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
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