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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 ZGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, 00002 $ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO ) 00003 * 00004 * -- LAPACK driver 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 JOBVL, JOBVR 00011 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N 00012 * .. 00013 * .. Array Arguments .. 00014 DOUBLE PRECISION RWORK( * ) 00015 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), 00016 $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ), 00017 $ WORK( * ) 00018 * .. 00019 * 00020 * Purpose 00021 * ======= 00022 * 00023 * ZGGEV computes for a pair of N-by-N complex nonsymmetric matrices 00024 * (A,B), the generalized eigenvalues, and optionally, the left and/or 00025 * right generalized eigenvectors. 00026 * 00027 * A generalized eigenvalue for a pair of matrices (A,B) is a scalar 00028 * lambda or a ratio alpha/beta = lambda, such that A - lambda*B is 00029 * singular. It is usually represented as the pair (alpha,beta), as 00030 * there is a reasonable interpretation for beta=0, and even for both 00031 * being zero. 00032 * 00033 * The right generalized eigenvector v(j) corresponding to the 00034 * generalized eigenvalue lambda(j) of (A,B) satisfies 00035 * 00036 * A * v(j) = lambda(j) * B * v(j). 00037 * 00038 * The left generalized eigenvector u(j) corresponding to the 00039 * generalized eigenvalues lambda(j) of (A,B) satisfies 00040 * 00041 * u(j)**H * A = lambda(j) * u(j)**H * B 00042 * 00043 * where u(j)**H is the conjugate-transpose of u(j). 00044 * 00045 * Arguments 00046 * ========= 00047 * 00048 * JOBVL (input) CHARACTER*1 00049 * = 'N': do not compute the left generalized eigenvectors; 00050 * = 'V': compute the left generalized eigenvectors. 00051 * 00052 * JOBVR (input) CHARACTER*1 00053 * = 'N': do not compute the right generalized eigenvectors; 00054 * = 'V': compute the right generalized eigenvectors. 00055 * 00056 * N (input) INTEGER 00057 * The order of the matrices A, B, VL, and VR. N >= 0. 00058 * 00059 * A (input/output) COMPLEX*16 array, dimension (LDA, N) 00060 * On entry, the matrix A in the pair (A,B). 00061 * On exit, A has been overwritten. 00062 * 00063 * LDA (input) INTEGER 00064 * The leading dimension of A. LDA >= max(1,N). 00065 * 00066 * B (input/output) COMPLEX*16 array, dimension (LDB, N) 00067 * On entry, the matrix B in the pair (A,B). 00068 * On exit, B has been overwritten. 00069 * 00070 * LDB (input) INTEGER 00071 * The leading dimension of B. LDB >= max(1,N). 00072 * 00073 * ALPHA (output) COMPLEX*16 array, dimension (N) 00074 * BETA (output) COMPLEX*16 array, dimension (N) 00075 * On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the 00076 * generalized eigenvalues. 00077 * 00078 * Note: the quotients ALPHA(j)/BETA(j) may easily over- or 00079 * underflow, and BETA(j) may even be zero. Thus, the user 00080 * should avoid naively computing the ratio alpha/beta. 00081 * However, ALPHA will be always less than and usually 00082 * comparable with norm(A) in magnitude, and BETA always less 00083 * than and usually comparable with norm(B). 00084 * 00085 * VL (output) COMPLEX*16 array, dimension (LDVL,N) 00086 * If JOBVL = 'V', the left generalized eigenvectors u(j) are 00087 * stored one after another in the columns of VL, in the same 00088 * order as their eigenvalues. 00089 * Each eigenvector is scaled so the largest component has 00090 * abs(real part) + abs(imag. part) = 1. 00091 * Not referenced if JOBVL = 'N'. 00092 * 00093 * LDVL (input) INTEGER 00094 * The leading dimension of the matrix VL. LDVL >= 1, and 00095 * if JOBVL = 'V', LDVL >= N. 00096 * 00097 * VR (output) COMPLEX*16 array, dimension (LDVR,N) 00098 * If JOBVR = 'V', the right generalized eigenvectors v(j) are 00099 * stored one after another in the columns of VR, in the same 00100 * order as their eigenvalues. 00101 * Each eigenvector is scaled so the largest component has 00102 * abs(real part) + abs(imag. part) = 1. 00103 * Not referenced if JOBVR = 'N'. 00104 * 00105 * LDVR (input) INTEGER 00106 * The leading dimension of the matrix VR. LDVR >= 1, and 00107 * if JOBVR = 'V', LDVR >= N. 00108 * 00109 * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) 00110 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00111 * 00112 * LWORK (input) INTEGER 00113 * The dimension of the array WORK. LWORK >= max(1,2*N). 00114 * For good performance, LWORK must generally be larger. 00115 * 00116 * If LWORK = -1, then a workspace query is assumed; the routine 00117 * only calculates the optimal size of the WORK array, returns 00118 * this value as the first entry of the WORK array, and no error 00119 * message related to LWORK is issued by XERBLA. 00120 * 00121 * RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N) 00122 * 00123 * INFO (output) INTEGER 00124 * = 0: successful exit 00125 * < 0: if INFO = -i, the i-th argument had an illegal value. 00126 * =1,...,N: 00127 * The QZ iteration failed. No eigenvectors have been 00128 * calculated, but ALPHA(j) and BETA(j) should be 00129 * correct for j=INFO+1,...,N. 00130 * > N: =N+1: other then QZ iteration failed in DHGEQZ, 00131 * =N+2: error return from DTGEVC. 00132 * 00133 * ===================================================================== 00134 * 00135 * .. Parameters .. 00136 DOUBLE PRECISION ZERO, ONE 00137 PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) 00138 COMPLEX*16 CZERO, CONE 00139 PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), 00140 $ CONE = ( 1.0D0, 0.0D0 ) ) 00141 * .. 00142 * .. Local Scalars .. 00143 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY 00144 CHARACTER CHTEMP 00145 INTEGER ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, ILO, 00146 $ IN, IRIGHT, IROWS, IRWRK, ITAU, IWRK, JC, JR, 00147 $ LWKMIN, LWKOPT 00148 DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, 00149 $ SMLNUM, TEMP 00150 COMPLEX*16 X 00151 * .. 00152 * .. Local Arrays .. 00153 LOGICAL LDUMMA( 1 ) 00154 * .. 00155 * .. External Subroutines .. 00156 EXTERNAL DLABAD, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, 00157 $ ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR, 00158 $ ZUNMQR 00159 * .. 00160 * .. External Functions .. 00161 LOGICAL LSAME 00162 INTEGER ILAENV 00163 DOUBLE PRECISION DLAMCH, ZLANGE 00164 EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE 00165 * .. 00166 * .. Intrinsic Functions .. 00167 INTRINSIC ABS, DBLE, DIMAG, MAX, SQRT 00168 * .. 00169 * .. Statement Functions .. 00170 DOUBLE PRECISION ABS1 00171 * .. 00172 * .. Statement Function definitions .. 00173 ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) ) 00174 * .. 00175 * .. Executable Statements .. 00176 * 00177 * Decode the input arguments 00178 * 00179 IF( LSAME( JOBVL, 'N' ) ) THEN 00180 IJOBVL = 1 00181 ILVL = .FALSE. 00182 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN 00183 IJOBVL = 2 00184 ILVL = .TRUE. 00185 ELSE 00186 IJOBVL = -1 00187 ILVL = .FALSE. 00188 END IF 00189 * 00190 IF( LSAME( JOBVR, 'N' ) ) THEN 00191 IJOBVR = 1 00192 ILVR = .FALSE. 00193 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN 00194 IJOBVR = 2 00195 ILVR = .TRUE. 00196 ELSE 00197 IJOBVR = -1 00198 ILVR = .FALSE. 00199 END IF 00200 ILV = ILVL .OR. ILVR 00201 * 00202 * Test the input arguments 00203 * 00204 INFO = 0 00205 LQUERY = ( LWORK.EQ.-1 ) 00206 IF( IJOBVL.LE.0 ) THEN 00207 INFO = -1 00208 ELSE IF( IJOBVR.LE.0 ) THEN 00209 INFO = -2 00210 ELSE IF( N.LT.0 ) THEN 00211 INFO = -3 00212 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00213 INFO = -5 00214 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00215 INFO = -7 00216 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN 00217 INFO = -11 00218 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN 00219 INFO = -13 00220 END IF 00221 * 00222 * Compute workspace 00223 * (Note: Comments in the code beginning "Workspace:" describe the 00224 * minimal amount of workspace needed at that point in the code, 00225 * as well as the preferred amount for good performance. 00226 * NB refers to the optimal block size for the immediately 00227 * following subroutine, as returned by ILAENV. The workspace is 00228 * computed assuming ILO = 1 and IHI = N, the worst case.) 00229 * 00230 IF( INFO.EQ.0 ) THEN 00231 LWKMIN = MAX( 1, 2*N ) 00232 LWKOPT = MAX( 1, N + N*ILAENV( 1, 'ZGEQRF', ' ', N, 1, N, 0 ) ) 00233 LWKOPT = MAX( LWKOPT, N + 00234 $ N*ILAENV( 1, 'ZUNMQR', ' ', N, 1, N, 0 ) ) 00235 IF( ILVL ) THEN 00236 LWKOPT = MAX( LWKOPT, N + 00237 $ N*ILAENV( 1, 'ZUNGQR', ' ', N, 1, N, -1 ) ) 00238 END IF 00239 WORK( 1 ) = LWKOPT 00240 * 00241 IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) 00242 $ INFO = -15 00243 END IF 00244 * 00245 IF( INFO.NE.0 ) THEN 00246 CALL XERBLA( 'ZGGEV ', -INFO ) 00247 RETURN 00248 ELSE IF( LQUERY ) THEN 00249 RETURN 00250 END IF 00251 * 00252 * Quick return if possible 00253 * 00254 IF( N.EQ.0 ) 00255 $ RETURN 00256 * 00257 * Get machine constants 00258 * 00259 EPS = DLAMCH( 'E' )*DLAMCH( 'B' ) 00260 SMLNUM = DLAMCH( 'S' ) 00261 BIGNUM = ONE / SMLNUM 00262 CALL DLABAD( SMLNUM, BIGNUM ) 00263 SMLNUM = SQRT( SMLNUM ) / EPS 00264 BIGNUM = ONE / SMLNUM 00265 * 00266 * Scale A if max element outside range [SMLNUM,BIGNUM] 00267 * 00268 ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK ) 00269 ILASCL = .FALSE. 00270 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 00271 ANRMTO = SMLNUM 00272 ILASCL = .TRUE. 00273 ELSE IF( ANRM.GT.BIGNUM ) THEN 00274 ANRMTO = BIGNUM 00275 ILASCL = .TRUE. 00276 END IF 00277 IF( ILASCL ) 00278 $ CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) 00279 * 00280 * Scale B if max element outside range [SMLNUM,BIGNUM] 00281 * 00282 BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK ) 00283 ILBSCL = .FALSE. 00284 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN 00285 BNRMTO = SMLNUM 00286 ILBSCL = .TRUE. 00287 ELSE IF( BNRM.GT.BIGNUM ) THEN 00288 BNRMTO = BIGNUM 00289 ILBSCL = .TRUE. 00290 END IF 00291 IF( ILBSCL ) 00292 $ CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) 00293 * 00294 * Permute the matrices A, B to isolate eigenvalues if possible 00295 * (Real Workspace: need 6*N) 00296 * 00297 ILEFT = 1 00298 IRIGHT = N + 1 00299 IRWRK = IRIGHT + N 00300 CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ), 00301 $ RWORK( IRIGHT ), RWORK( IRWRK ), IERR ) 00302 * 00303 * Reduce B to triangular form (QR decomposition of B) 00304 * (Complex Workspace: need N, prefer N*NB) 00305 * 00306 IROWS = IHI + 1 - ILO 00307 IF( ILV ) THEN 00308 ICOLS = N + 1 - ILO 00309 ELSE 00310 ICOLS = IROWS 00311 END IF 00312 ITAU = 1 00313 IWRK = ITAU + IROWS 00314 CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), 00315 $ WORK( IWRK ), LWORK+1-IWRK, IERR ) 00316 * 00317 * Apply the orthogonal transformation to matrix A 00318 * (Complex Workspace: need N, prefer N*NB) 00319 * 00320 CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, 00321 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), 00322 $ LWORK+1-IWRK, IERR ) 00323 * 00324 * Initialize VL 00325 * (Complex Workspace: need N, prefer N*NB) 00326 * 00327 IF( ILVL ) THEN 00328 CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL ) 00329 IF( IROWS.GT.1 ) THEN 00330 CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, 00331 $ VL( ILO+1, ILO ), LDVL ) 00332 END IF 00333 CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, 00334 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) 00335 END IF 00336 * 00337 * Initialize VR 00338 * 00339 IF( ILVR ) 00340 $ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR ) 00341 * 00342 * Reduce to generalized Hessenberg form 00343 * 00344 IF( ILV ) THEN 00345 * 00346 * Eigenvectors requested -- work on whole matrix. 00347 * 00348 CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, 00349 $ LDVL, VR, LDVR, IERR ) 00350 ELSE 00351 CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, 00352 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR ) 00353 END IF 00354 * 00355 * Perform QZ algorithm (Compute eigenvalues, and optionally, the 00356 * Schur form and Schur vectors) 00357 * (Complex Workspace: need N) 00358 * (Real Workspace: need N) 00359 * 00360 IWRK = ITAU 00361 IF( ILV ) THEN 00362 CHTEMP = 'S' 00363 ELSE 00364 CHTEMP = 'E' 00365 END IF 00366 CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, 00367 $ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWRK ), 00368 $ LWORK+1-IWRK, RWORK( IRWRK ), IERR ) 00369 IF( IERR.NE.0 ) THEN 00370 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN 00371 INFO = IERR 00372 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN 00373 INFO = IERR - N 00374 ELSE 00375 INFO = N + 1 00376 END IF 00377 GO TO 70 00378 END IF 00379 * 00380 * Compute Eigenvectors 00381 * (Real Workspace: need 2*N) 00382 * (Complex Workspace: need 2*N) 00383 * 00384 IF( ILV ) THEN 00385 IF( ILVL ) THEN 00386 IF( ILVR ) THEN 00387 CHTEMP = 'B' 00388 ELSE 00389 CHTEMP = 'L' 00390 END IF 00391 ELSE 00392 CHTEMP = 'R' 00393 END IF 00394 * 00395 CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL, 00396 $ VR, LDVR, N, IN, WORK( IWRK ), RWORK( IRWRK ), 00397 $ IERR ) 00398 IF( IERR.NE.0 ) THEN 00399 INFO = N + 2 00400 GO TO 70 00401 END IF 00402 * 00403 * Undo balancing on VL and VR and normalization 00404 * (Workspace: none needed) 00405 * 00406 IF( ILVL ) THEN 00407 CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ), 00408 $ RWORK( IRIGHT ), N, VL, LDVL, IERR ) 00409 DO 30 JC = 1, N 00410 TEMP = ZERO 00411 DO 10 JR = 1, N 00412 TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) ) 00413 10 CONTINUE 00414 IF( TEMP.LT.SMLNUM ) 00415 $ GO TO 30 00416 TEMP = ONE / TEMP 00417 DO 20 JR = 1, N 00418 VL( JR, JC ) = VL( JR, JC )*TEMP 00419 20 CONTINUE 00420 30 CONTINUE 00421 END IF 00422 IF( ILVR ) THEN 00423 CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ), 00424 $ RWORK( IRIGHT ), N, VR, LDVR, IERR ) 00425 DO 60 JC = 1, N 00426 TEMP = ZERO 00427 DO 40 JR = 1, N 00428 TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) ) 00429 40 CONTINUE 00430 IF( TEMP.LT.SMLNUM ) 00431 $ GO TO 60 00432 TEMP = ONE / TEMP 00433 DO 50 JR = 1, N 00434 VR( JR, JC ) = VR( JR, JC )*TEMP 00435 50 CONTINUE 00436 60 CONTINUE 00437 END IF 00438 END IF 00439 * 00440 * Undo scaling if necessary 00441 * 00442 IF( ILASCL ) 00443 $ CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) 00444 * 00445 IF( ILBSCL ) 00446 $ CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) 00447 * 00448 70 CONTINUE 00449 WORK( 1 ) = LWKOPT 00450 * 00451 RETURN 00452 * 00453 * End of ZGGEV 00454 * 00455 END
1.7.3 
