1 SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI,
2 $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER COMPQ, JOB
11 INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
12 DOUBLE PRECISION S, SEP
13 * ..
14 * .. Array Arguments ..
15 LOGICAL SELECT( * )
16 INTEGER IWORK( * )
17 DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
18 $ WR( * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DTRSEN reorders the real Schur factorization of a real matrix
25 * A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
26 * the leading diagonal blocks of the upper quasi-triangular matrix T,
27 * and the leading columns of Q form an orthonormal basis of the
28 * corresponding right invariant subspace.
29 *
30 * Optionally the routine computes the reciprocal condition numbers of
31 * the cluster of eigenvalues and/or the invariant subspace.
32 *
33 * T must be in Schur canonical form (as returned by DHSEQR), that is,
34 * block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
35 * 2-by-2 diagonal block has its diagonal elemnts equal and its
36 * off-diagonal elements of opposite sign.
37 *
38 * Arguments
39 * =========
40 *
41 * JOB (input) CHARACTER*1
42 * Specifies whether condition numbers are required for the
43 * cluster of eigenvalues (S) or the invariant subspace (SEP):
44 * = 'N': none;
45 * = 'E': for eigenvalues only (S);
46 * = 'V': for invariant subspace only (SEP);
47 * = 'B': for both eigenvalues and invariant subspace (S and
48 * SEP).
49 *
50 * COMPQ (input) CHARACTER*1
51 * = 'V': update the matrix Q of Schur vectors;
52 * = 'N': do not update Q.
53 *
54 * SELECT (input) LOGICAL array, dimension (N)
55 * SELECT specifies the eigenvalues in the selected cluster. To
56 * select a real eigenvalue w(j), SELECT(j) must be set to
57 * .TRUE.. To select a complex conjugate pair of eigenvalues
58 * w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
59 * either SELECT(j) or SELECT(j+1) or both must be set to
60 * .TRUE.; a complex conjugate pair of eigenvalues must be
61 * either both included in the cluster or both excluded.
62 *
63 * N (input) INTEGER
64 * The order of the matrix T. N >= 0.
65 *
66 * T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
67 * On entry, the upper quasi-triangular matrix T, in Schur
68 * canonical form.
69 * On exit, T is overwritten by the reordered matrix T, again in
70 * Schur canonical form, with the selected eigenvalues in the
71 * leading diagonal blocks.
72 *
73 * LDT (input) INTEGER
74 * The leading dimension of the array T. LDT >= max(1,N).
75 *
76 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
77 * On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
78 * On exit, if COMPQ = 'V', Q has been postmultiplied by the
79 * orthogonal transformation matrix which reorders T; the
80 * leading M columns of Q form an orthonormal basis for the
81 * specified invariant subspace.
82 * If COMPQ = 'N', Q is not referenced.
83 *
84 * LDQ (input) INTEGER
85 * The leading dimension of the array Q.
86 * LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
87 *
88 * WR (output) DOUBLE PRECISION array, dimension (N)
89 * WI (output) DOUBLE PRECISION array, dimension (N)
90 * The real and imaginary parts, respectively, of the reordered
91 * eigenvalues of T. The eigenvalues are stored in the same
92 * order as on the diagonal of T, with WR(i) = T(i,i) and, if
93 * T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
94 * WI(i+1) = -WI(i). Note that if a complex eigenvalue is
95 * sufficiently ill-conditioned, then its value may differ
96 * significantly from its value before reordering.
97 *
98 * M (output) INTEGER
99 * 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 * T * Z = ( T11 T12 ) n1
160 * ( 0 T22 ) n2
161 * n1 n2
162 *
163 * where N = n1+n2 and Z**T 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**T, then the reordered real Schur factorization of A is given
168 * by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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**T*R - R*T22**T = 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
2 $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER COMPQ, JOB
11 INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
12 DOUBLE PRECISION S, SEP
13 * ..
14 * .. Array Arguments ..
15 LOGICAL SELECT( * )
16 INTEGER IWORK( * )
17 DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
18 $ WR( * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DTRSEN reorders the real Schur factorization of a real matrix
25 * A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
26 * the leading diagonal blocks of the upper quasi-triangular matrix T,
27 * and the leading columns of Q form an orthonormal basis of the
28 * corresponding right invariant subspace.
29 *
30 * Optionally the routine computes the reciprocal condition numbers of
31 * the cluster of eigenvalues and/or the invariant subspace.
32 *
33 * T must be in Schur canonical form (as returned by DHSEQR), that is,
34 * block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
35 * 2-by-2 diagonal block has its diagonal elemnts equal and its
36 * off-diagonal elements of opposite sign.
37 *
38 * Arguments
39 * =========
40 *
41 * JOB (input) CHARACTER*1
42 * Specifies whether condition numbers are required for the
43 * cluster of eigenvalues (S) or the invariant subspace (SEP):
44 * = 'N': none;
45 * = 'E': for eigenvalues only (S);
46 * = 'V': for invariant subspace only (SEP);
47 * = 'B': for both eigenvalues and invariant subspace (S and
48 * SEP).
49 *
50 * COMPQ (input) CHARACTER*1
51 * = 'V': update the matrix Q of Schur vectors;
52 * = 'N': do not update Q.
53 *
54 * SELECT (input) LOGICAL array, dimension (N)
55 * SELECT specifies the eigenvalues in the selected cluster. To
56 * select a real eigenvalue w(j), SELECT(j) must be set to
57 * .TRUE.. To select a complex conjugate pair of eigenvalues
58 * w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
59 * either SELECT(j) or SELECT(j+1) or both must be set to
60 * .TRUE.; a complex conjugate pair of eigenvalues must be
61 * either both included in the cluster or both excluded.
62 *
63 * N (input) INTEGER
64 * The order of the matrix T. N >= 0.
65 *
66 * T (input/output) DOUBLE PRECISION array, dimension (LDT,N)
67 * On entry, the upper quasi-triangular matrix T, in Schur
68 * canonical form.
69 * On exit, T is overwritten by the reordered matrix T, again in
70 * Schur canonical form, with the selected eigenvalues in the
71 * leading diagonal blocks.
72 *
73 * LDT (input) INTEGER
74 * The leading dimension of the array T. LDT >= max(1,N).
75 *
76 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
77 * On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
78 * On exit, if COMPQ = 'V', Q has been postmultiplied by the
79 * orthogonal transformation matrix which reorders T; the
80 * leading M columns of Q form an orthonormal basis for the
81 * specified invariant subspace.
82 * If COMPQ = 'N', Q is not referenced.
83 *
84 * LDQ (input) INTEGER
85 * The leading dimension of the array Q.
86 * LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
87 *
88 * WR (output) DOUBLE PRECISION array, dimension (N)
89 * WI (output) DOUBLE PRECISION array, dimension (N)
90 * The real and imaginary parts, respectively, of the reordered
91 * eigenvalues of T. The eigenvalues are stored in the same
92 * order as on the diagonal of T, with WR(i) = T(i,i) and, if
93 * T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
94 * WI(i+1) = -WI(i). Note that if a complex eigenvalue is
95 * sufficiently ill-conditioned, then its value may differ
96 * significantly from its value before reordering.
97 *
98 * M (output) INTEGER
99 * 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 * T * Z = ( T11 T12 ) n1
160 * ( 0 T22 ) n2
161 * n1 n2
162 *
163 * where N = n1+n2 and Z**T 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**T, then the reordered real Schur factorization of A is given
168 * by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, 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**T*R - R*T22**T = 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