1: *> \brief \b DTRSEN
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DTRSEN + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrsen.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrsen.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrsen.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
22: * M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * CHARACTER COMPQ, JOB
26: * INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
27: * DOUBLE PRECISION S, SEP
28: * ..
29: * .. Array Arguments ..
30: * LOGICAL SELECT( * )
31: * INTEGER IWORK( * )
32: * DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
33: * $ WR( * )
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: *> DTRSEN reorders the real Schur factorization of a real matrix
43: *> A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
44: *> the leading diagonal blocks of the upper quasi-triangular matrix T,
45: *> and the leading columns of Q form an orthonormal basis of the
46: *> corresponding right invariant subspace.
47: *>
48: *> Optionally the routine computes the reciprocal condition numbers of
49: *> the cluster of eigenvalues and/or the invariant subspace.
50: *>
51: *> T must be in Schur canonical form (as returned by DHSEQR), that is,
52: *> block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
53: *> 2-by-2 diagonal block has its diagonal elemnts equal and its
54: *> off-diagonal elements of opposite sign.
55: *> \endverbatim
56: *
57: * Arguments:
58: * ==========
59: *
60: *> \param[in] JOB
61: *> \verbatim
62: *> JOB is CHARACTER*1
63: *> Specifies whether condition numbers are required for the
64: *> cluster of eigenvalues (S) or the invariant subspace (SEP):
65: *> = 'N': none;
66: *> = 'E': for eigenvalues only (S);
67: *> = 'V': for invariant subspace only (SEP);
68: *> = 'B': for both eigenvalues and invariant subspace (S and
69: *> SEP).
70: *> \endverbatim
71: *>
72: *> \param[in] COMPQ
73: *> \verbatim
74: *> COMPQ is CHARACTER*1
75: *> = 'V': update the matrix Q of Schur vectors;
76: *> = 'N': do not update Q.
77: *> \endverbatim
78: *>
79: *> \param[in] SELECT
80: *> \verbatim
81: *> SELECT is LOGICAL array, dimension (N)
82: *> SELECT specifies the eigenvalues in the selected cluster. To
83: *> select a real eigenvalue w(j), SELECT(j) must be set to
84: *> .TRUE.. To select a complex conjugate pair of eigenvalues
85: *> w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
86: *> either SELECT(j) or SELECT(j+1) or both must be set to
87: *> .TRUE.; a complex conjugate pair of eigenvalues must be
88: *> either both included in the cluster or both excluded.
89: *> \endverbatim
90: *>
91: *> \param[in] N
92: *> \verbatim
93: *> N is INTEGER
94: *> The order of the matrix T. N >= 0.
95: *> \endverbatim
96: *>
97: *> \param[in,out] T
98: *> \verbatim
99: *> T is DOUBLE PRECISION array, dimension (LDT,N)
100: *> On entry, the upper quasi-triangular matrix T, in Schur
101: *> canonical form.
102: *> On exit, T is overwritten by the reordered matrix T, again in
103: *> Schur canonical form, with the selected eigenvalues in the
104: *> leading diagonal blocks.
105: *> \endverbatim
106: *>
107: *> \param[in] LDT
108: *> \verbatim
109: *> LDT is INTEGER
110: *> The leading dimension of the array T. LDT >= max(1,N).
111: *> \endverbatim
112: *>
113: *> \param[in,out] Q
114: *> \verbatim
115: *> Q is DOUBLE PRECISION array, dimension (LDQ,N)
116: *> On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
117: *> On exit, if COMPQ = 'V', Q has been postmultiplied by the
118: *> orthogonal transformation matrix which reorders T; the
119: *> leading M columns of Q form an orthonormal basis for the
120: *> specified invariant subspace.
121: *> If COMPQ = 'N', Q is not referenced.
122: *> \endverbatim
123: *>
124: *> \param[in] LDQ
125: *> \verbatim
126: *> LDQ is INTEGER
127: *> The leading dimension of the array Q.
128: *> LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
129: *> \endverbatim
130: *>
131: *> \param[out] WR
132: *> \verbatim
133: *> WR is DOUBLE PRECISION array, dimension (N)
134: *> \endverbatim
135: *> \param[out] WI
136: *> \verbatim
137: *> WI is DOUBLE PRECISION array, dimension (N)
138: *>
139: *> The real and imaginary parts, respectively, of the reordered
140: *> eigenvalues of T. The eigenvalues are stored in the same
141: *> order as on the diagonal of T, with WR(i) = T(i,i) and, if
142: *> T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
143: *> WI(i+1) = -WI(i). Note that if a complex eigenvalue is
144: *> sufficiently ill-conditioned, then its value may differ
145: *> significantly from its value before reordering.
146: *> \endverbatim
147: *>
148: *> \param[out] M
149: *> \verbatim
150: *> M is INTEGER
151: *> The dimension of the specified invariant subspace.
152: *> 0 < = M <= N.
153: *> \endverbatim
154: *>
155: *> \param[out] S
156: *> \verbatim
157: *> S is DOUBLE PRECISION
158: *> If JOB = 'E' or 'B', S is a lower bound on the reciprocal
159: *> condition number for the selected cluster of eigenvalues.
160: *> S cannot underestimate the true reciprocal condition number
161: *> by more than a factor of sqrt(N). If M = 0 or N, S = 1.
162: *> If JOB = 'N' or 'V', S is not referenced.
163: *> \endverbatim
164: *>
165: *> \param[out] SEP
166: *> \verbatim
167: *> SEP is DOUBLE PRECISION
168: *> If JOB = 'V' or 'B', SEP is the estimated reciprocal
169: *> condition number of the specified invariant subspace. If
170: *> M = 0 or N, SEP = norm(T).
171: *> If JOB = 'N' or 'E', SEP is not referenced.
172: *> \endverbatim
173: *>
174: *> \param[out] WORK
175: *> \verbatim
176: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
177: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
178: *> \endverbatim
179: *>
180: *> \param[in] LWORK
181: *> \verbatim
182: *> LWORK is INTEGER
183: *> The dimension of the array WORK.
184: *> If JOB = 'N', LWORK >= max(1,N);
185: *> if JOB = 'E', LWORK >= max(1,M*(N-M));
186: *> if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
187: *>
188: *> If LWORK = -1, then a workspace query is assumed; the routine
189: *> only calculates the optimal size of the WORK array, returns
190: *> this value as the first entry of the WORK array, and no error
191: *> message related to LWORK is issued by XERBLA.
192: *> \endverbatim
193: *>
194: *> \param[out] IWORK
195: *> \verbatim
196: *> IWORK is INTEGER array, dimension (MAX(1,LIWORK))
197: *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
198: *> \endverbatim
199: *>
200: *> \param[in] LIWORK
201: *> \verbatim
202: *> LIWORK is INTEGER
203: *> The dimension of the array IWORK.
204: *> If JOB = 'N' or 'E', LIWORK >= 1;
205: *> if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
206: *>
207: *> If LIWORK = -1, then a workspace query is assumed; the
208: *> routine only calculates the optimal size of the IWORK array,
209: *> returns this value as the first entry of the IWORK array, and
210: *> no error message related to LIWORK is issued by XERBLA.
211: *> \endverbatim
212: *>
213: *> \param[out] INFO
214: *> \verbatim
215: *> INFO is INTEGER
216: *> = 0: successful exit
217: *> < 0: if INFO = -i, the i-th argument had an illegal value
218: *> = 1: reordering of T failed because some eigenvalues are too
219: *> close to separate (the problem is very ill-conditioned);
220: *> T may have been partially reordered, and WR and WI
221: *> contain the eigenvalues in the same order as in T; S and
222: *> SEP (if requested) are set to zero.
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 doubleOTHERcomputational
236: *
237: *> \par Further Details:
238: * =====================
239: *>
240: *> \verbatim
241: *>
242: *> DTRSEN first collects the selected eigenvalues by computing an
243: *> orthogonal transformation Z to move them to the top left corner of T.
244: *> In other words, the selected eigenvalues are the eigenvalues of T11
245: *> in:
246: *>
247: *> Z**T * T * Z = ( T11 T12 ) n1
248: *> ( 0 T22 ) n2
249: *> n1 n2
250: *>
251: *> where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
252: *> of Z span the specified invariant subspace of T.
253: *>
254: *> If T has been obtained from the real Schur factorization of a matrix
255: *> A = Q*T*Q**T, then the reordered real Schur factorization of A is given
256: *> by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
257: *> the corresponding invariant subspace of A.
258: *>
259: *> The reciprocal condition number of the average of the eigenvalues of
260: *> T11 may be returned in S. S lies between 0 (very badly conditioned)
261: *> and 1 (very well conditioned). It is computed as follows. First we
262: *> compute R so that
263: *>
264: *> P = ( I R ) n1
265: *> ( 0 0 ) n2
266: *> n1 n2
267: *>
268: *> is the projector on the invariant subspace associated with T11.
269: *> R is the solution of the Sylvester equation:
270: *>
271: *> T11*R - R*T22 = T12.
272: *>
273: *> Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
274: *> the two-norm of M. Then S is computed as the lower bound
275: *>
276: *> (1 + F-norm(R)**2)**(-1/2)
277: *>
278: *> on the reciprocal of 2-norm(P), the true reciprocal condition number.
279: *> S cannot underestimate 1 / 2-norm(P) by more than a factor of
280: *> sqrt(N).
281: *>
282: *> An approximate error bound for the computed average of the
283: *> eigenvalues of T11 is
284: *>
285: *> EPS * norm(T) / S
286: *>
287: *> where EPS is the machine precision.
288: *>
289: *> The reciprocal condition number of the right invariant subspace
290: *> spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
291: *> SEP is defined as the separation of T11 and T22:
292: *>
293: *> sep( T11, T22 ) = sigma-min( C )
294: *>
295: *> where sigma-min(C) is the smallest singular value of the
296: *> n1*n2-by-n1*n2 matrix
297: *>
298: *> C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
299: *>
300: *> I(m) is an m by m identity matrix, and kprod denotes the Kronecker
301: *> product. We estimate sigma-min(C) by the reciprocal of an estimate of
302: *> the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
303: *> cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
304: *>
305: *> When SEP is small, small changes in T can cause large changes in
306: *> the invariant subspace. An approximate bound on the maximum angular
307: *> error in the computed right invariant subspace is
308: *>
309: *> EPS * norm(T) / SEP
310: *> \endverbatim
311: *>
312: * =====================================================================
313: SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
314: $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
315: *
316: * -- LAPACK computational routine (version 3.4.0) --
317: * -- LAPACK is a software package provided by Univ. of Tennessee, --
318: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
319: * November 2011
320: *
321: * .. Scalar Arguments ..
322: CHARACTER COMPQ, JOB
323: INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
324: DOUBLE PRECISION S, SEP
325: * ..
326: * .. Array Arguments ..
327: LOGICAL SELECT( * )
328: INTEGER IWORK( * )
329: DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
330: $ WR( * )
331: * ..
332: *
333: * =====================================================================
334: *
335: * .. Parameters ..
336: DOUBLE PRECISION ZERO, ONE
337: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
338: * ..
339: * .. Local Scalars ..
340: LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
341: $ WANTSP
342: INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
343: $ NN
344: DOUBLE PRECISION EST, RNORM, SCALE
345: * ..
346: * .. Local Arrays ..
347: INTEGER ISAVE( 3 )
348: * ..
349: * .. External Functions ..
350: LOGICAL LSAME
351: DOUBLE PRECISION DLANGE
352: EXTERNAL LSAME, DLANGE
353: * ..
354: * .. External Subroutines ..
355: EXTERNAL DLACN2, DLACPY, DTREXC, DTRSYL, XERBLA
356: * ..
357: * .. Intrinsic Functions ..
358: INTRINSIC ABS, MAX, SQRT
359: * ..
360: * .. Executable Statements ..
361: *
362: * Decode and test the input parameters
363: *
364: WANTBH = LSAME( JOB, 'B' )
365: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
366: WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
367: WANTQ = LSAME( COMPQ, 'V' )
368: *
369: INFO = 0
370: LQUERY = ( LWORK.EQ.-1 )
371: IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
372: $ THEN
373: INFO = -1
374: ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
375: INFO = -2
376: ELSE IF( N.LT.0 ) THEN
377: INFO = -4
378: ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
379: INFO = -6
380: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
381: INFO = -8
382: ELSE
383: *
384: * Set M to the dimension of the specified invariant subspace,
385: * and test LWORK and LIWORK.
386: *
387: M = 0
388: PAIR = .FALSE.
389: DO 10 K = 1, N
390: IF( PAIR ) THEN
391: PAIR = .FALSE.
392: ELSE
393: IF( K.LT.N ) THEN
394: IF( T( K+1, K ).EQ.ZERO ) THEN
395: IF( SELECT( K ) )
396: $ M = M + 1
397: ELSE
398: PAIR = .TRUE.
399: IF( SELECT( K ) .OR. SELECT( K+1 ) )
400: $ M = M + 2
401: END IF
402: ELSE
403: IF( SELECT( N ) )
404: $ M = M + 1
405: END IF
406: END IF
407: 10 CONTINUE
408: *
409: N1 = M
410: N2 = N - M
411: NN = N1*N2
412: *
413: IF( WANTSP ) THEN
414: LWMIN = MAX( 1, 2*NN )
415: LIWMIN = MAX( 1, NN )
416: ELSE IF( LSAME( JOB, 'N' ) ) THEN
417: LWMIN = MAX( 1, N )
418: LIWMIN = 1
419: ELSE IF( LSAME( JOB, 'E' ) ) THEN
420: LWMIN = MAX( 1, NN )
421: LIWMIN = 1
422: END IF
423: *
424: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
425: INFO = -15
426: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
427: INFO = -17
428: END IF
429: END IF
430: *
431: IF( INFO.EQ.0 ) THEN
432: WORK( 1 ) = LWMIN
433: IWORK( 1 ) = LIWMIN
434: END IF
435: *
436: IF( INFO.NE.0 ) THEN
437: CALL XERBLA( 'DTRSEN', -INFO )
438: RETURN
439: ELSE IF( LQUERY ) THEN
440: RETURN
441: END IF
442: *
443: * Quick return if possible.
444: *
445: IF( M.EQ.N .OR. M.EQ.0 ) THEN
446: IF( WANTS )
447: $ S = ONE
448: IF( WANTSP )
449: $ SEP = DLANGE( '1', N, N, T, LDT, WORK )
450: GO TO 40
451: END IF
452: *
453: * Collect the selected blocks at the top-left corner of T.
454: *
455: KS = 0
456: PAIR = .FALSE.
457: DO 20 K = 1, N
458: IF( PAIR ) THEN
459: PAIR = .FALSE.
460: ELSE
461: SWAP = SELECT( K )
462: IF( K.LT.N ) THEN
463: IF( T( K+1, K ).NE.ZERO ) THEN
464: PAIR = .TRUE.
465: SWAP = SWAP .OR. SELECT( K+1 )
466: END IF
467: END IF
468: IF( SWAP ) THEN
469: KS = KS + 1
470: *
471: * Swap the K-th block to position KS.
472: *
473: IERR = 0
474: KK = K
475: IF( K.NE.KS )
476: $ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
477: $ IERR )
478: IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
479: *
480: * Blocks too close to swap: exit.
481: *
482: INFO = 1
483: IF( WANTS )
484: $ S = ZERO
485: IF( WANTSP )
486: $ SEP = ZERO
487: GO TO 40
488: END IF
489: IF( PAIR )
490: $ KS = KS + 1
491: END IF
492: END IF
493: 20 CONTINUE
494: *
495: IF( WANTS ) THEN
496: *
497: * Solve Sylvester equation for R:
498: *
499: * T11*R - R*T22 = scale*T12
500: *
501: CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
502: CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
503: $ LDT, WORK, N1, SCALE, IERR )
504: *
505: * Estimate the reciprocal of the condition number of the cluster
506: * of eigenvalues.
507: *
508: RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
509: IF( RNORM.EQ.ZERO ) THEN
510: S = ONE
511: ELSE
512: S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
513: $ SQRT( RNORM ) )
514: END IF
515: END IF
516: *
517: IF( WANTSP ) THEN
518: *
519: * Estimate sep(T11,T22).
520: *
521: EST = ZERO
522: KASE = 0
523: 30 CONTINUE
524: CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE )
525: IF( KASE.NE.0 ) THEN
526: IF( KASE.EQ.1 ) THEN
527: *
528: * Solve T11*R - R*T22 = scale*X.
529: *
530: CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
531: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
532: $ IERR )
533: ELSE
534: *
535: * Solve T11**T*R - R*T22**T = scale*X.
536: *
537: CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
538: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
539: $ IERR )
540: END IF
541: GO TO 30
542: END IF
543: *
544: SEP = SCALE / EST
545: END IF
546: *
547: 40 CONTINUE
548: *
549: * Store the output eigenvalues in WR and WI.
550: *
551: DO 50 K = 1, N
552: WR( K ) = T( K, K )
553: WI( K ) = ZERO
554: 50 CONTINUE
555: DO 60 K = 1, N - 1
556: IF( T( K+1, K ).NE.ZERO ) THEN
557: WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
558: $ SQRT( ABS( T( K+1, K ) ) )
559: WI( K+1 ) = -WI( K )
560: END IF
561: 60 CONTINUE
562: *
563: WORK( 1 ) = LWMIN
564: IWORK( 1 ) = LIWMIN
565: *
566: RETURN
567: *
568: * End of DTRSEN
569: *
570: END
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