1 SUBROUTINE ZHEEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
2 $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
3 $ IWORK, IFAIL, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 * @precisions normal z -> c
10 *
11 * .. Scalar Arguments ..
12 CHARACTER JOBZ, RANGE, UPLO
13 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
14 DOUBLE PRECISION ABSTOL, VL, VU
15 * ..
16 * .. Array Arguments ..
17 INTEGER IFAIL( * ), IWORK( * )
18 DOUBLE PRECISION RWORK( * ), W( * )
19 COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHEEVX computes selected eigenvalues and, optionally, eigenvectors
26 * of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
27 * be selected by specifying either a range of values or a range of
28 * indices for the desired eigenvalues.
29 *
30 * Arguments
31 * =========
32 *
33 * JOBZ (input) CHARACTER*1
34 * = 'N': Compute eigenvalues only;
35 * = 'V': Compute eigenvalues and eigenvectors.
36 *
37 * RANGE (input) CHARACTER*1
38 * = 'A': all eigenvalues will be found.
39 * = 'V': all eigenvalues in the half-open interval (VL,VU]
40 * will be found.
41 * = 'I': the IL-th through IU-th eigenvalues will be found.
42 *
43 * UPLO (input) CHARACTER*1
44 * = 'U': Upper triangle of A is stored;
45 * = 'L': Lower triangle of A is stored.
46 *
47 * N (input) INTEGER
48 * The order of the matrix A. N >= 0.
49 *
50 * A (input/output) COMPLEX*16 array, dimension (LDA, N)
51 * On entry, the Hermitian matrix A. If UPLO = 'U', the
52 * leading N-by-N upper triangular part of A contains the
53 * upper triangular part of the matrix A. If UPLO = 'L',
54 * the leading N-by-N lower triangular part of A contains
55 * the lower triangular part of the matrix A.
56 * On exit, the lower triangle (if UPLO='L') or the upper
57 * triangle (if UPLO='U') of A, including the diagonal, is
58 * destroyed.
59 *
60 * LDA (input) INTEGER
61 * The leading dimension of the array A. LDA >= max(1,N).
62 *
63 * VL (input) DOUBLE PRECISION
64 * VU (input) DOUBLE PRECISION
65 * If RANGE='V', the lower and upper bounds of the interval to
66 * be searched for eigenvalues. VL < VU.
67 * Not referenced if RANGE = 'A' or 'I'.
68 *
69 * IL (input) INTEGER
70 * IU (input) INTEGER
71 * If RANGE='I', the indices (in ascending order) of the
72 * smallest and largest eigenvalues to be returned.
73 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
74 * Not referenced if RANGE = 'A' or 'V'.
75 *
76 * ABSTOL (input) DOUBLE PRECISION
77 * The absolute error tolerance for the eigenvalues.
78 * An approximate eigenvalue is accepted as converged
79 * when it is determined to lie in an interval [a,b]
80 * of width less than or equal to
81 *
82 * ABSTOL + EPS * max( |a|,|b| ) ,
83 *
84 * where EPS is the machine precision. If ABSTOL is less than
85 * or equal to zero, then EPS*|T| will be used in its place,
86 * where |T| is the 1-norm of the tridiagonal matrix obtained
87 * by reducing A to tridiagonal form.
88 *
89 * Eigenvalues will be computed most accurately when ABSTOL is
90 * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
91 * If this routine returns with INFO>0, indicating that some
92 * eigenvectors did not converge, try setting ABSTOL to
93 * 2*DLAMCH('S').
94 *
95 * See "Computing Small Singular Values of Bidiagonal Matrices
96 * with Guaranteed High Relative Accuracy," by Demmel and
97 * Kahan, LAPACK Working Note #3.
98 *
99 * M (output) INTEGER
100 * The total number of eigenvalues found. 0 <= M <= N.
101 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
102 *
103 * W (output) DOUBLE PRECISION array, dimension (N)
104 * On normal exit, the first M elements contain the selected
105 * eigenvalues in ascending order.
106 *
107 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
108 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
109 * contain the orthonormal eigenvectors of the matrix A
110 * corresponding to the selected eigenvalues, with the i-th
111 * column of Z holding the eigenvector associated with W(i).
112 * If an eigenvector fails to converge, then that column of Z
113 * contains the latest approximation to the eigenvector, and the
114 * index of the eigenvector is returned in IFAIL.
115 * If JOBZ = 'N', then Z is not referenced.
116 * Note: the user must ensure that at least max(1,M) columns are
117 * supplied in the array Z; if RANGE = 'V', the exact value of M
118 * is not known in advance and an upper bound must be used.
119 *
120 * LDZ (input) INTEGER
121 * The leading dimension of the array Z. LDZ >= 1, and if
122 * JOBZ = 'V', LDZ >= max(1,N).
123 *
124 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
125 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
126 *
127 * LWORK (input) INTEGER
128 * The length of the array WORK. LWORK >= 1, when N <= 1;
129 * otherwise 2*N.
130 * For optimal efficiency, LWORK >= (NB+1)*N,
131 * where NB is the max of the blocksize for ZHETRD and for
132 * ZUNMTR as returned by ILAENV.
133 *
134 * If LWORK = -1, then a workspace query is assumed; the routine
135 * only calculates the optimal size of the WORK array, returns
136 * this value as the first entry of the WORK array, and no error
137 * message related to LWORK is issued by XERBLA.
138 *
139 * RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
140 *
141 * IWORK (workspace) INTEGER array, dimension (5*N)
142 *
143 * IFAIL (output) INTEGER array, dimension (N)
144 * If JOBZ = 'V', then if INFO = 0, the first M elements of
145 * IFAIL are zero. If INFO > 0, then IFAIL contains the
146 * indices of the eigenvectors that failed to converge.
147 * If JOBZ = 'N', then IFAIL is not referenced.
148 *
149 * INFO (output) INTEGER
150 * = 0: successful exit
151 * < 0: if INFO = -i, the i-th argument had an illegal value
152 * > 0: if INFO = i, then i eigenvectors failed to converge.
153 * Their indices are stored in array IFAIL.
154 *
155 * =====================================================================
156 *
157 * .. Parameters ..
158 DOUBLE PRECISION ZERO, ONE
159 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
160 COMPLEX*16 CONE
161 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
162 * ..
163 * .. Local Scalars ..
164 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
165 $ WANTZ
166 CHARACTER ORDER
167 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
168 $ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
169 $ ITMP1, J, JJ, LLWORK, LWKMIN, LWKOPT, NB,
170 $ NSPLIT
171 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
172 $ SIGMA, SMLNUM, TMP1, VLL, VUU
173 * ..
174 * .. External Functions ..
175 LOGICAL LSAME
176 INTEGER ILAENV
177 DOUBLE PRECISION DLAMCH, ZLANHE
178 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE
179 * ..
180 * .. External Subroutines ..
181 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
182 $ ZHETRD, ZLACPY, ZSTEIN, ZSTEQR, ZSWAP, ZUNGTR,
183 $ ZUNMTR
184 * ..
185 * .. Intrinsic Functions ..
186 INTRINSIC DBLE, MAX, MIN, SQRT
187 * ..
188 * .. Executable Statements ..
189 *
190 * Test the input parameters.
191 *
192 LOWER = LSAME( UPLO, 'L' )
193 WANTZ = LSAME( JOBZ, 'V' )
194 ALLEIG = LSAME( RANGE, 'A' )
195 VALEIG = LSAME( RANGE, 'V' )
196 INDEIG = LSAME( RANGE, 'I' )
197 LQUERY = ( LWORK.EQ.-1 )
198 *
199 INFO = 0
200 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
201 INFO = -1
202 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
203 INFO = -2
204 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
205 INFO = -3
206 ELSE IF( N.LT.0 ) THEN
207 INFO = -4
208 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
209 INFO = -6
210 ELSE
211 IF( VALEIG ) THEN
212 IF( N.GT.0 .AND. VU.LE.VL )
213 $ INFO = -8
214 ELSE IF( INDEIG ) THEN
215 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
216 INFO = -9
217 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
218 INFO = -10
219 END IF
220 END IF
221 END IF
222 IF( INFO.EQ.0 ) THEN
223 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
224 INFO = -15
225 END IF
226 END IF
227 *
228 IF( INFO.EQ.0 ) THEN
229 IF( N.LE.1 ) THEN
230 LWKMIN = 1
231 WORK( 1 ) = LWKMIN
232 ELSE
233 LWKMIN = 2*N
234 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
235 NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
236 LWKOPT = MAX( 1, ( NB + 1 )*N )
237 WORK( 1 ) = LWKOPT
238 END IF
239 *
240 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
241 $ INFO = -17
242 END IF
243 *
244 IF( INFO.NE.0 ) THEN
245 CALL XERBLA( 'ZHEEVX', -INFO )
246 RETURN
247 ELSE IF( LQUERY ) THEN
248 RETURN
249 END IF
250 *
251 * Quick return if possible
252 *
253 M = 0
254 IF( N.EQ.0 ) THEN
255 RETURN
256 END IF
257 *
258 IF( N.EQ.1 ) THEN
259 IF( ALLEIG .OR. INDEIG ) THEN
260 M = 1
261 W( 1 ) = A( 1, 1 )
262 ELSE IF( VALEIG ) THEN
263 IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
264 $ THEN
265 M = 1
266 W( 1 ) = A( 1, 1 )
267 END IF
268 END IF
269 IF( WANTZ )
270 $ Z( 1, 1 ) = CONE
271 RETURN
272 END IF
273 *
274 * Get machine constants.
275 *
276 SAFMIN = DLAMCH( 'Safe minimum' )
277 EPS = DLAMCH( 'Precision' )
278 SMLNUM = SAFMIN / EPS
279 BIGNUM = ONE / SMLNUM
280 RMIN = SQRT( SMLNUM )
281 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
282 *
283 * Scale matrix to allowable range, if necessary.
284 *
285 ISCALE = 0
286 ABSTLL = ABSTOL
287 IF( VALEIG ) THEN
288 VLL = VL
289 VUU = VU
290 END IF
291 ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK )
292 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
293 ISCALE = 1
294 SIGMA = RMIN / ANRM
295 ELSE IF( ANRM.GT.RMAX ) THEN
296 ISCALE = 1
297 SIGMA = RMAX / ANRM
298 END IF
299 IF( ISCALE.EQ.1 ) THEN
300 IF( LOWER ) THEN
301 DO 10 J = 1, N
302 CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
303 10 CONTINUE
304 ELSE
305 DO 20 J = 1, N
306 CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
307 20 CONTINUE
308 END IF
309 IF( ABSTOL.GT.0 )
310 $ ABSTLL = ABSTOL*SIGMA
311 IF( VALEIG ) THEN
312 VLL = VL*SIGMA
313 VUU = VU*SIGMA
314 END IF
315 END IF
316 *
317 * Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
318 *
319 INDD = 1
320 INDE = INDD + N
321 INDRWK = INDE + N
322 INDTAU = 1
323 INDWRK = INDTAU + N
324 LLWORK = LWORK - INDWRK + 1
325 CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDD ), RWORK( INDE ),
326 $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
327 *
328 * If all eigenvalues are desired and ABSTOL is less than or equal to
329 * zero, then call DSTERF or ZUNGTR and ZSTEQR. If this fails for
330 * some eigenvalue, then try DSTEBZ.
331 *
332 TEST = .FALSE.
333 IF( INDEIG ) THEN
334 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
335 TEST = .TRUE.
336 END IF
337 END IF
338 IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
339 CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
340 INDEE = INDRWK + 2*N
341 IF( .NOT.WANTZ ) THEN
342 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
343 CALL DSTERF( N, W, RWORK( INDEE ), INFO )
344 ELSE
345 CALL ZLACPY( 'A', N, N, A, LDA, Z, LDZ )
346 CALL ZUNGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
347 $ WORK( INDWRK ), LLWORK, IINFO )
348 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
349 CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
350 $ RWORK( INDRWK ), INFO )
351 IF( INFO.EQ.0 ) THEN
352 DO 30 I = 1, N
353 IFAIL( I ) = 0
354 30 CONTINUE
355 END IF
356 END IF
357 IF( INFO.EQ.0 ) THEN
358 M = N
359 GO TO 40
360 END IF
361 INFO = 0
362 END IF
363 *
364 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
365 *
366 IF( WANTZ ) THEN
367 ORDER = 'B'
368 ELSE
369 ORDER = 'E'
370 END IF
371 INDIBL = 1
372 INDISP = INDIBL + N
373 INDIWK = INDISP + N
374 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
375 $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
376 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
377 $ IWORK( INDIWK ), INFO )
378 *
379 IF( WANTZ ) THEN
380 CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
381 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
382 $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
383 *
384 * Apply unitary matrix used in reduction to tridiagonal
385 * form to eigenvectors returned by ZSTEIN.
386 *
387 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
388 $ LDZ, WORK( INDWRK ), LLWORK, IINFO )
389 END IF
390 *
391 * If matrix was scaled, then rescale eigenvalues appropriately.
392 *
393 40 CONTINUE
394 IF( ISCALE.EQ.1 ) THEN
395 IF( INFO.EQ.0 ) THEN
396 IMAX = M
397 ELSE
398 IMAX = INFO - 1
399 END IF
400 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
401 END IF
402 *
403 * If eigenvalues are not in order, then sort them, along with
404 * eigenvectors.
405 *
406 IF( WANTZ ) THEN
407 DO 60 J = 1, M - 1
408 I = 0
409 TMP1 = W( J )
410 DO 50 JJ = J + 1, M
411 IF( W( JJ ).LT.TMP1 ) THEN
412 I = JJ
413 TMP1 = W( JJ )
414 END IF
415 50 CONTINUE
416 *
417 IF( I.NE.0 ) THEN
418 ITMP1 = IWORK( INDIBL+I-1 )
419 W( I ) = W( J )
420 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
421 W( J ) = TMP1
422 IWORK( INDIBL+J-1 ) = ITMP1
423 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
424 IF( INFO.NE.0 ) THEN
425 ITMP1 = IFAIL( I )
426 IFAIL( I ) = IFAIL( J )
427 IFAIL( J ) = ITMP1
428 END IF
429 END IF
430 60 CONTINUE
431 END IF
432 *
433 * Set WORK(1) to optimal complex workspace size.
434 *
435 WORK( 1 ) = LWKOPT
436 *
437 RETURN
438 *
439 * End of ZHEEVX
440 *
441 END
2 $ ABSTOL, M, W, Z, LDZ, WORK, LWORK, RWORK,
3 $ IWORK, IFAIL, INFO )
4 *
5 * -- LAPACK driver routine (version 3.3.1) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * -- April 2011 --
9 * @precisions normal z -> c
10 *
11 * .. Scalar Arguments ..
12 CHARACTER JOBZ, RANGE, UPLO
13 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
14 DOUBLE PRECISION ABSTOL, VL, VU
15 * ..
16 * .. Array Arguments ..
17 INTEGER IFAIL( * ), IWORK( * )
18 DOUBLE PRECISION RWORK( * ), W( * )
19 COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHEEVX computes selected eigenvalues and, optionally, eigenvectors
26 * of a complex Hermitian matrix A. Eigenvalues and eigenvectors can
27 * be selected by specifying either a range of values or a range of
28 * indices for the desired eigenvalues.
29 *
30 * Arguments
31 * =========
32 *
33 * JOBZ (input) CHARACTER*1
34 * = 'N': Compute eigenvalues only;
35 * = 'V': Compute eigenvalues and eigenvectors.
36 *
37 * RANGE (input) CHARACTER*1
38 * = 'A': all eigenvalues will be found.
39 * = 'V': all eigenvalues in the half-open interval (VL,VU]
40 * will be found.
41 * = 'I': the IL-th through IU-th eigenvalues will be found.
42 *
43 * UPLO (input) CHARACTER*1
44 * = 'U': Upper triangle of A is stored;
45 * = 'L': Lower triangle of A is stored.
46 *
47 * N (input) INTEGER
48 * The order of the matrix A. N >= 0.
49 *
50 * A (input/output) COMPLEX*16 array, dimension (LDA, N)
51 * On entry, the Hermitian matrix A. If UPLO = 'U', the
52 * leading N-by-N upper triangular part of A contains the
53 * upper triangular part of the matrix A. If UPLO = 'L',
54 * the leading N-by-N lower triangular part of A contains
55 * the lower triangular part of the matrix A.
56 * On exit, the lower triangle (if UPLO='L') or the upper
57 * triangle (if UPLO='U') of A, including the diagonal, is
58 * destroyed.
59 *
60 * LDA (input) INTEGER
61 * The leading dimension of the array A. LDA >= max(1,N).
62 *
63 * VL (input) DOUBLE PRECISION
64 * VU (input) DOUBLE PRECISION
65 * If RANGE='V', the lower and upper bounds of the interval to
66 * be searched for eigenvalues. VL < VU.
67 * Not referenced if RANGE = 'A' or 'I'.
68 *
69 * IL (input) INTEGER
70 * IU (input) INTEGER
71 * If RANGE='I', the indices (in ascending order) of the
72 * smallest and largest eigenvalues to be returned.
73 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
74 * Not referenced if RANGE = 'A' or 'V'.
75 *
76 * ABSTOL (input) DOUBLE PRECISION
77 * The absolute error tolerance for the eigenvalues.
78 * An approximate eigenvalue is accepted as converged
79 * when it is determined to lie in an interval [a,b]
80 * of width less than or equal to
81 *
82 * ABSTOL + EPS * max( |a|,|b| ) ,
83 *
84 * where EPS is the machine precision. If ABSTOL is less than
85 * or equal to zero, then EPS*|T| will be used in its place,
86 * where |T| is the 1-norm of the tridiagonal matrix obtained
87 * by reducing A to tridiagonal form.
88 *
89 * Eigenvalues will be computed most accurately when ABSTOL is
90 * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
91 * If this routine returns with INFO>0, indicating that some
92 * eigenvectors did not converge, try setting ABSTOL to
93 * 2*DLAMCH('S').
94 *
95 * See "Computing Small Singular Values of Bidiagonal Matrices
96 * with Guaranteed High Relative Accuracy," by Demmel and
97 * Kahan, LAPACK Working Note #3.
98 *
99 * M (output) INTEGER
100 * The total number of eigenvalues found. 0 <= M <= N.
101 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
102 *
103 * W (output) DOUBLE PRECISION array, dimension (N)
104 * On normal exit, the first M elements contain the selected
105 * eigenvalues in ascending order.
106 *
107 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
108 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
109 * contain the orthonormal eigenvectors of the matrix A
110 * corresponding to the selected eigenvalues, with the i-th
111 * column of Z holding the eigenvector associated with W(i).
112 * If an eigenvector fails to converge, then that column of Z
113 * contains the latest approximation to the eigenvector, and the
114 * index of the eigenvector is returned in IFAIL.
115 * If JOBZ = 'N', then Z is not referenced.
116 * Note: the user must ensure that at least max(1,M) columns are
117 * supplied in the array Z; if RANGE = 'V', the exact value of M
118 * is not known in advance and an upper bound must be used.
119 *
120 * LDZ (input) INTEGER
121 * The leading dimension of the array Z. LDZ >= 1, and if
122 * JOBZ = 'V', LDZ >= max(1,N).
123 *
124 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
125 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
126 *
127 * LWORK (input) INTEGER
128 * The length of the array WORK. LWORK >= 1, when N <= 1;
129 * otherwise 2*N.
130 * For optimal efficiency, LWORK >= (NB+1)*N,
131 * where NB is the max of the blocksize for ZHETRD and for
132 * ZUNMTR as returned by ILAENV.
133 *
134 * If LWORK = -1, then a workspace query is assumed; the routine
135 * only calculates the optimal size of the WORK array, returns
136 * this value as the first entry of the WORK array, and no error
137 * message related to LWORK is issued by XERBLA.
138 *
139 * RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
140 *
141 * IWORK (workspace) INTEGER array, dimension (5*N)
142 *
143 * IFAIL (output) INTEGER array, dimension (N)
144 * If JOBZ = 'V', then if INFO = 0, the first M elements of
145 * IFAIL are zero. If INFO > 0, then IFAIL contains the
146 * indices of the eigenvectors that failed to converge.
147 * If JOBZ = 'N', then IFAIL is not referenced.
148 *
149 * INFO (output) INTEGER
150 * = 0: successful exit
151 * < 0: if INFO = -i, the i-th argument had an illegal value
152 * > 0: if INFO = i, then i eigenvectors failed to converge.
153 * Their indices are stored in array IFAIL.
154 *
155 * =====================================================================
156 *
157 * .. Parameters ..
158 DOUBLE PRECISION ZERO, ONE
159 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
160 COMPLEX*16 CONE
161 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) )
162 * ..
163 * .. Local Scalars ..
164 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
165 $ WANTZ
166 CHARACTER ORDER
167 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
168 $ INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
169 $ ITMP1, J, JJ, LLWORK, LWKMIN, LWKOPT, NB,
170 $ NSPLIT
171 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
172 $ SIGMA, SMLNUM, TMP1, VLL, VUU
173 * ..
174 * .. External Functions ..
175 LOGICAL LSAME
176 INTEGER ILAENV
177 DOUBLE PRECISION DLAMCH, ZLANHE
178 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE
179 * ..
180 * .. External Subroutines ..
181 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZDSCAL,
182 $ ZHETRD, ZLACPY, ZSTEIN, ZSTEQR, ZSWAP, ZUNGTR,
183 $ ZUNMTR
184 * ..
185 * .. Intrinsic Functions ..
186 INTRINSIC DBLE, MAX, MIN, SQRT
187 * ..
188 * .. Executable Statements ..
189 *
190 * Test the input parameters.
191 *
192 LOWER = LSAME( UPLO, 'L' )
193 WANTZ = LSAME( JOBZ, 'V' )
194 ALLEIG = LSAME( RANGE, 'A' )
195 VALEIG = LSAME( RANGE, 'V' )
196 INDEIG = LSAME( RANGE, 'I' )
197 LQUERY = ( LWORK.EQ.-1 )
198 *
199 INFO = 0
200 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
201 INFO = -1
202 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
203 INFO = -2
204 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
205 INFO = -3
206 ELSE IF( N.LT.0 ) THEN
207 INFO = -4
208 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
209 INFO = -6
210 ELSE
211 IF( VALEIG ) THEN
212 IF( N.GT.0 .AND. VU.LE.VL )
213 $ INFO = -8
214 ELSE IF( INDEIG ) THEN
215 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
216 INFO = -9
217 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
218 INFO = -10
219 END IF
220 END IF
221 END IF
222 IF( INFO.EQ.0 ) THEN
223 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
224 INFO = -15
225 END IF
226 END IF
227 *
228 IF( INFO.EQ.0 ) THEN
229 IF( N.LE.1 ) THEN
230 LWKMIN = 1
231 WORK( 1 ) = LWKMIN
232 ELSE
233 LWKMIN = 2*N
234 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
235 NB = MAX( NB, ILAENV( 1, 'ZUNMTR', UPLO, N, -1, -1, -1 ) )
236 LWKOPT = MAX( 1, ( NB + 1 )*N )
237 WORK( 1 ) = LWKOPT
238 END IF
239 *
240 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY )
241 $ INFO = -17
242 END IF
243 *
244 IF( INFO.NE.0 ) THEN
245 CALL XERBLA( 'ZHEEVX', -INFO )
246 RETURN
247 ELSE IF( LQUERY ) THEN
248 RETURN
249 END IF
250 *
251 * Quick return if possible
252 *
253 M = 0
254 IF( N.EQ.0 ) THEN
255 RETURN
256 END IF
257 *
258 IF( N.EQ.1 ) THEN
259 IF( ALLEIG .OR. INDEIG ) THEN
260 M = 1
261 W( 1 ) = A( 1, 1 )
262 ELSE IF( VALEIG ) THEN
263 IF( VL.LT.DBLE( A( 1, 1 ) ) .AND. VU.GE.DBLE( A( 1, 1 ) ) )
264 $ THEN
265 M = 1
266 W( 1 ) = A( 1, 1 )
267 END IF
268 END IF
269 IF( WANTZ )
270 $ Z( 1, 1 ) = CONE
271 RETURN
272 END IF
273 *
274 * Get machine constants.
275 *
276 SAFMIN = DLAMCH( 'Safe minimum' )
277 EPS = DLAMCH( 'Precision' )
278 SMLNUM = SAFMIN / EPS
279 BIGNUM = ONE / SMLNUM
280 RMIN = SQRT( SMLNUM )
281 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
282 *
283 * Scale matrix to allowable range, if necessary.
284 *
285 ISCALE = 0
286 ABSTLL = ABSTOL
287 IF( VALEIG ) THEN
288 VLL = VL
289 VUU = VU
290 END IF
291 ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK )
292 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
293 ISCALE = 1
294 SIGMA = RMIN / ANRM
295 ELSE IF( ANRM.GT.RMAX ) THEN
296 ISCALE = 1
297 SIGMA = RMAX / ANRM
298 END IF
299 IF( ISCALE.EQ.1 ) THEN
300 IF( LOWER ) THEN
301 DO 10 J = 1, N
302 CALL ZDSCAL( N-J+1, SIGMA, A( J, J ), 1 )
303 10 CONTINUE
304 ELSE
305 DO 20 J = 1, N
306 CALL ZDSCAL( J, SIGMA, A( 1, J ), 1 )
307 20 CONTINUE
308 END IF
309 IF( ABSTOL.GT.0 )
310 $ ABSTLL = ABSTOL*SIGMA
311 IF( VALEIG ) THEN
312 VLL = VL*SIGMA
313 VUU = VU*SIGMA
314 END IF
315 END IF
316 *
317 * Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
318 *
319 INDD = 1
320 INDE = INDD + N
321 INDRWK = INDE + N
322 INDTAU = 1
323 INDWRK = INDTAU + N
324 LLWORK = LWORK - INDWRK + 1
325 CALL ZHETRD( UPLO, N, A, LDA, RWORK( INDD ), RWORK( INDE ),
326 $ WORK( INDTAU ), WORK( INDWRK ), LLWORK, IINFO )
327 *
328 * If all eigenvalues are desired and ABSTOL is less than or equal to
329 * zero, then call DSTERF or ZUNGTR and ZSTEQR. If this fails for
330 * some eigenvalue, then try DSTEBZ.
331 *
332 TEST = .FALSE.
333 IF( INDEIG ) THEN
334 IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
335 TEST = .TRUE.
336 END IF
337 END IF
338 IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
339 CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
340 INDEE = INDRWK + 2*N
341 IF( .NOT.WANTZ ) THEN
342 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
343 CALL DSTERF( N, W, RWORK( INDEE ), INFO )
344 ELSE
345 CALL ZLACPY( 'A', N, N, A, LDA, Z, LDZ )
346 CALL ZUNGTR( UPLO, N, Z, LDZ, WORK( INDTAU ),
347 $ WORK( INDWRK ), LLWORK, IINFO )
348 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
349 CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
350 $ RWORK( INDRWK ), INFO )
351 IF( INFO.EQ.0 ) THEN
352 DO 30 I = 1, N
353 IFAIL( I ) = 0
354 30 CONTINUE
355 END IF
356 END IF
357 IF( INFO.EQ.0 ) THEN
358 M = N
359 GO TO 40
360 END IF
361 INFO = 0
362 END IF
363 *
364 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
365 *
366 IF( WANTZ ) THEN
367 ORDER = 'B'
368 ELSE
369 ORDER = 'E'
370 END IF
371 INDIBL = 1
372 INDISP = INDIBL + N
373 INDIWK = INDISP + N
374 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
375 $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
376 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
377 $ IWORK( INDIWK ), INFO )
378 *
379 IF( WANTZ ) THEN
380 CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
381 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
382 $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
383 *
384 * Apply unitary matrix used in reduction to tridiagonal
385 * form to eigenvectors returned by ZSTEIN.
386 *
387 CALL ZUNMTR( 'L', UPLO, 'N', N, M, A, LDA, WORK( INDTAU ), Z,
388 $ LDZ, WORK( INDWRK ), LLWORK, IINFO )
389 END IF
390 *
391 * If matrix was scaled, then rescale eigenvalues appropriately.
392 *
393 40 CONTINUE
394 IF( ISCALE.EQ.1 ) THEN
395 IF( INFO.EQ.0 ) THEN
396 IMAX = M
397 ELSE
398 IMAX = INFO - 1
399 END IF
400 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
401 END IF
402 *
403 * If eigenvalues are not in order, then sort them, along with
404 * eigenvectors.
405 *
406 IF( WANTZ ) THEN
407 DO 60 J = 1, M - 1
408 I = 0
409 TMP1 = W( J )
410 DO 50 JJ = J + 1, M
411 IF( W( JJ ).LT.TMP1 ) THEN
412 I = JJ
413 TMP1 = W( JJ )
414 END IF
415 50 CONTINUE
416 *
417 IF( I.NE.0 ) THEN
418 ITMP1 = IWORK( INDIBL+I-1 )
419 W( I ) = W( J )
420 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
421 W( J ) = TMP1
422 IWORK( INDIBL+J-1 ) = ITMP1
423 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
424 IF( INFO.NE.0 ) THEN
425 ITMP1 = IFAIL( I )
426 IFAIL( I ) = IFAIL( J )
427 IFAIL( J ) = ITMP1
428 END IF
429 END IF
430 60 CONTINUE
431 END IF
432 *
433 * Set WORK(1) to optimal complex workspace size.
434 *
435 WORK( 1 ) = LWKOPT
436 *
437 RETURN
438 *
439 * End of ZHEEVX
440 *
441 END