LAPACK  3.7.1
LAPACK: Linear Algebra PACKage
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
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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 *> \date June 2016
262 *
263 *> \ingroup realOTHEReigen
264 *
265 *> \par Contributors:
266 * ==================
267 *>
268 *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
269 *
270 * =====================================================================
271  SUBROUTINE sspgvx( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU,
272  $ IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
273  $ IFAIL, INFO )
274 *
275 * -- LAPACK driver routine (version 3.7.0) --
276 * -- LAPACK is a software package provided by Univ. of Tennessee, --
277 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
278 * June 2016
279 *
280 * .. Scalar Arguments ..
281  CHARACTER JOBZ, RANGE, UPLO
282  INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
283  REAL ABSTOL, VL, VU
284 * ..
285 * .. Array Arguments ..
286  INTEGER IFAIL( * ), IWORK( * )
287  REAL AP( * ), BP( * ), W( * ), WORK( * ),
288  $ z( ldz, * )
289 * ..
290 *
291 * =====================================================================
292 *
293 * .. Local Scalars ..
294  LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
295  CHARACTER TRANS
296  INTEGER J
297 * ..
298 * .. External Functions ..
299  LOGICAL LSAME
300  EXTERNAL lsame
301 * ..
302 * .. External Subroutines ..
303  EXTERNAL spptrf, sspevx, sspgst, stpmv, stpsv, xerbla
304 * ..
305 * .. Intrinsic Functions ..
306  INTRINSIC min
307 * ..
308 * .. Executable Statements ..
309 *
310 * Test the input parameters.
311 *
312  upper = lsame( uplo, 'U' )
313  wantz = lsame( jobz, 'V' )
314  alleig = lsame( range, 'A' )
315  valeig = lsame( range, 'V' )
316  indeig = lsame( range, 'I' )
317 *
318  info = 0
319  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
320  info = -1
321  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
322  info = -2
323  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
324  info = -3
325  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
326  info = -4
327  ELSE IF( n.LT.0 ) THEN
328  info = -5
329  ELSE
330  IF( valeig ) THEN
331  IF( n.GT.0 .AND. vu.LE.vl ) THEN
332  info = -9
333  END IF
334  ELSE IF( indeig ) THEN
335  IF( il.LT.1 ) THEN
336  info = -10
337  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
338  info = -11
339  END IF
340  END IF
341  END IF
342  IF( info.EQ.0 ) THEN
343  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
344  info = -16
345  END IF
346  END IF
347 *
348  IF( info.NE.0 ) THEN
349  CALL xerbla( 'SSPGVX', -info )
350  RETURN
351  END IF
352 *
353 * Quick return if possible
354 *
355  m = 0
356  IF( n.EQ.0 )
357  $ RETURN
358 *
359 * Form a Cholesky factorization of B.
360 *
361  CALL spptrf( uplo, n, bp, info )
362  IF( info.NE.0 ) THEN
363  info = n + info
364  RETURN
365  END IF
366 *
367 * Transform problem to standard eigenvalue problem and solve.
368 *
369  CALL sspgst( itype, uplo, n, ap, bp, info )
370  CALL sspevx( jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m,
371  $ w, z, ldz, work, iwork, ifail, info )
372 *
373  IF( wantz ) THEN
374 *
375 * Backtransform eigenvectors to the original problem.
376 *
377  IF( info.GT.0 )
378  $ m = info - 1
379  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
380 *
381 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
382 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
383 *
384  IF( upper ) THEN
385  trans = 'N'
386  ELSE
387  trans = 'T'
388  END IF
389 *
390  DO 10 j = 1, m
391  CALL stpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
392  $ 1 )
393  10 CONTINUE
394 *
395  ELSE IF( itype.EQ.3 ) THEN
396 *
397 * For B*A*x=(lambda)*x;
398 * backtransform eigenvectors: x = L*y or U**T*y
399 *
400  IF( upper ) THEN
401  trans = 'T'
402  ELSE
403  trans = 'N'
404  END IF
405 *
406  DO 20 j = 1, m
407  CALL stpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
408  $ 1 )
409  20 CONTINUE
410  END IF
411  END IF
412 *
413  RETURN
414 *
415 * End of SSPGVX
416 *
417  END
spptrf
subroutine spptrf(UPLO, N, AP, INFO)
SPPTRF
Definition: spptrf.f:121
stpmv
subroutine stpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPMV
Definition: stpmv.f:144
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 matric...
Definition: sspevx.f:236
xerbla
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
stpsv
subroutine stpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPSV
Definition: stpsv.f:146
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:274
sspgst
subroutine sspgst(ITYPE, UPLO, N, AP, BP, INFO)
SSPGST
Definition: sspgst.f:115