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 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
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*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
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