1: *> \brief <b> ZHEEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices</b>
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZHEEVD + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zheevd.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zheevd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZHEEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK,
22: * LRWORK, IWORK, LIWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER JOBZ, UPLO
26: * INTEGER INFO, LDA, LIWORK, LRWORK, LWORK, N
27: * ..
28: * .. Array Arguments ..
29: * INTEGER IWORK( * )
30: * DOUBLE PRECISION RWORK( * ), W( * )
31: * COMPLEX*16 A( LDA, * ), WORK( * )
32: * ..
33: *
34: *
35: *> \par Purpose:
36: * =============
37: *>
38: *> \verbatim
39: *>
40: *> ZHEEVD computes all eigenvalues and, optionally, eigenvectors of a
41: *> complex Hermitian matrix A. If eigenvectors are desired, it uses a
42: *> divide and conquer algorithm.
43: *>
44: *> The divide and conquer algorithm makes very mild assumptions about
45: *> floating point arithmetic. It will work on machines with a guard
46: *> digit in add/subtract, or on those binary machines without guard
47: *> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
48: *> Cray-2. It could conceivably fail on hexadecimal or decimal machines
49: *> without guard digits, but we know of none.
50: *> \endverbatim
51: *
52: * Arguments:
53: * ==========
54: *
55: *> \param[in] JOBZ
56: *> \verbatim
57: *> JOBZ is CHARACTER*1
58: *> = 'N': Compute eigenvalues only;
59: *> = 'V': Compute eigenvalues and eigenvectors.
60: *> \endverbatim
61: *>
62: *> \param[in] UPLO
63: *> \verbatim
64: *> UPLO is CHARACTER*1
65: *> = 'U': Upper triangle of A is stored;
66: *> = 'L': Lower triangle of A is stored.
67: *> \endverbatim
68: *>
69: *> \param[in] N
70: *> \verbatim
71: *> N is INTEGER
72: *> The order of the matrix A. N >= 0.
73: *> \endverbatim
74: *>
75: *> \param[in,out] A
76: *> \verbatim
77: *> A is COMPLEX*16 array, dimension (LDA, N)
78: *> On entry, the Hermitian matrix A. If UPLO = 'U', the
79: *> leading N-by-N upper triangular part of A contains the
80: *> upper triangular part of the matrix A. If UPLO = 'L',
81: *> the leading N-by-N lower triangular part of A contains
82: *> the lower triangular part of the matrix A.
83: *> On exit, if JOBZ = 'V', then if INFO = 0, A contains the
84: *> orthonormal eigenvectors of the matrix A.
85: *> If JOBZ = 'N', then on exit the lower triangle (if UPLO='L')
86: *> or the upper triangle (if UPLO='U') of A, including the
87: *> diagonal, is destroyed.
88: *> \endverbatim
89: *>
90: *> \param[in] LDA
91: *> \verbatim
92: *> LDA is INTEGER
93: *> The leading dimension of the array A. LDA >= max(1,N).
94: *> \endverbatim
95: *>
96: *> \param[out] W
97: *> \verbatim
98: *> W is DOUBLE PRECISION array, dimension (N)
99: *> If INFO = 0, the eigenvalues in ascending order.
100: *> \endverbatim
101: *>
102: *> \param[out] WORK
103: *> \verbatim
104: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
105: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
106: *> \endverbatim
107: *>
108: *> \param[in] LWORK
109: *> \verbatim
110: *> LWORK is INTEGER
111: *> The length of the array WORK.
112: *> If N <= 1, LWORK must be at least 1.
113: *> If JOBZ = 'N' and N > 1, LWORK must be at least N + 1.
114: *> If JOBZ = 'V' and N > 1, LWORK must be at least 2*N + N**2.
115: *>
116: *> If LWORK = -1, then a workspace query is assumed; the routine
117: *> only calculates the optimal sizes of the WORK, RWORK and
118: *> IWORK arrays, returns these values as the first entries of
119: *> the WORK, RWORK and IWORK arrays, and no error message
120: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
121: *> \endverbatim
122: *>
123: *> \param[out] RWORK
124: *> \verbatim
125: *> RWORK is DOUBLE PRECISION array,
126: *> dimension (LRWORK)
127: *> On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
128: *> \endverbatim
129: *>
130: *> \param[in] LRWORK
131: *> \verbatim
132: *> LRWORK is INTEGER
133: *> The dimension of the array RWORK.
134: *> If N <= 1, LRWORK must be at least 1.
135: *> If JOBZ = 'N' and N > 1, LRWORK must be at least N.
136: *> If JOBZ = 'V' and N > 1, LRWORK must be at least
137: *> 1 + 5*N + 2*N**2.
138: *>
139: *> If LRWORK = -1, then a workspace query is assumed; the
140: *> routine only calculates the optimal sizes of the WORK, RWORK
141: *> and IWORK arrays, returns these values as the first entries
142: *> of the WORK, RWORK and IWORK arrays, and no error message
143: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
144: *> \endverbatim
145: *>
146: *> \param[out] IWORK
147: *> \verbatim
148: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
149: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
150: *> \endverbatim
151: *>
152: *> \param[in] LIWORK
153: *> \verbatim
154: *> LIWORK is INTEGER
155: *> The dimension of the array IWORK.
156: *> If N <= 1, LIWORK must be at least 1.
157: *> If JOBZ = 'N' and N > 1, LIWORK must be at least 1.
158: *> If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
159: *>
160: *> If LIWORK = -1, then a workspace query is assumed; the
161: *> routine only calculates the optimal sizes of the WORK, RWORK
162: *> and IWORK arrays, returns these values as the first entries
163: *> of the WORK, RWORK and IWORK arrays, and no error message
164: *> related to LWORK or LRWORK or LIWORK is issued by XERBLA.
165: *> \endverbatim
166: *>
167: *> \param[out] INFO
168: *> \verbatim
169: *> INFO is INTEGER
170: *> = 0: successful exit
171: *> < 0: if INFO = -i, the i-th argument had an illegal value
172: *> > 0: if INFO = i and JOBZ = 'N', then the algorithm failed
173: *> to converge; i off-diagonal elements of an intermediate
174: *> tridiagonal form did not converge to zero;
175: *> if INFO = i and JOBZ = 'V', then the algorithm failed
176: *> to compute an eigenvalue while working on the submatrix
177: *> lying in rows and columns INFO/(N+1) through
178: *> mod(INFO,N+1).
179: *> \endverbatim
180: *
181: * Authors:
182: * ========
183: *
184: *> \author Univ. of Tennessee
185: *> \author Univ. of California Berkeley
186: *> \author Univ. of Colorado Denver
187: *> \author NAG Ltd.
188: *
189: *> \ingroup complex16HEeigen
190: *
191: *> \par Further Details:
192: * =====================
193: *>
194: *> Modified description of INFO. Sven, 16 Feb 05.
195: *
196: *> \par Contributors:
197: * ==================
198: *>
199: *> Jeff Rutter, Computer Science Division, University of California
200: *> at Berkeley, USA
201: *>
202: * =====================================================================
203: SUBROUTINE ZHEEVD( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, RWORK,
204: $ LRWORK, IWORK, LIWORK, INFO )
205: *
206: * -- LAPACK driver routine --
207: * -- LAPACK is a software package provided by Univ. of Tennessee, --
208: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
209: *
210: * .. Scalar Arguments ..
211: CHARACTER JOBZ, UPLO
212: INTEGER INFO, LDA, LIWORK, LRWORK, LWORK, N
213: * ..
214: * .. Array Arguments ..
215: INTEGER IWORK( * )
216: DOUBLE PRECISION RWORK( * ), W( * )
217: COMPLEX*16 A( LDA, * ), WORK( * )
218: * ..
219: *
220: * =====================================================================
221: *
222: * .. Parameters ..
223: DOUBLE PRECISION ZERO, ONE
224: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
225: COMPLEX*16 CONE
226: PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) )
227: * ..
228: * .. Local Scalars ..
229: LOGICAL LOWER, LQUERY, WANTZ
230: INTEGER IINFO, IMAX, INDE, INDRWK, INDTAU, INDWK2,
231: $ INDWRK, ISCALE, LIOPT, LIWMIN, LLRWK, LLWORK,
232: $ LLWRK2, LOPT, LROPT, LRWMIN, LWMIN
233: DOUBLE PRECISION ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
234: $ SMLNUM
235: * ..
236: * .. External Functions ..
237: LOGICAL LSAME
238: INTEGER ILAENV
239: DOUBLE PRECISION DLAMCH, ZLANHE
240: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANHE
241: * ..
242: * .. External Subroutines ..
243: EXTERNAL DSCAL, DSTERF, XERBLA, ZHETRD, ZLACPY, ZLASCL,
244: $ ZSTEDC, ZUNMTR
245: * ..
246: * .. Intrinsic Functions ..
247: INTRINSIC MAX, SQRT
248: * ..
249: * .. Executable Statements ..
250: *
251: * Test the input parameters.
252: *
253: WANTZ = LSAME( JOBZ, 'V' )
254: LOWER = LSAME( UPLO, 'L' )
255: LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
256: *
257: INFO = 0
258: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
259: INFO = -1
260: ELSE IF( .NOT.( LOWER .OR. LSAME( UPLO, 'U' ) ) ) THEN
261: INFO = -2
262: ELSE IF( N.LT.0 ) THEN
263: INFO = -3
264: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
265: INFO = -5
266: END IF
267: *
268: IF( INFO.EQ.0 ) THEN
269: IF( N.LE.1 ) THEN
270: LWMIN = 1
271: LRWMIN = 1
272: LIWMIN = 1
273: LOPT = LWMIN
274: LROPT = LRWMIN
275: LIOPT = LIWMIN
276: ELSE
277: IF( WANTZ ) THEN
278: LWMIN = 2*N + N*N
279: LRWMIN = 1 + 5*N + 2*N**2
280: LIWMIN = 3 + 5*N
281: ELSE
282: LWMIN = N + 1
283: LRWMIN = N
284: LIWMIN = 1
285: END IF
286: LOPT = MAX( LWMIN, N +
287: $ N*ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) )
288: LROPT = LRWMIN
289: LIOPT = LIWMIN
290: END IF
291: WORK( 1 ) = LOPT
292: RWORK( 1 ) = LROPT
293: IWORK( 1 ) = LIOPT
294: *
295: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
296: INFO = -8
297: ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
298: INFO = -10
299: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
300: INFO = -12
301: END IF
302: END IF
303: *
304: IF( INFO.NE.0 ) THEN
305: CALL XERBLA( 'ZHEEVD', -INFO )
306: RETURN
307: ELSE IF( LQUERY ) THEN
308: RETURN
309: END IF
310: *
311: * Quick return if possible
312: *
313: IF( N.EQ.0 )
314: $ RETURN
315: *
316: IF( N.EQ.1 ) THEN
317: W( 1 ) = DBLE( A( 1, 1 ) )
318: IF( WANTZ )
319: $ A( 1, 1 ) = CONE
320: RETURN
321: END IF
322: *
323: * Get machine constants.
324: *
325: SAFMIN = DLAMCH( 'Safe minimum' )
326: EPS = DLAMCH( 'Precision' )
327: SMLNUM = SAFMIN / EPS
328: BIGNUM = ONE / SMLNUM
329: RMIN = SQRT( SMLNUM )
330: RMAX = SQRT( BIGNUM )
331: *
332: * Scale matrix to allowable range, if necessary.
333: *
334: ANRM = ZLANHE( 'M', UPLO, N, A, LDA, RWORK )
335: ISCALE = 0
336: IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
337: ISCALE = 1
338: SIGMA = RMIN / ANRM
339: ELSE IF( ANRM.GT.RMAX ) THEN
340: ISCALE = 1
341: SIGMA = RMAX / ANRM
342: END IF
343: IF( ISCALE.EQ.1 )
344: $ CALL ZLASCL( UPLO, 0, 0, ONE, SIGMA, N, N, A, LDA, INFO )
345: *
346: * Call ZHETRD to reduce Hermitian matrix to tridiagonal form.
347: *
348: INDE = 1
349: INDTAU = 1
350: INDWRK = INDTAU + N
351: INDRWK = INDE + N
352: INDWK2 = INDWRK + N*N
353: LLWORK = LWORK - INDWRK + 1
354: LLWRK2 = LWORK - INDWK2 + 1
355: LLRWK = LRWORK - INDRWK + 1
356: CALL ZHETRD( UPLO, N, A, LDA, W, RWORK( INDE ), WORK( INDTAU ),
357: $ WORK( INDWRK ), LLWORK, IINFO )
358: *
359: * For eigenvalues only, call DSTERF. For eigenvectors, first call
360: * ZSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
361: * tridiagonal matrix, then call ZUNMTR to multiply it to the
362: * Householder transformations represented as Householder vectors in
363: * A.
364: *
365: IF( .NOT.WANTZ ) THEN
366: CALL DSTERF( N, W, RWORK( INDE ), INFO )
367: ELSE
368: CALL ZSTEDC( 'I', N, W, RWORK( INDE ), WORK( INDWRK ), N,
369: $ WORK( INDWK2 ), LLWRK2, RWORK( INDRWK ), LLRWK,
370: $ IWORK, LIWORK, INFO )
371: CALL ZUNMTR( 'L', UPLO, 'N', N, N, A, LDA, WORK( INDTAU ),
372: $ WORK( INDWRK ), N, WORK( INDWK2 ), LLWRK2, IINFO )
373: CALL ZLACPY( 'A', N, N, WORK( INDWRK ), N, A, LDA )
374: END IF
375: *
376: * If matrix was scaled, then rescale eigenvalues appropriately.
377: *
378: IF( ISCALE.EQ.1 ) THEN
379: IF( INFO.EQ.0 ) THEN
380: IMAX = N
381: ELSE
382: IMAX = INFO - 1
383: END IF
384: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
385: END IF
386: *
387: WORK( 1 ) = LOPT
388: RWORK( 1 ) = LROPT
389: IWORK( 1 ) = LIOPT
390: *
391: RETURN
392: *
393: * End of ZHEEVD
394: *
395: END
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