LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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zhpgvx.f
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1*> \brief \b ZHPGVX
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZHPGVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhpgvx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhpgvx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhpgvx.f">
15*> [TXT]</a>
16*> \endhtmlonly
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*> principal minor of order i of B is not positive.
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*> \ingroup hpgvx
267*
268*> \par Contributors:
269* ==================
270*>
271*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
272*
273* =====================================================================
274 SUBROUTINE zhpgvx( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
275 $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
276 $ IWORK, IFAIL, INFO )
277*
278* -- LAPACK driver routine --
279* -- LAPACK is a software package provided by Univ. of Tennessee, --
280* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
281*
282* .. Scalar Arguments ..
283 CHARACTER JOBZ, RANGE, UPLO
284 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
285 DOUBLE PRECISION ABSTOL, VL, VU
286* ..
287* .. Array Arguments ..
288 INTEGER IFAIL( * ), IWORK( * )
289 DOUBLE PRECISION RWORK( * ), W( * )
290 COMPLEX*16 AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
291* ..
292*
293* =====================================================================
294*
295* .. Local Scalars ..
296 LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
297 CHARACTER TRANS
298 INTEGER J
299* ..
300* .. External Functions ..
301 LOGICAL LSAME
302 EXTERNAL LSAME
303* ..
304* .. External Subroutines ..
305 EXTERNAL xerbla, zhpevx, zhpgst, zpptrf, ztpmv, ztpsv
306* ..
307* .. Intrinsic Functions ..
308 INTRINSIC min
309* ..
310* .. Executable Statements ..
311*
312* Test the input parameters.
313*
314 wantz = lsame( jobz, 'V' )
315 upper = lsame( uplo, 'U' )
316 alleig = lsame( range, 'A' )
317 valeig = lsame( range, 'V' )
318 indeig = lsame( range, 'I' )
319*
320 info = 0
321 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
322 info = -1
323 ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
324 info = -2
325 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
326 info = -3
327 ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
328 info = -4
329 ELSE IF( n.LT.0 ) THEN
330 info = -5
331 ELSE
332 IF( valeig ) THEN
333 IF( n.GT.0 .AND. vu.LE.vl ) THEN
334 info = -9
335 END IF
336 ELSE IF( indeig ) THEN
337 IF( il.LT.1 ) THEN
338 info = -10
339 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
340 info = -11
341 END IF
342 END IF
343 END IF
344 IF( info.EQ.0 ) THEN
345 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
346 info = -16
347 END IF
348 END IF
349*
350 IF( info.NE.0 ) THEN
351 CALL xerbla( 'ZHPGVX', -info )
352 RETURN
353 END IF
354*
355* Quick return if possible
356*
357 IF( n.EQ.0 )
358 $ RETURN
359*
360* Form a Cholesky factorization of B.
361*
362 CALL zpptrf( uplo, n, bp, info )
363 IF( info.NE.0 ) THEN
364 info = n + info
365 RETURN
366 END IF
367*
368* Transform problem to standard eigenvalue problem and solve.
369*
370 CALL zhpgst( itype, uplo, n, ap, bp, info )
371 CALL zhpevx( jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m,
372 $ w, z, ldz, work, rwork, iwork, ifail, info )
373*
374 IF( wantz ) THEN
375*
376* Backtransform eigenvectors to the original problem.
377*
378 IF( info.GT.0 )
379 $ m = info - 1
380 IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
381*
382* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
383* backtransform eigenvectors: x = inv(L)**H *y or inv(U)*y
384*
385 IF( upper ) THEN
386 trans = 'N'
387 ELSE
388 trans = 'C'
389 END IF
390*
391 DO 10 j = 1, m
392 CALL ztpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
393 $ 1 )
394 10 CONTINUE
395*
396 ELSE IF( itype.EQ.3 ) THEN
397*
398* For B*A*x=(lambda)*x;
399* backtransform eigenvectors: x = L*y or U**H *y
400*
401 IF( upper ) THEN
402 trans = 'C'
403 ELSE
404 trans = 'N'
405 END IF
406*
407 DO 20 j = 1, m
408 CALL ztpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
409 $ 1 )
410 20 CONTINUE
411 END IF
412 END IF
413*
414 RETURN
415*
416* End of ZHPGVX
417*
418 END
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
zhpevx
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 matrice...
Definition zhpevx.f:240
zhpgst
subroutine zhpgst(itype, uplo, n, ap, bp, info)
ZHPGST
Definition zhpgst.f:113
zhpgvx
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:277
zpptrf
subroutine zpptrf(uplo, n, ap, info)
ZPPTRF
Definition zpptrf.f:119
ztpmv
subroutine ztpmv(uplo, trans, diag, n, ap, x, incx)
ZTPMV
Definition ztpmv.f:142
ztpsv
subroutine ztpsv(uplo, trans, diag, n, ap, x, incx)
ZTPSV
Definition ztpsv.f:144