LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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chpgvx.f
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1*> \brief \b CHPGVX
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CHPGVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chpgvx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chpgvx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chpgvx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CHPGVX( 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* REAL ABSTOL, VL, VU
29* ..
30* .. Array Arguments ..
31* INTEGER IFAIL( * ), IWORK( * )
32* REAL RWORK( * ), W( * )
33* COMPLEX AP( * ), BP( * ), WORK( * ), Z( LDZ, * )
34* ..
35*
36*
37*> \par Purpose:
38* =============
39*>
40*> \verbatim
41*>
42*> CHPGVX 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 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 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 REAL
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 REAL
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 REAL
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*SLAMCH('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*SLAMCH('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 REAL 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 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 array, dimension (2*N)
222*> \endverbatim
223*>
224*> \param[out] RWORK
225*> \verbatim
226*> RWORK is REAL 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: CPPTRF or CHPEVX returned an error code:
249*> <= N: if INFO = i, CHPEVX 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*> \ingroup complexOTHEReigen
267*
268*> \par Contributors:
269* ==================
270*>
271*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
272*
273* =====================================================================
274 SUBROUTINE chpgvx( 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 REAL ABSTOL, VL, VU
286* ..
287* .. Array Arguments ..
288 INTEGER IFAIL( * ), IWORK( * )
289 REAL RWORK( * ), W( * )
290 COMPLEX 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 chpevx, chpgst, cpptrf, ctpmv, ctpsv, xerbla
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( 'CHPGVX', -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 cpptrf( 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 chpgst( itype, uplo, n, ap, bp, info )
371 CALL chpevx( 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 ctpsv( 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 ctpmv( 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 CHPGVX
417*
418 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ctpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
CTPSV
Definition: ctpsv.f:144
subroutine ctpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
CTPMV
Definition: ctpmv.f:142
subroutine cpptrf(UPLO, N, AP, INFO)
CPPTRF
Definition: cpptrf.f:119
subroutine chpgst(ITYPE, UPLO, N, AP, BP, INFO)
CHPGST
Definition: chpgst.f:113
subroutine chpgvx(ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
CHPGVX
Definition: chpgvx.f:277
subroutine chpevx(JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrice...
Definition: chpevx.f:240