1: *> \brief <b> DSBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSBEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbevx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
22: * VU, 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, IU, KD, LDAB, LDQ, LDZ, M, N
28: * DOUBLE PRECISION ABSTOL, VL, VU
29: * ..
30: * .. Array Arguments ..
31: * INTEGER IFAIL( * ), IWORK( * )
32: * DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
33: * $ Z( LDZ, * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> DSBEVX computes selected eigenvalues and, optionally, eigenvectors
43: *> of a real symmetric band matrix A. Eigenvalues and eigenvectors can
44: *> be selected by specifying either a range of values or a range of
45: *> indices for the desired eigenvalues.
46: *> \endverbatim
47: *
48: * Arguments:
49: * ==========
50: *
51: *> \param[in] JOBZ
52: *> \verbatim
53: *> JOBZ is CHARACTER*1
54: *> = 'N': Compute eigenvalues only;
55: *> = 'V': Compute eigenvalues and eigenvectors.
56: *> \endverbatim
57: *>
58: *> \param[in] RANGE
59: *> \verbatim
60: *> RANGE is CHARACTER*1
61: *> = 'A': all eigenvalues will be found;
62: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
63: *> will be found;
64: *> = 'I': the IL-th through IU-th eigenvalues will be found.
65: *> \endverbatim
66: *>
67: *> \param[in] UPLO
68: *> \verbatim
69: *> UPLO is CHARACTER*1
70: *> = 'U': Upper triangle of A is stored;
71: *> = 'L': Lower triangle of A is stored.
72: *> \endverbatim
73: *>
74: *> \param[in] N
75: *> \verbatim
76: *> N is INTEGER
77: *> The order of the matrix A. N >= 0.
78: *> \endverbatim
79: *>
80: *> \param[in] KD
81: *> \verbatim
82: *> KD is INTEGER
83: *> The number of superdiagonals of the matrix A if UPLO = 'U',
84: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
85: *> \endverbatim
86: *>
87: *> \param[in,out] AB
88: *> \verbatim
89: *> AB is DOUBLE PRECISION array, dimension (LDAB, N)
90: *> On entry, the upper or lower triangle of the symmetric band
91: *> matrix A, stored in the first KD+1 rows of the array. The
92: *> j-th column of A is stored in the j-th column of the array AB
93: *> as follows:
94: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
95: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
96: *>
97: *> On exit, AB is overwritten by values generated during the
98: *> reduction to tridiagonal form. If UPLO = 'U', the first
99: *> superdiagonal and the diagonal of the tridiagonal matrix T
100: *> are returned in rows KD and KD+1 of AB, and if UPLO = 'L',
101: *> the diagonal and first subdiagonal of T are returned in the
102: *> first two rows of AB.
103: *> \endverbatim
104: *>
105: *> \param[in] LDAB
106: *> \verbatim
107: *> LDAB is INTEGER
108: *> The leading dimension of the array AB. LDAB >= KD + 1.
109: *> \endverbatim
110: *>
111: *> \param[out] Q
112: *> \verbatim
113: *> Q is DOUBLE PRECISION array, dimension (LDQ, N)
114: *> If JOBZ = 'V', the N-by-N orthogonal matrix used in the
115: *> reduction to tridiagonal form.
116: *> If JOBZ = 'N', the array Q is not referenced.
117: *> \endverbatim
118: *>
119: *> \param[in] LDQ
120: *> \verbatim
121: *> LDQ is INTEGER
122: *> The leading dimension of the array Q. If JOBZ = 'V', then
123: *> LDQ >= max(1,N).
124: *> \endverbatim
125: *>
126: *> \param[in] VL
127: *> \verbatim
128: *> VL is DOUBLE PRECISION
129: *> \endverbatim
130: *>
131: *> \param[in] VU
132: *> \verbatim
133: *> VU is DOUBLE PRECISION
134: *> If RANGE='V', the lower and upper bounds of the interval to
135: *> be searched for eigenvalues. VL < VU.
136: *> Not referenced if RANGE = 'A' or 'I'.
137: *> \endverbatim
138: *>
139: *> \param[in] IL
140: *> \verbatim
141: *> IL is INTEGER
142: *> \endverbatim
143: *>
144: *> \param[in] IU
145: *> \verbatim
146: *> IU is INTEGER
147: *> If RANGE='I', the indices (in ascending order) of the
148: *> smallest and largest eigenvalues to be returned.
149: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
150: *> Not referenced if RANGE = 'A' or 'V'.
151: *> \endverbatim
152: *>
153: *> \param[in] ABSTOL
154: *> \verbatim
155: *> ABSTOL is DOUBLE PRECISION
156: *> The absolute error tolerance for the eigenvalues.
157: *> An approximate eigenvalue is accepted as converged
158: *> when it is determined to lie in an interval [a,b]
159: *> of width less than or equal to
160: *>
161: *> ABSTOL + EPS * max( |a|,|b| ) ,
162: *>
163: *> where EPS is the machine precision. If ABSTOL is less than
164: *> or equal to zero, then EPS*|T| will be used in its place,
165: *> where |T| is the 1-norm of the tridiagonal matrix obtained
166: *> by reducing AB to tridiagonal form.
167: *>
168: *> Eigenvalues will be computed most accurately when ABSTOL is
169: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
170: *> If this routine returns with INFO>0, indicating that some
171: *> eigenvectors did not converge, try setting ABSTOL to
172: *> 2*DLAMCH('S').
173: *>
174: *> See "Computing Small Singular Values of Bidiagonal Matrices
175: *> with Guaranteed High Relative Accuracy," by Demmel and
176: *> Kahan, LAPACK Working Note #3.
177: *> \endverbatim
178: *>
179: *> \param[out] M
180: *> \verbatim
181: *> M is INTEGER
182: *> The total number of eigenvalues found. 0 <= M <= N.
183: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
184: *> \endverbatim
185: *>
186: *> \param[out] W
187: *> \verbatim
188: *> W is DOUBLE PRECISION array, dimension (N)
189: *> The first M elements contain the selected eigenvalues in
190: *> ascending order.
191: *> \endverbatim
192: *>
193: *> \param[out] Z
194: *> \verbatim
195: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
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: *> If an eigenvector fails to converge, then that column of Z
201: *> contains the latest approximation to the eigenvector, and the
202: *> index of the eigenvector is returned in IFAIL.
203: *> If JOBZ = 'N', then Z is not referenced.
204: *> Note: the user must ensure that at least max(1,M) columns are
205: *> supplied in the array Z; if RANGE = 'V', the exact value of M
206: *> is not known in advance and an upper bound must be used.
207: *> \endverbatim
208: *>
209: *> \param[in] LDZ
210: *> \verbatim
211: *> LDZ is INTEGER
212: *> The leading dimension of the array Z. LDZ >= 1, and if
213: *> JOBZ = 'V', LDZ >= max(1,N).
214: *> \endverbatim
215: *>
216: *> \param[out] WORK
217: *> \verbatim
218: *> WORK is DOUBLE PRECISION array, dimension (7*N)
219: *> \endverbatim
220: *>
221: *> \param[out] IWORK
222: *> \verbatim
223: *> IWORK is INTEGER array, dimension (5*N)
224: *> \endverbatim
225: *>
226: *> \param[out] IFAIL
227: *> \verbatim
228: *> IFAIL is INTEGER array, dimension (N)
229: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
230: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
231: *> indices of the eigenvectors that failed to converge.
232: *> If JOBZ = 'N', then IFAIL is not referenced.
233: *> \endverbatim
234: *>
235: *> \param[out] INFO
236: *> \verbatim
237: *> INFO is INTEGER
238: *> = 0: successful exit.
239: *> < 0: if INFO = -i, the i-th argument had an illegal value.
240: *> > 0: if INFO = i, then i eigenvectors failed to converge.
241: *> Their indices are stored in array IFAIL.
242: *> \endverbatim
243: *
244: * Authors:
245: * ========
246: *
247: *> \author Univ. of Tennessee
248: *> \author Univ. of California Berkeley
249: *> \author Univ. of Colorado Denver
250: *> \author NAG Ltd.
251: *
252: *> \date November 2011
253: *
254: *> \ingroup doubleOTHEReigen
255: *
256: * =====================================================================
257: SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
258: $ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK,
259: $ IFAIL, INFO )
260: *
261: * -- LAPACK driver routine (version 3.4.0) --
262: * -- LAPACK is a software package provided by Univ. of Tennessee, --
263: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
264: * November 2011
265: *
266: * .. Scalar Arguments ..
267: CHARACTER JOBZ, RANGE, UPLO
268: INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N
269: DOUBLE PRECISION ABSTOL, VL, VU
270: * ..
271: * .. Array Arguments ..
272: INTEGER IFAIL( * ), IWORK( * )
273: DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ),
274: $ Z( LDZ, * )
275: * ..
276: *
277: * =====================================================================
278: *
279: * .. Parameters ..
280: DOUBLE PRECISION ZERO, ONE
281: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
282: * ..
283: * .. Local Scalars ..
284: LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
285: CHARACTER ORDER
286: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
287: $ INDISP, INDIWO, INDWRK, ISCALE, ITMP1, J, JJ,
288: $ NSPLIT
289: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
290: $ SIGMA, SMLNUM, TMP1, VLL, VUU
291: * ..
292: * .. External Functions ..
293: LOGICAL LSAME
294: DOUBLE PRECISION DLAMCH, DLANSB
295: EXTERNAL LSAME, DLAMCH, DLANSB
296: * ..
297: * .. External Subroutines ..
298: EXTERNAL DCOPY, DGEMV, DLACPY, DLASCL, DSBTRD, DSCAL,
299: $ DSTEBZ, DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
300: * ..
301: * .. Intrinsic Functions ..
302: INTRINSIC MAX, MIN, SQRT
303: * ..
304: * .. Executable Statements ..
305: *
306: * Test the input parameters.
307: *
308: WANTZ = LSAME( JOBZ, 'V' )
309: ALLEIG = LSAME( RANGE, 'A' )
310: VALEIG = LSAME( RANGE, 'V' )
311: INDEIG = LSAME( RANGE, 'I' )
312: LOWER = LSAME( UPLO, 'L' )
313: *
314: INFO = 0
315: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
316: INFO = -1
317: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
318: INFO = -2
319: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
320: INFO = -3
321: ELSE IF( N.LT.0 ) THEN
322: INFO = -4
323: ELSE IF( KD.LT.0 ) THEN
324: INFO = -5
325: ELSE IF( LDAB.LT.KD+1 ) THEN
326: INFO = -7
327: ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
328: INFO = -9
329: ELSE
330: IF( VALEIG ) THEN
331: IF( N.GT.0 .AND. VU.LE.VL )
332: $ INFO = -11
333: ELSE IF( INDEIG ) THEN
334: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
335: INFO = -12
336: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
337: INFO = -13
338: END IF
339: END IF
340: END IF
341: IF( INFO.EQ.0 ) THEN
342: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
343: $ INFO = -18
344: END IF
345: *
346: IF( INFO.NE.0 ) THEN
347: CALL XERBLA( 'DSBEVX', -INFO )
348: RETURN
349: END IF
350: *
351: * Quick return if possible
352: *
353: M = 0
354: IF( N.EQ.0 )
355: $ RETURN
356: *
357: IF( N.EQ.1 ) THEN
358: M = 1
359: IF( LOWER ) THEN
360: TMP1 = AB( 1, 1 )
361: ELSE
362: TMP1 = AB( KD+1, 1 )
363: END IF
364: IF( VALEIG ) THEN
365: IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
366: $ M = 0
367: END IF
368: IF( M.EQ.1 ) THEN
369: W( 1 ) = TMP1
370: IF( WANTZ )
371: $ Z( 1, 1 ) = ONE
372: END IF
373: RETURN
374: END IF
375: *
376: * Get machine constants.
377: *
378: SAFMIN = DLAMCH( 'Safe minimum' )
379: EPS = DLAMCH( 'Precision' )
380: SMLNUM = SAFMIN / EPS
381: BIGNUM = ONE / SMLNUM
382: RMIN = SQRT( SMLNUM )
383: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
384: *
385: * Scale matrix to allowable range, if necessary.
386: *
387: ISCALE = 0
388: ABSTLL = ABSTOL
389: IF( VALEIG ) THEN
390: VLL = VL
391: VUU = VU
392: ELSE
393: VLL = ZERO
394: VUU = ZERO
395: END IF
396: ANRM = DLANSB( 'M', UPLO, N, KD, AB, LDAB, WORK )
397: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
398: ISCALE = 1
399: SIGMA = RMIN / ANRM
400: ELSE IF( ANRM.GT.RMAX ) THEN
401: ISCALE = 1
402: SIGMA = RMAX / ANRM
403: END IF
404: IF( ISCALE.EQ.1 ) THEN
405: IF( LOWER ) THEN
406: CALL DLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
407: ELSE
408: CALL DLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
409: END IF
410: IF( ABSTOL.GT.0 )
411: $ ABSTLL = ABSTOL*SIGMA
412: IF( VALEIG ) THEN
413: VLL = VL*SIGMA
414: VUU = VU*SIGMA
415: END IF
416: END IF
417: *
418: * Call DSBTRD to reduce symmetric band matrix to tridiagonal form.
419: *
420: INDD = 1
421: INDE = INDD + N
422: INDWRK = INDE + N
423: CALL DSBTRD( JOBZ, UPLO, N, KD, AB, LDAB, WORK( INDD ),
424: $ WORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
425: *
426: * If all eigenvalues are desired and ABSTOL is less than or equal
427: * to zero, then call DSTERF or SSTEQR. If this fails for some
428: * eigenvalue, then try DSTEBZ.
429: *
430: TEST = .FALSE.
431: IF (INDEIG) THEN
432: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
433: TEST = .TRUE.
434: END IF
435: END IF
436: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
437: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
438: INDEE = INDWRK + 2*N
439: IF( .NOT.WANTZ ) THEN
440: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
441: CALL DSTERF( N, W, WORK( INDEE ), INFO )
442: ELSE
443: CALL DLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
444: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
445: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
446: $ WORK( INDWRK ), INFO )
447: IF( INFO.EQ.0 ) THEN
448: DO 10 I = 1, N
449: IFAIL( I ) = 0
450: 10 CONTINUE
451: END IF
452: END IF
453: IF( INFO.EQ.0 ) THEN
454: M = N
455: GO TO 30
456: END IF
457: INFO = 0
458: END IF
459: *
460: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
461: *
462: IF( WANTZ ) THEN
463: ORDER = 'B'
464: ELSE
465: ORDER = 'E'
466: END IF
467: INDIBL = 1
468: INDISP = INDIBL + N
469: INDIWO = INDISP + N
470: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
471: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
472: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
473: $ IWORK( INDIWO ), INFO )
474: *
475: IF( WANTZ ) THEN
476: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
477: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
478: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
479: *
480: * Apply orthogonal matrix used in reduction to tridiagonal
481: * form to eigenvectors returned by DSTEIN.
482: *
483: DO 20 J = 1, M
484: CALL DCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
485: CALL DGEMV( 'N', N, N, ONE, Q, LDQ, WORK, 1, ZERO,
486: $ Z( 1, J ), 1 )
487: 20 CONTINUE
488: END IF
489: *
490: * If matrix was scaled, then rescale eigenvalues appropriately.
491: *
492: 30 CONTINUE
493: IF( ISCALE.EQ.1 ) THEN
494: IF( INFO.EQ.0 ) THEN
495: IMAX = M
496: ELSE
497: IMAX = INFO - 1
498: END IF
499: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
500: END IF
501: *
502: * If eigenvalues are not in order, then sort them, along with
503: * eigenvectors.
504: *
505: IF( WANTZ ) THEN
506: DO 50 J = 1, M - 1
507: I = 0
508: TMP1 = W( J )
509: DO 40 JJ = J + 1, M
510: IF( W( JJ ).LT.TMP1 ) THEN
511: I = JJ
512: TMP1 = W( JJ )
513: END IF
514: 40 CONTINUE
515: *
516: IF( I.NE.0 ) THEN
517: ITMP1 = IWORK( INDIBL+I-1 )
518: W( I ) = W( J )
519: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
520: W( J ) = TMP1
521: IWORK( INDIBL+J-1 ) = ITMP1
522: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
523: IF( INFO.NE.0 ) THEN
524: ITMP1 = IFAIL( I )
525: IFAIL( I ) = IFAIL( J )
526: IFAIL( J ) = ITMP1
527: END IF
528: END IF
529: 50 CONTINUE
530: END IF
531: *
532: RETURN
533: *
534: * End of DSBEVX
535: *
536: END
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