Annotation of rpl/lapack/lapack/dspevx.f, revision 1.13
1.8 bertrand 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
1.13 ! bertrand 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'.
1.8 bertrand 102: *> \endverbatim
103: *>
104: *> \param[in] VU
105: *> \verbatim
106: *> VU is DOUBLE PRECISION
1.13 ! bertrand 107: *> If RANGE='V', the upper bound of the interval to
1.8 bertrand 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
1.13 ! bertrand 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'.
1.8 bertrand 119: *> \endverbatim
120: *>
121: *> \param[in] IU
122: *> \verbatim
123: *> IU is INTEGER
1.13 ! bertrand 124: *> If RANGE='I', the index of the
! 125: *> largest eigenvalue to be returned.
1.8 bertrand 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: *
1.13 ! bertrand 228: *> \date June 2016
1.8 bertrand 229: *
230: *> \ingroup doubleOTHEReigen
231: *
232: * =====================================================================
1.1 bertrand 233: SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
234: $ ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
235: $ INFO )
236: *
1.13 ! bertrand 237: * -- LAPACK driver routine (version 3.6.1) --
1.1 bertrand 238: * -- LAPACK is a software package provided by Univ. of Tennessee, --
239: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.13 ! bertrand 240: * June 2016
1.1 bertrand 241: *
242: * .. Scalar Arguments ..
243: CHARACTER JOBZ, RANGE, UPLO
244: INTEGER IL, INFO, IU, LDZ, M, N
245: DOUBLE PRECISION ABSTOL, VL, VU
246: * ..
247: * .. Array Arguments ..
248: INTEGER IFAIL( * ), IWORK( * )
249: DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
250: * ..
251: *
252: * =====================================================================
253: *
254: * .. Parameters ..
255: DOUBLE PRECISION ZERO, ONE
256: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
257: * ..
258: * .. Local Scalars ..
259: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
260: CHARACTER ORDER
261: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
262: $ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
263: $ J, JJ, NSPLIT
264: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
265: $ SIGMA, SMLNUM, TMP1, VLL, VUU
266: * ..
267: * .. External Functions ..
268: LOGICAL LSAME
269: DOUBLE PRECISION DLAMCH, DLANSP
270: EXTERNAL LSAME, DLAMCH, DLANSP
271: * ..
272: * .. External Subroutines ..
273: EXTERNAL DCOPY, DOPGTR, DOPMTR, DSCAL, DSPTRD, DSTEBZ,
274: $ DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
275: * ..
276: * .. Intrinsic Functions ..
277: INTRINSIC MAX, MIN, SQRT
278: * ..
279: * .. Executable Statements ..
280: *
281: * Test the input parameters.
282: *
283: WANTZ = LSAME( JOBZ, 'V' )
284: ALLEIG = LSAME( RANGE, 'A' )
285: VALEIG = LSAME( RANGE, 'V' )
286: INDEIG = LSAME( RANGE, 'I' )
287: *
288: INFO = 0
289: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
290: INFO = -1
291: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
292: INFO = -2
293: ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
294: $ THEN
295: INFO = -3
296: ELSE IF( N.LT.0 ) THEN
297: INFO = -4
298: ELSE
299: IF( VALEIG ) THEN
300: IF( N.GT.0 .AND. VU.LE.VL )
301: $ INFO = -7
302: ELSE IF( INDEIG ) THEN
303: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
304: INFO = -8
305: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
306: INFO = -9
307: END IF
308: END IF
309: END IF
310: IF( INFO.EQ.0 ) THEN
311: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
312: $ INFO = -14
313: END IF
314: *
315: IF( INFO.NE.0 ) THEN
316: CALL XERBLA( 'DSPEVX', -INFO )
317: RETURN
318: END IF
319: *
320: * Quick return if possible
321: *
322: M = 0
323: IF( N.EQ.0 )
324: $ RETURN
325: *
326: IF( N.EQ.1 ) THEN
327: IF( ALLEIG .OR. INDEIG ) THEN
328: M = 1
329: W( 1 ) = AP( 1 )
330: ELSE
331: IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
332: M = 1
333: W( 1 ) = AP( 1 )
334: END IF
335: END IF
336: IF( WANTZ )
337: $ Z( 1, 1 ) = ONE
338: RETURN
339: END IF
340: *
341: * Get machine constants.
342: *
343: SAFMIN = DLAMCH( 'Safe minimum' )
344: EPS = DLAMCH( 'Precision' )
345: SMLNUM = SAFMIN / EPS
346: BIGNUM = ONE / SMLNUM
347: RMIN = SQRT( SMLNUM )
348: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
349: *
350: * Scale matrix to allowable range, if necessary.
351: *
352: ISCALE = 0
353: ABSTLL = ABSTOL
354: IF( VALEIG ) THEN
355: VLL = VL
356: VUU = VU
357: ELSE
358: VLL = ZERO
359: VUU = ZERO
360: END IF
361: ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
362: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
363: ISCALE = 1
364: SIGMA = RMIN / ANRM
365: ELSE IF( ANRM.GT.RMAX ) THEN
366: ISCALE = 1
367: SIGMA = RMAX / ANRM
368: END IF
369: IF( ISCALE.EQ.1 ) THEN
370: CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
371: IF( ABSTOL.GT.0 )
372: $ ABSTLL = ABSTOL*SIGMA
373: IF( VALEIG ) THEN
374: VLL = VL*SIGMA
375: VUU = VU*SIGMA
376: END IF
377: END IF
378: *
379: * Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
380: *
381: INDTAU = 1
382: INDE = INDTAU + N
383: INDD = INDE + N
384: INDWRK = INDD + N
385: CALL DSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
386: $ WORK( INDTAU ), IINFO )
387: *
388: * If all eigenvalues are desired and ABSTOL is less than or equal
389: * to zero, then call DSTERF or DOPGTR and SSTEQR. If this fails
390: * for some eigenvalue, then try DSTEBZ.
391: *
392: TEST = .FALSE.
393: IF (INDEIG) THEN
394: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
395: TEST = .TRUE.
396: END IF
397: END IF
398: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
399: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
400: INDEE = INDWRK + 2*N
401: IF( .NOT.WANTZ ) THEN
402: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
403: CALL DSTERF( N, W, WORK( INDEE ), INFO )
404: ELSE
405: CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
406: $ WORK( INDWRK ), IINFO )
407: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
408: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
409: $ WORK( INDWRK ), INFO )
410: IF( INFO.EQ.0 ) THEN
411: DO 10 I = 1, N
412: IFAIL( I ) = 0
413: 10 CONTINUE
414: END IF
415: END IF
416: IF( INFO.EQ.0 ) THEN
417: M = N
418: GO TO 20
419: END IF
420: INFO = 0
421: END IF
422: *
423: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
424: *
425: IF( WANTZ ) THEN
426: ORDER = 'B'
427: ELSE
428: ORDER = 'E'
429: END IF
430: INDIBL = 1
431: INDISP = INDIBL + N
432: INDIWO = INDISP + N
433: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
434: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
435: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
436: $ IWORK( INDIWO ), INFO )
437: *
438: IF( WANTZ ) THEN
439: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
440: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
441: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
442: *
443: * Apply orthogonal matrix used in reduction to tridiagonal
444: * form to eigenvectors returned by DSTEIN.
445: *
446: CALL DOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
447: $ WORK( INDWRK ), IINFO )
448: END IF
449: *
450: * If matrix was scaled, then rescale eigenvalues appropriately.
451: *
452: 20 CONTINUE
453: IF( ISCALE.EQ.1 ) THEN
454: IF( INFO.EQ.0 ) THEN
455: IMAX = M
456: ELSE
457: IMAX = INFO - 1
458: END IF
459: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
460: END IF
461: *
462: * If eigenvalues are not in order, then sort them, along with
463: * eigenvectors.
464: *
465: IF( WANTZ ) THEN
466: DO 40 J = 1, M - 1
467: I = 0
468: TMP1 = W( J )
469: DO 30 JJ = J + 1, M
470: IF( W( JJ ).LT.TMP1 ) THEN
471: I = JJ
472: TMP1 = W( JJ )
473: END IF
474: 30 CONTINUE
475: *
476: IF( I.NE.0 ) THEN
477: ITMP1 = IWORK( INDIBL+I-1 )
478: W( I ) = W( J )
479: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
480: W( J ) = TMP1
481: IWORK( INDIBL+J-1 ) = ITMP1
482: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
483: IF( INFO.NE.0 ) THEN
484: ITMP1 = IFAIL( I )
485: IFAIL( I ) = IFAIL( J )
486: IFAIL( J ) = ITMP1
487: END IF
488: END IF
489: 40 CONTINUE
490: END IF
491: *
492: RETURN
493: *
494: * End of DSPEVX
495: *
496: END
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