Annotation of rpl/lapack/lapack/zgeesx.f, revision 1.11
1.9 bertrand 1: *> \brief <b> ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE 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 ZGEESX + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgeesx.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgeesx.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgeesx.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
22: * VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
23: * BWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * CHARACTER JOBVS, SENSE, SORT
27: * INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
28: * DOUBLE PRECISION RCONDE, RCONDV
29: * ..
30: * .. Array Arguments ..
31: * LOGICAL BWORK( * )
32: * DOUBLE PRECISION RWORK( * )
33: * COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
34: * ..
35: * .. Function Arguments ..
36: * LOGICAL SELECT
37: * EXTERNAL SELECT
38: * ..
39: *
40: *
41: *> \par Purpose:
42: * =============
43: *>
44: *> \verbatim
45: *>
46: *> ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the
47: *> eigenvalues, the Schur form T, and, optionally, the matrix of Schur
48: *> vectors Z. This gives the Schur factorization A = Z*T*(Z**H).
49: *>
50: *> Optionally, it also orders the eigenvalues on the diagonal of the
51: *> Schur form so that selected eigenvalues are at the top left;
52: *> computes a reciprocal condition number for the average of the
53: *> selected eigenvalues (RCONDE); and computes a reciprocal condition
54: *> number for the right invariant subspace corresponding to the
55: *> selected eigenvalues (RCONDV). The leading columns of Z form an
56: *> orthonormal basis for this invariant subspace.
57: *>
58: *> For further explanation of the reciprocal condition numbers RCONDE
59: *> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
60: *> these quantities are called s and sep respectively).
61: *>
62: *> A complex matrix is in Schur form if it is upper triangular.
63: *> \endverbatim
64: *
65: * Arguments:
66: * ==========
67: *
68: *> \param[in] JOBVS
69: *> \verbatim
70: *> JOBVS is CHARACTER*1
71: *> = 'N': Schur vectors are not computed;
72: *> = 'V': Schur vectors are computed.
73: *> \endverbatim
74: *>
75: *> \param[in] SORT
76: *> \verbatim
77: *> SORT is CHARACTER*1
78: *> Specifies whether or not to order the eigenvalues on the
79: *> diagonal of the Schur form.
80: *> = 'N': Eigenvalues are not ordered;
81: *> = 'S': Eigenvalues are ordered (see SELECT).
82: *> \endverbatim
83: *>
84: *> \param[in] SELECT
85: *> \verbatim
86: *> SELECT is procedure) LOGICAL FUNCTION of one COMPLEX*16 argument
87: *> SELECT must be declared EXTERNAL in the calling subroutine.
88: *> If SORT = 'S', SELECT is used to select eigenvalues to order
89: *> to the top left of the Schur form.
90: *> If SORT = 'N', SELECT is not referenced.
91: *> An eigenvalue W(j) is selected if SELECT(W(j)) is true.
92: *> \endverbatim
93: *>
94: *> \param[in] SENSE
95: *> \verbatim
96: *> SENSE is CHARACTER*1
97: *> Determines which reciprocal condition numbers are computed.
98: *> = 'N': None are computed;
99: *> = 'E': Computed for average of selected eigenvalues only;
100: *> = 'V': Computed for selected right invariant subspace only;
101: *> = 'B': Computed for both.
102: *> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
103: *> \endverbatim
104: *>
105: *> \param[in] N
106: *> \verbatim
107: *> N is INTEGER
108: *> The order of the matrix A. N >= 0.
109: *> \endverbatim
110: *>
111: *> \param[in,out] A
112: *> \verbatim
113: *> A is COMPLEX*16 array, dimension (LDA, N)
114: *> On entry, the N-by-N matrix A.
115: *> On exit, A is overwritten by its Schur form T.
116: *> \endverbatim
117: *>
118: *> \param[in] LDA
119: *> \verbatim
120: *> LDA is INTEGER
121: *> The leading dimension of the array A. LDA >= max(1,N).
122: *> \endverbatim
123: *>
124: *> \param[out] SDIM
125: *> \verbatim
126: *> SDIM is INTEGER
127: *> If SORT = 'N', SDIM = 0.
128: *> If SORT = 'S', SDIM = number of eigenvalues for which
129: *> SELECT is true.
130: *> \endverbatim
131: *>
132: *> \param[out] W
133: *> \verbatim
134: *> W is COMPLEX*16 array, dimension (N)
135: *> W contains the computed eigenvalues, in the same order
136: *> that they appear on the diagonal of the output Schur form T.
137: *> \endverbatim
138: *>
139: *> \param[out] VS
140: *> \verbatim
141: *> VS is COMPLEX*16 array, dimension (LDVS,N)
142: *> If JOBVS = 'V', VS contains the unitary matrix Z of Schur
143: *> vectors.
144: *> If JOBVS = 'N', VS is not referenced.
145: *> \endverbatim
146: *>
147: *> \param[in] LDVS
148: *> \verbatim
149: *> LDVS is INTEGER
150: *> The leading dimension of the array VS. LDVS >= 1, and if
151: *> JOBVS = 'V', LDVS >= N.
152: *> \endverbatim
153: *>
154: *> \param[out] RCONDE
155: *> \verbatim
156: *> RCONDE is DOUBLE PRECISION
157: *> If SENSE = 'E' or 'B', RCONDE contains the reciprocal
158: *> condition number for the average of the selected eigenvalues.
159: *> Not referenced if SENSE = 'N' or 'V'.
160: *> \endverbatim
161: *>
162: *> \param[out] RCONDV
163: *> \verbatim
164: *> RCONDV is DOUBLE PRECISION
165: *> If SENSE = 'V' or 'B', RCONDV contains the reciprocal
166: *> condition number for the selected right invariant subspace.
167: *> Not referenced if SENSE = 'N' or 'E'.
168: *> \endverbatim
169: *>
170: *> \param[out] WORK
171: *> \verbatim
172: *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
173: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
174: *> \endverbatim
175: *>
176: *> \param[in] LWORK
177: *> \verbatim
178: *> LWORK is INTEGER
179: *> The dimension of the array WORK. LWORK >= max(1,2*N).
180: *> Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
181: *> where SDIM is the number of selected eigenvalues computed by
182: *> this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
183: *> that an error is only returned if LWORK < max(1,2*N), but if
184: *> SENSE = 'E' or 'V' or 'B' this may not be large enough.
185: *> For good performance, LWORK must generally be larger.
186: *>
187: *> If LWORK = -1, then a workspace query is assumed; the routine
188: *> only calculates upper bound on the optimal size of the
189: *> array WORK, returns this value as the first entry of the WORK
190: *> array, and no error message related to LWORK is issued by
191: *> XERBLA.
192: *> \endverbatim
193: *>
194: *> \param[out] RWORK
195: *> \verbatim
196: *> RWORK is DOUBLE PRECISION array, dimension (N)
197: *> \endverbatim
198: *>
199: *> \param[out] BWORK
200: *> \verbatim
201: *> BWORK is LOGICAL array, dimension (N)
202: *> Not referenced if SORT = 'N'.
203: *> \endverbatim
204: *>
205: *> \param[out] INFO
206: *> \verbatim
207: *> INFO is INTEGER
208: *> = 0: successful exit
209: *> < 0: if INFO = -i, the i-th argument had an illegal value.
210: *> > 0: if INFO = i, and i is
211: *> <= N: the QR algorithm failed to compute all the
212: *> eigenvalues; elements 1:ILO-1 and i+1:N of W
213: *> contain those eigenvalues which have converged; if
214: *> JOBVS = 'V', VS contains the transformation which
215: *> reduces A to its partially converged Schur form.
216: *> = N+1: the eigenvalues could not be reordered because some
217: *> eigenvalues were too close to separate (the problem
218: *> is very ill-conditioned);
219: *> = N+2: after reordering, roundoff changed values of some
220: *> complex eigenvalues so that leading eigenvalues in
221: *> the Schur form no longer satisfy SELECT=.TRUE. This
222: *> could also be caused by underflow due to scaling.
223: *> \endverbatim
224: *
225: * Authors:
226: * ========
227: *
228: *> \author Univ. of Tennessee
229: *> \author Univ. of California Berkeley
230: *> \author Univ. of Colorado Denver
231: *> \author NAG Ltd.
232: *
233: *> \date November 2011
234: *
235: *> \ingroup complex16GEeigen
236: *
237: * =====================================================================
1.1 bertrand 238: SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W,
239: $ VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK,
240: $ BWORK, INFO )
241: *
1.9 bertrand 242: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 243: * -- LAPACK is a software package provided by Univ. of Tennessee, --
244: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 bertrand 245: * November 2011
1.1 bertrand 246: *
247: * .. Scalar Arguments ..
248: CHARACTER JOBVS, SENSE, SORT
249: INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
250: DOUBLE PRECISION RCONDE, RCONDV
251: * ..
252: * .. Array Arguments ..
253: LOGICAL BWORK( * )
254: DOUBLE PRECISION RWORK( * )
255: COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
256: * ..
257: * .. Function Arguments ..
258: LOGICAL SELECT
259: EXTERNAL SELECT
260: * ..
261: *
262: * =====================================================================
263: *
264: * .. Parameters ..
265: DOUBLE PRECISION ZERO, ONE
266: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
267: * ..
268: * .. Local Scalars ..
1.5 bertrand 269: LOGICAL LQUERY, SCALEA, WANTSB, WANTSE, WANTSN, WANTST,
270: $ WANTSV, WANTVS
1.1 bertrand 271: INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO,
272: $ ITAU, IWRK, LWRK, MAXWRK, MINWRK
273: DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM
274: * ..
275: * .. Local Arrays ..
276: DOUBLE PRECISION DUM( 1 )
277: * ..
278: * .. External Subroutines ..
279: EXTERNAL DLABAD, DLASCL, XERBLA, ZCOPY, ZGEBAK, ZGEBAL,
280: $ ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZTRSEN, ZUNGHR
281: * ..
282: * .. External Functions ..
283: LOGICAL LSAME
284: INTEGER ILAENV
285: DOUBLE PRECISION DLAMCH, ZLANGE
286: EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
287: * ..
288: * .. Intrinsic Functions ..
289: INTRINSIC MAX, SQRT
290: * ..
291: * .. Executable Statements ..
292: *
293: * Test the input arguments
294: *
295: INFO = 0
296: WANTVS = LSAME( JOBVS, 'V' )
297: WANTST = LSAME( SORT, 'S' )
298: WANTSN = LSAME( SENSE, 'N' )
299: WANTSE = LSAME( SENSE, 'E' )
300: WANTSV = LSAME( SENSE, 'V' )
301: WANTSB = LSAME( SENSE, 'B' )
1.5 bertrand 302: LQUERY = ( LWORK.EQ.-1 )
303: *
1.1 bertrand 304: IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
305: INFO = -1
306: ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
307: INFO = -2
308: ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
309: $ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN
310: INFO = -4
311: ELSE IF( N.LT.0 ) THEN
312: INFO = -5
313: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
314: INFO = -7
315: ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
316: INFO = -11
317: END IF
318: *
319: * Compute workspace
320: * (Note: Comments in the code beginning "Workspace:" describe the
321: * minimal amount of real workspace needed at that point in the
322: * code, as well as the preferred amount for good performance.
323: * CWorkspace refers to complex workspace, and RWorkspace to real
324: * workspace. NB refers to the optimal block size for the
325: * immediately following subroutine, as returned by ILAENV.
326: * HSWORK refers to the workspace preferred by ZHSEQR, as
327: * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
328: * the worst case.
329: * If SENSE = 'E', 'V' or 'B', then the amount of workspace needed
330: * depends on SDIM, which is computed by the routine ZTRSEN later
331: * in the code.)
332: *
333: IF( INFO.EQ.0 ) THEN
334: IF( N.EQ.0 ) THEN
335: MINWRK = 1
336: LWRK = 1
337: ELSE
338: MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 )
339: MINWRK = 2*N
340: *
341: CALL ZHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS,
342: $ WORK, -1, IEVAL )
343: HSWORK = WORK( 1 )
344: *
345: IF( .NOT.WANTVS ) THEN
346: MAXWRK = MAX( MAXWRK, HSWORK )
347: ELSE
348: MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR',
349: $ ' ', N, 1, N, -1 ) )
350: MAXWRK = MAX( MAXWRK, HSWORK )
351: END IF
352: LWRK = MAXWRK
353: IF( .NOT.WANTSN )
354: $ LWRK = MAX( LWRK, ( N*N )/2 )
355: END IF
356: WORK( 1 ) = LWRK
357: *
1.5 bertrand 358: IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
1.1 bertrand 359: INFO = -15
360: END IF
361: END IF
362: *
363: IF( INFO.NE.0 ) THEN
364: CALL XERBLA( 'ZGEESX', -INFO )
365: RETURN
1.5 bertrand 366: ELSE IF( LQUERY ) THEN
367: RETURN
1.1 bertrand 368: END IF
369: *
370: * Quick return if possible
371: *
372: IF( N.EQ.0 ) THEN
373: SDIM = 0
374: RETURN
375: END IF
376: *
377: * Get machine constants
378: *
379: EPS = DLAMCH( 'P' )
380: SMLNUM = DLAMCH( 'S' )
381: BIGNUM = ONE / SMLNUM
382: CALL DLABAD( SMLNUM, BIGNUM )
383: SMLNUM = SQRT( SMLNUM ) / EPS
384: BIGNUM = ONE / SMLNUM
385: *
386: * Scale A if max element outside range [SMLNUM,BIGNUM]
387: *
388: ANRM = ZLANGE( 'M', N, N, A, LDA, DUM )
389: SCALEA = .FALSE.
390: IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
391: SCALEA = .TRUE.
392: CSCALE = SMLNUM
393: ELSE IF( ANRM.GT.BIGNUM ) THEN
394: SCALEA = .TRUE.
395: CSCALE = BIGNUM
396: END IF
397: IF( SCALEA )
398: $ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
399: *
400: *
401: * Permute the matrix to make it more nearly triangular
402: * (CWorkspace: none)
403: * (RWorkspace: need N)
404: *
405: IBAL = 1
406: CALL ZGEBAL( 'P', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR )
407: *
408: * Reduce to upper Hessenberg form
409: * (CWorkspace: need 2*N, prefer N+N*NB)
410: * (RWorkspace: none)
411: *
412: ITAU = 1
413: IWRK = N + ITAU
414: CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
415: $ LWORK-IWRK+1, IERR )
416: *
417: IF( WANTVS ) THEN
418: *
419: * Copy Householder vectors to VS
420: *
421: CALL ZLACPY( 'L', N, N, A, LDA, VS, LDVS )
422: *
423: * Generate unitary matrix in VS
424: * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB)
425: * (RWorkspace: none)
426: *
427: CALL ZUNGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ),
428: $ LWORK-IWRK+1, IERR )
429: END IF
430: *
431: SDIM = 0
432: *
433: * Perform QR iteration, accumulating Schur vectors in VS if desired
434: * (CWorkspace: need 1, prefer HSWORK (see comments) )
435: * (RWorkspace: none)
436: *
437: IWRK = ITAU
438: CALL ZHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, VS, LDVS,
439: $ WORK( IWRK ), LWORK-IWRK+1, IEVAL )
440: IF( IEVAL.GT.0 )
441: $ INFO = IEVAL
442: *
443: * Sort eigenvalues if desired
444: *
445: IF( WANTST .AND. INFO.EQ.0 ) THEN
446: IF( SCALEA )
447: $ CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR )
448: DO 10 I = 1, N
449: BWORK( I ) = SELECT( W( I ) )
450: 10 CONTINUE
451: *
452: * Reorder eigenvalues, transform Schur vectors, and compute
453: * reciprocal condition numbers
454: * (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM)
455: * otherwise, need none )
456: * (RWorkspace: none)
457: *
458: CALL ZTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM,
459: $ RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1,
460: $ ICOND )
461: IF( .NOT.WANTSN )
462: $ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) )
463: IF( ICOND.EQ.-14 ) THEN
464: *
465: * Not enough complex workspace
466: *
467: INFO = -15
468: END IF
469: END IF
470: *
471: IF( WANTVS ) THEN
472: *
473: * Undo balancing
474: * (CWorkspace: none)
475: * (RWorkspace: need N)
476: *
477: CALL ZGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS,
478: $ IERR )
479: END IF
480: *
481: IF( SCALEA ) THEN
482: *
483: * Undo scaling for the Schur form of A
484: *
485: CALL ZLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
486: CALL ZCOPY( N, A, LDA+1, W, 1 )
487: IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN
488: DUM( 1 ) = RCONDV
489: CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR )
490: RCONDV = DUM( 1 )
491: END IF
492: END IF
493: *
494: WORK( 1 ) = MAXWRK
495: RETURN
496: *
497: * End of ZGEESX
498: *
499: END
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