001:       SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
002:      $                   SEP, WORK, LWORK, 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: *     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
010: *
011: *     .. Scalar Arguments ..
012:       CHARACTER          COMPQ, JOB
013:       INTEGER            INFO, LDQ, LDT, LWORK, M, N
014:       DOUBLE PRECISION   S, SEP
015: *     ..
016: *     .. Array Arguments ..
017:       LOGICAL            SELECT( * )
018:       COMPLEX*16         Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
019: *     ..
020: *
021: *  Purpose
022: *  =======
023: *
024: *  ZTRSEN reorders the Schur factorization of a complex matrix
025: *  A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
026: *  the leading positions on the diagonal of the upper triangular matrix
027: *  T, 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: *  Arguments
034: *  =========
035: *
036: *  JOB     (input) CHARACTER*1
037: *          Specifies whether condition numbers are required for the
038: *          cluster of eigenvalues (S) or the invariant subspace (SEP):
039: *          = 'N': none;
040: *          = 'E': for eigenvalues only (S);
041: *          = 'V': for invariant subspace only (SEP);
042: *          = 'B': for both eigenvalues and invariant subspace (S and
043: *                 SEP).
044: *
045: *  COMPQ   (input) CHARACTER*1
046: *          = 'V': update the matrix Q of Schur vectors;
047: *          = 'N': do not update Q.
048: *
049: *  SELECT  (input) LOGICAL array, dimension (N)
050: *          SELECT specifies the eigenvalues in the selected cluster. To
051: *          select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
052: *
053: *  N       (input) INTEGER
054: *          The order of the matrix T. N >= 0.
055: *
056: *  T       (input/output) COMPLEX*16 array, dimension (LDT,N)
057: *          On entry, the upper triangular matrix T.
058: *          On exit, T is overwritten by the reordered matrix T, with the
059: *          selected eigenvalues as the leading diagonal elements.
060: *
061: *  LDT     (input) INTEGER
062: *          The leading dimension of the array T. LDT >= max(1,N).
063: *
064: *  Q       (input/output) COMPLEX*16 array, dimension (LDQ,N)
065: *          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
066: *          On exit, if COMPQ = 'V', Q has been postmultiplied by the
067: *          unitary transformation matrix which reorders T; the leading M
068: *          columns of Q form an orthonormal basis for the specified
069: *          invariant subspace.
070: *          If COMPQ = 'N', Q is not referenced.
071: *
072: *  LDQ     (input) INTEGER
073: *          The leading dimension of the array Q.
074: *          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
075: *
076: *  W       (output) COMPLEX*16 array, dimension (N)
077: *          The reordered eigenvalues of T, in the same order as they
078: *          appear on the diagonal of T.
079: *
080: *  M       (output) INTEGER
081: *          The dimension of the specified invariant subspace.
082: *          0 <= M <= N.
083: *
084: *  S       (output) DOUBLE PRECISION
085: *          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
086: *          condition number for the selected cluster of eigenvalues.
087: *          S cannot underestimate the true reciprocal condition number
088: *          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
089: *          If JOB = 'N' or 'V', S is not referenced.
090: *
091: *  SEP     (output) DOUBLE PRECISION
092: *          If JOB = 'V' or 'B', SEP is the estimated reciprocal
093: *          condition number of the specified invariant subspace. If
094: *          M = 0 or N, SEP = norm(T).
095: *          If JOB = 'N' or 'E', SEP is not referenced.
096: *
097: *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
098: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
099: *
100: *  LWORK   (input) INTEGER
101: *          The dimension of the array WORK.
102: *          If JOB = 'N', LWORK >= 1;
103: *          if JOB = 'E', LWORK = max(1,M*(N-M));
104: *          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
105: *
106: *          If LWORK = -1, then a workspace query is assumed; the routine
107: *          only calculates the optimal size of the WORK array, returns
108: *          this value as the first entry of the WORK array, and no error
109: *          message related to LWORK is issued by XERBLA.
110: *
111: *  INFO    (output) INTEGER
112: *          = 0:  successful exit
113: *          < 0:  if INFO = -i, the i-th argument had an illegal value
114: *
115: *  Further Details
116: *  ===============
117: *
118: *  ZTRSEN first collects the selected eigenvalues by computing a unitary
119: *  transformation Z to move them to the top left corner of T. In other
120: *  words, the selected eigenvalues are the eigenvalues of T11 in:
121: *
122: *                Z'*T*Z = ( T11 T12 ) n1
123: *                         (  0  T22 ) n2
124: *                            n1  n2
125: *
126: *  where N = n1+n2 and Z' means the conjugate transpose of Z. The first
127: *  n1 columns of Z span the specified invariant subspace of T.
128: *
129: *  If T has been obtained from the Schur factorization of a matrix
130: *  A = Q*T*Q', then the reordered Schur factorization of A is given by
131: *  A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the
132: *  corresponding invariant subspace of A.
133: *
134: *  The reciprocal condition number of the average of the eigenvalues of
135: *  T11 may be returned in S. S lies between 0 (very badly conditioned)
136: *  and 1 (very well conditioned). It is computed as follows. First we
137: *  compute R so that
138: *
139: *                         P = ( I  R ) n1
140: *                             ( 0  0 ) n2
141: *                               n1 n2
142: *
143: *  is the projector on the invariant subspace associated with T11.
144: *  R is the solution of the Sylvester equation:
145: *
146: *                        T11*R - R*T22 = T12.
147: *
148: *  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
149: *  the two-norm of M. Then S is computed as the lower bound
150: *
151: *                      (1 + F-norm(R)**2)**(-1/2)
152: *
153: *  on the reciprocal of 2-norm(P), the true reciprocal condition number.
154: *  S cannot underestimate 1 / 2-norm(P) by more than a factor of
155: *  sqrt(N).
156: *
157: *  An approximate error bound for the computed average of the
158: *  eigenvalues of T11 is
159: *
160: *                         EPS * norm(T) / S
161: *
162: *  where EPS is the machine precision.
163: *
164: *  The reciprocal condition number of the right invariant subspace
165: *  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
166: *  SEP is defined as the separation of T11 and T22:
167: *
168: *                     sep( T11, T22 ) = sigma-min( C )
169: *
170: *  where sigma-min(C) is the smallest singular value of the
171: *  n1*n2-by-n1*n2 matrix
172: *
173: *     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
174: *
175: *  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
176: *  product. We estimate sigma-min(C) by the reciprocal of an estimate of
177: *  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
178: *  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
179: *
180: *  When SEP is small, small changes in T can cause large changes in
181: *  the invariant subspace. An approximate bound on the maximum angular
182: *  error in the computed right invariant subspace is
183: *
184: *                      EPS * norm(T) / SEP
185: *
186: *  =====================================================================
187: *
188: *     .. Parameters ..
189:       DOUBLE PRECISION   ZERO, ONE
190:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
191: *     ..
192: *     .. Local Scalars ..
193:       LOGICAL            LQUERY, WANTBH, WANTQ, WANTS, WANTSP
194:       INTEGER            IERR, K, KASE, KS, LWMIN, N1, N2, NN
195:       DOUBLE PRECISION   EST, RNORM, SCALE
196: *     ..
197: *     .. Local Arrays ..
198:       INTEGER            ISAVE( 3 )
199:       DOUBLE PRECISION   RWORK( 1 )
200: *     ..
201: *     .. External Functions ..
202:       LOGICAL            LSAME
203:       DOUBLE PRECISION   ZLANGE
204:       EXTERNAL           LSAME, ZLANGE
205: *     ..
206: *     .. External Subroutines ..
207:       EXTERNAL           XERBLA, ZLACN2, ZLACPY, ZTREXC, ZTRSYL
208: *     ..
209: *     .. Intrinsic Functions ..
210:       INTRINSIC          MAX, SQRT
211: *     ..
212: *     .. Executable Statements ..
213: *
214: *     Decode and test the input parameters.
215: *
216:       WANTBH = LSAME( JOB, 'B' )
217:       WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
218:       WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
219:       WANTQ = LSAME( COMPQ, 'V' )
220: *
221: *     Set M to the number of selected eigenvalues.
222: *
223:       M = 0
224:       DO 10 K = 1, N
225:          IF( SELECT( K ) )
226:      $      M = M + 1
227:    10 CONTINUE
228: *
229:       N1 = M
230:       N2 = N - M
231:       NN = N1*N2
232: *
233:       INFO = 0
234:       LQUERY = ( LWORK.EQ.-1 )
235: *
236:       IF( WANTSP ) THEN
237:          LWMIN = MAX( 1, 2*NN )
238:       ELSE IF( LSAME( JOB, 'N' ) ) THEN
239:          LWMIN = 1
240:       ELSE IF( LSAME( JOB, 'E' ) ) THEN
241:          LWMIN = MAX( 1, NN )
242:       END IF
243: *
244:       IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
245:      $     THEN
246:          INFO = -1
247:       ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
248:          INFO = -2
249:       ELSE IF( N.LT.0 ) THEN
250:          INFO = -4
251:       ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
252:          INFO = -6
253:       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
254:          INFO = -8
255:       ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
256:          INFO = -14
257:       END IF
258: *
259:       IF( INFO.EQ.0 ) THEN
260:          WORK( 1 ) = LWMIN
261:       END IF
262: *
263:       IF( INFO.NE.0 ) THEN
264:          CALL XERBLA( 'ZTRSEN', -INFO )
265:          RETURN
266:       ELSE IF( LQUERY ) THEN
267:          RETURN
268:       END IF
269: *
270: *     Quick return if possible
271: *
272:       IF( M.EQ.N .OR. M.EQ.0 ) THEN
273:          IF( WANTS )
274:      $      S = ONE
275:          IF( WANTSP )
276:      $      SEP = ZLANGE( '1', N, N, T, LDT, RWORK )
277:          GO TO 40
278:       END IF
279: *
280: *     Collect the selected eigenvalues at the top left corner of T.
281: *
282:       KS = 0
283:       DO 20 K = 1, N
284:          IF( SELECT( K ) ) THEN
285:             KS = KS + 1
286: *
287: *           Swap the K-th eigenvalue to position KS.
288: *
289:             IF( K.NE.KS )
290:      $         CALL ZTREXC( COMPQ, N, T, LDT, Q, LDQ, K, KS, IERR )
291:          END IF
292:    20 CONTINUE
293: *
294:       IF( WANTS ) THEN
295: *
296: *        Solve the Sylvester equation for R:
297: *
298: *           T11*R - R*T22 = scale*T12
299: *
300:          CALL ZLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
301:          CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
302:      $                LDT, WORK, N1, SCALE, IERR )
303: *
304: *        Estimate the reciprocal of the condition number of the cluster
305: *        of eigenvalues.
306: *
307:          RNORM = ZLANGE( 'F', N1, N2, WORK, N1, RWORK )
308:          IF( RNORM.EQ.ZERO ) THEN
309:             S = ONE
310:          ELSE
311:             S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
312:      $          SQRT( RNORM ) )
313:          END IF
314:       END IF
315: *
316:       IF( WANTSP ) THEN
317: *
318: *        Estimate sep(T11,T22).
319: *
320:          EST = ZERO
321:          KASE = 0
322:    30    CONTINUE
323:          CALL ZLACN2( NN, WORK( NN+1 ), WORK, EST, KASE, ISAVE )
324:          IF( KASE.NE.0 ) THEN
325:             IF( KASE.EQ.1 ) THEN
326: *
327: *              Solve T11*R - R*T22 = scale*X.
328: *
329:                CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
330:      $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
331:      $                      IERR )
332:             ELSE
333: *
334: *              Solve T11'*R - R*T22' = scale*X.
335: *
336:                CALL ZTRSYL( 'C', 'C', -1, N1, N2, T, LDT,
337:      $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
338:      $                      IERR )
339:             END IF
340:             GO TO 30
341:          END IF
342: *
343:          SEP = SCALE / EST
344:       END IF
345: *
346:    40 CONTINUE
347: *
348: *     Copy reordered eigenvalues to W.
349: *
350:       DO 50 K = 1, N
351:          W( K ) = T( K, K )
352:    50 CONTINUE
353: *
354:       WORK( 1 ) = LWMIN
355: *
356:       RETURN
357: *
358: *     End of ZTRSEN
359: *
360:       END
361: