1: SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
2: $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
3: $ IFAIL, INFO )
4: *
5: * -- LAPACK driver routine (version 3.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * November 2006
9: *
10: * .. Scalar Arguments ..
11: CHARACTER JOBZ, RANGE, UPLO
12: INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
13: DOUBLE PRECISION ABSTOL, VL, VU
14: * ..
15: * .. Array Arguments ..
16: INTEGER IFAIL( * ), IWORK( * )
17: DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
18: * ..
19: *
20: * Purpose
21: * =======
22: *
23: * DSYEVX computes selected eigenvalues and, optionally, eigenvectors
24: * of a real symmetric matrix A. Eigenvalues and eigenvectors can be
25: * selected by specifying either a range of values or a range of indices
26: * for the desired eigenvalues.
27: *
28: * Arguments
29: * =========
30: *
31: * JOBZ (input) CHARACTER*1
32: * = 'N': Compute eigenvalues only;
33: * = 'V': Compute eigenvalues and eigenvectors.
34: *
35: * RANGE (input) CHARACTER*1
36: * = 'A': all eigenvalues will be found.
37: * = 'V': all eigenvalues in the half-open interval (VL,VU]
38: * will be found.
39: * = 'I': the IL-th through IU-th eigenvalues will be found.
40: *
41: * UPLO (input) CHARACTER*1
42: * = 'U': Upper triangle of A is stored;
43: * = 'L': Lower triangle of A is stored.
44: *
45: * N (input) INTEGER
46: * The order of the matrix A. N >= 0.
47: *
48: * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
49: * On entry, the symmetric matrix A. If UPLO = 'U', the
50: * leading N-by-N upper triangular part of A contains the
51: * upper triangular part of the matrix A. If UPLO = 'L',
52: * the leading N-by-N lower triangular part of A contains
53: * the lower triangular part of the matrix A.
54: * On exit, the lower triangle (if UPLO='L') or the upper
55: * triangle (if UPLO='U') of A, including the diagonal, is
56: * destroyed.
57: *
58: * LDA (input) INTEGER
59: * The leading dimension of the array A. LDA >= max(1,N).
60: *
61: * VL (input) DOUBLE PRECISION
62: * VU (input) DOUBLE PRECISION
63: * If RANGE='V', the lower and upper bounds of the interval to
64: * be searched for eigenvalues. VL < VU.
65: * Not referenced if RANGE = 'A' or 'I'.
66: *
67: * IL (input) INTEGER
68: * IU (input) INTEGER
69: * If RANGE='I', the indices (in ascending order) of the
70: * smallest and largest eigenvalues to be returned.
71: * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
72: * Not referenced if RANGE = 'A' or 'V'.
73: *
74: * ABSTOL (input) DOUBLE PRECISION
75: * The absolute error tolerance for the eigenvalues.
76: * An approximate eigenvalue is accepted as converged
77: * when it is determined to lie in an interval [a,b]
78: * of width less than or equal to
79: *
80: * ABSTOL + EPS * max( |a|,|b| ) ,
81: *
82: * where EPS is the machine precision. If ABSTOL is less than
83: * or equal to zero, then EPS*|T| will be used in its place,
84: * where |T| is the 1-norm of the tridiagonal matrix obtained
85: * by reducing A to tridiagonal form.
86: *
87: * Eigenvalues will be computed most accurately when ABSTOL is
88: * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
89: * If this routine returns with INFO>0, indicating that some
90: * eigenvectors did not converge, try setting ABSTOL to
91: * 2*DLAMCH('S').
92: *
93: * See "Computing Small Singular Values of Bidiagonal Matrices
94: * with Guaranteed High Relative Accuracy," by Demmel and
95: * Kahan, LAPACK Working Note #3.
96: *
97: * M (output) INTEGER
98: * The total number of eigenvalues found. 0 <= M <= N.
99: * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
100: *
101: * W (output) DOUBLE PRECISION array, dimension (N)
102: * On normal exit, the first M elements contain the selected
103: * eigenvalues in ascending order.
104: *
105: * Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
106: * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
107: * contain the orthonormal eigenvectors of the matrix A
108: * corresponding to the selected eigenvalues, with the i-th
109: * column of Z holding the eigenvector associated with W(i).
110: * If an eigenvector fails to converge, then that column of Z
111: * contains the latest approximation to the eigenvector, and the
112: * index of the eigenvector is returned in IFAIL.
113: * If JOBZ = 'N', then Z is not referenced.
114: * Note: the user must ensure that at least max(1,M) columns are
115: * supplied in the array Z; if RANGE = 'V', the exact value of M
116: * is not known in advance and an upper bound must be used.
117: *
118: * LDZ (input) INTEGER
119: * The leading dimension of the array Z. LDZ >= 1, and if
120: * JOBZ = 'V', LDZ >= max(1,N).
121: *
122: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
123: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
124: *
125: * LWORK (input) INTEGER
126: * The length of the array WORK. LWORK >= 1, when N <= 1;
127: * otherwise 8*N.
128: * For optimal efficiency, LWORK >= (NB+3)*N,
129: * where NB is the max of the blocksize for DSYTRD and DORMTR
130: * returned by ILAENV.
131: *
132: * If LWORK = -1, then a workspace query is assumed; the routine
133: * only calculates the optimal size of the WORK array, returns
134: * this value as the first entry of the WORK array, and no error
135: * message related to LWORK is issued by XERBLA.
136: *
137: * IWORK (workspace) INTEGER array, dimension (5*N)
138: *
139: * IFAIL (output) INTEGER array, dimension (N)
140: * If JOBZ = 'V', then if INFO = 0, the first M elements of
141: * IFAIL are zero. If INFO > 0, then IFAIL contains the
142: * indices of the eigenvectors that failed to converge.
143: * If JOBZ = 'N', then IFAIL is not referenced.
144: *
145: * INFO (output) INTEGER
146: * = 0: successful exit
147: * < 0: if INFO = -i, the i-th argument had an illegal value
148: * > 0: if INFO = i, then i eigenvectors failed to converge.
149: * Their indices are stored in array IFAIL.
150: *
151: * =====================================================================
152: *
153: * .. Parameters ..
154: DOUBLE PRECISION ZERO, ONE
155: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
156: * ..
157: * .. Local Scalars ..
158: LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
159: $ WANTZ
160: CHARACTER ORDER
161: INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
162: $ INDISP, INDIWO, INDTAU, INDWKN, INDWRK, ISCALE,
163: $ ITMP1, J, JJ, LLWORK, LLWRKN, LWKMIN,
164: $ LWKOPT, NB, NSPLIT
165: DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
166: $ SIGMA, SMLNUM, TMP1, VLL, VUU
167: * ..
168: * .. External Functions ..
169: LOGICAL LSAME
170: INTEGER ILAENV
171: DOUBLE PRECISION DLAMCH, DLANSY
172: EXTERNAL LSAME, ILAENV, DLAMCH, DLANSY
173: * ..
174: * .. External Subroutines ..
175: EXTERNAL DCOPY, DLACPY, DORGTR, DORMTR, DSCAL, DSTEBZ,
176: $ DSTEIN, DSTEQR, DSTERF, DSWAP, DSYTRD, XERBLA
177: * ..
178: * .. Intrinsic Functions ..
179: INTRINSIC MAX, MIN, SQRT
180: * ..
181: * .. Executable Statements ..
182: *
183: * Test the input parameters.
184: *
185: LOWER = LSAME( UPLO, 'L' )
186: WANTZ = LSAME( JOBZ, 'V' )
187: ALLEIG = LSAME( RANGE, 'A' )
188: VALEIG = LSAME( RANGE, 'V' )
189: INDEIG = LSAME( RANGE, 'I' )
190: LQUERY = ( LWORK.EQ.-1 )
191: *
192: INFO = 0
193: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
194: INFO = -1
195: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
196: INFO = -2
197: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
198: INFO = -3
199: ELSE IF( N.LT.0 ) THEN
200: INFO = -4
201: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
202: INFO = -6
203: ELSE
204: IF( VALEIG ) THEN
205: IF( N.GT.0 .AND. VU.LE.VL )
206: $ INFO = -8
207: ELSE IF( INDEIG ) THEN
208: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
209: INFO = -9
210: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
211: INFO = -10
212: END IF
213: END IF
214: END IF
215: IF( INFO.EQ.0 ) THEN
216: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
217: INFO = -15
218: END IF
219: END IF
220: *
221: IF( INFO.EQ.0 ) THEN
222: IF( N.LE.1 ) THEN
223: LWKMIN = 1
224: WORK( 1 ) = LWKMIN
225: ELSE
226: LWKMIN = 8*N
227: NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 )
228: NB = MAX( NB, ILAENV( 1, 'DORMTR', UPLO, N, -1, -1, -1 ) )
229: LWKOPT = MAX( LWKMIN, ( NB + 3 )*N )
230: WORK( 1 ) = LWKOPT
231: END IF
232: *
233: IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
234: $ INFO = -17
235: END IF
236: *
237: IF( INFO.NE.0 ) THEN
238: CALL XERBLA( 'DSYEVX', -INFO )
239: RETURN
240: ELSE IF( LQUERY ) THEN
241: RETURN
242: END IF
243: *
244: * Quick return if possible
245: *
246: M = 0
247: IF( N.EQ.0 ) THEN
248: RETURN
249: END IF
250: *
251: IF( N.EQ.1 ) THEN
252: IF( ALLEIG .OR. INDEIG ) THEN
253: M = 1
254: W( 1 ) = A( 1, 1 )
255: ELSE
256: IF( VL.LT.A( 1, 1 ) .AND. VU.GE.A( 1, 1 ) ) THEN
257: M = 1
258: W( 1 ) = A( 1, 1 )
259: END IF
260: END IF
261: IF( WANTZ )
262: $ Z( 1, 1 ) = ONE
263: RETURN
264: END IF
265: *
266: * Get machine constants.
267: *
268: SAFMIN = DLAMCH( 'Safe minimum' )
269: EPS = DLAMCH( 'Precision' )
270: SMLNUM = SAFMIN / EPS
271: BIGNUM = ONE / SMLNUM
272: RMIN = SQRT( SMLNUM )
273: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
274: *
275: * Scale matrix to allowable range, if necessary.
276: *
277: ISCALE = 0
278: ABSTLL = ABSTOL
279: IF( VALEIG ) THEN
280: VLL = VL
281: VUU = VU
282: END IF
283: ANRM = DLANSY( 'M', UPLO, N, A, LDA, WORK )
284: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
285: ISCALE = 1
286: SIGMA = RMIN / ANRM
287: ELSE IF( ANRM.GT.RMAX ) THEN
288: ISCALE = 1
289: SIGMA = RMAX / ANRM
290: END IF
291: IF( ISCALE.EQ.1 ) THEN
292: IF( LOWER ) THEN
293: DO 10 J = 1, N
294: CALL DSCAL( N-J+1, SIGMA, A( J, J ), 1 )
295: 10 CONTINUE
296: ELSE
297: DO 20 J = 1, N
298: CALL DSCAL( J, SIGMA, A( 1, J ), 1 )
299: 20 CONTINUE
300: END IF
301: IF( ABSTOL.GT.0 )
302: $ ABSTLL = ABSTOL*SIGMA
303: IF( VALEIG ) THEN
304: VLL = VL*SIGMA
305: VUU = VU*SIGMA
306: END IF
307: END IF
308: *
309: * Call DSYTRD to reduce symmetric matrix to tridiagonal form.
310: *
311: INDTAU = 1
312: INDE = INDTAU + N
313: INDD = INDE + N
314: INDWRK = INDD + N
315: LLWORK = LWORK - INDWRK + 1
316: CALL DSYTRD( UPLO, N, A, LDA, WORK( INDD ), WORK( INDE ),
317: $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
318: *
319: * If all eigenvalues are desired and ABSTOL is less than or equal to
320: * zero, then call DSTERF or DORGTR and SSTEQR. If this fails for
321: * some eigenvalue, then try DSTEBZ.
322: *
323: TEST = .FALSE.
324: IF( INDEIG ) THEN
325: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
326: TEST = .TRUE.
327: END IF
328: END IF
329: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
330: CALL DCOPY( N, WORK( INDD ), 1, W, 1 )
331: INDEE = INDWRK + 2*N
332: IF( .NOT.WANTZ ) THEN
333: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
334: CALL DSTERF( N, W, WORK( INDEE ), INFO )
335: ELSE
336: CALL DLACPY( 'A', N, N, A, LDA, Z, LDZ )
337: CALL DORGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
338: $ WORK( INDWRK ), LLWORK, IINFO )
339: CALL DCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
340: CALL DSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
341: $ WORK( INDWRK ), INFO )
342: IF( INFO.EQ.0 ) THEN
343: DO 30 I = 1, N
344: IFAIL( I ) = 0
345: 30 CONTINUE
346: END IF
347: END IF
348: IF( INFO.EQ.0 ) THEN
349: M = N
350: GO TO 40
351: END IF
352: INFO = 0
353: END IF
354: *
355: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
356: *
357: IF( WANTZ ) THEN
358: ORDER = 'B'
359: ELSE
360: ORDER = 'E'
361: END IF
362: INDIBL = 1
363: INDISP = INDIBL + N
364: INDIWO = INDISP + N
365: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
366: $ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
367: $ IWORK( INDIBL ), IWORK( INDISP ), WORK( INDWRK ),
368: $ IWORK( INDIWO ), INFO )
369: *
370: IF( WANTZ ) THEN
371: CALL DSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
372: $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
373: $ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
374: *
375: * Apply orthogonal matrix used in reduction to tridiagonal
376: * form to eigenvectors returned by DSTEIN.
377: *
378: INDWKN = INDE
379: LLWRKN = LWORK - INDWKN + 1
380: CALL DORMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
381: $ LDZ, WORK( INDWKN ), LLWRKN, IINFO )
382: END IF
383: *
384: * If matrix was scaled, then rescale eigenvalues appropriately.
385: *
386: 40 CONTINUE
387: IF( ISCALE.EQ.1 ) THEN
388: IF( INFO.EQ.0 ) THEN
389: IMAX = M
390: ELSE
391: IMAX = INFO - 1
392: END IF
393: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
394: END IF
395: *
396: * If eigenvalues are not in order, then sort them, along with
397: * eigenvectors.
398: *
399: IF( WANTZ ) THEN
400: DO 60 J = 1, M - 1
401: I = 0
402: TMP1 = W( J )
403: DO 50 JJ = J + 1, M
404: IF( W( JJ ).LT.TMP1 ) THEN
405: I = JJ
406: TMP1 = W( JJ )
407: END IF
408: 50 CONTINUE
409: *
410: IF( I.NE.0 ) THEN
411: ITMP1 = IWORK( INDIBL+I-1 )
412: W( I ) = W( J )
413: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
414: W( J ) = TMP1
415: IWORK( INDIBL+J-1 ) = ITMP1
416: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
417: IF( INFO.NE.0 ) THEN
418: ITMP1 = IFAIL( I )
419: IFAIL( I ) = IFAIL( J )
420: IFAIL( J ) = ITMP1
421: END IF
422: END IF
423: 60 CONTINUE
424: END IF
425: *
426: * Set WORK(1) to optimal workspace size.
427: *
428: WORK( 1 ) = LWKOPT
429: *
430: RETURN
431: *
432: * End of DSYEVX
433: *
434: END
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