![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief <b> ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b> 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZGEGS + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgegs.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgegs.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgegs.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, 00022 * VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, 00023 * INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER JOBVSL, JOBVSR 00027 * INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N 00028 * .. 00029 * .. Array Arguments .. 00030 * DOUBLE PRECISION RWORK( * ) 00031 * COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), 00032 * $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), 00033 * $ WORK( * ) 00034 * .. 00035 * 00036 * 00037 *> \par Purpose: 00038 * ============= 00039 *> 00040 *> \verbatim 00041 *> 00042 *> This routine is deprecated and has been replaced by routine ZGGES. 00043 *> 00044 *> ZGEGS computes the eigenvalues, Schur form, and, optionally, the 00045 *> left and or/right Schur vectors of a complex matrix pair (A,B). 00046 *> Given two square matrices A and B, the generalized Schur 00047 *> factorization has the form 00048 *> 00049 *> A = Q*S*Z**H, B = Q*T*Z**H 00050 *> 00051 *> where Q and Z are unitary matrices and S and T are upper triangular. 00052 *> The columns of Q are the left Schur vectors 00053 *> and the columns of Z are the right Schur vectors. 00054 *> 00055 *> If only the eigenvalues of (A,B) are needed, the driver routine 00056 *> ZGEGV should be used instead. See ZGEGV for a description of the 00057 *> eigenvalues of the generalized nonsymmetric eigenvalue problem 00058 *> (GNEP). 00059 *> \endverbatim 00060 * 00061 * Arguments: 00062 * ========== 00063 * 00064 *> \param[in] JOBVSL 00065 *> \verbatim 00066 *> JOBVSL is CHARACTER*1 00067 *> = 'N': do not compute the left Schur vectors; 00068 *> = 'V': compute the left Schur vectors (returned in VSL). 00069 *> \endverbatim 00070 *> 00071 *> \param[in] JOBVSR 00072 *> \verbatim 00073 *> JOBVSR is CHARACTER*1 00074 *> = 'N': do not compute the right Schur vectors; 00075 *> = 'V': compute the right Schur vectors (returned in VSR). 00076 *> \endverbatim 00077 *> 00078 *> \param[in] N 00079 *> \verbatim 00080 *> N is INTEGER 00081 *> The order of the matrices A, B, VSL, and VSR. N >= 0. 00082 *> \endverbatim 00083 *> 00084 *> \param[in,out] A 00085 *> \verbatim 00086 *> A is COMPLEX*16 array, dimension (LDA, N) 00087 *> On entry, the matrix A. 00088 *> On exit, the upper triangular matrix S from the generalized 00089 *> Schur factorization. 00090 *> \endverbatim 00091 *> 00092 *> \param[in] LDA 00093 *> \verbatim 00094 *> LDA is INTEGER 00095 *> The leading dimension of A. LDA >= max(1,N). 00096 *> \endverbatim 00097 *> 00098 *> \param[in,out] B 00099 *> \verbatim 00100 *> B is COMPLEX*16 array, dimension (LDB, N) 00101 *> On entry, the matrix B. 00102 *> On exit, the upper triangular matrix T from the generalized 00103 *> Schur factorization. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] LDB 00107 *> \verbatim 00108 *> LDB is INTEGER 00109 *> The leading dimension of B. LDB >= max(1,N). 00110 *> \endverbatim 00111 *> 00112 *> \param[out] ALPHA 00113 *> \verbatim 00114 *> ALPHA is COMPLEX*16 array, dimension (N) 00115 *> The complex scalars alpha that define the eigenvalues of 00116 *> GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur 00117 *> form of A. 00118 *> \endverbatim 00119 *> 00120 *> \param[out] BETA 00121 *> \verbatim 00122 *> BETA is COMPLEX*16 array, dimension (N) 00123 *> The non-negative real scalars beta that define the 00124 *> eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element 00125 *> of the triangular factor T. 00126 *> 00127 *> Together, the quantities alpha = ALPHA(j) and beta = BETA(j) 00128 *> represent the j-th eigenvalue of the matrix pair (A,B), in 00129 *> one of the forms lambda = alpha/beta or mu = beta/alpha. 00130 *> Since either lambda or mu may overflow, they should not, 00131 *> in general, be computed. 00132 *> \endverbatim 00133 *> 00134 *> \param[out] VSL 00135 *> \verbatim 00136 *> VSL is COMPLEX*16 array, dimension (LDVSL,N) 00137 *> If JOBVSL = 'V', the matrix of left Schur vectors Q. 00138 *> Not referenced if JOBVSL = 'N'. 00139 *> \endverbatim 00140 *> 00141 *> \param[in] LDVSL 00142 *> \verbatim 00143 *> LDVSL is INTEGER 00144 *> The leading dimension of the matrix VSL. LDVSL >= 1, and 00145 *> if JOBVSL = 'V', LDVSL >= N. 00146 *> \endverbatim 00147 *> 00148 *> \param[out] VSR 00149 *> \verbatim 00150 *> VSR is COMPLEX*16 array, dimension (LDVSR,N) 00151 *> If JOBVSR = 'V', the matrix of right Schur vectors Z. 00152 *> Not referenced if JOBVSR = 'N'. 00153 *> \endverbatim 00154 *> 00155 *> \param[in] LDVSR 00156 *> \verbatim 00157 *> LDVSR is INTEGER 00158 *> The leading dimension of the matrix VSR. LDVSR >= 1, and 00159 *> if JOBVSR = 'V', LDVSR >= N. 00160 *> \endverbatim 00161 *> 00162 *> \param[out] WORK 00163 *> \verbatim 00164 *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 00165 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00166 *> \endverbatim 00167 *> 00168 *> \param[in] LWORK 00169 *> \verbatim 00170 *> LWORK is INTEGER 00171 *> The dimension of the array WORK. LWORK >= max(1,2*N). 00172 *> For good performance, LWORK must generally be larger. 00173 *> To compute the optimal value of LWORK, call ILAENV to get 00174 *> blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.) Then compute: 00175 *> NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR; 00176 *> the optimal LWORK is N*(NB+1). 00177 *> 00178 *> If LWORK = -1, then a workspace query is assumed; the routine 00179 *> only calculates the optimal size of the WORK array, returns 00180 *> this value as the first entry of the WORK array, and no error 00181 *> message related to LWORK is issued by XERBLA. 00182 *> \endverbatim 00183 *> 00184 *> \param[out] RWORK 00185 *> \verbatim 00186 *> RWORK is DOUBLE PRECISION array, dimension (3*N) 00187 *> \endverbatim 00188 *> 00189 *> \param[out] INFO 00190 *> \verbatim 00191 *> INFO is INTEGER 00192 *> = 0: successful exit 00193 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00194 *> =1,...,N: 00195 *> The QZ iteration failed. (A,B) are not in Schur 00196 *> form, but ALPHA(j) and BETA(j) should be correct for 00197 *> j=INFO+1,...,N. 00198 *> > N: errors that usually indicate LAPACK problems: 00199 *> =N+1: error return from ZGGBAL 00200 *> =N+2: error return from ZGEQRF 00201 *> =N+3: error return from ZUNMQR 00202 *> =N+4: error return from ZUNGQR 00203 *> =N+5: error return from ZGGHRD 00204 *> =N+6: error return from ZHGEQZ (other than failed 00205 *> iteration) 00206 *> =N+7: error return from ZGGBAK (computing VSL) 00207 *> =N+8: error return from ZGGBAK (computing VSR) 00208 *> =N+9: error return from ZLASCL (various places) 00209 *> \endverbatim 00210 * 00211 * Authors: 00212 * ======== 00213 * 00214 *> \author Univ. of Tennessee 00215 *> \author Univ. of California Berkeley 00216 *> \author Univ. of Colorado Denver 00217 *> \author NAG Ltd. 00218 * 00219 *> \date November 2011 00220 * 00221 *> \ingroup complex16GEeigen 00222 * 00223 * ===================================================================== 00224 SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA, 00225 $ VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, 00226 $ INFO ) 00227 * 00228 * -- LAPACK driver routine (version 3.4.0) -- 00229 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00230 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00231 * November 2011 00232 * 00233 * .. Scalar Arguments .. 00234 CHARACTER JOBVSL, JOBVSR 00235 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N 00236 * .. 00237 * .. Array Arguments .. 00238 DOUBLE PRECISION RWORK( * ) 00239 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), 00240 $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), 00241 $ WORK( * ) 00242 * .. 00243 * 00244 * ===================================================================== 00245 * 00246 * .. Parameters .. 00247 DOUBLE PRECISION ZERO, ONE 00248 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 00249 COMPLEX*16 CZERO, CONE 00250 PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), 00251 $ CONE = ( 1.0D0, 0.0D0 ) ) 00252 * .. 00253 * .. Local Scalars .. 00254 LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY 00255 INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO, 00256 $ IRIGHT, IROWS, IRWORK, ITAU, IWORK, LOPT, 00257 $ LWKMIN, LWKOPT, NB, NB1, NB2, NB3 00258 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, 00259 $ SAFMIN, SMLNUM 00260 * .. 00261 * .. External Subroutines .. 00262 EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ, 00263 $ ZLACPY, ZLASCL, ZLASET, ZUNGQR, ZUNMQR 00264 * .. 00265 * .. External Functions .. 00266 LOGICAL LSAME 00267 INTEGER ILAENV 00268 DOUBLE PRECISION DLAMCH, ZLANGE 00269 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE 00270 * .. 00271 * .. Intrinsic Functions .. 00272 INTRINSIC INT, MAX 00273 * .. 00274 * .. Executable Statements .. 00275 * 00276 * Decode the input arguments 00277 * 00278 IF( LSAME( JOBVSL, 'N' ) ) THEN 00279 IJOBVL = 1 00280 ILVSL = .FALSE. 00281 ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN 00282 IJOBVL = 2 00283 ILVSL = .TRUE. 00284 ELSE 00285 IJOBVL = -1 00286 ILVSL = .FALSE. 00287 END IF 00288 * 00289 IF( LSAME( JOBVSR, 'N' ) ) THEN 00290 IJOBVR = 1 00291 ILVSR = .FALSE. 00292 ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN 00293 IJOBVR = 2 00294 ILVSR = .TRUE. 00295 ELSE 00296 IJOBVR = -1 00297 ILVSR = .FALSE. 00298 END IF 00299 * 00300 * Test the input arguments 00301 * 00302 LWKMIN = MAX( 2*N, 1 ) 00303 LWKOPT = LWKMIN 00304 WORK( 1 ) = LWKOPT 00305 LQUERY = ( LWORK.EQ.-1 ) 00306 INFO = 0 00307 IF( IJOBVL.LE.0 ) THEN 00308 INFO = -1 00309 ELSE IF( IJOBVR.LE.0 ) THEN 00310 INFO = -2 00311 ELSE IF( N.LT.0 ) THEN 00312 INFO = -3 00313 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00314 INFO = -5 00315 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00316 INFO = -7 00317 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN 00318 INFO = -11 00319 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN 00320 INFO = -13 00321 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN 00322 INFO = -15 00323 END IF 00324 * 00325 IF( INFO.EQ.0 ) THEN 00326 NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 ) 00327 NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 ) 00328 NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 ) 00329 NB = MAX( NB1, NB2, NB3 ) 00330 LOPT = N*( NB+1 ) 00331 WORK( 1 ) = LOPT 00332 END IF 00333 * 00334 IF( INFO.NE.0 ) THEN 00335 CALL XERBLA( 'ZGEGS ', -INFO ) 00336 RETURN 00337 ELSE IF( LQUERY ) THEN 00338 RETURN 00339 END IF 00340 * 00341 * Quick return if possible 00342 * 00343 IF( N.EQ.0 ) 00344 $ RETURN 00345 * 00346 * Get machine constants 00347 * 00348 EPS = DLAMCH( 'E' )*DLAMCH( 'B' ) 00349 SAFMIN = DLAMCH( 'S' ) 00350 SMLNUM = N*SAFMIN / EPS 00351 BIGNUM = ONE / SMLNUM 00352 * 00353 * Scale A if max element outside range [SMLNUM,BIGNUM] 00354 * 00355 ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK ) 00356 ILASCL = .FALSE. 00357 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 00358 ANRMTO = SMLNUM 00359 ILASCL = .TRUE. 00360 ELSE IF( ANRM.GT.BIGNUM ) THEN 00361 ANRMTO = BIGNUM 00362 ILASCL = .TRUE. 00363 END IF 00364 * 00365 IF( ILASCL ) THEN 00366 CALL ZLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO ) 00367 IF( IINFO.NE.0 ) THEN 00368 INFO = N + 9 00369 RETURN 00370 END IF 00371 END IF 00372 * 00373 * Scale B if max element outside range [SMLNUM,BIGNUM] 00374 * 00375 BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK ) 00376 ILBSCL = .FALSE. 00377 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN 00378 BNRMTO = SMLNUM 00379 ILBSCL = .TRUE. 00380 ELSE IF( BNRM.GT.BIGNUM ) THEN 00381 BNRMTO = BIGNUM 00382 ILBSCL = .TRUE. 00383 END IF 00384 * 00385 IF( ILBSCL ) THEN 00386 CALL ZLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO ) 00387 IF( IINFO.NE.0 ) THEN 00388 INFO = N + 9 00389 RETURN 00390 END IF 00391 END IF 00392 * 00393 * Permute the matrix to make it more nearly triangular 00394 * 00395 ILEFT = 1 00396 IRIGHT = N + 1 00397 IRWORK = IRIGHT + N 00398 IWORK = 1 00399 CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ), 00400 $ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO ) 00401 IF( IINFO.NE.0 ) THEN 00402 INFO = N + 1 00403 GO TO 10 00404 END IF 00405 * 00406 * Reduce B to triangular form, and initialize VSL and/or VSR 00407 * 00408 IROWS = IHI + 1 - ILO 00409 ICOLS = N + 1 - ILO 00410 ITAU = IWORK 00411 IWORK = ITAU + IROWS 00412 CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), 00413 $ WORK( IWORK ), LWORK+1-IWORK, IINFO ) 00414 IF( IINFO.GE.0 ) 00415 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00416 IF( IINFO.NE.0 ) THEN 00417 INFO = N + 2 00418 GO TO 10 00419 END IF 00420 * 00421 CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, 00422 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ), 00423 $ LWORK+1-IWORK, IINFO ) 00424 IF( IINFO.GE.0 ) 00425 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00426 IF( IINFO.NE.0 ) THEN 00427 INFO = N + 3 00428 GO TO 10 00429 END IF 00430 * 00431 IF( ILVSL ) THEN 00432 CALL ZLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL ) 00433 CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, 00434 $ VSL( ILO+1, ILO ), LDVSL ) 00435 CALL ZUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, 00436 $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK, 00437 $ IINFO ) 00438 IF( IINFO.GE.0 ) 00439 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00440 IF( IINFO.NE.0 ) THEN 00441 INFO = N + 4 00442 GO TO 10 00443 END IF 00444 END IF 00445 * 00446 IF( ILVSR ) 00447 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR ) 00448 * 00449 * Reduce to generalized Hessenberg form 00450 * 00451 CALL ZGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL, 00452 $ LDVSL, VSR, LDVSR, IINFO ) 00453 IF( IINFO.NE.0 ) THEN 00454 INFO = N + 5 00455 GO TO 10 00456 END IF 00457 * 00458 * Perform QZ algorithm, computing Schur vectors if desired 00459 * 00460 IWORK = ITAU 00461 CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, 00462 $ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWORK ), 00463 $ LWORK+1-IWORK, RWORK( IRWORK ), IINFO ) 00464 IF( IINFO.GE.0 ) 00465 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00466 IF( IINFO.NE.0 ) THEN 00467 IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN 00468 INFO = IINFO 00469 ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN 00470 INFO = IINFO - N 00471 ELSE 00472 INFO = N + 6 00473 END IF 00474 GO TO 10 00475 END IF 00476 * 00477 * Apply permutation to VSL and VSR 00478 * 00479 IF( ILVSL ) THEN 00480 CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ), 00481 $ RWORK( IRIGHT ), N, VSL, LDVSL, IINFO ) 00482 IF( IINFO.NE.0 ) THEN 00483 INFO = N + 7 00484 GO TO 10 00485 END IF 00486 END IF 00487 IF( ILVSR ) THEN 00488 CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ), 00489 $ RWORK( IRIGHT ), N, VSR, LDVSR, IINFO ) 00490 IF( IINFO.NE.0 ) THEN 00491 INFO = N + 8 00492 GO TO 10 00493 END IF 00494 END IF 00495 * 00496 * Undo scaling 00497 * 00498 IF( ILASCL ) THEN 00499 CALL ZLASCL( 'U', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO ) 00500 IF( IINFO.NE.0 ) THEN 00501 INFO = N + 9 00502 RETURN 00503 END IF 00504 CALL ZLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHA, N, IINFO ) 00505 IF( IINFO.NE.0 ) THEN 00506 INFO = N + 9 00507 RETURN 00508 END IF 00509 END IF 00510 * 00511 IF( ILBSCL ) THEN 00512 CALL ZLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO ) 00513 IF( IINFO.NE.0 ) THEN 00514 INFO = N + 9 00515 RETURN 00516 END IF 00517 CALL ZLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO ) 00518 IF( IINFO.NE.0 ) THEN 00519 INFO = N + 9 00520 RETURN 00521 END IF 00522 END IF 00523 * 00524 10 CONTINUE 00525 WORK( 1 ) = LWKOPT 00526 * 00527 RETURN 00528 * 00529 * End of ZGEGS 00530 * 00531 END