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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE STRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, 00002 $ M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.3.1) -- 00005 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00006 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00007 * -- April 2011 -- 00008 * 00009 * Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH. 00010 * 00011 * .. Scalar Arguments .. 00012 CHARACTER COMPQ, JOB 00013 INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N 00014 REAL S, SEP 00015 * .. 00016 * .. Array Arguments .. 00017 LOGICAL SELECT( * ) 00018 INTEGER IWORK( * ) 00019 REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ), 00020 $ WR( * ) 00021 * .. 00022 * 00023 * Purpose 00024 * ======= 00025 * 00026 * STRSEN reorders the real Schur factorization of a real matrix 00027 * A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in 00028 * the leading diagonal blocks of the upper quasi-triangular matrix T, 00029 * and the leading columns of Q form an orthonormal basis of the 00030 * corresponding right invariant subspace. 00031 * 00032 * Optionally the routine computes the reciprocal condition numbers of 00033 * the cluster of eigenvalues and/or the invariant subspace. 00034 * 00035 * T must be in Schur canonical form (as returned by SHSEQR), that is, 00036 * block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 00037 * 2-by-2 diagonal block has its diagonal elemnts equal and its 00038 * off-diagonal elements of opposite sign. 00039 * 00040 * Arguments 00041 * ========= 00042 * 00043 * JOB (input) CHARACTER*1 00044 * Specifies whether condition numbers are required for the 00045 * cluster of eigenvalues (S) or the invariant subspace (SEP): 00046 * = 'N': none; 00047 * = 'E': for eigenvalues only (S); 00048 * = 'V': for invariant subspace only (SEP); 00049 * = 'B': for both eigenvalues and invariant subspace (S and 00050 * SEP). 00051 * 00052 * COMPQ (input) CHARACTER*1 00053 * = 'V': update the matrix Q of Schur vectors; 00054 * = 'N': do not update Q. 00055 * 00056 * SELECT (input) LOGICAL array, dimension (N) 00057 * SELECT specifies the eigenvalues in the selected cluster. To 00058 * select a real eigenvalue w(j), SELECT(j) must be set to 00059 * .TRUE.. To select a complex conjugate pair of eigenvalues 00060 * w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, 00061 * either SELECT(j) or SELECT(j+1) or both must be set to 00062 * .TRUE.; a complex conjugate pair of eigenvalues must be 00063 * either both included in the cluster or both excluded. 00064 * 00065 * N (input) INTEGER 00066 * The order of the matrix T. N >= 0. 00067 * 00068 * T (input/output) REAL array, dimension (LDT,N) 00069 * On entry, the upper quasi-triangular matrix T, in Schur 00070 * canonical form. 00071 * On exit, T is overwritten by the reordered matrix T, again in 00072 * Schur canonical form, with the selected eigenvalues in the 00073 * leading diagonal blocks. 00074 * 00075 * LDT (input) INTEGER 00076 * The leading dimension of the array T. LDT >= max(1,N). 00077 * 00078 * Q (input/output) REAL array, dimension (LDQ,N) 00079 * On entry, if COMPQ = 'V', the matrix Q of Schur vectors. 00080 * On exit, if COMPQ = 'V', Q has been postmultiplied by the 00081 * orthogonal transformation matrix which reorders T; the 00082 * leading M columns of Q form an orthonormal basis for the 00083 * specified invariant subspace. 00084 * If COMPQ = 'N', Q is not referenced. 00085 * 00086 * LDQ (input) INTEGER 00087 * The leading dimension of the array Q. 00088 * LDQ >= 1; and if COMPQ = 'V', LDQ >= N. 00089 * 00090 * WR (output) REAL array, dimension (N) 00091 * WI (output) REAL array, dimension (N) 00092 * The real and imaginary parts, respectively, of the reordered 00093 * eigenvalues of T. The eigenvalues are stored in the same 00094 * order as on the diagonal of T, with WR(i) = T(i,i) and, if 00095 * T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and 00096 * WI(i+1) = -WI(i). Note that if a complex eigenvalue is 00097 * sufficiently ill-conditioned, then its value may differ 00098 * significantly from its value before reordering. 00099 * 00100 * M (output) INTEGER 00101 * The dimension of the specified invariant subspace. 00102 * 0 < = M <= N. 00103 * 00104 * S (output) REAL 00105 * If JOB = 'E' or 'B', S is a lower bound on the reciprocal 00106 * condition number for the selected cluster of eigenvalues. 00107 * S cannot underestimate the true reciprocal condition number 00108 * by more than a factor of sqrt(N). If M = 0 or N, S = 1. 00109 * If JOB = 'N' or 'V', S is not referenced. 00110 * 00111 * SEP (output) REAL 00112 * If JOB = 'V' or 'B', SEP is the estimated reciprocal 00113 * condition number of the specified invariant subspace. If 00114 * M = 0 or N, SEP = norm(T). 00115 * If JOB = 'N' or 'E', SEP is not referenced. 00116 * 00117 * WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) 00118 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00119 * 00120 * LWORK (input) INTEGER 00121 * The dimension of the array WORK. 00122 * If JOB = 'N', LWORK >= max(1,N); 00123 * if JOB = 'E', LWORK >= max(1,M*(N-M)); 00124 * if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)). 00125 * 00126 * If LWORK = -1, then a workspace query is assumed; the routine 00127 * only calculates the optimal size of the WORK array, returns 00128 * this value as the first entry of the WORK array, and no error 00129 * message related to LWORK is issued by XERBLA. 00130 * 00131 * IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK)) 00132 * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. 00133 * 00134 * LIWORK (input) INTEGER 00135 * The dimension of the array IWORK. 00136 * If JOB = 'N' or 'E', LIWORK >= 1; 00137 * if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)). 00138 * 00139 * If LIWORK = -1, then a workspace query is assumed; the 00140 * routine only calculates the optimal size of the IWORK array, 00141 * returns this value as the first entry of the IWORK array, and 00142 * no error message related to LIWORK is issued by XERBLA. 00143 * 00144 * INFO (output) INTEGER 00145 * = 0: successful exit 00146 * < 0: if INFO = -i, the i-th argument had an illegal value 00147 * = 1: reordering of T failed because some eigenvalues are too 00148 * close to separate (the problem is very ill-conditioned); 00149 * T may have been partially reordered, and WR and WI 00150 * contain the eigenvalues in the same order as in T; S and 00151 * SEP (if requested) are set to zero. 00152 * 00153 * Further Details 00154 * =============== 00155 * 00156 * STRSEN first collects the selected eigenvalues by computing an 00157 * orthogonal transformation Z to move them to the top left corner of T. 00158 * In other words, the selected eigenvalues are the eigenvalues of T11 00159 * in: 00160 * 00161 * Z**T * T * Z = ( T11 T12 ) n1 00162 * ( 0 T22 ) n2 00163 * n1 n2 00164 * 00165 * where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns 00166 * of Z span the specified invariant subspace of T. 00167 * 00168 * If T has been obtained from the real Schur factorization of a matrix 00169 * A = Q*T*Q**T, then the reordered real Schur factorization of A is given 00170 * by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span 00171 * the corresponding invariant subspace of A. 00172 * 00173 * The reciprocal condition number of the average of the eigenvalues of 00174 * T11 may be returned in S. S lies between 0 (very badly conditioned) 00175 * and 1 (very well conditioned). It is computed as follows. First we 00176 * compute R so that 00177 * 00178 * P = ( I R ) n1 00179 * ( 0 0 ) n2 00180 * n1 n2 00181 * 00182 * is the projector on the invariant subspace associated with T11. 00183 * R is the solution of the Sylvester equation: 00184 * 00185 * T11*R - R*T22 = T12. 00186 * 00187 * Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote 00188 * the two-norm of M. Then S is computed as the lower bound 00189 * 00190 * (1 + F-norm(R)**2)**(-1/2) 00191 * 00192 * on the reciprocal of 2-norm(P), the true reciprocal condition number. 00193 * S cannot underestimate 1 / 2-norm(P) by more than a factor of 00194 * sqrt(N). 00195 * 00196 * An approximate error bound for the computed average of the 00197 * eigenvalues of T11 is 00198 * 00199 * EPS * norm(T) / S 00200 * 00201 * where EPS is the machine precision. 00202 * 00203 * The reciprocal condition number of the right invariant subspace 00204 * spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP. 00205 * SEP is defined as the separation of T11 and T22: 00206 * 00207 * sep( T11, T22 ) = sigma-min( C ) 00208 * 00209 * where sigma-min(C) is the smallest singular value of the 00210 * n1*n2-by-n1*n2 matrix 00211 * 00212 * C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) ) 00213 * 00214 * I(m) is an m by m identity matrix, and kprod denotes the Kronecker 00215 * product. We estimate sigma-min(C) by the reciprocal of an estimate of 00216 * the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C) 00217 * cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2). 00218 * 00219 * When SEP is small, small changes in T can cause large changes in 00220 * the invariant subspace. An approximate bound on the maximum angular 00221 * error in the computed right invariant subspace is 00222 * 00223 * EPS * norm(T) / SEP 00224 * 00225 * ===================================================================== 00226 * 00227 * .. Parameters .. 00228 REAL ZERO, ONE 00229 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00230 * .. 00231 * .. Local Scalars .. 00232 LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS, 00233 $ WANTSP 00234 INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2, 00235 $ NN 00236 REAL EST, RNORM, SCALE 00237 * .. 00238 * .. Local Arrays .. 00239 INTEGER ISAVE( 3 ) 00240 * .. 00241 * .. External Functions .. 00242 LOGICAL LSAME 00243 REAL SLANGE 00244 EXTERNAL LSAME, SLANGE 00245 * .. 00246 * .. External Subroutines .. 00247 EXTERNAL SLACN2, SLACPY, STREXC, STRSYL, XERBLA 00248 * .. 00249 * .. Intrinsic Functions .. 00250 INTRINSIC ABS, MAX, SQRT 00251 * .. 00252 * .. Executable Statements .. 00253 * 00254 * Decode and test the input parameters 00255 * 00256 WANTBH = LSAME( JOB, 'B' ) 00257 WANTS = LSAME( JOB, 'E' ) .OR. WANTBH 00258 WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH 00259 WANTQ = LSAME( COMPQ, 'V' ) 00260 * 00261 INFO = 0 00262 LQUERY = ( LWORK.EQ.-1 ) 00263 IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP ) 00264 $ THEN 00265 INFO = -1 00266 ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN 00267 INFO = -2 00268 ELSE IF( N.LT.0 ) THEN 00269 INFO = -4 00270 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN 00271 INFO = -6 00272 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN 00273 INFO = -8 00274 ELSE 00275 * 00276 * Set M to the dimension of the specified invariant subspace, 00277 * and test LWORK and LIWORK. 00278 * 00279 M = 0 00280 PAIR = .FALSE. 00281 DO 10 K = 1, N 00282 IF( PAIR ) THEN 00283 PAIR = .FALSE. 00284 ELSE 00285 IF( K.LT.N ) THEN 00286 IF( T( K+1, K ).EQ.ZERO ) THEN 00287 IF( SELECT( K ) ) 00288 $ M = M + 1 00289 ELSE 00290 PAIR = .TRUE. 00291 IF( SELECT( K ) .OR. SELECT( K+1 ) ) 00292 $ M = M + 2 00293 END IF 00294 ELSE 00295 IF( SELECT( N ) ) 00296 $ M = M + 1 00297 END IF 00298 END IF 00299 10 CONTINUE 00300 * 00301 N1 = M 00302 N2 = N - M 00303 NN = N1*N2 00304 * 00305 IF( WANTSP ) THEN 00306 LWMIN = MAX( 1, 2*NN ) 00307 LIWMIN = MAX( 1, NN ) 00308 ELSE IF( LSAME( JOB, 'N' ) ) THEN 00309 LWMIN = MAX( 1, N ) 00310 LIWMIN = 1 00311 ELSE IF( LSAME( JOB, 'E' ) ) THEN 00312 LWMIN = MAX( 1, NN ) 00313 LIWMIN = 1 00314 END IF 00315 * 00316 IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN 00317 INFO = -15 00318 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN 00319 INFO = -17 00320 END IF 00321 END IF 00322 * 00323 IF( INFO.EQ.0 ) THEN 00324 WORK( 1 ) = LWMIN 00325 IWORK( 1 ) = LIWMIN 00326 END IF 00327 * 00328 IF( INFO.NE.0 ) THEN 00329 CALL XERBLA( 'STRSEN', -INFO ) 00330 RETURN 00331 ELSE IF( LQUERY ) THEN 00332 RETURN 00333 END IF 00334 * 00335 * Quick return if possible. 00336 * 00337 IF( M.EQ.N .OR. M.EQ.0 ) THEN 00338 IF( WANTS ) 00339 $ S = ONE 00340 IF( WANTSP ) 00341 $ SEP = SLANGE( '1', N, N, T, LDT, WORK ) 00342 GO TO 40 00343 END IF 00344 * 00345 * Collect the selected blocks at the top-left corner of T. 00346 * 00347 KS = 0 00348 PAIR = .FALSE. 00349 DO 20 K = 1, N 00350 IF( PAIR ) THEN 00351 PAIR = .FALSE. 00352 ELSE 00353 SWAP = SELECT( K ) 00354 IF( K.LT.N ) THEN 00355 IF( T( K+1, K ).NE.ZERO ) THEN 00356 PAIR = .TRUE. 00357 SWAP = SWAP .OR. SELECT( K+1 ) 00358 END IF 00359 END IF 00360 IF( SWAP ) THEN 00361 KS = KS + 1 00362 * 00363 * Swap the K-th block to position KS. 00364 * 00365 IERR = 0 00366 KK = K 00367 IF( K.NE.KS ) 00368 $ CALL STREXC( COMPQ, N, T, LDT, Q, LDQ, KK, KS, WORK, 00369 $ IERR ) 00370 IF( IERR.EQ.1 .OR. IERR.EQ.2 ) THEN 00371 * 00372 * Blocks too close to swap: exit. 00373 * 00374 INFO = 1 00375 IF( WANTS ) 00376 $ S = ZERO 00377 IF( WANTSP ) 00378 $ SEP = ZERO 00379 GO TO 40 00380 END IF 00381 IF( PAIR ) 00382 $ KS = KS + 1 00383 END IF 00384 END IF 00385 20 CONTINUE 00386 * 00387 IF( WANTS ) THEN 00388 * 00389 * Solve Sylvester equation for R: 00390 * 00391 * T11*R - R*T22 = scale*T12 00392 * 00393 CALL SLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 ) 00394 CALL STRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ), 00395 $ LDT, WORK, N1, SCALE, IERR ) 00396 * 00397 * Estimate the reciprocal of the condition number of the cluster 00398 * of eigenvalues. 00399 * 00400 RNORM = SLANGE( 'F', N1, N2, WORK, N1, WORK ) 00401 IF( RNORM.EQ.ZERO ) THEN 00402 S = ONE 00403 ELSE 00404 S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )* 00405 $ SQRT( RNORM ) ) 00406 END IF 00407 END IF 00408 * 00409 IF( WANTSP ) THEN 00410 * 00411 * Estimate sep(T11,T22). 00412 * 00413 EST = ZERO 00414 KASE = 0 00415 30 CONTINUE 00416 CALL SLACN2( NN, WORK( NN+1 ), WORK, IWORK, EST, KASE, ISAVE ) 00417 IF( KASE.NE.0 ) THEN 00418 IF( KASE.EQ.1 ) THEN 00419 * 00420 * Solve T11*R - R*T22 = scale*X. 00421 * 00422 CALL STRSYL( 'N', 'N', -1, N1, N2, T, LDT, 00423 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, 00424 $ IERR ) 00425 ELSE 00426 * 00427 * Solve T11**T*R - R*T22**T = scale*X. 00428 * 00429 CALL STRSYL( 'T', 'T', -1, N1, N2, T, LDT, 00430 $ T( N1+1, N1+1 ), LDT, WORK, N1, SCALE, 00431 $ IERR ) 00432 END IF 00433 GO TO 30 00434 END IF 00435 * 00436 SEP = SCALE / EST 00437 END IF 00438 * 00439 40 CONTINUE 00440 * 00441 * Store the output eigenvalues in WR and WI. 00442 * 00443 DO 50 K = 1, N 00444 WR( K ) = T( K, K ) 00445 WI( K ) = ZERO 00446 50 CONTINUE 00447 DO 60 K = 1, N - 1 00448 IF( T( K+1, K ).NE.ZERO ) THEN 00449 WI( K ) = SQRT( ABS( T( K, K+1 ) ) )* 00450 $ SQRT( ABS( T( K+1, K ) ) ) 00451 WI( K+1 ) = -WI( K ) 00452 END IF 00453 60 CONTINUE 00454 * 00455 WORK( 1 ) = LWMIN 00456 IWORK( 1 ) = LIWMIN 00457 * 00458 RETURN 00459 * 00460 * End of STRSEN 00461 * 00462 END
1.7.3 
