001:       SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
002:      $                   M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
003: *
004: *  -- LAPACK routine (version 3.2) --
005: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
006: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
007: *     November 2006
008: *
009: *     .. Scalar Arguments ..
010:       CHARACTER          COMPQ, JOB
011:       INTEGER            INFO, LDQ, LDT, LIWORK, LWORK, M, N
012:       DOUBLE PRECISION   S, SEP
013: *     ..
014: *     .. Array Arguments ..
015:       LOGICAL            SELECT( * )
016:       INTEGER            IWORK( * )
017:       DOUBLE PRECISION   Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
018:      $                   WR( * )
019: *     ..
020: *
021: *  Purpose
022: *  =======
023: *
024: *  DTRSEN reorders the real Schur factorization of a real matrix
025: *  A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
026: *  the leading diagonal blocks of the upper quasi-triangular matrix T,
027: *  and the leading columns of Q form an orthonormal basis of the
028: *  corresponding right invariant subspace.
029: *
030: *  Optionally the routine computes the reciprocal condition numbers of
031: *  the cluster of eigenvalues and/or the invariant subspace.
032: *
033: *  T must be in Schur canonical form (as returned by DHSEQR), that is,
034: *  block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
035: *  2-by-2 diagonal block has its diagonal elemnts equal and its
036: *  off-diagonal elements of opposite sign.
037: *
038: *  Arguments
039: *  =========
040: *
041: *  JOB     (input) CHARACTER*1
042: *          Specifies whether condition numbers are required for the
043: *          cluster of eigenvalues (S) or the invariant subspace (SEP):
044: *          = 'N': none;
045: *          = 'E': for eigenvalues only (S);
046: *          = 'V': for invariant subspace only (SEP);
047: *          = 'B': for both eigenvalues and invariant subspace (S and
048: *                 SEP).
049: *
050: *  COMPQ   (input) CHARACTER*1
051: *          = 'V': update the matrix Q of Schur vectors;
052: *          = 'N': do not update Q.
053: *
054: *  SELECT  (input) LOGICAL array, dimension (N)
055: *          SELECT specifies the eigenvalues in the selected cluster. To
056: *          select a real eigenvalue w(j), SELECT(j) must be set to
057: *          .TRUE.. To select a complex conjugate pair of eigenvalues
058: *          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
059: *          either SELECT(j) or SELECT(j+1) or both must be set to
060: *          .TRUE.; a complex conjugate pair of eigenvalues must be
061: *          either both included in the cluster or both excluded.
062: *
063: *  N       (input) INTEGER
064: *          The order of the matrix T. N >= 0.
065: *
066: *  T       (input/output) DOUBLE PRECISION array, dimension (LDT,N)
067: *          On entry, the upper quasi-triangular matrix T, in Schur
068: *          canonical form.
069: *          On exit, T is overwritten by the reordered matrix T, again in
070: *          Schur canonical form, with the selected eigenvalues in the
071: *          leading diagonal blocks.
072: *
073: *  LDT     (input) INTEGER
074: *          The leading dimension of the array T. LDT >= max(1,N).
075: *
076: *  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
077: *          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
078: *          On exit, if COMPQ = 'V', Q has been postmultiplied by the
079: *          orthogonal transformation matrix which reorders T; the
080: *          leading M columns of Q form an orthonormal basis for the
081: *          specified invariant subspace.
082: *          If COMPQ = 'N', Q is not referenced.
083: *
084: *  LDQ     (input) INTEGER
085: *          The leading dimension of the array Q.
086: *          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
087: *
088: *  WR      (output) DOUBLE PRECISION array, dimension (N)
089: *  WI      (output) DOUBLE PRECISION array, dimension (N)
090: *          The real and imaginary parts, respectively, of the reordered
091: *          eigenvalues of T. The eigenvalues are stored in the same
092: *          order as on the diagonal of T, with WR(i) = T(i,i) and, if
093: *          T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
094: *          WI(i+1) = -WI(i). Note that if a complex eigenvalue is
095: *          sufficiently ill-conditioned, then its value may differ
096: *          significantly from its value before reordering.
097: *
098: *  M       (output) INTEGER
099: *          The dimension of the specified invariant subspace.
100: *          0 < = M <= N.
101: *
102: *  S       (output) DOUBLE PRECISION
103: *          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
104: *          condition number for the selected cluster of eigenvalues.
105: *          S cannot underestimate the true reciprocal condition number
106: *          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
107: *          If JOB = 'N' or 'V', S is not referenced.
108: *
109: *  SEP     (output) DOUBLE PRECISION
110: *          If JOB = 'V' or 'B', SEP is the estimated reciprocal
111: *          condition number of the specified invariant subspace. If
112: *          M = 0 or N, SEP = norm(T).
113: *          If JOB = 'N' or 'E', SEP is not referenced.
114: *
115: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
116: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
117: *
118: *  LWORK   (input) INTEGER
119: *          The dimension of the array WORK.
120: *          If JOB = 'N', LWORK >= max(1,N);
121: *          if JOB = 'E', LWORK >= max(1,M*(N-M));
122: *          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
123: *
124: *          If LWORK = -1, then a workspace query is assumed; the routine
125: *          only calculates the optimal size of the WORK array, returns
126: *          this value as the first entry of the WORK array, and no error
127: *          message related to LWORK is issued by XERBLA.
128: *
129: *  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
130: *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
131: *
132: *  LIWORK  (input) INTEGER
133: *          The dimension of the array IWORK.
134: *          If JOB = 'N' or 'E', LIWORK >= 1;
135: *          if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).
136: *
137: *          If LIWORK = -1, then a workspace query is assumed; the
138: *          routine only calculates the optimal size of the IWORK array,
139: *          returns this value as the first entry of the IWORK array, and
140: *          no error message related to LIWORK is issued by XERBLA.
141: *
142: *  INFO    (output) INTEGER
143: *          = 0: successful exit
144: *          < 0: if INFO = -i, the i-th argument had an illegal value
145: *          = 1: reordering of T failed because some eigenvalues are too
146: *               close to separate (the problem is very ill-conditioned);
147: *               T may have been partially reordered, and WR and WI
148: *               contain the eigenvalues in the same order as in T; S and
149: *               SEP (if requested) are set to zero.
150: *
151: *  Further Details
152: *  ===============
153: *
154: *  DTRSEN first collects the selected eigenvalues by computing an
155: *  orthogonal transformation Z to move them to the top left corner of T.
156: *  In other words, the selected eigenvalues are the eigenvalues of T11
157: *  in:
158: *
159: *                Z'*T*Z = ( T11 T12 ) n1
160: *                         (  0  T22 ) n2
161: *                            n1  n2
162: *
163: *  where N = n1+n2 and Z' means the transpose of Z. The first n1 columns
164: *  of Z span the specified invariant subspace of T.
165: *
166: *  If T has been obtained from the real Schur factorization of a matrix
167: *  A = Q*T*Q', then the reordered real Schur factorization of A is given
168: *  by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span
169: *  the corresponding invariant subspace of A.
170: *
171: *  The reciprocal condition number of the average of the eigenvalues of
172: *  T11 may be returned in S. S lies between 0 (very badly conditioned)
173: *  and 1 (very well conditioned). It is computed as follows. First we
174: *  compute R so that
175: *
176: *                         P = ( I  R ) n1
177: *                             ( 0  0 ) n2
178: *                               n1 n2
179: *
180: *  is the projector on the invariant subspace associated with T11.
181: *  R is the solution of the Sylvester equation:
182: *
183: *                        T11*R - R*T22 = T12.
184: *
185: *  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
186: *  the two-norm of M. Then S is computed as the lower bound
187: *
188: *                      (1 + F-norm(R)**2)**(-1/2)
189: *
190: *  on the reciprocal of 2-norm(P), the true reciprocal condition number.
191: *  S cannot underestimate 1 / 2-norm(P) by more than a factor of
192: *  sqrt(N).
193: *
194: *  An approximate error bound for the computed average of the
195: *  eigenvalues of T11 is
196: *
197: *                         EPS * norm(T) / S
198: *
199: *  where EPS is the machine precision.
200: *
201: *  The reciprocal condition number of the right invariant subspace
202: *  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
203: *  SEP is defined as the separation of T11 and T22:
204: *
205: *                     sep( T11, T22 ) = sigma-min( C )
206: *
207: *  where sigma-min(C) is the smallest singular value of the
208: *  n1*n2-by-n1*n2 matrix
209: *
210: *     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
211: *
212: *  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
213: *  product. We estimate sigma-min(C) by the reciprocal of an estimate of
214: *  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
215: *  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
216: *
217: *  When SEP is small, small changes in T can cause large changes in
218: *  the invariant subspace. An approximate bound on the maximum angular
219: *  error in the computed right invariant subspace is
220: *
221: *                      EPS * norm(T) / SEP
222: *
223: *  =====================================================================
224: *
225: *     .. Parameters ..
226:       DOUBLE PRECISION   ZERO, ONE
227:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
228: *     ..
229: *     .. Local Scalars ..
230:       LOGICAL            LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
231:      $                   WANTSP
232:       INTEGER            IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
233:      $                   NN
234:       DOUBLE PRECISION   EST, RNORM, SCALE
235: *     ..
236: *     .. Local Arrays ..
237:       INTEGER            ISAVE( 3 )
238: *     ..
239: *     .. External Functions ..
240:       LOGICAL            LSAME
241:       DOUBLE PRECISION   DLANGE
242:       EXTERNAL           LSAME, DLANGE
243: *     ..
244: *     .. External Subroutines ..
245:       EXTERNAL           DLACN2, DLACPY, DTREXC, DTRSYL, XERBLA
246: *     ..
247: *     .. Intrinsic Functions ..
248:       INTRINSIC          ABS, MAX, SQRT
249: *     ..
250: *     .. Executable Statements ..
251: *
252: *     Decode and test the input parameters
253: *
254:       WANTBH = LSAME( JOB, 'B' )
255:       WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
256:       WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
257:       WANTQ = LSAME( COMPQ, 'V' )
258: *
259:       INFO = 0
260:       LQUERY = ( LWORK.EQ.-1 )
261:       IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
262:      $     THEN
263:          INFO = -1
264:       ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
265:          INFO = -2
266:       ELSE IF( N.LT.0 ) THEN
267:          INFO = -4
268:       ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
269:          INFO = -6
270:       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
271:          INFO = -8
272:       ELSE
273: *
274: *        Set M to the dimension of the specified invariant subspace,
275: *        and test LWORK and LIWORK.
276: *
277:          M = 0
278:          PAIR = .FALSE.
279:          DO 10 K = 1, N
280:             IF( PAIR ) THEN
281:                PAIR = .FALSE.
282:             ELSE
283:                IF( K.LT.N ) THEN
284:                   IF( T( K+1, K ).EQ.ZERO ) THEN
285:                      IF( SELECT( K ) )
286:      $                  M = M + 1
287:                   ELSE
288:                      PAIR = .TRUE.
289:                      IF( SELECT( K ) .OR. SELECT( K+1 ) )
290:      $                  M = M + 2
291:                   END IF
292:                ELSE
293:                   IF( SELECT( N ) )
294:      $               M = M + 1
295:                END IF
296:             END IF
297:    10    CONTINUE
298: *
299:          N1 = M
300:          N2 = N - M
301:          NN = N1*N2
302: *
303:          IF( WANTSP ) THEN
304:             LWMIN = MAX( 1, 2*NN )
305:             LIWMIN = MAX( 1, NN )
306:          ELSE IF( LSAME( JOB, 'N' ) ) THEN
307:             LWMIN = MAX( 1, N )
308:             LIWMIN = 1
309:          ELSE IF( LSAME( JOB, 'E' ) ) THEN
310:             LWMIN = MAX( 1, NN )
311:             LIWMIN = 1
312:          END IF
313: *
314:          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
315:             INFO = -15
316:          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
317:             INFO = -17
318:          END IF
319:       END IF
320: *
321:       IF( INFO.EQ.0 ) THEN
322:          WORK( 1 ) = LWMIN
323:          IWORK( 1 ) = LIWMIN
324:       END IF
325: *
326:       IF( INFO.NE.0 ) THEN
327:          CALL XERBLA( 'DTRSEN', -INFO )
328:          RETURN
329:       ELSE IF( LQUERY ) THEN
330:          RETURN
331:       END IF
332: *
333: *     Quick return if possible.
334: *
335:       IF( M.EQ.N .OR. M.EQ.0 ) THEN
336:          IF( WANTS )
337:      $      S = ONE
338:          IF( WANTSP )
339:      $      SEP = DLANGE( '1', N, N, T, LDT, WORK )
340:          GO TO 40
341:       END IF
342: *
343: *     Collect the selected blocks at the top-left corner of T.
344: *
345:       KS = 0
346:       PAIR = .FALSE.
347:       DO 20 K = 1, N
348:          IF( PAIR ) THEN
349:             PAIR = .FALSE.
350:          ELSE
351:             SWAP = SELECT( K )
352:             IF( K.LT.N ) THEN
353:                IF( T( K+1, K ).NE.ZERO ) THEN
354:                   PAIR = .TRUE.
355:                   SWAP = SWAP .OR. SELECT( K+1 )
356:                END IF
357:             END IF
358:             IF( SWAP ) THEN
359:                KS = KS + 1
360: *
361: *              Swap the K-th block to position KS.
362: *
363:                IERR = 0
364:                KK = K
365:                IF( K.NE.KS )
366:      $            CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK,
367:      $                         IERR )
368:                IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN
369: *
370: *                 Blocks too close to swap: exit.
371: *
372:                   INFO = 1
373:                   IF( WANTS )
374:      $               S = ZERO
375:                   IF( WANTSP )
376:      $               SEP = ZERO
377:                   GO TO 40
378:                END IF
379:                IF( PAIR )
380:      $            KS = KS + 1
381:             END IF
382:          END IF
383:    20 CONTINUE
384: *
385:       IF( WANTS ) THEN
386: *
387: *        Solve Sylvester equation for R:
388: *
389: *           T11*R - R*T22 = scale*T12
390: *
391:          CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
392:          CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
393:      $                LDT, WORK, N1, SCALE, IERR )
394: *
395: *        Estimate the reciprocal of the condition number of the cluster
396: *        of eigenvalues.
397: *
398:          RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK )
399:          IF( RNORM.EQ.ZERO ) THEN
400:             S = ONE
401:          ELSE
402:             S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
403:      $          SQRT( RNORM ) )
404:          END IF
405:       END IF
406: *
407:       IF( WANTSP ) THEN
408: *
409: *        Estimate sep(T11,T22).
410: *
411:          EST = ZERO
412:          KASE = 0
413:    30    CONTINUE
414:          CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE )
415:          IF( KASE.NE.0 ) THEN
416:             IF( KASE.EQ.1 ) THEN
417: *
418: *              Solve  T11*R - R*T22 = scale*X.
419: *
420:                CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
421:      $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
422:      $                      IERR )
423:             ELSE
424: *
425: *              Solve  T11'*R - R*T22' = scale*X.
426: *
427:                CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT,
428:      $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
429:      $                      IERR )
430:             END IF
431:             GO TO 30
432:          END IF
433: *
434:          SEP = SCALE / EST
435:       END IF
436: *
437:    40 CONTINUE
438: *
439: *     Store the output eigenvalues in WR and WI.
440: *
441:       DO 50 K = 1, N
442:          WR( K ) = T( K, K )
443:          WI( K ) = ZERO
444:    50 CONTINUE
445:       DO 60 K = 1, N - 1
446:          IF( T( K+1, K ).NE.ZERO ) THEN
447:             WI( K ) = SQRT( ABS( T( K, K+1 ) ) )*
448:      $                SQRT( ABS( T( K+1, K ) ) )
449:             WI( K+1 ) = -WI( K )
450:          END IF
451:    60 CONTINUE
452: *
453:       WORK( 1 ) = LWMIN
454:       IWORK( 1 ) = LIWMIN
455: *
456:       RETURN
457: *
458: *     End of DTRSEN
459: *
460:       END
461: