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sspgvx.f
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1*> \brief \b SSPGVX
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download SSPGVX + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sspgvx.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sspgvx.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sspgvx.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
22* IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
23* 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 AP( * ), BP( * ), W( * ), WORK( * ),
33* $ Z( LDZ, * )
34* ..
35*
36*
37*> \par Purpose:
38* =============
39*>
40*> \verbatim
41*>
42*> SSPGVX computes selected eigenvalues, and optionally, eigenvectors
43*> of a real generalized symmetric-definite eigenproblem, of the form
44*> A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A
45*> and B are assumed to be symmetric, stored in packed storage, and B
46*> is also positive definite. Eigenvalues and eigenvectors can be
47*> selected by specifying either a range of values or a range of indices
48*> 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 triangle of A and B are stored;
83*> = 'L': Lower triangle of A and B are stored.
84*> \endverbatim
85*>
86*> \param[in] N
87*> \verbatim
88*> N is INTEGER
89*> The order of the matrix pencil (A,B). N >= 0.
90*> \endverbatim
91*>
92*> \param[in,out] AP
93*> \verbatim
94*> AP is REAL array, dimension (N*(N+1)/2)
95*> On entry, the upper or lower triangle of the symmetric 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 REAL array, dimension (N*(N+1)/2)
107*> On entry, the upper or lower triangle of the symmetric 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**T*U or B = L*L**T, 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 A 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 REAL array, dimension (LDZ, max(1,M))
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**T*B*Z = I;
202*> if ITYPE = 3, Z**T*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 REAL array, dimension (8*N)
222*> \endverbatim
223*>
224*> \param[out] IWORK
225*> \verbatim
226*> IWORK is INTEGER array, dimension (5*N)
227*> \endverbatim
228*>
229*> \param[out] IFAIL
230*> \verbatim
231*> IFAIL is INTEGER array, dimension (N)
232*> If JOBZ = 'V', then if INFO = 0, the first M elements of
233*> IFAIL are zero. If INFO > 0, then IFAIL contains the
234*> indices of the eigenvectors that failed to converge.
235*> If JOBZ = 'N', then IFAIL is not referenced.
236*> \endverbatim
237*>
238*> \param[out] INFO
239*> \verbatim
240*> INFO is INTEGER
241*> = 0: successful exit
242*> < 0: if INFO = -i, the i-th argument had an illegal value
243*> > 0: SPPTRF or SSPEVX returned an error code:
244*> <= N: if INFO = i, SSPEVX failed to converge;
245*> i eigenvectors failed to converge. Their indices
246*> are stored in array IFAIL.
247*> > N: if INFO = N + i, for 1 <= i <= N, then the leading
248*> minor of order i of B is not positive definite.
249*> The factorization of B could not be completed and
250*> no eigenvalues or eigenvectors were computed.
251*> \endverbatim
252*
253* Authors:
254* ========
255*
256*> \author Univ. of Tennessee
257*> \author Univ. of California Berkeley
258*> \author Univ. of Colorado Denver
259*> \author NAG Ltd.
260*
261*> \ingroup realOTHEReigen
262*
263*> \par Contributors:
264* ==================
265*>
266*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
267*
268* =====================================================================
269 SUBROUTINE sspgvx( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
270 $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
271 $ IFAIL, INFO )
272*
273* -- LAPACK driver routine --
274* -- LAPACK is a software package provided by Univ. of Tennessee, --
275* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
276*
277* .. Scalar Arguments ..
278 CHARACTER JOBZ, RANGE, UPLO
279 INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
280 REAL ABSTOL, VL, VU
281* ..
282* .. Array Arguments ..
283 INTEGER IFAIL( * ), IWORK( * )
284 REAL AP( * ), BP( * ), W( * ), WORK( * ),
285 $ z( ldz, * )
286* ..
287*
288* =====================================================================
289*
290* .. Local Scalars ..
291 LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
292 CHARACTER TRANS
293 INTEGER J
294* ..
295* .. External Functions ..
296 LOGICAL LSAME
297 EXTERNAL LSAME
298* ..
299* .. External Subroutines ..
300 EXTERNAL spptrf, sspevx, sspgst, stpmv, stpsv, xerbla
301* ..
302* .. Intrinsic Functions ..
303 INTRINSIC min
304* ..
305* .. Executable Statements ..
306*
307* Test the input parameters.
308*
309 upper = lsame( uplo, 'U' )
310 wantz = lsame( jobz, 'v' )
311 ALLEIG = LSAME( RANGE, 'a' )
312 VALEIG = LSAME( RANGE, 'v' )
313 INDEIG = LSAME( RANGE, 'i' )
314*
315 INFO = 0
316.LT..OR..GT. IF( ITYPE1 ITYPE3 ) THEN
317 INFO = -1
318.NOT..OR. ELSE IF( ( WANTZ LSAME( JOBZ, 'n' ) ) ) THEN
319 INFO = -2
320.NOT..OR..OR. ELSE IF( ( ALLEIG VALEIG INDEIG ) ) THEN
321 INFO = -3
322.NOT..OR. ELSE IF( ( UPPER LSAME( UPLO, 'l' ) ) ) THEN
323 INFO = -4
324.LT. ELSE IF( N0 ) THEN
325 INFO = -5
326 ELSE
327 IF( VALEIG ) THEN
328.GT..AND..LE. IF( N0 VUVL ) THEN
329 INFO = -9
330 END IF
331 ELSE IF( INDEIG ) THEN
332.LT. IF( IL1 ) THEN
333 INFO = -10
334.LT..OR..GT. ELSE IF( IUMIN( N, IL ) IUN ) THEN
335 INFO = -11
336 END IF
337 END IF
338 END IF
339.EQ. IF( INFO0 ) THEN
340.LT..OR..AND..LT. IF( LDZ1 ( WANTZ LDZN ) ) THEN
341 INFO = -16
342 END IF
343 END IF
344*
345.NE. IF( INFO0 ) THEN
346 CALL XERBLA( 'sspgvx', -INFO )
347 RETURN
348 END IF
349*
350* Quick return if possible
351*
352 M = 0
353.EQ. IF( N0 )
354 $ RETURN
355*
356* Form a Cholesky factorization of B.
357*
358 CALL SPPTRF( UPLO, N, BP, INFO )
359.NE. IF( INFO0 ) THEN
360 INFO = N + INFO
361 RETURN
362 END IF
363*
364* Transform problem to standard eigenvalue problem and solve.
365*
366 CALL SSPGST( ITYPE, UPLO, N, AP, BP, INFO )
367 CALL SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M,
368 $ W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
369*
370 IF( WANTZ ) THEN
371*
372* Backtransform eigenvectors to the original problem.
373*
374.GT. IF( INFO0 )
375 $ M = INFO - 1
376.EQ..OR..EQ. IF( ITYPE1 ITYPE2 ) THEN
377*
378* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
379* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
380*
381 IF( UPPER ) THEN
382 TRANS = 'n'
383 ELSE
384 TRANS = 't'
385 END IF
386*
387 DO 10 J = 1, M
388 CALL STPSV( UPLO, TRANS, 'non-unit', N, BP, Z( 1, J ),
389 $ 1 )
390 10 CONTINUE
391*
392.EQ. ELSE IF( ITYPE3 ) THEN
393*
394* For B*A*x=(lambda)*x;
395* backtransform eigenvectors: x = L*y or U**T*y
396*
397 IF( UPPER ) THEN
398 TRANS = 't'
399 ELSE
400 TRANS = 'n'
401 END IF
402*
403 DO 20 J = 1, M
404 CALL STPMV( UPLO, TRANS, 'non-unit', N, BP, Z( 1, J ),
405 $ 1 )
406 20 CONTINUE
407 END IF
408 END IF
409*
410 RETURN
411*
412* End of SSPGVX
413*
414 END
xerbla
subroutine xerbla(srname, info)
XERBLA
Definition xerbla.f:60
spptrf
subroutine spptrf(uplo, n, ap, info)
SPPTRF
Definition spptrf.f:119
sspgst
subroutine sspgst(itype, uplo, n, ap, bp, info)
SSPGST
Definition sspgst.f:113
sspgvx
subroutine sspgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSPGVX
Definition sspgvx.f:272
sspevx
subroutine sspevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m, w, z, ldz, work, iwork, ifail, info)
SSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrice...
Definition sspevx.f:234
stpmv
subroutine stpmv(uplo, trans, diag, n, ap, x, incx)
STPMV
Definition stpmv.f:142
stpsv
subroutine stpsv(uplo, trans, diag, n, ap, x, incx)
STPSV
Definition stpsv.f:144
min
#define min(a, b)
Definition macros.h:20