1: SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
2: $ SEP, WORK, LWORK, 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: * Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
10: *
11: * .. Scalar Arguments ..
12: CHARACTER COMPQ, JOB
13: INTEGER INFO, LDQ, LDT, LWORK, M, N
14: DOUBLE PRECISION S, SEP
15: * ..
16: * .. Array Arguments ..
17: LOGICAL SELECT( * )
18: COMPLEX*16 Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
19: * ..
20: *
21: * Purpose
22: * =======
23: *
24: * ZTRSEN reorders the Schur factorization of a complex matrix
25: * A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
26: * the leading positions on the diagonal of the upper triangular matrix
27: * T, 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: * Arguments
34: * =========
35: *
36: * JOB (input) CHARACTER*1
37: * Specifies whether condition numbers are required for the
38: * cluster of eigenvalues (S) or the invariant subspace (SEP):
39: * = 'N': none;
40: * = 'E': for eigenvalues only (S);
41: * = 'V': for invariant subspace only (SEP);
42: * = 'B': for both eigenvalues and invariant subspace (S and
43: * SEP).
44: *
45: * COMPQ (input) CHARACTER*1
46: * = 'V': update the matrix Q of Schur vectors;
47: * = 'N': do not update Q.
48: *
49: * SELECT (input) LOGICAL array, dimension (N)
50: * SELECT specifies the eigenvalues in the selected cluster. To
51: * select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
52: *
53: * N (input) INTEGER
54: * The order of the matrix T. N >= 0.
55: *
56: * T (input/output) COMPLEX*16 array, dimension (LDT,N)
57: * On entry, the upper triangular matrix T.
58: * On exit, T is overwritten by the reordered matrix T, with the
59: * selected eigenvalues as the leading diagonal elements.
60: *
61: * LDT (input) INTEGER
62: * The leading dimension of the array T. LDT >= max(1,N).
63: *
64: * Q (input/output) COMPLEX*16 array, dimension (LDQ,N)
65: * On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
66: * On exit, if COMPQ = 'V', Q has been postmultiplied by the
67: * unitary transformation matrix which reorders T; the leading M
68: * columns of Q form an orthonormal basis for the specified
69: * invariant subspace.
70: * If COMPQ = 'N', Q is not referenced.
71: *
72: * LDQ (input) INTEGER
73: * The leading dimension of the array Q.
74: * LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
75: *
76: * W (output) COMPLEX*16 array, dimension (N)
77: * The reordered eigenvalues of T, in the same order as they
78: * appear on the diagonal of T.
79: *
80: * M (output) INTEGER
81: * The dimension of the specified invariant subspace.
82: * 0 <= M <= N.
83: *
84: * S (output) DOUBLE PRECISION
85: * If JOB = 'E' or 'B', S is a lower bound on the reciprocal
86: * condition number for the selected cluster of eigenvalues.
87: * S cannot underestimate the true reciprocal condition number
88: * by more than a factor of sqrt(N). If M = 0 or N, S = 1.
89: * If JOB = 'N' or 'V', S is not referenced.
90: *
91: * SEP (output) DOUBLE PRECISION
92: * If JOB = 'V' or 'B', SEP is the estimated reciprocal
93: * condition number of the specified invariant subspace. If
94: * M = 0 or N, SEP = norm(T).
95: * If JOB = 'N' or 'E', SEP is not referenced.
96: *
97: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
98: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
99: *
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
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