1 SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL,
2 $ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
3 $ IWORK, 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, KD, LDAB, LDQ, LDZ, M, N
13 DOUBLE PRECISION ABSTOL, VL, VU
14 * ..
15 * .. Array Arguments ..
16 INTEGER IFAIL( * ), IWORK( * )
17 DOUBLE PRECISION RWORK( * ), W( * )
18 COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
19 $ Z( LDZ, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHBEVX computes selected eigenvalues and, optionally, eigenvectors
26 * of a complex Hermitian band matrix A. Eigenvalues and eigenvectors
27 * can 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 * KD (input) INTEGER
51 * The number of superdiagonals of the matrix A if UPLO = 'U',
52 * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
53 *
54 * AB (input/output) COMPLEX*16 array, dimension (LDAB, N)
55 * On entry, the upper or lower triangle of the Hermitian band
56 * matrix A, stored in the first KD+1 rows of the array. The
57 * j-th column of A is stored in the j-th column of the array AB
58 * as follows:
59 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
60 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
61 *
62 * On exit, AB is overwritten by values generated during the
63 * reduction to tridiagonal form.
64 *
65 * LDAB (input) INTEGER
66 * The leading dimension of the array AB. LDAB >= KD + 1.
67 *
68 * Q (output) COMPLEX*16 array, dimension (LDQ, N)
69 * If JOBZ = 'V', the N-by-N unitary matrix used in the
70 * reduction to tridiagonal form.
71 * If JOBZ = 'N', the array Q is not referenced.
72 *
73 * LDQ (input) INTEGER
74 * The leading dimension of the array Q. If JOBZ = 'V', then
75 * LDQ >= max(1,N).
76 *
77 * VL (input) DOUBLE PRECISION
78 * VU (input) DOUBLE PRECISION
79 * If RANGE='V', the lower and upper bounds of the interval to
80 * be searched for eigenvalues. VL < VU.
81 * Not referenced if RANGE = 'A' or 'I'.
82 *
83 * IL (input) INTEGER
84 * IU (input) INTEGER
85 * If RANGE='I', the indices (in ascending order) of the
86 * smallest and largest eigenvalues to be returned.
87 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
88 * Not referenced if RANGE = 'A' or 'V'.
89 *
90 * ABSTOL (input) DOUBLE PRECISION
91 * The absolute error tolerance for the eigenvalues.
92 * An approximate eigenvalue is accepted as converged
93 * when it is determined to lie in an interval [a,b]
94 * of width less than or equal to
95 *
96 * ABSTOL + EPS * max( |a|,|b| ) ,
97 *
98 * where EPS is the machine precision. If ABSTOL is less than
99 * or equal to zero, then EPS*|T| will be used in its place,
100 * where |T| is the 1-norm of the tridiagonal matrix obtained
101 * by reducing AB to tridiagonal form.
102 *
103 * Eigenvalues will be computed most accurately when ABSTOL is
104 * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
105 * If this routine returns with INFO>0, indicating that some
106 * eigenvectors did not converge, try setting ABSTOL to
107 * 2*DLAMCH('S').
108 *
109 * See "Computing Small Singular Values of Bidiagonal Matrices
110 * with Guaranteed High Relative Accuracy," by Demmel and
111 * Kahan, LAPACK Working Note #3.
112 *
113 * M (output) INTEGER
114 * The total number of eigenvalues found. 0 <= M <= N.
115 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
116 *
117 * W (output) DOUBLE PRECISION array, dimension (N)
118 * The first M elements contain the selected eigenvalues in
119 * ascending order.
120 *
121 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
122 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
123 * contain the orthonormal eigenvectors of the matrix A
124 * corresponding to the selected eigenvalues, with the i-th
125 * column of Z holding the eigenvector associated with W(i).
126 * If an eigenvector fails to converge, then that column of Z
127 * contains the latest approximation to the eigenvector, and the
128 * index of the eigenvector is returned in IFAIL.
129 * If JOBZ = 'N', then Z is not referenced.
130 * Note: the user must ensure that at least max(1,M) columns are
131 * supplied in the array Z; if RANGE = 'V', the exact value of M
132 * is not known in advance and an upper bound must be used.
133 *
134 * LDZ (input) INTEGER
135 * The leading dimension of the array Z. LDZ >= 1, and if
136 * JOBZ = 'V', LDZ >= max(1,N).
137 *
138 * WORK (workspace) COMPLEX*16 array, dimension (N)
139 *
140 * RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
141 *
142 * IWORK (workspace) INTEGER array, dimension (5*N)
143 *
144 * IFAIL (output) INTEGER array, dimension (N)
145 * If JOBZ = 'V', then if INFO = 0, the first M elements of
146 * IFAIL are zero. If INFO > 0, then IFAIL contains the
147 * indices of the eigenvectors that failed to converge.
148 * If JOBZ = 'N', then IFAIL is not referenced.
149 *
150 * INFO (output) INTEGER
151 * = 0: successful exit
152 * < 0: if INFO = -i, the i-th argument had an illegal value
153 * > 0: if INFO = i, then i eigenvectors failed to converge.
154 * Their indices are stored in array IFAIL.
155 *
156 * =====================================================================
157 *
158 * .. Parameters ..
159 DOUBLE PRECISION ZERO, ONE
160 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
161 COMPLEX*16 CZERO, CONE
162 PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
163 $ CONE = ( 1.0D0, 0.0D0 ) )
164 * ..
165 * .. Local Scalars ..
166 LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
167 CHARACTER ORDER
168 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
169 $ INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1,
170 $ J, JJ, NSPLIT
171 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
172 $ SIGMA, SMLNUM, TMP1, VLL, VUU
173 COMPLEX*16 CTMP1
174 * ..
175 * .. External Functions ..
176 LOGICAL LSAME
177 DOUBLE PRECISION DLAMCH, ZLANHB
178 EXTERNAL LSAME, DLAMCH, ZLANHB
179 * ..
180 * .. External Subroutines ..
181 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZCOPY,
182 $ ZGEMV, ZHBTRD, ZLACPY, ZLASCL, ZSTEIN, ZSTEQR,
183 $ ZSWAP
184 * ..
185 * .. Intrinsic Functions ..
186 INTRINSIC DBLE, MAX, MIN, SQRT
187 * ..
188 * .. Executable Statements ..
189 *
190 * Test the input parameters.
191 *
192 WANTZ = LSAME( JOBZ, 'V' )
193 ALLEIG = LSAME( RANGE, 'A' )
194 VALEIG = LSAME( RANGE, 'V' )
195 INDEIG = LSAME( RANGE, 'I' )
196 LOWER = LSAME( UPLO, 'L' )
197 *
198 INFO = 0
199 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
200 INFO = -1
201 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
202 INFO = -2
203 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
204 INFO = -3
205 ELSE IF( N.LT.0 ) THEN
206 INFO = -4
207 ELSE IF( KD.LT.0 ) THEN
208 INFO = -5
209 ELSE IF( LDAB.LT.KD+1 ) THEN
210 INFO = -7
211 ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
212 INFO = -9
213 ELSE
214 IF( VALEIG ) THEN
215 IF( N.GT.0 .AND. VU.LE.VL )
216 $ INFO = -11
217 ELSE IF( INDEIG ) THEN
218 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
219 INFO = -12
220 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
221 INFO = -13
222 END IF
223 END IF
224 END IF
225 IF( INFO.EQ.0 ) THEN
226 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
227 $ INFO = -18
228 END IF
229 *
230 IF( INFO.NE.0 ) THEN
231 CALL XERBLA( 'ZHBEVX', -INFO )
232 RETURN
233 END IF
234 *
235 * Quick return if possible
236 *
237 M = 0
238 IF( N.EQ.0 )
239 $ RETURN
240 *
241 IF( N.EQ.1 ) THEN
242 M = 1
243 IF( LOWER ) THEN
244 CTMP1 = AB( 1, 1 )
245 ELSE
246 CTMP1 = AB( KD+1, 1 )
247 END IF
248 TMP1 = DBLE( CTMP1 )
249 IF( VALEIG ) THEN
250 IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
251 $ M = 0
252 END IF
253 IF( M.EQ.1 ) THEN
254 W( 1 ) = CTMP1
255 IF( WANTZ )
256 $ Z( 1, 1 ) = CONE
257 END IF
258 RETURN
259 END IF
260 *
261 * Get machine constants.
262 *
263 SAFMIN = DLAMCH( 'Safe minimum' )
264 EPS = DLAMCH( 'Precision' )
265 SMLNUM = SAFMIN / EPS
266 BIGNUM = ONE / SMLNUM
267 RMIN = SQRT( SMLNUM )
268 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
269 *
270 * Scale matrix to allowable range, if necessary.
271 *
272 ISCALE = 0
273 ABSTLL = ABSTOL
274 IF( VALEIG ) THEN
275 VLL = VL
276 VUU = VU
277 ELSE
278 VLL = ZERO
279 VUU = ZERO
280 END IF
281 ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
282 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
283 ISCALE = 1
284 SIGMA = RMIN / ANRM
285 ELSE IF( ANRM.GT.RMAX ) THEN
286 ISCALE = 1
287 SIGMA = RMAX / ANRM
288 END IF
289 IF( ISCALE.EQ.1 ) THEN
290 IF( LOWER ) THEN
291 CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
292 ELSE
293 CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
294 END IF
295 IF( ABSTOL.GT.0 )
296 $ ABSTLL = ABSTOL*SIGMA
297 IF( VALEIG ) THEN
298 VLL = VL*SIGMA
299 VUU = VU*SIGMA
300 END IF
301 END IF
302 *
303 * Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
304 *
305 INDD = 1
306 INDE = INDD + N
307 INDRWK = INDE + N
308 INDWRK = 1
309 CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, RWORK( INDD ),
310 $ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
311 *
312 * If all eigenvalues are desired and ABSTOL is less than or equal
313 * to zero, then call DSTERF or ZSTEQR. If this fails for some
314 * eigenvalue, then try DSTEBZ.
315 *
316 TEST = .FALSE.
317 IF (INDEIG) THEN
318 IF (IL.EQ.1 .AND. IU.EQ.N) THEN
319 TEST = .TRUE.
320 END IF
321 END IF
322 IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
323 CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
324 INDEE = INDRWK + 2*N
325 IF( .NOT.WANTZ ) THEN
326 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
327 CALL DSTERF( N, W, RWORK( INDEE ), INFO )
328 ELSE
329 CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
330 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
331 CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
332 $ RWORK( INDRWK ), INFO )
333 IF( INFO.EQ.0 ) THEN
334 DO 10 I = 1, N
335 IFAIL( I ) = 0
336 10 CONTINUE
337 END IF
338 END IF
339 IF( INFO.EQ.0 ) THEN
340 M = N
341 GO TO 30
342 END IF
343 INFO = 0
344 END IF
345 *
346 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
347 *
348 IF( WANTZ ) THEN
349 ORDER = 'B'
350 ELSE
351 ORDER = 'E'
352 END IF
353 INDIBL = 1
354 INDISP = INDIBL + N
355 INDIWK = INDISP + N
356 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
357 $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
358 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
359 $ IWORK( INDIWK ), INFO )
360 *
361 IF( WANTZ ) THEN
362 CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
363 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
364 $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
365 *
366 * Apply unitary matrix used in reduction to tridiagonal
367 * form to eigenvectors returned by ZSTEIN.
368 *
369 DO 20 J = 1, M
370 CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
371 CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
372 $ Z( 1, J ), 1 )
373 20 CONTINUE
374 END IF
375 *
376 * If matrix was scaled, then rescale eigenvalues appropriately.
377 *
378 30 CONTINUE
379 IF( ISCALE.EQ.1 ) THEN
380 IF( INFO.EQ.0 ) THEN
381 IMAX = M
382 ELSE
383 IMAX = INFO - 1
384 END IF
385 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
386 END IF
387 *
388 * If eigenvalues are not in order, then sort them, along with
389 * eigenvectors.
390 *
391 IF( WANTZ ) THEN
392 DO 50 J = 1, M - 1
393 I = 0
394 TMP1 = W( J )
395 DO 40 JJ = J + 1, M
396 IF( W( JJ ).LT.TMP1 ) THEN
397 I = JJ
398 TMP1 = W( JJ )
399 END IF
400 40 CONTINUE
401 *
402 IF( I.NE.0 ) THEN
403 ITMP1 = IWORK( INDIBL+I-1 )
404 W( I ) = W( J )
405 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
406 W( J ) = TMP1
407 IWORK( INDIBL+J-1 ) = ITMP1
408 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
409 IF( INFO.NE.0 ) THEN
410 ITMP1 = IFAIL( I )
411 IFAIL( I ) = IFAIL( J )
412 IFAIL( J ) = ITMP1
413 END IF
414 END IF
415 50 CONTINUE
416 END IF
417 *
418 RETURN
419 *
420 * End of ZHBEVX
421 *
422 END
2 $ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK,
3 $ IWORK, 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, KD, LDAB, LDQ, LDZ, M, N
13 DOUBLE PRECISION ABSTOL, VL, VU
14 * ..
15 * .. Array Arguments ..
16 INTEGER IFAIL( * ), IWORK( * )
17 DOUBLE PRECISION RWORK( * ), W( * )
18 COMPLEX*16 AB( LDAB, * ), Q( LDQ, * ), WORK( * ),
19 $ Z( LDZ, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHBEVX computes selected eigenvalues and, optionally, eigenvectors
26 * of a complex Hermitian band matrix A. Eigenvalues and eigenvectors
27 * can 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 * KD (input) INTEGER
51 * The number of superdiagonals of the matrix A if UPLO = 'U',
52 * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
53 *
54 * AB (input/output) COMPLEX*16 array, dimension (LDAB, N)
55 * On entry, the upper or lower triangle of the Hermitian band
56 * matrix A, stored in the first KD+1 rows of the array. The
57 * j-th column of A is stored in the j-th column of the array AB
58 * as follows:
59 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
60 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
61 *
62 * On exit, AB is overwritten by values generated during the
63 * reduction to tridiagonal form.
64 *
65 * LDAB (input) INTEGER
66 * The leading dimension of the array AB. LDAB >= KD + 1.
67 *
68 * Q (output) COMPLEX*16 array, dimension (LDQ, N)
69 * If JOBZ = 'V', the N-by-N unitary matrix used in the
70 * reduction to tridiagonal form.
71 * If JOBZ = 'N', the array Q is not referenced.
72 *
73 * LDQ (input) INTEGER
74 * The leading dimension of the array Q. If JOBZ = 'V', then
75 * LDQ >= max(1,N).
76 *
77 * VL (input) DOUBLE PRECISION
78 * VU (input) DOUBLE PRECISION
79 * If RANGE='V', the lower and upper bounds of the interval to
80 * be searched for eigenvalues. VL < VU.
81 * Not referenced if RANGE = 'A' or 'I'.
82 *
83 * IL (input) INTEGER
84 * IU (input) INTEGER
85 * If RANGE='I', the indices (in ascending order) of the
86 * smallest and largest eigenvalues to be returned.
87 * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
88 * Not referenced if RANGE = 'A' or 'V'.
89 *
90 * ABSTOL (input) DOUBLE PRECISION
91 * The absolute error tolerance for the eigenvalues.
92 * An approximate eigenvalue is accepted as converged
93 * when it is determined to lie in an interval [a,b]
94 * of width less than or equal to
95 *
96 * ABSTOL + EPS * max( |a|,|b| ) ,
97 *
98 * where EPS is the machine precision. If ABSTOL is less than
99 * or equal to zero, then EPS*|T| will be used in its place,
100 * where |T| is the 1-norm of the tridiagonal matrix obtained
101 * by reducing AB to tridiagonal form.
102 *
103 * Eigenvalues will be computed most accurately when ABSTOL is
104 * set to twice the underflow threshold 2*DLAMCH('S'), not zero.
105 * If this routine returns with INFO>0, indicating that some
106 * eigenvectors did not converge, try setting ABSTOL to
107 * 2*DLAMCH('S').
108 *
109 * See "Computing Small Singular Values of Bidiagonal Matrices
110 * with Guaranteed High Relative Accuracy," by Demmel and
111 * Kahan, LAPACK Working Note #3.
112 *
113 * M (output) INTEGER
114 * The total number of eigenvalues found. 0 <= M <= N.
115 * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
116 *
117 * W (output) DOUBLE PRECISION array, dimension (N)
118 * The first M elements contain the selected eigenvalues in
119 * ascending order.
120 *
121 * Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M))
122 * If JOBZ = 'V', then if INFO = 0, the first M columns of Z
123 * contain the orthonormal eigenvectors of the matrix A
124 * corresponding to the selected eigenvalues, with the i-th
125 * column of Z holding the eigenvector associated with W(i).
126 * If an eigenvector fails to converge, then that column of Z
127 * contains the latest approximation to the eigenvector, and the
128 * index of the eigenvector is returned in IFAIL.
129 * If JOBZ = 'N', then Z is not referenced.
130 * Note: the user must ensure that at least max(1,M) columns are
131 * supplied in the array Z; if RANGE = 'V', the exact value of M
132 * is not known in advance and an upper bound must be used.
133 *
134 * LDZ (input) INTEGER
135 * The leading dimension of the array Z. LDZ >= 1, and if
136 * JOBZ = 'V', LDZ >= max(1,N).
137 *
138 * WORK (workspace) COMPLEX*16 array, dimension (N)
139 *
140 * RWORK (workspace) DOUBLE PRECISION array, dimension (7*N)
141 *
142 * IWORK (workspace) INTEGER array, dimension (5*N)
143 *
144 * IFAIL (output) INTEGER array, dimension (N)
145 * If JOBZ = 'V', then if INFO = 0, the first M elements of
146 * IFAIL are zero. If INFO > 0, then IFAIL contains the
147 * indices of the eigenvectors that failed to converge.
148 * If JOBZ = 'N', then IFAIL is not referenced.
149 *
150 * INFO (output) INTEGER
151 * = 0: successful exit
152 * < 0: if INFO = -i, the i-th argument had an illegal value
153 * > 0: if INFO = i, then i eigenvectors failed to converge.
154 * Their indices are stored in array IFAIL.
155 *
156 * =====================================================================
157 *
158 * .. Parameters ..
159 DOUBLE PRECISION ZERO, ONE
160 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
161 COMPLEX*16 CZERO, CONE
162 PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
163 $ CONE = ( 1.0D0, 0.0D0 ) )
164 * ..
165 * .. Local Scalars ..
166 LOGICAL ALLEIG, INDEIG, LOWER, TEST, VALEIG, WANTZ
167 CHARACTER ORDER
168 INTEGER I, IINFO, IMAX, INDD, INDE, INDEE, INDIBL,
169 $ INDISP, INDIWK, INDRWK, INDWRK, ISCALE, ITMP1,
170 $ J, JJ, NSPLIT
171 DOUBLE PRECISION ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
172 $ SIGMA, SMLNUM, TMP1, VLL, VUU
173 COMPLEX*16 CTMP1
174 * ..
175 * .. External Functions ..
176 LOGICAL LSAME
177 DOUBLE PRECISION DLAMCH, ZLANHB
178 EXTERNAL LSAME, DLAMCH, ZLANHB
179 * ..
180 * .. External Subroutines ..
181 EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTERF, XERBLA, ZCOPY,
182 $ ZGEMV, ZHBTRD, ZLACPY, ZLASCL, ZSTEIN, ZSTEQR,
183 $ ZSWAP
184 * ..
185 * .. Intrinsic Functions ..
186 INTRINSIC DBLE, MAX, MIN, SQRT
187 * ..
188 * .. Executable Statements ..
189 *
190 * Test the input parameters.
191 *
192 WANTZ = LSAME( JOBZ, 'V' )
193 ALLEIG = LSAME( RANGE, 'A' )
194 VALEIG = LSAME( RANGE, 'V' )
195 INDEIG = LSAME( RANGE, 'I' )
196 LOWER = LSAME( UPLO, 'L' )
197 *
198 INFO = 0
199 IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
200 INFO = -1
201 ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
202 INFO = -2
203 ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
204 INFO = -3
205 ELSE IF( N.LT.0 ) THEN
206 INFO = -4
207 ELSE IF( KD.LT.0 ) THEN
208 INFO = -5
209 ELSE IF( LDAB.LT.KD+1 ) THEN
210 INFO = -7
211 ELSE IF( WANTZ .AND. LDQ.LT.MAX( 1, N ) ) THEN
212 INFO = -9
213 ELSE
214 IF( VALEIG ) THEN
215 IF( N.GT.0 .AND. VU.LE.VL )
216 $ INFO = -11
217 ELSE IF( INDEIG ) THEN
218 IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
219 INFO = -12
220 ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
221 INFO = -13
222 END IF
223 END IF
224 END IF
225 IF( INFO.EQ.0 ) THEN
226 IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
227 $ INFO = -18
228 END IF
229 *
230 IF( INFO.NE.0 ) THEN
231 CALL XERBLA( 'ZHBEVX', -INFO )
232 RETURN
233 END IF
234 *
235 * Quick return if possible
236 *
237 M = 0
238 IF( N.EQ.0 )
239 $ RETURN
240 *
241 IF( N.EQ.1 ) THEN
242 M = 1
243 IF( LOWER ) THEN
244 CTMP1 = AB( 1, 1 )
245 ELSE
246 CTMP1 = AB( KD+1, 1 )
247 END IF
248 TMP1 = DBLE( CTMP1 )
249 IF( VALEIG ) THEN
250 IF( .NOT.( VL.LT.TMP1 .AND. VU.GE.TMP1 ) )
251 $ M = 0
252 END IF
253 IF( M.EQ.1 ) THEN
254 W( 1 ) = CTMP1
255 IF( WANTZ )
256 $ Z( 1, 1 ) = CONE
257 END IF
258 RETURN
259 END IF
260 *
261 * Get machine constants.
262 *
263 SAFMIN = DLAMCH( 'Safe minimum' )
264 EPS = DLAMCH( 'Precision' )
265 SMLNUM = SAFMIN / EPS
266 BIGNUM = ONE / SMLNUM
267 RMIN = SQRT( SMLNUM )
268 RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
269 *
270 * Scale matrix to allowable range, if necessary.
271 *
272 ISCALE = 0
273 ABSTLL = ABSTOL
274 IF( VALEIG ) THEN
275 VLL = VL
276 VUU = VU
277 ELSE
278 VLL = ZERO
279 VUU = ZERO
280 END IF
281 ANRM = ZLANHB( 'M', UPLO, N, KD, AB, LDAB, RWORK )
282 IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
283 ISCALE = 1
284 SIGMA = RMIN / ANRM
285 ELSE IF( ANRM.GT.RMAX ) THEN
286 ISCALE = 1
287 SIGMA = RMAX / ANRM
288 END IF
289 IF( ISCALE.EQ.1 ) THEN
290 IF( LOWER ) THEN
291 CALL ZLASCL( 'B', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
292 ELSE
293 CALL ZLASCL( 'Q', KD, KD, ONE, SIGMA, N, N, AB, LDAB, INFO )
294 END IF
295 IF( ABSTOL.GT.0 )
296 $ ABSTLL = ABSTOL*SIGMA
297 IF( VALEIG ) THEN
298 VLL = VL*SIGMA
299 VUU = VU*SIGMA
300 END IF
301 END IF
302 *
303 * Call ZHBTRD to reduce Hermitian band matrix to tridiagonal form.
304 *
305 INDD = 1
306 INDE = INDD + N
307 INDRWK = INDE + N
308 INDWRK = 1
309 CALL ZHBTRD( JOBZ, UPLO, N, KD, AB, LDAB, RWORK( INDD ),
310 $ RWORK( INDE ), Q, LDQ, WORK( INDWRK ), IINFO )
311 *
312 * If all eigenvalues are desired and ABSTOL is less than or equal
313 * to zero, then call DSTERF or ZSTEQR. If this fails for some
314 * eigenvalue, then try DSTEBZ.
315 *
316 TEST = .FALSE.
317 IF (INDEIG) THEN
318 IF (IL.EQ.1 .AND. IU.EQ.N) THEN
319 TEST = .TRUE.
320 END IF
321 END IF
322 IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
323 CALL DCOPY( N, RWORK( INDD ), 1, W, 1 )
324 INDEE = INDRWK + 2*N
325 IF( .NOT.WANTZ ) THEN
326 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
327 CALL DSTERF( N, W, RWORK( INDEE ), INFO )
328 ELSE
329 CALL ZLACPY( 'A', N, N, Q, LDQ, Z, LDZ )
330 CALL DCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
331 CALL ZSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
332 $ RWORK( INDRWK ), INFO )
333 IF( INFO.EQ.0 ) THEN
334 DO 10 I = 1, N
335 IFAIL( I ) = 0
336 10 CONTINUE
337 END IF
338 END IF
339 IF( INFO.EQ.0 ) THEN
340 M = N
341 GO TO 30
342 END IF
343 INFO = 0
344 END IF
345 *
346 * Otherwise, call DSTEBZ and, if eigenvectors are desired, ZSTEIN.
347 *
348 IF( WANTZ ) THEN
349 ORDER = 'B'
350 ELSE
351 ORDER = 'E'
352 END IF
353 INDIBL = 1
354 INDISP = INDIBL + N
355 INDIWK = INDISP + N
356 CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
357 $ RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
358 $ IWORK( INDIBL ), IWORK( INDISP ), RWORK( INDRWK ),
359 $ IWORK( INDIWK ), INFO )
360 *
361 IF( WANTZ ) THEN
362 CALL ZSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
363 $ IWORK( INDIBL ), IWORK( INDISP ), Z, LDZ,
364 $ RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
365 *
366 * Apply unitary matrix used in reduction to tridiagonal
367 * form to eigenvectors returned by ZSTEIN.
368 *
369 DO 20 J = 1, M
370 CALL ZCOPY( N, Z( 1, J ), 1, WORK( 1 ), 1 )
371 CALL ZGEMV( 'N', N, N, CONE, Q, LDQ, WORK, 1, CZERO,
372 $ Z( 1, J ), 1 )
373 20 CONTINUE
374 END IF
375 *
376 * If matrix was scaled, then rescale eigenvalues appropriately.
377 *
378 30 CONTINUE
379 IF( ISCALE.EQ.1 ) THEN
380 IF( INFO.EQ.0 ) THEN
381 IMAX = M
382 ELSE
383 IMAX = INFO - 1
384 END IF
385 CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
386 END IF
387 *
388 * If eigenvalues are not in order, then sort them, along with
389 * eigenvectors.
390 *
391 IF( WANTZ ) THEN
392 DO 50 J = 1, M - 1
393 I = 0
394 TMP1 = W( J )
395 DO 40 JJ = J + 1, M
396 IF( W( JJ ).LT.TMP1 ) THEN
397 I = JJ
398 TMP1 = W( JJ )
399 END IF
400 40 CONTINUE
401 *
402 IF( I.NE.0 ) THEN
403 ITMP1 = IWORK( INDIBL+I-1 )
404 W( I ) = W( J )
405 IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
406 W( J ) = TMP1
407 IWORK( INDIBL+J-1 ) = ITMP1
408 CALL ZSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
409 IF( INFO.NE.0 ) THEN
410 ITMP1 = IFAIL( I )
411 IFAIL( I ) = IFAIL( J )
412 IFAIL( J ) = ITMP1
413 END IF
414 END IF
415 50 CONTINUE
416 END IF
417 *
418 RETURN
419 *
420 * End of ZHBEVX
421 *
422 END