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 elements 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: *> \ingroup doubleOTHERcomputational
234: *
235: *> \par Further Details:
236: * =====================
237: *>
238: *> \verbatim
239: *>
240: *> DTRSEN first collects the selected eigenvalues by computing an
241: *> orthogonal transformation Z to move them to the top left corner of T.
242: *> In other words, the selected eigenvalues are the eigenvalues of T11
243: *> in:
244: *>
245: *> Z**T * T * Z = ( T11 T12 ) n1
246: *> ( 0 T22 ) n2
247: *> n1 n2
248: *>
249: *> where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
250: *> of Z span the specified invariant subspace of T.
251: *>
252: *> If T has been obtained from the real Schur factorization of a matrix
253: *> A = Q*T*Q**T, then the reordered real Schur factorization of A is given
254: *> by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
255: *> the corresponding invariant subspace of A.
256: *>
257: *> The reciprocal condition number of the average of the eigenvalues of
258: *> T11 may be returned in S. S lies between 0 (very badly conditioned)
259: *> and 1 (very well conditioned). It is computed as follows. First we
260: *> compute R so that
261: *>
262: *> P = ( I R ) n1
263: *> ( 0 0 ) n2
264: *> n1 n2
265: *>
266: *> is the projector on the invariant subspace associated with T11.
267: *> R is the solution of the Sylvester equation:
268: *>
269: *> T11*R - R*T22 = T12.
270: *>
271: *> Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
272: *> the two-norm of M. Then S is computed as the lower bound
273: *>
274: *> (1 + F-norm(R)**2)**(-1/2)
275: *>
276: *> on the reciprocal of 2-norm(P), the true reciprocal condition number.
277: *> S cannot underestimate 1 / 2-norm(P) by more than a factor of
278: *> sqrt(N).
279: *>
280: *> An approximate error bound for the computed average of the
281: *> eigenvalues of T11 is
282: *>
283: *> EPS * norm(T) / S
284: *>
285: *> where EPS is the machine precision.
286: *>
287: *> The reciprocal condition number of the right invariant subspace
288: *> spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
289: *> SEP is defined as the separation of T11 and T22:
290: *>
291: *> sep( T11, T22 ) = sigma-min( C )
292: *>
293: *> where sigma-min(C) is the smallest singular value of the
294: *> n1*n2-by-n1*n2 matrix
295: *>
296: *> C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
297: *>
298: *> I(m) is an m by m identity matrix, and kprod denotes the Kronecker
299: *> product. We estimate sigma-min(C) by the reciprocal of an estimate of
300: *> the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
301: *> cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
302: *>
303: *> When SEP is small, small changes in T can cause large changes in
304: *> the invariant subspace. An approximate bound on the maximum angular
305: *> error in the computed right invariant subspace is
306: *>
307: *> EPS * norm(T) / SEP
308: *> \endverbatim
309: *>
310: * =====================================================================
311: SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
312: $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
313: *
314: * -- LAPACK computational routine --
315: * -- LAPACK is a software package provided by Univ. of Tennessee, --
316: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
317: *
318: * .. Scalar Arguments ..
319: CHARACTER COMPQ, JOB
320: INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
321: DOUBLE PRECISION S, SEP
322: * ..
323: * .. Array Arguments ..
324: LOGICAL SELECT( * )
325: INTEGER IWORK( * )
326: DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
327: $ WR( * )
328: * ..
329: *
330: * =====================================================================
331: *
332: * .. Parameters ..
333: DOUBLE PRECISION ZERO, ONE
334: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
335: * ..
336: * .. Local Scalars ..
337: LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
338: $ WANTSP
339: INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
340: $ NN
341: DOUBLE PRECISION EST, RNORM, SCALE
342: * ..
343: * .. Local Arrays ..
344: INTEGER ISAVE( 3 )
345: * ..
346: * .. External Functions ..
347: LOGICAL LSAME
348: DOUBLE PRECISION DLANGE
349: EXTERNAL LSAME, DLANGE
350: * ..
351: * .. External Subroutines ..
352: EXTERNAL DLACN2, DLACPY, DTREXC, DTRSYL, XERBLA
353: * ..
354: * .. Intrinsic Functions ..
355: INTRINSIC ABS, MAX, SQRT
356: * ..
357: * .. Executable Statements ..
358: *
359: * Decode and test the input parameters
360: *
361: WANTBH = LSAME( JOB, 'B' )
362: WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
363: WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
364: WANTQ = LSAME( COMPQ, 'V' )
365: *
366: INFO = 0
367: LQUERY = ( LWORK.EQ.-1 )
368: IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
369: $ THEN
370: INFO = -1
371: ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
372: INFO = -2
373: ELSE IF( N.LT.0 ) THEN
374: INFO = -4
375: ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
376: INFO = -6
377: ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
378: INFO = -8
379: ELSE
380: *
381: * Set M to the dimension of the specified invariant subspace,
382: * and test LWORK and LIWORK.
383: *
384: M = 0
385: PAIR = .FALSE.
386: DO 10 K = 1, N
387: IF( PAIR ) THEN
388: PAIR = .FALSE.
389: ELSE
390: IF( K.LT.N ) THEN
391: IF( T( K+1, K ).EQ.ZERO ) THEN
392: IF( SELECT( K ) )
393: $ M = M + 1
394: ELSE
395: PAIR = .TRUE.
396: IF( SELECT( K ) .OR. SELECT( K+1 ) )
397: $ M = M + 2
398: END IF
399: ELSE
400: IF( SELECT( N ) )
401: $ M = M + 1
402: END IF
403: END IF
404: 10 CONTINUE
405: *
406: N1 = M
407: N2 = N - M
408: NN = N1*N2
409: *
410: IF( WANTSP ) THEN
411: LWMIN = MAX( 1, 2*NN )
412: LIWMIN = MAX( 1, NN )
413: ELSE IF( LSAME( JOB, 'N' ) ) THEN
414: LWMIN = MAX( 1, N )
415: LIWMIN = 1
416: ELSE IF( LSAME( JOB, 'E' ) ) THEN
417: LWMIN = MAX( 1, NN )
418: LIWMIN = 1
419: END IF
420: *
421: IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
422: INFO = -15
423: ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
424: INFO = -17
425: END IF
426: END IF
427: *
428: IF( INFO.EQ.0 ) THEN
429: WORK( 1 ) = LWMIN
430: IWORK( 1 ) = LIWMIN
431: END IF
432: *
433: IF( INFO.NE.0 ) THEN
434: CALL XERBLA( 'DTRSEN', -INFO )
435: RETURN
436: ELSE IF( LQUERY ) THEN
437: RETURN
438: END IF
439: *
440: * Quick return if possible.
441: *
442: IF( M.EQ.N .OR. M.EQ.0 ) THEN
443: IF( WANTS )
444: $ S = ONE
445: IF( WANTSP )
446: $ SEP = DLANGE( '1', N, N, T, LDT, WORK )
447: GO TO 40
448: END IF
449: *
450: * Collect the selected blocks at the top-left corner of T.
451: *
452: KS = 0
453: PAIR = .FALSE.
454: DO 20 K = 1, N
455: IF( PAIR ) THEN
456: PAIR = .FALSE.
457: ELSE
458: SWAP = SELECT( K )
459: IF( K.LT.N ) THEN
460: IF( T( K+1, K ).NE.ZERO ) THEN
461: PAIR = .TRUE.
462: SWAP = SWAP .OR. SELECT( K+1 )
463: END IF
464: END IF
465: IF( SWAP ) THEN
466: KS = KS + 1
467: *
468: * Swap the K-th block to position KS.
469: *
470: IERR = 0
471: KK = K
472: IF( K.NE.KS )
473: $ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
474: $ IERR )
475: IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
476: *
477: * Blocks too close to swap: exit.
478: *
479: INFO = 1
480: IF( WANTS )
481: $ S = ZERO
482: IF( WANTSP )
483: $ SEP = ZERO
484: GO TO 40
485: END IF
486: IF( PAIR )
487: $ KS = KS + 1
488: END IF
489: END IF
490: 20 CONTINUE
491: *
492: IF( WANTS ) THEN
493: *
494: * Solve Sylvester equation for R:
495: *
496: * T11*R - R*T22 = scale*T12
497: *
498: CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
499: CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
500: $ LDT, WORK, N1, SCALE, IERR )
501: *
502: * Estimate the reciprocal of the condition number of the cluster
503: * of eigenvalues.
504: *
505: RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
506: IF( RNORM.EQ.ZERO ) THEN
507: S = ONE
508: ELSE
509: S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
510: $ SQRT( RNORM ) )
511: END IF
512: END IF
513: *
514: IF( WANTSP ) THEN
515: *
516: * Estimate sep(T11,T22).
517: *
518: EST = ZERO
519: KASE = 0
520: 30 CONTINUE
521: CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE )
522: IF( KASE.NE.0 ) THEN
523: IF( KASE.EQ.1 ) THEN
524: *
525: * Solve T11*R - R*T22 = scale*X.
526: *
527: CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
528: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
529: $ IERR )
530: ELSE
531: *
532: * Solve T11**T*R - R*T22**T = scale*X.
533: *
534: CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
535: $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
536: $ IERR )
537: END IF
538: GO TO 30
539: END IF
540: *
541: SEP = SCALE / EST
542: END IF
543: *
544: 40 CONTINUE
545: *
546: * Store the output eigenvalues in WR and WI.
547: *
548: DO 50 K = 1, N
549: WR( K ) = T( K, K )
550: WI( K ) = ZERO
551: 50 CONTINUE
552: DO 60 K = 1, N - 1
553: IF( T( K+1, K ).NE.ZERO ) THEN
554: WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
555: $ SQRT( ABS( T( K+1, K ) ) )
556: WI( K+1 ) = -WI( K )
557: END IF
558: 60 CONTINUE
559: *
560: WORK( 1 ) = LWMIN
561: IWORK( 1 ) = LIWMIN
562: *
563: RETURN
564: *
565: * End of DTRSEN
566: *
567: END
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