00001 SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, 00002 $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.2) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * November 2006 00008 * 00009 * .. Scalar Arguments .. 00010 CHARACTER COMPQ, JOB 00011 INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N 00012 DOUBLE PRECISION S, SEP 00013 * .. 00014 * .. Array Arguments .. 00015 LOGICAL SELECT( * ) 00016 INTEGER IWORK( * ) 00017 DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), 00018 $ WR( * ) 00019 * .. 00020 * 00021 * Purpose 00022 * ======= 00023 * 00024 * DTRSEN reorders the real Schur factorization of a real matrix 00025 * A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in 00026 * the leading diagonal blocks of the upper quasi-triangular matrix T, 00027 * and the leading columns of Q form an orthonormal basis of the 00028 * corresponding right invariant subspace. 00029 * 00030 * Optionally the routine computes the reciprocal condition numbers of 00031 * the cluster of eigenvalues and/or the invariant subspace. 00032 * 00033 * T must be in Schur canonical form (as returned by DHSEQR), that is, 00034 * block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 00035 * 2-by-2 diagonal block has its diagonal elemnts equal and its 00036 * off-diagonal elements of opposite sign. 00037 * 00038 * Arguments 00039 * ========= 00040 * 00041 * JOB (input) CHARACTER*1 00042 * Specifies whether condition numbers are required for the 00043 * cluster of eigenvalues (S) or the invariant subspace (SEP): 00044 * = 'N': none; 00045 * = 'E': for eigenvalues only (S); 00046 * = 'V': for invariant subspace only (SEP); 00047 * = 'B': for both eigenvalues and invariant subspace (S and 00048 * SEP). 00049 * 00050 * COMPQ (input) CHARACTER*1 00051 * = 'V': update the matrix Q of Schur vectors; 00052 * = 'N': do not update Q. 00053 * 00054 * SELECT (input) LOGICAL array, dimension (N) 00055 * SELECT specifies the eigenvalues in the selected cluster. To 00056 * select a real eigenvalue w(j), SELECT(j) must be set to 00057 * .TRUE.. To select a complex conjugate pair of eigenvalues 00058 * w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, 00059 * either SELECT(j) or SELECT(j+1) or both must be set to 00060 * .TRUE.; a complex conjugate pair of eigenvalues must be 00061 * either both included in the cluster or both excluded. 00062 * 00063 * N (input) INTEGER 00064 * The order of the matrix T. N >= 0. 00065 * 00066 * T (input/output) DOUBLE PRECISION array, dimension (LDT,N) 00067 * On entry, the upper quasi-triangular matrix T, in Schur 00068 * canonical form. 00069 * On exit, T is overwritten by the reordered matrix T, again in 00070 * Schur canonical form, with the selected eigenvalues in the 00071 * leading diagonal blocks. 00072 * 00073 * LDT (input) INTEGER 00074 * The leading dimension of the array T. LDT >= max(1,N). 00075 * 00076 * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) 00077 * On entry, if COMPQ = 'V', the matrix Q of Schur vectors. 00078 * On exit, if COMPQ = 'V', Q has been postmultiplied by the 00079 * orthogonal transformation matrix which reorders T; the 00080 * leading M columns of Q form an orthonormal basis for the 00081 * specified invariant subspace. 00082 * If COMPQ = 'N', Q is not referenced. 00083 * 00084 * LDQ (input) INTEGER 00085 * The leading dimension of the array Q. 00086 * LDQ >= 1; and if COMPQ = 'V', LDQ >= N. 00087 * 00088 * WR (output) DOUBLE PRECISION array, dimension (N) 00089 * WI (output) DOUBLE PRECISION array, dimension (N) 00090 * The real and imaginary parts, respectively, of the reordered 00091 * eigenvalues of T. The eigenvalues are stored in the same 00092 * order as on the diagonal of T, with WR(i) = T(i,i) and, if 00093 * T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and 00094 * WI(i+1) = -WI(i). Note that if a complex eigenvalue is 00095 * sufficiently ill-conditioned, then its value may differ 00096 * significantly from its value before reordering. 00097 * 00098 * M (output) INTEGER 00099 * The dimension of the specified invariant subspace. 00100 * 0 < = M <= N. 00101 * 00102 * S (output) DOUBLE PRECISION 00103 * If JOB = 'E' or 'B', S is a lower bound on the reciprocal 00104 * condition number for the selected cluster of eigenvalues. 00105 * S cannot underestimate the true reciprocal condition number 00106 * by more than a factor of sqrt(N). If M = 0 or N, S = 1. 00107 * If JOB = 'N' or 'V', S is not referenced. 00108 * 00109 * SEP (output) DOUBLE PRECISION 00110 * If JOB = 'V' or 'B', SEP is the estimated reciprocal 00111 * condition number of the specified invariant subspace. If 00112 * M = 0 or N, SEP = norm(T). 00113 * If JOB = 'N' or 'E', SEP is not referenced. 00114 * 00115 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 00116 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00117 * 00118 * LWORK (input) INTEGER 00119 * The dimension of the array WORK. 00120 * If JOB = 'N', LWORK >= max(1,N); 00121 * if JOB = 'E', LWORK >= max(1,M*(N-M)); 00122 * if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)). 00123 * 00124 * If LWORK = -1, then a workspace query is assumed; the routine 00125 * only calculates the optimal size of the WORK array, returns 00126 * this value as the first entry of the WORK array, and no error 00127 * message related to LWORK is issued by XERBLA. 00128 * 00129 * IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK)) 00130 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00131 * 00132 * LIWORK (input) INTEGER 00133 * The dimension of the array IWORK. 00134 * If JOB = 'N' or 'E', LIWORK >= 1; 00135 * if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)). 00136 * 00137 * If LIWORK = -1, then a workspace query is assumed; the 00138 * routine only calculates the optimal size of the IWORK array, 00139 * returns this value as the first entry of the IWORK array, and 00140 * no error message related to LIWORK is issued by XERBLA. 00141 * 00142 * INFO (output) INTEGER 00143 * = 0: successful exit 00144 * < 0: if INFO = -i, the i-th argument had an illegal value 00145 * = 1: reordering of T failed because some eigenvalues are too 00146 * close to separate (the problem is very ill-conditioned); 00147 * T may have been partially reordered, and WR and WI 00148 * contain the eigenvalues in the same order as in T; S and 00149 * SEP (if requested) are set to zero. 00150 * 00151 * Further Details 00152 * =============== 00153 * 00154 * DTRSEN first collects the selected eigenvalues by computing an 00155 * orthogonal transformation Z to move them to the top left corner of T. 00156 * In other words, the selected eigenvalues are the eigenvalues of T11 00157 * in: 00158 * 00159 * Z'*T*Z = ( T11 T12 ) n1 00160 * ( 0 T22 ) n2 00161 * n1 n2 00162 * 00163 * where N = n1+n2 and Z' means the transpose of Z. The first n1 columns 00164 * of Z span the specified invariant subspace of T. 00165 * 00166 * If T has been obtained from the real Schur factorization of a matrix 00167 * A = Q*T*Q', then the reordered real Schur factorization of A is given 00168 * by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span 00169 * the corresponding invariant subspace of A. 00170 * 00171 * The reciprocal condition number of the average of the eigenvalues of 00172 * T11 may be returned in S. S lies between 0 (very badly conditioned) 00173 * and 1 (very well conditioned). It is computed as follows. First we 00174 * compute R so that 00175 * 00176 * P = ( I R ) n1 00177 * ( 0 0 ) n2 00178 * n1 n2 00179 * 00180 * is the projector on the invariant subspace associated with T11. 00181 * R is the solution of the Sylvester equation: 00182 * 00183 * T11*R - R*T22 = T12. 00184 * 00185 * Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote 00186 * the two-norm of M. Then S is computed as the lower bound 00187 * 00188 * (1 + F-norm(R)**2)**(-1/2) 00189 * 00190 * on the reciprocal of 2-norm(P), the true reciprocal condition number. 00191 * S cannot underestimate 1 / 2-norm(P) by more than a factor of 00192 * sqrt(N). 00193 * 00194 * An approximate error bound for the computed average of the 00195 * eigenvalues of T11 is 00196 * 00197 * EPS * norm(T) / S 00198 * 00199 * where EPS is the machine precision. 00200 * 00201 * The reciprocal condition number of the right invariant subspace 00202 * spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. 00203 * SEP is defined as the separation of T11 and T22: 00204 * 00205 * sep( T11, T22 ) = sigma-min( C ) 00206 * 00207 * where sigma-min(C) is the smallest singular value of the 00208 * n1*n2-by-n1*n2 matrix 00209 * 00210 * C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) 00211 * 00212 * I(m) is an m by m identity matrix, and kprod denotes the Kronecker 00213 * product. We estimate sigma-min(C) by the reciprocal of an estimate of 00214 * the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) 00215 * cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). 00216 * 00217 * When SEP is small, small changes in T can cause large changes in 00218 * the invariant subspace. An approximate bound on the maximum angular 00219 * error in the computed right invariant subspace is 00220 * 00221 * EPS * norm(T) / SEP 00222 * 00223 * ===================================================================== 00224 * 00225 * .. Parameters .. 00226 DOUBLE PRECISION ZERO, ONE 00227 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00228 * .. 00229 * .. Local Scalars .. 00230 LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS, 00231 $ WANTSP 00232 INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2, 00233 $ NN 00234 DOUBLE PRECISION EST, RNORM, SCALE 00235 * .. 00236 * .. Local Arrays .. 00237 INTEGER ISAVE( 3 ) 00238 * .. 00239 * .. External Functions .. 00240 LOGICAL LSAME 00241 DOUBLE PRECISION DLANGE 00242 EXTERNAL LSAME, DLANGE 00243 * .. 00244 * .. External Subroutines .. 00245 EXTERNAL DLACN2, DLACPY, DTREXC, DTRSYL, XERBLA 00246 * .. 00247 * .. Intrinsic Functions .. 00248 INTRINSIC ABS, MAX, SQRT 00249 * .. 00250 * .. Executable Statements .. 00251 * 00252 * Decode and test the input parameters 00253 * 00254 WANTBH = LSAME( JOB, 'B' ) 00255 WANTS = LSAME( JOB, 'E' ) .OR. WANTBH 00256 WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH 00257 WANTQ = LSAME( COMPQ, 'V' ) 00258 * 00259 INFO = 0 00260 LQUERY = ( LWORK.EQ.-1 ) 00261 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP ) 00262 $ THEN 00263 INFO = -1 00264 ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN 00265 INFO = -2 00266 ELSE IF( N.LT.0 ) THEN 00267 INFO = -4 00268 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN 00269 INFO = -6 00270 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN 00271 INFO = -8 00272 ELSE 00273 * 00274 * Set M to the dimension of the specified invariant subspace, 00275 * and test LWORK and LIWORK. 00276 * 00277 M = 0 00278 PAIR = .FALSE. 00279 DO 10 K = 1, N 00280 IF( PAIR ) THEN 00281 PAIR = .FALSE. 00282 ELSE 00283 IF( K.LT.N ) THEN 00284 IF( T( K+1, K ).EQ.ZERO ) THEN 00285 IF( SELECT( K ) ) 00286 $ M = M + 1 00287 ELSE 00288 PAIR = .TRUE. 00289 IF( SELECT( K ) .OR. SELECT( K+1 ) ) 00290 $ M = M + 2 00291 END IF 00292 ELSE 00293 IF( SELECT( N ) ) 00294 $ M = M + 1 00295 END IF 00296 END IF 00297 10 CONTINUE 00298 * 00299 N1 = M 00300 N2 = N - M 00301 NN = N1*N2 00302 * 00303 IF( WANTSP ) THEN 00304 LWMIN = MAX( 1, 2*NN ) 00305 LIWMIN = MAX( 1, NN ) 00306 ELSE IF( LSAME( JOB, 'N' ) ) THEN 00307 LWMIN = MAX( 1, N ) 00308 LIWMIN = 1 00309 ELSE IF( LSAME( JOB, 'E' ) ) THEN 00310 LWMIN = MAX( 1, NN ) 00311 LIWMIN = 1 00312 END IF 00313 * 00314 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 00315 INFO = -15 00316 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 00317 INFO = -17 00318 END IF 00319 END IF 00320 * 00321 IF( INFO.EQ.0 ) THEN 00322 WORK( 1 ) = LWMIN 00323 IWORK( 1 ) = LIWMIN 00324 END IF 00325 * 00326 IF( INFO.NE.0 ) THEN 00327 CALL XERBLA( 'DTRSEN', -INFO ) 00328 RETURN 00329 ELSE IF( LQUERY ) THEN 00330 RETURN 00331 END IF 00332 * 00333 * Quick return if possible. 00334 * 00335 IF( M.EQ.N .OR. M.EQ.0 ) THEN 00336 IF( WANTS ) 00337 $ S = ONE 00338 IF( WANTSP ) 00339 $ SEP = DLANGE( '1', N, N, T, LDT, WORK ) 00340 GO TO 40 00341 END IF 00342 * 00343 * Collect the selected blocks at the top-left corner of T. 00344 * 00345 KS = 0 00346 PAIR = .FALSE. 00347 DO 20 K = 1, N 00348 IF( PAIR ) THEN 00349 PAIR = .FALSE. 00350 ELSE 00351 SWAP = SELECT( K ) 00352 IF( K.LT.N ) THEN 00353 IF( T( K+1, K ).NE.ZERO ) THEN 00354 PAIR = .TRUE. 00355 SWAP = SWAP .OR. SELECT( K+1 ) 00356 END IF 00357 END IF 00358 IF( SWAP ) THEN 00359 KS = KS + 1 00360 * 00361 * Swap the K-th block to position KS. 00362 * 00363 IERR = 0 00364 KK = K 00365 IF( K.NE.KS ) 00366 $ CALL DTREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK, 00367 $ IERR ) 00368 IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN 00369 * 00370 * Blocks too close to swap: exit. 00371 * 00372 INFO = 1 00373 IF( WANTS ) 00374 $ S = ZERO 00375 IF( WANTSP ) 00376 $ SEP = ZERO 00377 GO TO 40 00378 END IF 00379 IF( PAIR ) 00380 $ KS = KS + 1 00381 END IF 00382 END IF 00383 20 CONTINUE 00384 * 00385 IF( WANTS ) THEN 00386 * 00387 * Solve Sylvester equation for R: 00388 * 00389 * T11*R - R*T22 = scale*T12 00390 * 00391 CALL DLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 ) 00392 CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ), 00393 $ LDT, WORK, N1, SCALE, IERR ) 00394 * 00395 * Estimate the reciprocal of the condition number of the cluster 00396 * of eigenvalues. 00397 * 00398 RNORM = DLANGE( 'F', N1, N2, WORK, N1, WORK ) 00399 IF( RNORM.EQ.ZERO ) THEN 00400 S = ONE 00401 ELSE 00402 S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )* 00403 $ SQRT( RNORM ) ) 00404 END IF 00405 END IF 00406 * 00407 IF( WANTSP ) THEN 00408 * 00409 * Estimate sep(T11,T22). 00410 * 00411 EST = ZERO 00412 KASE = 0 00413 30 CONTINUE 00414 CALL DLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE ) 00415 IF( KASE.NE.0 ) THEN 00416 IF( KASE.EQ.1 ) THEN 00417 * 00418 * Solve T11*R - R*T22 = scale*X. 00419 * 00420 CALL DTRSYL( 'N', 'N', -1, N1, N2, T, LDT, 00421 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, 00422 $ IERR ) 00423 ELSE 00424 * 00425 * Solve T11'*R - R*T22' = scale*X. 00426 * 00427 CALL DTRSYL( 'T', 'T', -1, N1, N2, T, LDT, 00428 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, 00429 $ IERR ) 00430 END IF 00431 GO TO 30 00432 END IF 00433 * 00434 SEP = SCALE / EST 00435 END IF 00436 * 00437 40 CONTINUE 00438 * 00439 * Store the output eigenvalues in WR and WI. 00440 * 00441 DO 50 K = 1, N 00442 WR( K ) = T( K, K ) 00443 WI( K ) = ZERO 00444 50 CONTINUE 00445 DO 60 K = 1, N - 1 00446 IF( T( K+1, K ).NE.ZERO ) THEN 00447 WI( K ) = SQRT( ABS( T( K, K+1 ) ) )* 00448 $ SQRT( ABS( T( K+1, K ) ) ) 00449 WI( K+1 ) = -WI( K ) 00450 END IF 00451 60 CONTINUE 00452 * 00453 WORK( 1 ) = LWMIN 00454 IWORK( 1 ) = LIWMIN 00455 * 00456 RETURN 00457 * 00458 * End of DTRSEN 00459 * 00460 END
1.7.2