Annotation of rpl/lapack/lapack/dspevx.f, revision 1.12
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
99: *> \endverbatim
100: *>
101: *> \param[in] VU
102: *> \verbatim
103: *> VU is DOUBLE PRECISION
104: *> If RANGE='V', the lower and upper bounds of the interval to
105: *> be searched for eigenvalues. VL < VU.
106: *> Not referenced if RANGE = 'A' or 'I'.
107: *> \endverbatim
108: *>
109: *> \param[in] IL
110: *> \verbatim
111: *> IL is INTEGER
112: *> \endverbatim
113: *>
114: *> \param[in] IU
115: *> \verbatim
116: *> IU is INTEGER
117: *> If RANGE='I', the indices (in ascending order) of the
118: *> smallest and largest eigenvalues to be returned.
119: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
120: *> Not referenced if RANGE = 'A' or 'V'.
121: *> \endverbatim
122: *>
123: *> \param[in] ABSTOL
124: *> \verbatim
125: *> ABSTOL is DOUBLE PRECISION
126: *> The absolute error tolerance for the eigenvalues.
127: *> An approximate eigenvalue is accepted as converged
128: *> when it is determined to lie in an interval [a,b]
129: *> of width less than or equal to
130: *>
131: *> ABSTOL + EPS * max( |a|,|b| ) ,
132: *>
133: *> where EPS is the machine precision. If ABSTOL is less than
134: *> or equal to zero, then EPS*|T| will be used in its place,
135: *> where |T| is the 1-norm of the tridiagonal matrix obtained
136: *> by reducing AP to tridiagonal form.
137: *>
138: *> Eigenvalues will be computed most accurately when ABSTOL is
139: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
140: *> If this routine returns with INFO>0, indicating that some
141: *> eigenvectors did not converge, try setting ABSTOL to
142: *> 2*DLAMCH('S').
143: *>
144: *> See "Computing Small Singular Values of Bidiagonal Matrices
145: *> with Guaranteed High Relative Accuracy," by Demmel and
146: *> Kahan, LAPACK Working Note #3.
147: *> \endverbatim
148: *>
149: *> \param[out] M
150: *> \verbatim
151: *> M is INTEGER
152: *> The total number of eigenvalues found. 0 <= M <= N.
153: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
154: *> \endverbatim
155: *>
156: *> \param[out] W
157: *> \verbatim
158: *> W is DOUBLE PRECISION array, dimension (N)
159: *> If INFO = 0, the selected eigenvalues in ascending order.
160: *> \endverbatim
161: *>
162: *> \param[out] Z
163: *> \verbatim
164: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M))
165: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
166: *> contain the orthonormal eigenvectors of the matrix A
167: *> corresponding to the selected eigenvalues, with the i-th
168: *> column of Z holding the eigenvector associated with W(i).
169: *> If an eigenvector fails to converge, then that column of Z
170: *> contains the latest approximation to the eigenvector, and the
171: *> index of the eigenvector is returned in IFAIL.
172: *> If JOBZ = 'N', then Z is not referenced.
173: *> Note: the user must ensure that at least max(1,M) columns are
174: *> supplied in the array Z; if RANGE = 'V', the exact value of M
175: *> is not known in advance and an upper bound must be used.
176: *> \endverbatim
177: *>
178: *> \param[in] LDZ
179: *> \verbatim
180: *> LDZ is INTEGER
181: *> The leading dimension of the array Z. LDZ >= 1, and if
182: *> JOBZ = 'V', LDZ >= max(1,N).
183: *> \endverbatim
184: *>
185: *> \param[out] WORK
186: *> \verbatim
187: *> WORK is DOUBLE PRECISION array, dimension (8*N)
188: *> \endverbatim
189: *>
190: *> \param[out] IWORK
191: *> \verbatim
192: *> IWORK is INTEGER array, dimension (5*N)
193: *> \endverbatim
194: *>
195: *> \param[out] IFAIL
196: *> \verbatim
197: *> IFAIL is INTEGER array, dimension (N)
198: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
199: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
200: *> indices of the eigenvectors that failed to converge.
201: *> If JOBZ = 'N', then IFAIL is not referenced.
202: *> \endverbatim
203: *>
204: *> \param[out] INFO
205: *> \verbatim
206: *> INFO is INTEGER
207: *> = 0: successful exit
208: *> < 0: if INFO = -i, the i-th argument had an illegal value
209: *> > 0: if INFO = i, then i eigenvectors failed to converge.
210: *> Their indices are stored in array IFAIL.
211: *> \endverbatim
212: *
213: * Authors:
214: * ========
215: *
216: *> \author Univ. of Tennessee
217: *> \author Univ. of California Berkeley
218: *> \author Univ. of Colorado Denver
219: *> \author NAG Ltd.
220: *
221: *> \date November 2011
222: *
223: *> \ingroup doubleOTHEReigen
224: *
225: * =====================================================================
1.1 bertrand 226: SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
227: $ ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
228: $ INFO )
229: *
1.8 bertrand 230: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 231: * -- LAPACK is a software package provided by Univ. of Tennessee, --
232: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 bertrand 233: * November 2011
1.1 bertrand 234: *
235: * .. Scalar Arguments ..
236: CHARACTER JOBZ, RANGE, UPLO
237: INTEGER IL, INFO, IU, LDZ, M, N
238: DOUBLE PRECISION ABSTOL, VL, VU
239: * ..
240: * .. Array Arguments ..
241: INTEGER IFAIL( * ), IWORK( * )
242: DOUBLE PRECISION AP( * ), W( * ), WORK( * ), Z( LDZ, * )
243: * ..
244: *
245: * =====================================================================
246: *
247: * .. Parameters ..
248: DOUBLE PRECISION ZERO, ONE
249: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
250: * ..
251: * .. Local Scalars ..
252: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
253: CHARACTER ORDER
254: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
255: $ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
256: $ J, JJ, NSPLIT
257: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
258: $ SIGMA, SMLNUM, TMP1, VLL, VUU
259: * ..
260: * .. External Functions ..
261: LOGICAL LSAME
262: DOUBLE PRECISION DLAMCH, DLANSP
263: EXTERNAL LSAME, DLAMCH, DLANSP
264: * ..
265: * .. External Subroutines ..
266: EXTERNAL DCOPY, DOPGTR, DOPMTR, DSCAL, DSPTRD, DSTEBZ,
267: $ DSTEIN, DSTEQR, DSTERF, DSWAP, XERBLA
268: * ..
269: * .. Intrinsic Functions ..
270: INTRINSIC MAX, MIN, SQRT
271: * ..
272: * .. Executable Statements ..
273: *
274: * Test the input parameters.
275: *
276: WANTZ = LSAME( JOBZ, 'V' )
277: ALLEIG = LSAME( RANGE, 'A' )
278: VALEIG = LSAME( RANGE, 'V' )
279: INDEIG = LSAME( RANGE, 'I' )
280: *
281: INFO = 0
282: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
283: INFO = -1
284: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
285: INFO = -2
286: ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
287: $ THEN
288: INFO = -3
289: ELSE IF( N.LT.0 ) THEN
290: INFO = -4
291: ELSE
292: IF( VALEIG ) THEN
293: IF( N.GT.0 .AND. VU.LE.VL )
294: $ INFO = -7
295: ELSE IF( INDEIG ) THEN
296: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
297: INFO = -8
298: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
299: INFO = -9
300: END IF
301: END IF
302: END IF
303: IF( INFO.EQ.0 ) THEN
304: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
305: $ INFO = -14
306: END IF
307: *
308: IF( INFO.NE.0 ) THEN
309: CALL XERBLA( 'DSPEVX', -INFO )
310: RETURN
311: END IF
312: *
313: * Quick return if possible
314: *
315: M = 0
316: IF( N.EQ.0 )
317: $ RETURN
318: *
319: IF( N.EQ.1 ) THEN
320: IF( ALLEIG .OR. INDEIG ) THEN
321: M = 1
322: W( 1 ) = AP( 1 )
323: ELSE
324: IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
325: M = 1
326: W( 1 ) = AP( 1 )
327: END IF
328: END IF
329: IF( WANTZ )
330: $ Z( 1, 1 ) = ONE
331: RETURN
332: END IF
333: *
334: * Get machine constants.
335: *
336: SAFMIN = DLAMCH( 'Safe minimum' )
337: EPS = DLAMCH( 'Precision' )
338: SMLNUM = SAFMIN / EPS
339: BIGNUM = ONE / SMLNUM
340: RMIN = SQRT( SMLNUM )
341: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
342: *
343: * Scale matrix to allowable range, if necessary.
344: *
345: ISCALE = 0
346: ABSTLL = ABSTOL
347: IF( VALEIG ) THEN
348: VLL = VL
349: VUU = VU
350: ELSE
351: VLL = ZERO
352: VUU = ZERO
353: END IF
354: ANRM = DLANSP( 'M', UPLO, N, AP, WORK )
355: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
356: ISCALE = 1
357: SIGMA = RMIN / ANRM
358: ELSE IF( ANRM.GT.RMAX ) THEN
359: ISCALE = 1
360: SIGMA = RMAX / ANRM
361: END IF
362: IF( ISCALE.EQ.1 ) THEN
363: CALL DSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
364: IF( ABSTOL.GT.0 )
365: $ ABSTLL = ABSTOL*SIGMA
366: IF( VALEIG ) THEN
367: VLL = VL*SIGMA
368: VUU = VU*SIGMA
369: END IF
370: END IF
371: *
372: * Call DSPTRD to reduce symmetric packed matrix to tridiagonal form.
373: *
374: INDTAU = 1
375: INDE = INDTAU + N
376: INDD = INDE + N
377: INDWRK = INDD + N
378: CALL DSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
379: $ WORK( INDTAU ), IINFO )
380: *
381: * If all eigenvalues are desired and ABSTOL is less than or equal
382: * to zero, then call DSTERF or DOPGTR and SSTEQR. If this fails
383: * for some eigenvalue, then try DSTEBZ.
384: *
385: TEST = .FALSE.
386: IF (INDEIG) THEN
387: IF (IL.EQ.1 .AND. IU.EQ.N) THEN
388: TEST = .TRUE.
389: END IF
390: END IF
391: IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
392: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
393: INDEE = INDWRK + 2*N
394: IF( .NOT.WANTZ ) THEN
395: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
396: CALL DSTERF( N, W, WORK( INDEE ), INFO )
397: ELSE
398: CALL DOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
399: $ WORK( INDWRK ), IINFO )
400: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
401: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
402: $ WORK( INDWRK ), INFO )
403: IF( INFO.EQ.0 ) THEN
404: DO 10 I = 1, N
405: IFAIL( I ) = 0
406: 10 CONTINUE
407: END IF
408: END IF
409: IF( INFO.EQ.0 ) THEN
410: M = N
411: GO TO 20
412: END IF
413: INFO = 0
414: END IF
415: *
416: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
417: *
418: IF( WANTZ ) THEN
419: ORDER = 'B'
420: ELSE
421: ORDER = 'E'
422: END IF
423: INDIBL = 1
424: INDISP = INDIBL + N
425: INDIWO = INDISP + N
426: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
427: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
428: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
429: $ IWORK( INDIWO ), INFO )
430: *
431: IF( WANTZ ) THEN
432: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
433: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
434: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
435: *
436: * Apply orthogonal matrix used in reduction to tridiagonal
437: * form to eigenvectors returned by DSTEIN.
438: *
439: CALL DOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
440: $ WORK( INDWRK ), IINFO )
441: END IF
442: *
443: * If matrix was scaled, then rescale eigenvalues appropriately.
444: *
445: 20 CONTINUE
446: IF( ISCALE.EQ.1 ) THEN
447: IF( INFO.EQ.0 ) THEN
448: IMAX = M
449: ELSE
450: IMAX = INFO - 1
451: END IF
452: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
453: END IF
454: *
455: * If eigenvalues are not in order, then sort them, along with
456: * eigenvectors.
457: *
458: IF( WANTZ ) THEN
459: DO 40 J = 1, M - 1
460: I = 0
461: TMP1 = W( J )
462: DO 30 JJ = J + 1, M
463: IF( W( JJ ).LT.TMP1 ) THEN
464: I = JJ
465: TMP1 = W( JJ )
466: END IF
467: 30 CONTINUE
468: *
469: IF( I.NE.0 ) THEN
470: ITMP1 = IWORK( INDIBL+I-1 )
471: W( I ) = W( J )
472: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
473: W( J ) = TMP1
474: IWORK( INDIBL+J-1 ) = ITMP1
475: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
476: IF( INFO.NE.0 ) THEN
477: ITMP1 = IFAIL( I )
478: IFAIL( I ) = IFAIL( J )
479: IFAIL( J ) = ITMP1
480: END IF
481: END IF
482: 40 CONTINUE
483: END IF
484: *
485: RETURN
486: *
487: * End of DSPEVX
488: *
489: END
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