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
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