 <= M <= N.
182 *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
183 *> \endverbatim
184 *>
185 *> \param[out] W
186 *> \verbatim
187 *> W is DOUBLE PRECISION array, dimension (N)
188 *> On normal exit, the first M elements contain the selected
189 *> eigenvalues in ascending order.
190 *> \endverbatim
191 *>
192 *> \param[out] Z
193 *> \verbatim
194 *> Z is COMPLEX*16 array, dimension (LDZ, N)
195 *> If JOBZ = 'N', then Z is not referenced.
196 *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
197 *> contain the orthonormal eigenvectors of the matrix A
198 *> corresponding to the selected eigenvalues, with the i-th
199 *> column of Z holding the eigenvector associated with W(i).
200 *> The eigenvectors are normalized as follows:
201 *> if ITYPE = 1 or 2, Z**H*B*Z = I;
202 *> if ITYPE = 3, Z**H*inv(B)*Z = I.
203 *>
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 *> Note: the user must ensure that at least max(1,M) columns are
208 *> supplied in the array Z; if RANGE = 'V', the exact value of M
209 *> is not known in advance and an upper bound must be used.
210 *> \endverbatim
211 *>
212 *> \param[in] LDZ
213 *> \verbatim
214 *> LDZ is INTEGER
215 *> The leading dimension of the array Z. LDZ >= 1, and if
216 *> JOBZ = 'V', LDZ >= max(1,N).
217 *> \endverbatim
218 *>
219 *> \param[out] WORK
220 *> \verbatim
221 *> WORK is COMPLEX*16 array, dimension (2*N)
222 *> \endverbatim
223 *>
224 *> \param[out] RWORK
225 *> \verbatim
226 *> RWORK is DOUBLE PRECISION array, dimension (7*N)
227 *> \endverbatim
228 *>
229 *> \param[out] IWORK
230 *> \verbatim
231 *> IWORK is INTEGER array, dimension (5*N)
232 *> \endverbatim
233 *>
234 *> \param[out] IFAIL
235 *> \verbatim
236 *> IFAIL is INTEGER array, dimension (N)
237 *> If JOBZ = 'V', then if INFO = 0, the first M elements of
238 *> IFAIL are zero. If INFO > 0, then IFAIL contains the
239 *> indices of the eigenvectors that failed to converge.
240 *> If JOBZ = 'N', then IFAIL is not referenced.
241 *> \endverbatim
242 *>
243 *> \param[out] INFO
244 *> \verbatim
245 *> INFO is INTEGER
246 *> = 0: successful exit
247 *> < 0: if INFO = -i, the i-th argument had an illegal value
248 *> > 0: ZPPTRF or ZHPEVX returned an error code:
249 *> <= N: if INFO = i, ZHPEVX failed to converge;
250 *> i eigenvectors failed to converge. Their indices
251 *> are stored in array IFAIL.
252 *> > N: if INFO = N + i, for 1 <= i <= n, then the leading
253 *> minor of order i of B is not positive definite.
254 *> The factorization of B could not be completed and
255 *> no eigenvalues or eigenvectors were computed.
256 *> \endverbatim
257 *
258 * Authors:
259 * ========
260 *
261 *> \author Univ. of Tennessee
262 *> \author Univ. of California Berkeley
263 *> \author Univ. of Colorado Denver
264 *> \author NAG Ltd.
265 *
266 *> \date June 2016
267 *
268 *> \ingroup complex16OTHEReigen
269 *
270 *> \par Contributors:
271 * ==================
272 *>
273 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
274 *
275 * =====================================================================
276  SUBROUTINE zhpgvx( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
277  \$ il, iu, abstol, m, w, z, ldz, work, rwork,
278  \$ iwork, ifail, info )
279 *
280 * -- LAPACK driver routine (version 3.6.1) --
281 * -- LAPACK is a software package provided by Univ. of Tennessee, --
282 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
283 * June 2016
284 *
285 * .. Scalar Arguments ..
286  CHARACTER JOBZ, RANGE, UPLO
287  INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
288  DOUBLE PRECISION ABSTOL, VL, VU
289 * ..
290 * .. Array Arguments ..
291  INTEGER IFAIL( * ), IWORK( * )
292  DOUBLE PRECISION RWORK( * ), W( * )
293  COMPLEX*16 AP( * ), BP( * ), WORK( * ), Z( ldz, * )
294 * ..
295 *
296 * =====================================================================
297 *
298 * .. Local Scalars ..
299  LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
300  CHARACTER TRANS
301  INTEGER J
302 * ..
303 * .. External Functions ..
304  LOGICAL LSAME
305  EXTERNAL lsame
306 * ..
307 * .. External Subroutines ..
308  EXTERNAL xerbla, zhpevx, zhpgst, zpptrf, ztpmv, ztpsv
309 * ..
310 * .. Intrinsic Functions ..
311  INTRINSIC min
312 * ..
313 * .. Executable Statements ..
314 *
315 * Test the input parameters.
316 *
317  wantz = lsame( jobz, 'V' )
318  upper = lsame( uplo, 'U' )
319  alleig = lsame( range, 'A' )
320  valeig = lsame( range, 'V' )
321  indeig = lsame( range, 'I' )
322 *
323  info = 0
324  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
325  info = -1
326  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
327  info = -2
328  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
329  info = -3
330  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
331  info = -4
332  ELSE IF( n.LT.0 ) THEN
333  info = -5
334  ELSE
335  IF( valeig ) THEN
336  IF( n.GT.0 .AND. vu.LE.vl ) THEN
337  info = -9
338  END IF
339  ELSE IF( indeig ) THEN
340  IF( il.LT.1 ) THEN
341  info = -10
342  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
343  info = -11
344  END IF
345  END IF
346  END IF
347  IF( info.EQ.0 ) THEN
348  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
349  info = -16
350  END IF
351  END IF
352 *
353  IF( info.NE.0 ) THEN
354  CALL xerbla( 'ZHPGVX', -info )
355  RETURN
356  END IF
357 *
358 * Quick return if possible
359 *
360  IF( n.EQ.0 )
361  \$ RETURN
362 *
363 * Form a Cholesky factorization of B.
364 *
365  CALL zpptrf( uplo, n, bp, info )
366  IF( info.NE.0 ) THEN
367  info = n + info
368  RETURN
369  END IF
370 *
371 * Transform problem to standard eigenvalue problem and solve.
372 *
373  CALL zhpgst( itype, uplo, n, ap, bp, info )
374  CALL zhpevx( jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m,
375  \$ w, z, ldz, work, rwork, iwork, ifail, info )
376 *
377  IF( wantz ) THEN
378 *
379 * Backtransform eigenvectors to the original problem.
380 *
381  IF( info.GT.0 )
382  \$ m = info - 1
383  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
384 *
385 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
386 * backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
387 *
388  IF( upper ) THEN
389  trans = 'N'
390  ELSE
391  trans = 'C'
392  END IF
393 *
394  DO 10 j = 1, m
395  CALL ztpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
396  \$ 1 )
397  10 CONTINUE
398 *
399  ELSE IF( itype.EQ.3 ) THEN
400 *
401 * For B*A*x=(lambda)*x;
402 * backtransform eigenvectors: x = L*y or U**H *y
403 *
404  IF( upper ) THEN
405  trans = 'C'
406  ELSE
407  trans = 'N'
408  END IF
409 *
410  DO 20 j = 1, m
411  CALL ztpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
412  \$ 1 )
413  20 CONTINUE
414  END IF
415  END IF
416 *
417  RETURN
418 *
419 * End of ZHPGVX
420 *
421  END
subroutine zhpgvx(ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
ZHPGVX
Definition: zhpgvx.f:279
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine ztpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
ZTPSV
Definition: ztpsv.f:146
subroutine ztpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
ZTPMV
Definition: ztpmv.f:144
subroutine zhpevx(JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
ZHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matric...
Definition: zhpevx.f:242
subroutine zpptrf(UPLO, N, AP, INFO)
ZPPTRF
Definition: zpptrf.f:121
subroutine zhpgst(ITYPE, UPLO, N, AP, BP, INFO)
ZHPGST
Definition: zhpgst.f:115