1: *> \brief <b> DSPEVX 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 DSPEVX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dspevx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dspevx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dspevx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
22: * ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
23: * INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBZ, RANGE, UPLO
27: * INTEGER IL, INFO, IU, LDZ, M, N
28: * DOUBLE PRECISION ABSTOL, VL, VU
29: * ..
30: * .. Array Arguments ..
31: * INTEGER IFAIL( * ), IWORK( * )
32: * DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
33: * ..
34: *
35: *
36: *> \par Purpose:
37: * =============
38: *>
39: *> \verbatim
40: *>
41: *> DSPEVX computes selected eigenvalues and, optionally, eigenvectors
42: *> of a real symmetric matrix A in packed storage. Eigenvalues/vectors
43: *> can be selected by specifying either a range of values or a range of
44: *> indices for the desired eigenvalues.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] JOBZ
51: *> \verbatim
52: *> JOBZ is CHARACTER*1
53: *> = 'N': Compute eigenvalues only;
54: *> = 'V': Compute eigenvalues and eigenvectors.
55: *> \endverbatim
56: *>
57: *> \param[in] RANGE
58: *> \verbatim
59: *> RANGE is CHARACTER*1
60: *> = 'A': all eigenvalues will be found;
61: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
62: *> will be found;
63: *> = 'I': the IL-th through IU-th eigenvalues will be found.
64: *> \endverbatim
65: *>
66: *> \param[in] UPLO
67: *> \verbatim
68: *> UPLO is CHARACTER*1
69: *> = 'U': Upper triangle of A is stored;
70: *> = 'L': Lower triangle of A is stored.
71: *> \endverbatim
72: *>
73: *> \param[in] N
74: *> \verbatim
75: *> N is INTEGER
76: *> The order of the matrix A. N >= 0.
77: *> \endverbatim
78: *>
79: *> \param[in,out] AP
80: *> \verbatim
81: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
82: *> On entry, the upper or lower triangle of the symmetric matrix
83: *> A, packed columnwise in a linear array. The j-th column of A
84: *> is stored in the array AP as follows:
85: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
86: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
87: *>
88: *> On exit, AP is overwritten by values generated during the
89: *> reduction to tridiagonal form. If UPLO = 'U', the diagonal
90: *> and first superdiagonal of the tridiagonal matrix T overwrite
91: *> the corresponding elements of A, and if UPLO = 'L', the
92: *> diagonal and first subdiagonal of T overwrite the
93: *> corresponding elements of A.
94: *> \endverbatim
95: *>
96: *> \param[in] VL
97: *> \verbatim
98: *> VL is DOUBLE PRECISION
99: *> If RANGE='V', the lower bound of the interval to
100: *> be searched for eigenvalues. VL < VU.
101: *> Not referenced if RANGE = 'A' or 'I'.
102: *> \endverbatim
103: *>
104: *> \param[in] VU
105: *> \verbatim
106: *> VU is DOUBLE PRECISION
107: *> If RANGE='V', the upper bound of the interval to
108: *> be searched for eigenvalues. VL < VU.
109: *> Not referenced if RANGE = 'A' or 'I'.
110: *> \endverbatim
111: *>
112: *> \param[in] IL
113: *> \verbatim
114: *> IL is INTEGER
115: *> If RANGE='I', the index of the
116: *> smallest eigenvalue to be returned.
117: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
118: *> Not referenced if RANGE = 'A' or 'V'.
119: *> \endverbatim
120: *>
121: *> \param[in] IU
122: *> \verbatim
123: *> IU is INTEGER
124: *> If RANGE='I', the index of the
125: *> largest eigenvalue to be returned.
126: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
127: *> Not referenced if RANGE = 'A' or 'V'.
128: *> \endverbatim
129: *>
130: *> \param[in] ABSTOL
131: *> \verbatim
132: *> ABSTOL is DOUBLE PRECISION
133: *> The absolute error tolerance for the eigenvalues.
134: *> An approximate eigenvalue is accepted as converged
135: *> when it is determined to lie in an interval [a,b]
136: *> of width less than or equal to
137: *>
138: *> ABSTOL + EPS * max( |a|,|b| ) ,
139: *>
140: *> where EPS is the machine precision. If ABSTOL is less than
141: *> or equal to zero, then EPS*|T| will be used in its place,
142: *> where |T| is the 1-norm of the tridiagonal matrix obtained
143: *> by reducing AP to tridiagonal form.
144: *>
145: *> Eigenvalues will be computed most accurately when ABSTOL is
146: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
147: *> If this routine returns with INFO>0, indicating that some
148: *> eigenvectors did not converge, try setting ABSTOL to
149: *> 2*DLAMCH('S').
150: *>
151: *> See "Computing Small Singular Values of Bidiagonal Matrices
152: *> with Guaranteed High Relative Accuracy," by Demmel and
153: *> Kahan, LAPACK Working Note #3.
154: *> \endverbatim
155: *>
156: *> \param[out] M
157: *> \verbatim
158: *> M is INTEGER
159: *> The total number of eigenvalues found. 0 <= M <= N.
160: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
161: *> \endverbatim
162: *>
163: *> \param[out] W
164: *> \verbatim
165: *> W is DOUBLE PRECISION array, dimension (N)
166: *> If INFO = 0, the selected eigenvalues in ascending order.
167: *> \endverbatim
168: *>
169: *> \param[out] Z
170: *> \verbatim
171: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
172: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
173: *> contain the orthonormal eigenvectors of the matrix A
174: *> corresponding to the selected eigenvalues, with the i-th
175: *> column of Z holding the eigenvector associated with W(i).
176: *> If an eigenvector fails to converge, then that column of Z
177: *> contains the latest approximation to the eigenvector, and the
178: *> index of the eigenvector is returned in IFAIL.
179: *> If JOBZ = 'N', then Z is not referenced.
180: *> Note: the user must ensure that at least max(1,M) columns are
181: *> supplied in the array Z; if RANGE = 'V', the exact value of M
182: *> is not known in advance and an upper bound must be used.
183: *> \endverbatim
184: *>
185: *> \param[in] LDZ
186: *> \verbatim
187: *> LDZ is INTEGER
188: *> The leading dimension of the array Z. LDZ >= 1, and if
189: *> JOBZ = 'V', LDZ >= max(1,N).
190: *> \endverbatim
191: *>
192: *> \param[out] WORK
193: *> \verbatim
194: *> WORK is DOUBLE PRECISION array, dimension (8*N)
195: *> \endverbatim
196: *>
197: *> \param[out] IWORK
198: *> \verbatim
199: *> IWORK is INTEGER array, dimension (5*N)
200: *> \endverbatim
201: *>
202: *> \param[out] IFAIL
203: *> \verbatim
204: *> IFAIL is INTEGER array, dimension (N)
205: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
206: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
207: *> indices of the eigenvectors that failed to converge.
208: *> If JOBZ = 'N', then IFAIL is not referenced.
209: *> \endverbatim
210: *>
211: *> \param[out] INFO
212: *> \verbatim
213: *> INFO is INTEGER
214: *> = 0: successful exit
215: *> < 0: if INFO = -i, the i-th argument had an illegal value
216: *> > 0: if INFO = i, then i eigenvectors failed to converge.
217: *> Their indices are stored in array IFAIL.
218: *> \endverbatim
219: *
220: * Authors:
221: * ========
222: *
223: *> \author Univ. of Tennessee
224: *> \author Univ. of California Berkeley
225: *> \author Univ. of Colorado Denver
226: *> \author NAG Ltd.
227: *
228: *> \ingroup doubleOTHEReigen
229: *
230: * =====================================================================
231: SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
232: $ ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
233: $ INFO )
234: *
235: * -- LAPACK driver routine --
236: * -- LAPACK is a software package provided by Univ. of Tennessee, --
237: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
238: *
239: * .. Scalar Arguments ..
240: CHARACTER JOBZ, RANGE, UPLO
241: INTEGER IL, INFO, IU, LDZ, M, N
242: DOUBLE PRECISION ABSTOL, VL, VU
243: * ..
244: * .. Array Arguments ..
245: INTEGER IFAIL( * ), IWORK( * )
246: DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
247: * ..
248: *
249: * =====================================================================
250: *
251: * .. Parameters ..
252: DOUBLE PRECISION ZERO, ONE
253: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
254: * ..
255: * .. Local Scalars ..
256: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
257: CHARACTER ORDER
258: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
259: $ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
260: $ J, JJ, NSPLIT
261: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
262: $ SIGMA, SMLNUM, TMP1, VLL, VUU
263: * ..
264: * .. External Functions ..
265: LOGICAL LSAME
266: DOUBLE PRECISION DLAMCH, DLANSP
267: EXTERNAL LSAME, DLAMCH, DLANSP
268: * ..
269: * .. External Subroutines ..
270: EXTERNAL DCOPY, DOPGTR, DOPMTR, DSCAL, DSPTRD, DSTEBZ,
271: $ DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
272: * ..
273: * .. Intrinsic Functions ..
274: INTRINSIC MAX, MIN, SQRT
275: * ..
276: * .. Executable Statements ..
277: *
278: * Test the input parameters.
279: *
280: WANTZ = LSAME( JOBZ, 'V' )
281: ALLEIG = LSAME( RANGE, 'A' )
282: VALEIG = LSAME( RANGE, 'V' )
283: INDEIG = LSAME( RANGE, 'I' )
284: *
285: INFO = 0
286: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
287: INFO = -1
288: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
289: INFO = -2
290: ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
291: $ THEN
292: INFO = -3
293: ELSE IF( N.LT.0 ) THEN
294: INFO = -4
295: ELSE
296: IF( VALEIG ) THEN
297: IF( N.GT.0 .AND. VU.LE.VL )
298: $ INFO = -7
299: ELSE IF( INDEIG ) THEN
300: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
301: INFO = -8
302: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
303: INFO = -9
304: END IF
305: END IF
306: END IF
307: IF( INFO.EQ.0 ) THEN
308: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
309: $ INFO = -14
310: END IF
311: *
312: IF( INFO.NE.0 ) THEN
313: CALL XERBLA( 'DSPEVX', -INFO )
314: RETURN
315: END IF
316: *
317: * Quick return if possible
318: *
319: M = 0
320: IF( N.EQ.0 )
321: $ RETURN
322: *
323: IF( N.EQ.1 ) THEN
324: IF( ALLEIG .OR. INDEIG ) THEN
325: M = 1
326: W( 1 ) = AP( 1 )
327: ELSE
328: IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
329: M = 1
330: W( 1 ) = AP( 1 )
331: END IF
332: END IF
333: IF( WANTZ )
334: $ Z( 1, 1 ) = ONE
335: RETURN
336: END IF
337: *
338: * Get machine constants.
339: *
340: SAFMIN = DLAMCH( 'Safe minimum' )
341: EPS = DLAMCH( 'Precision' )
342: SMLNUM = SAFMIN / EPS
343: BIGNUM = ONE / SMLNUM
344: RMIN = SQRT( SMLNUM )
345: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
346: *
347: * Scale matrix to allowable range, if necessary.
348: *
349: ISCALE = 0
350: ABSTLL = ABSTOL
351: IF( VALEIG ) THEN
352: VLL = VL
353: VUU = VU
354: ELSE
355: VLL = ZERO
356: VUU = ZERO
357: END IF
358: ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
359: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
360: ISCALE = 1
361: SIGMA = RMIN / ANRM
362: ELSE IF( ANRM.GT.RMAX ) THEN
363: ISCALE = 1
364: SIGMA = RMAX / ANRM
365: END IF
366: IF( ISCALE.EQ.1 ) THEN
367: CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
368: IF( ABSTOL.GT.0 )
369: $ ABSTLL = ABSTOL*SIGMA
370: IF( VALEIG ) THEN
371: VLL = VL*SIGMA
372: VUU = VU*SIGMA
373: END IF
374: END IF
375: *
376: * Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
377: *
378: INDTAU = 1
379: INDE = INDTAU + N
380: INDD = INDE + N
381: INDWRK = INDD + N
382: CALL DSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
383: $ WORK( INDTAU ), IINFO )
384: *
385: * If all eigenvalues are desired and ABSTOL is less than or equal
386: * to zero, then call DSTERF or DOPGTR and SSTEQR. If this fails
387: * for some eigenvalue, then try DSTEBZ.
388: *
389: TEST = .FALSE.
390: IF (INDEIG) THEN
391: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
392: TEST = .TRUE.
393: END IF
394: END IF
395: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
396: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
397: INDEE = INDWRK + 2*N
398: IF( .NOT.WANTZ ) THEN
399: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
400: CALL DSTERF( N, W, WORK( INDEE ), INFO )
401: ELSE
402: CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
403: $ WORK( INDWRK ), IINFO )
404: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
405: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
406: $ WORK( INDWRK ), INFO )
407: IF( INFO.EQ.0 ) THEN
408: DO 10 I = 1, N
409: IFAIL( I ) = 0
410: 10 CONTINUE
411: END IF
412: END IF
413: IF( INFO.EQ.0 ) THEN
414: M = N
415: GO TO 20
416: END IF
417: INFO = 0
418: END IF
419: *
420: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
421: *
422: IF( WANTZ ) THEN
423: ORDER = 'B'
424: ELSE
425: ORDER = 'E'
426: END IF
427: INDIBL = 1
428: INDISP = INDIBL + N
429: INDIWO = INDISP + N
430: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
431: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
432: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
433: $ IWORK( INDIWO ), INFO )
434: *
435: IF( WANTZ ) THEN
436: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
437: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
438: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
439: *
440: * Apply orthogonal matrix used in reduction to tridiagonal
441: * form to eigenvectors returned by DSTEIN.
442: *
443: CALL DOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
444: $ WORK( INDWRK ), IINFO )
445: END IF
446: *
447: * If matrix was scaled, then rescale eigenvalues appropriately.
448: *
449: 20 CONTINUE
450: IF( ISCALE.EQ.1 ) THEN
451: IF( INFO.EQ.0 ) THEN
452: IMAX = M
453: ELSE
454: IMAX = INFO - 1
455: END IF
456: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
457: END IF
458: *
459: * If eigenvalues are not in order, then sort them, along with
460: * eigenvectors.
461: *
462: IF( WANTZ ) THEN
463: DO 40 J = 1, M - 1
464: I = 0
465: TMP1 = W( J )
466: DO 30 JJ = J + 1, M
467: IF( W( JJ ).LT.TMP1 ) THEN
468: I = JJ
469: TMP1 = W( JJ )
470: END IF
471: 30 CONTINUE
472: *
473: IF( I.NE.0 ) THEN
474: ITMP1 = IWORK( INDIBL+I-1 )
475: W( I ) = W( J )
476: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
477: W( J ) = TMP1
478: IWORK( INDIBL+J-1 ) = ITMP1
479: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
480: IF( INFO.NE.0 ) THEN
481: ITMP1 = IFAIL( I )
482: IFAIL( I ) = IFAIL( J )
483: IFAIL( J ) = ITMP1
484: END IF
485: END IF
486: 40 CONTINUE
487: END IF
488: *
489: RETURN
490: *
491: * End of DSPEVX
492: *
493: END
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