1: *> \brief \b DSBGVX
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSBGVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
22: * LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
23: * LDZ, WORK, IWORK, IFAIL, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE, UPLO
27: * INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
28: * $ N
29: * DOUBLE PRECISION ABSTOL, VL, VU
30: * ..
31: * .. Array Arguments ..
32: * INTEGER IFAIL( * ), IWORK( * )
33: * DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
34: * $ W( * ), WORK( * ), Z( LDZ, * )
35: * ..
36: *
37: *
38: *> \par Purpose:
39: * =============
40: *>
41: *> \verbatim
42: *>
43: *> DSBGVX computes selected eigenvalues, and optionally, eigenvectors
44: *> of a real generalized symmetric-definite banded eigenproblem, of
45: *> the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric
46: *> and banded, and B is also positive definite. Eigenvalues and
47: *> eigenvectors can be selected by specifying either all eigenvalues,
48: *> a range of values or a range of indices for the desired eigenvalues.
49: *> \endverbatim
50: *
51: * Arguments:
52: * ==========
53: *
54: *> \param[in] JOBZ
55: *> \verbatim
56: *> JOBZ is CHARACTER*1
57: *> = 'N': Compute eigenvalues only;
58: *> = 'V': Compute eigenvalues and eigenvectors.
59: *> \endverbatim
60: *>
61: *> \param[in] RANGE
62: *> \verbatim
63: *> RANGE is CHARACTER*1
64: *> = 'A': all eigenvalues will be found.
65: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
66: *> will be found.
67: *> = 'I': the IL-th through IU-th eigenvalues will be found.
68: *> \endverbatim
69: *>
70: *> \param[in] UPLO
71: *> \verbatim
72: *> UPLO is CHARACTER*1
73: *> = 'U': Upper triangles of A and B are stored;
74: *> = 'L': Lower triangles of A and B are stored.
75: *> \endverbatim
76: *>
77: *> \param[in] N
78: *> \verbatim
79: *> N is INTEGER
80: *> The order of the matrices A and B. N >= 0.
81: *> \endverbatim
82: *>
83: *> \param[in] KA
84: *> \verbatim
85: *> KA is INTEGER
86: *> The number of superdiagonals of the matrix A if UPLO = 'U',
87: *> or the number of subdiagonals if UPLO = 'L'. KA >= 0.
88: *> \endverbatim
89: *>
90: *> \param[in] KB
91: *> \verbatim
92: *> KB is INTEGER
93: *> The number of superdiagonals of the matrix B if UPLO = 'U',
94: *> or the number of subdiagonals if UPLO = 'L'. KB >= 0.
95: *> \endverbatim
96: *>
97: *> \param[in,out] AB
98: *> \verbatim
99: *> AB is DOUBLE PRECISION array, dimension (LDAB, N)
100: *> On entry, the upper or lower triangle of the symmetric band
101: *> matrix A, stored in the first ka+1 rows of the array. The
102: *> j-th column of A is stored in the j-th column of the array AB
103: *> as follows:
104: *> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j;
105: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka).
106: *>
107: *> On exit, the contents of AB are destroyed.
108: *> \endverbatim
109: *>
110: *> \param[in] LDAB
111: *> \verbatim
112: *> LDAB is INTEGER
113: *> The leading dimension of the array AB. LDAB >= KA+1.
114: *> \endverbatim
115: *>
116: *> \param[in,out] BB
117: *> \verbatim
118: *> BB is DOUBLE PRECISION array, dimension (LDBB, N)
119: *> On entry, the upper or lower triangle of the symmetric band
120: *> matrix B, stored in the first kb+1 rows of the array. The
121: *> j-th column of B is stored in the j-th column of the array BB
122: *> as follows:
123: *> if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j;
124: *> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb).
125: *>
126: *> On exit, the factor S from the split Cholesky factorization
127: *> B = S**T*S, as returned by DPBSTF.
128: *> \endverbatim
129: *>
130: *> \param[in] LDBB
131: *> \verbatim
132: *> LDBB is INTEGER
133: *> The leading dimension of the array BB. LDBB >= KB+1.
134: *> \endverbatim
135: *>
136: *> \param[out] Q
137: *> \verbatim
138: *> Q is DOUBLE PRECISION array, dimension (LDQ, N)
139: *> If JOBZ = 'V', the n-by-n matrix used in the reduction of
140: *> A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x,
141: *> and consequently C to tridiagonal form.
142: *> If JOBZ = 'N', the array Q is not referenced.
143: *> \endverbatim
144: *>
145: *> \param[in] LDQ
146: *> \verbatim
147: *> LDQ is INTEGER
148: *> The leading dimension of the array Q. If JOBZ = 'N',
149: *> LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
150: *> \endverbatim
151: *>
152: *> \param[in] VL
153: *> \verbatim
154: *> VL is DOUBLE PRECISION
155: *>
156: *> If RANGE='V', the lower bound of the interval to
157: *> be searched for eigenvalues. VL < VU.
158: *> Not referenced if RANGE = 'A' or 'I'.
159: *> \endverbatim
160: *>
161: *> \param[in] VU
162: *> \verbatim
163: *> VU is DOUBLE PRECISION
164: *>
165: *> If RANGE='V', the upper bound of the interval to
166: *> be searched for eigenvalues. VL < VU.
167: *> Not referenced if RANGE = 'A' or 'I'.
168: *> \endverbatim
169: *>
170: *> \param[in] IL
171: *> \verbatim
172: *> IL is INTEGER
173: *>
174: *> If RANGE='I', the index of the
175: *> smallest eigenvalue to be returned.
176: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
177: *> Not referenced if RANGE = 'A' or 'V'.
178: *> \endverbatim
179: *>
180: *> \param[in] IU
181: *> \verbatim
182: *> IU is INTEGER
183: *>
184: *> If RANGE='I', the index of the
185: *> largest eigenvalue to be returned.
186: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
187: *> Not referenced if RANGE = 'A' or 'V'.
188: *> \endverbatim
189: *>
190: *> \param[in] ABSTOL
191: *> \verbatim
192: *> ABSTOL is DOUBLE PRECISION
193: *> The absolute error tolerance for the eigenvalues.
194: *> An approximate eigenvalue is accepted as converged
195: *> when it is determined to lie in an interval [a,b]
196: *> of width less than or equal to
197: *>
198: *> ABSTOL + EPS * max( |a|,|b| ) ,
199: *>
200: *> where EPS is the machine precision. If ABSTOL is less than
201: *> or equal to zero, then EPS*|T| will be used in its place,
202: *> where |T| is the 1-norm of the tridiagonal matrix obtained
203: *> by reducing A to tridiagonal form.
204: *>
205: *> Eigenvalues will be computed most accurately when ABSTOL is
206: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
207: *> If this routine returns with INFO>0, indicating that some
208: *> eigenvectors did not converge, try setting ABSTOL to
209: *> 2*DLAMCH('S').
210: *> \endverbatim
211: *>
212: *> \param[out] M
213: *> \verbatim
214: *> M is INTEGER
215: *> The total number of eigenvalues found. 0 <= M <= N.
216: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
217: *> \endverbatim
218: *>
219: *> \param[out] W
220: *> \verbatim
221: *> W is DOUBLE PRECISION array, dimension (N)
222: *> If INFO = 0, the eigenvalues in ascending order.
223: *> \endverbatim
224: *>
225: *> \param[out] Z
226: *> \verbatim
227: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
228: *> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of
229: *> eigenvectors, with the i-th column of Z holding the
230: *> eigenvector associated with W(i). The eigenvectors are
231: *> normalized so Z**T*B*Z = I.
232: *> If JOBZ = 'N', then Z is not referenced.
233: *> \endverbatim
234: *>
235: *> \param[in] LDZ
236: *> \verbatim
237: *> LDZ is INTEGER
238: *> The leading dimension of the array Z. LDZ >= 1, and if
239: *> JOBZ = 'V', LDZ >= max(1,N).
240: *> \endverbatim
241: *>
242: *> \param[out] WORK
243: *> \verbatim
244: *> WORK is DOUBLE PRECISION array, dimension (7*N)
245: *> \endverbatim
246: *>
247: *> \param[out] IWORK
248: *> \verbatim
249: *> IWORK is INTEGER array, dimension (5*N)
250: *> \endverbatim
251: *>
252: *> \param[out] IFAIL
253: *> \verbatim
254: *> IFAIL is INTEGER array, dimension (M)
255: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
256: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
257: *> indices of the eigenvalues that failed to converge.
258: *> If JOBZ = 'N', then IFAIL is not referenced.
259: *> \endverbatim
260: *>
261: *> \param[out] INFO
262: *> \verbatim
263: *> INFO is INTEGER
264: *> = 0 : successful exit
265: *> < 0 : if INFO = -i, the i-th argument had an illegal value
266: *> <= N: if INFO = i, then i eigenvectors failed to converge.
267: *> Their indices are stored in IFAIL.
268: *> > N : DPBSTF returned an error code; i.e.,
269: *> if INFO = N + i, for 1 <= i <= N, then the leading
270: *> minor of order i of B is not positive definite.
271: *> The factorization of B could not be completed and
272: *> no eigenvalues or eigenvectors were computed.
273: *> \endverbatim
274: *
275: * Authors:
276: * ========
277: *
278: *> \author Univ. of Tennessee
279: *> \author Univ. of California Berkeley
280: *> \author Univ. of Colorado Denver
281: *> \author NAG Ltd.
282: *
283: *> \date June 2016
284: *
285: *> \ingroup doubleOTHEReigen
286: *
287: *> \par Contributors:
288: * ==================
289: *>
290: *> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
291: *
292: * =====================================================================
293: SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB,
294: $ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z,
295: $ LDZ, WORK, IWORK, IFAIL, INFO )
296: *
297: * -- LAPACK driver routine (version 3.7.0) --
298: * -- LAPACK is a software package provided by Univ. of Tennessee, --
299: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
300: * June 2016
301: *
302: * .. Scalar Arguments ..
303: CHARACTER JOBZ, RANGE, UPLO
304: INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M,
305: $ N
306: DOUBLE PRECISION ABSTOL, VL, VU
307: * ..
308: * .. Array Arguments ..
309: INTEGER IFAIL( * ), IWORK( * )
310: DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ),
311: $ W( * ), WORK( * ), Z( LDZ, * )
312: * ..
313: *
314: * =====================================================================
315: *
316: * .. Parameters ..
317: DOUBLE PRECISION ZERO, ONE
318: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
319: * ..
320: * .. Local Scalars ..
321: LOGICAL ALLEIG, INDEIG, TEST, UPPER, VALEIG, WANTZ
322: CHARACTER ORDER, VECT
323: INTEGER I, IINFO, INDD, INDE, INDEE, INDIBL, INDISP,
324: $ INDIWO, INDWRK, ITMP1, J, JJ, NSPLIT
325: DOUBLE PRECISION TMP1
326: * ..
327: * .. External Functions ..
328: LOGICAL LSAME
329: EXTERNAL LSAME
330: * ..
331: * .. External Subroutines ..
332: EXTERNAL DCOPY, DGEMV, DLACPY, DPBSTF, DSBGST, DSBTRD,
333: $ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
334: * ..
335: * .. Intrinsic Functions ..
336: INTRINSIC MIN
337: * ..
338: * .. Executable Statements ..
339: *
340: * Test the input parameters.
341: *
342: WANTZ = LSAME( JOBZ, 'V' )
343: UPPER = LSAME( UPLO, 'U' )
344: ALLEIG = LSAME( RANGE, 'A' )
345: VALEIG = LSAME( RANGE, 'V' )
346: INDEIG = LSAME( RANGE, 'I' )
347: *
348: INFO = 0
349: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
350: INFO = -1
351: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
352: INFO = -2
353: ELSE IF( .NOT.( UPPER .OR. LSAME( UPLO, 'L' ) ) ) THEN
354: INFO = -3
355: ELSE IF( N.LT.0 ) THEN
356: INFO = -4
357: ELSE IF( KA.LT.0 ) THEN
358: INFO = -5
359: ELSE IF( KB.LT.0 .OR. KB.GT.KA ) THEN
360: INFO = -6
361: ELSE IF( LDAB.LT.KA+1 ) THEN
362: INFO = -8
363: ELSE IF( LDBB.LT.KB+1 ) THEN
364: INFO = -10
365: ELSE IF( LDQ.LT.1 .OR. ( WANTZ .AND. LDQ.LT.N ) ) THEN
366: INFO = -12
367: ELSE
368: IF( VALEIG ) THEN
369: IF( N.GT.0 .AND. VU.LE.VL )
370: $ INFO = -14
371: ELSE IF( INDEIG ) THEN
372: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
373: INFO = -15
374: ELSE IF ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
375: INFO = -16
376: END IF
377: END IF
378: END IF
379: IF( INFO.EQ.0) THEN
380: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
381: INFO = -21
382: END IF
383: END IF
384: *
385: IF( INFO.NE.0 ) THEN
386: CALL XERBLA( 'DSBGVX', -INFO )
387: RETURN
388: END IF
389: *
390: * Quick return if possible
391: *
392: M = 0
393: IF( N.EQ.0 )
394: $ RETURN
395: *
396: * Form a split Cholesky factorization of B.
397: *
398: CALL DPBSTF( UPLO, N, KB, BB, LDBB, INFO )
399: IF( INFO.NE.0 ) THEN
400: INFO = N + INFO
401: RETURN
402: END IF
403: *
404: * Transform problem to standard eigenvalue problem.
405: *
406: CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ,
407: $ WORK, IINFO )
408: *
409: * Reduce symmetric band matrix to tridiagonal form.
410: *
411: INDD = 1
412: INDE = INDD + N
413: INDWRK = INDE + N
414: IF( WANTZ ) THEN
415: VECT = 'U'
416: ELSE
417: VECT = 'N'
418: END IF
419: CALL DSBTRD( VECT, UPLO, N, KA, AB, LDAB, WORK( INDD ),
420: $ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
421: *
422: * If all eigenvalues are desired and ABSTOL is less than or equal
423: * to zero, then call DSTERF or SSTEQR. If this fails for some
424: * eigenvalue, then try DSTEBZ.
425: *
426: TEST = .FALSE.
427: IF( INDEIG ) THEN
428: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
429: TEST = .TRUE.
430: END IF
431: END IF
432: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
433: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
434: INDEE = INDWRK + 2*N
435: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
436: IF( .NOT.WANTZ ) THEN
437: CALL DSTERF( N, W, WORK( INDEE ), INFO )
438: ELSE
439: CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
440: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
441: $ WORK( INDWRK ), INFO )
442: IF( INFO.EQ.0 ) THEN
443: DO 10 I = 1, N
444: IFAIL( I ) = 0
445: 10 CONTINUE
446: END IF
447: END IF
448: IF( INFO.EQ.0 ) THEN
449: M = N
450: GO TO 30
451: END IF
452: INFO = 0
453: END IF
454: *
455: * Otherwise, call DSTEBZ and, if eigenvectors are desired,
456: * call DSTEIN.
457: *
458: IF( WANTZ ) THEN
459: ORDER = 'B'
460: ELSE
461: ORDER = 'E'
462: END IF
463: INDIBL = 1
464: INDISP = INDIBL + N
465: INDIWO = INDISP + N
466: CALL DSTEBZ( RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL,
467: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
468: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
469: $ IWORK( INDIWO ), INFO )
470: *
471: IF( WANTZ ) THEN
472: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
473: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
474: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
475: *
476: * Apply transformation matrix used in reduction to tridiagonal
477: * form to eigenvectors returned by DSTEIN.
478: *
479: DO 20 J = 1, M
480: CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
481: CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
482: $ Z( 1, J ), 1 )
483: 20 CONTINUE
484: END IF
485: *
486: 30 CONTINUE
487: *
488: * If eigenvalues are not in order, then sort them, along with
489: * eigenvectors.
490: *
491: IF( WANTZ ) THEN
492: DO 50 J = 1, M - 1
493: I = 0
494: TMP1 = W( J )
495: DO 40 JJ = J + 1, M
496: IF( W( JJ ).LT.TMP1 ) THEN
497: I = JJ
498: TMP1 = W( JJ )
499: END IF
500: 40 CONTINUE
501: *
502: IF( I.NE.0 ) THEN
503: ITMP1 = IWORK( INDIBL+I-1 )
504: W( I ) = W( J )
505: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
506: W( J ) = TMP1
507: IWORK( INDIBL+J-1 ) = ITMP1
508: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
509: IF( INFO.NE.0 ) THEN
510: ITMP1 = IFAIL( I )
511: IFAIL( I ) = IFAIL( J )
512: IFAIL( J ) = ITMP1
513: END IF
514: END IF
515: 50 CONTINUE
516: END IF
517: *
518: RETURN
519: *
520: * End of DSBGVX
521: *
522: END
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