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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 SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, 00002 $ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, 00003 $ LWORK, INFO ) 00004 * 00005 * -- LAPACK driver routine (version 3.2) -- 00006 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00007 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00008 * November 2006 00009 * 00010 * .. Scalar Arguments .. 00011 CHARACTER JOBVSL, JOBVSR 00012 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N 00013 * .. 00014 * .. Array Arguments .. 00015 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), 00016 $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ), 00017 $ VSR( LDVSR, * ), WORK( * ) 00018 * .. 00019 * 00020 * Purpose 00021 * ======= 00022 * 00023 * This routine is deprecated and has been replaced by routine SGGES. 00024 * 00025 * SGEGS computes the eigenvalues, real Schur form, and, optionally, 00026 * left and or/right Schur vectors of a real matrix pair (A,B). 00027 * Given two square matrices A and B, the generalized real Schur 00028 * factorization has the form 00029 * 00030 * A = Q*S*Z**T, B = Q*T*Z**T 00031 * 00032 * where Q and Z are orthogonal matrices, T is upper triangular, and S 00033 * is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal 00034 * blocks, the 2-by-2 blocks corresponding to complex conjugate pairs 00035 * of eigenvalues of (A,B). The columns of Q are the left Schur vectors 00036 * and the columns of Z are the right Schur vectors. 00037 * 00038 * If only the eigenvalues of (A,B) are needed, the driver routine 00039 * SGEGV should be used instead. See SGEGV for a description of the 00040 * eigenvalues of the generalized nonsymmetric eigenvalue problem 00041 * (GNEP). 00042 * 00043 * Arguments 00044 * ========= 00045 * 00046 * JOBVSL (input) CHARACTER*1 00047 * = 'N': do not compute the left Schur vectors; 00048 * = 'V': compute the left Schur vectors (returned in VSL). 00049 * 00050 * JOBVSR (input) CHARACTER*1 00051 * = 'N': do not compute the right Schur vectors; 00052 * = 'V': compute the right Schur vectors (returned in VSR). 00053 * 00054 * N (input) INTEGER 00055 * The order of the matrices A, B, VSL, and VSR. N >= 0. 00056 * 00057 * A (input/output) REAL array, dimension (LDA, N) 00058 * On entry, the matrix A. 00059 * On exit, the upper quasi-triangular matrix S from the 00060 * generalized real Schur factorization. 00061 * 00062 * LDA (input) INTEGER 00063 * The leading dimension of A. LDA >= max(1,N). 00064 * 00065 * B (input/output) REAL array, dimension (LDB, N) 00066 * On entry, the matrix B. 00067 * On exit, the upper triangular matrix T from the generalized 00068 * real Schur factorization. 00069 * 00070 * LDB (input) INTEGER 00071 * The leading dimension of B. LDB >= max(1,N). 00072 * 00073 * ALPHAR (output) REAL array, dimension (N) 00074 * The real parts of each scalar alpha defining an eigenvalue 00075 * of GNEP. 00076 * 00077 * ALPHAI (output) REAL array, dimension (N) 00078 * The imaginary parts of each scalar alpha defining an 00079 * eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th 00080 * eigenvalue is real; if positive, then the j-th and (j+1)-st 00081 * eigenvalues are a complex conjugate pair, with 00082 * ALPHAI(j+1) = -ALPHAI(j). 00083 * 00084 * BETA (output) REAL array, dimension (N) 00085 * The scalars beta that define the eigenvalues of GNEP. 00086 * Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and 00087 * beta = BETA(j) represent the j-th eigenvalue of the matrix 00088 * pair (A,B), in one of the forms lambda = alpha/beta or 00089 * mu = beta/alpha. Since either lambda or mu may overflow, 00090 * they should not, in general, be computed. 00091 * 00092 * VSL (output) REAL array, dimension (LDVSL,N) 00093 * If JOBVSL = 'V', the matrix of left Schur vectors Q. 00094 * Not referenced if JOBVSL = 'N'. 00095 * 00096 * LDVSL (input) INTEGER 00097 * The leading dimension of the matrix VSL. LDVSL >=1, and 00098 * if JOBVSL = 'V', LDVSL >= N. 00099 * 00100 * VSR (output) REAL array, dimension (LDVSR,N) 00101 * If JOBVSR = 'V', the matrix of right Schur vectors Z. 00102 * Not referenced if JOBVSR = 'N'. 00103 * 00104 * LDVSR (input) INTEGER 00105 * The leading dimension of the matrix VSR. LDVSR >= 1, and 00106 * if JOBVSR = 'V', LDVSR >= N. 00107 * 00108 * WORK (workspace/output) REAL array, dimension (MAX(1,LWORK)) 00109 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00110 * 00111 * LWORK (input) INTEGER 00112 * The dimension of the array WORK. LWORK >= max(1,4*N). 00113 * For good performance, LWORK must generally be larger. 00114 * To compute the optimal value of LWORK, call ILAENV to get 00115 * blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute: 00116 * NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR 00117 * The optimal LWORK is 2*N + N*(NB+1). 00118 * 00119 * If LWORK = -1, then a workspace query is assumed; the routine 00120 * only calculates the optimal size of the WORK array, returns 00121 * this value as the first entry of the WORK array, and no error 00122 * message related to LWORK is issued by XERBLA. 00123 * 00124 * INFO (output) INTEGER 00125 * = 0: successful exit 00126 * < 0: if INFO = -i, the i-th argument had an illegal value. 00127 * = 1,...,N: 00128 * The QZ iteration failed. (A,B) are not in Schur 00129 * form, but ALPHAR(j), ALPHAI(j), and BETA(j) should 00130 * be correct for j=INFO+1,...,N. 00131 * > N: errors that usually indicate LAPACK problems: 00132 * =N+1: error return from SGGBAL 00133 * =N+2: error return from SGEQRF 00134 * =N+3: error return from SORMQR 00135 * =N+4: error return from SORGQR 00136 * =N+5: error return from SGGHRD 00137 * =N+6: error return from SHGEQZ (other than failed 00138 * iteration) 00139 * =N+7: error return from SGGBAK (computing VSL) 00140 * =N+8: error return from SGGBAK (computing VSR) 00141 * =N+9: error return from SLASCL (various places) 00142 * 00143 * ===================================================================== 00144 * 00145 * .. Parameters .. 00146 REAL ZERO, ONE 00147 PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 ) 00148 * .. 00149 * .. Local Scalars .. 00150 LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY 00151 INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, 00152 $ ILO, IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN, 00153 $ LWKOPT, NB, NB1, NB2, NB3 00154 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, 00155 $ SAFMIN, SMLNUM 00156 * .. 00157 * .. External Subroutines .. 00158 EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLACPY, 00159 $ SLASCL, SLASET, SORGQR, SORMQR, XERBLA 00160 * .. 00161 * .. External Functions .. 00162 LOGICAL LSAME 00163 INTEGER ILAENV 00164 REAL SLAMCH, SLANGE 00165 EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE 00166 * .. 00167 * .. Intrinsic Functions .. 00168 INTRINSIC INT, MAX 00169 * .. 00170 * .. Executable Statements .. 00171 * 00172 * Decode the input arguments 00173 * 00174 IF( LSAME( JOBVSL, 'N' ) ) THEN 00175 IJOBVL = 1 00176 ILVSL = .FALSE. 00177 ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN 00178 IJOBVL = 2 00179 ILVSL = .TRUE. 00180 ELSE 00181 IJOBVL = -1 00182 ILVSL = .FALSE. 00183 END IF 00184 * 00185 IF( LSAME( JOBVSR, 'N' ) ) THEN 00186 IJOBVR = 1 00187 ILVSR = .FALSE. 00188 ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN 00189 IJOBVR = 2 00190 ILVSR = .TRUE. 00191 ELSE 00192 IJOBVR = -1 00193 ILVSR = .FALSE. 00194 END IF 00195 * 00196 * Test the input arguments 00197 * 00198 LWKMIN = MAX( 4*N, 1 ) 00199 LWKOPT = LWKMIN 00200 WORK( 1 ) = LWKOPT 00201 LQUERY = ( LWORK.EQ.-1 ) 00202 INFO = 0 00203 IF( IJOBVL.LE.0 ) THEN 00204 INFO = -1 00205 ELSE IF( IJOBVR.LE.0 ) THEN 00206 INFO = -2 00207 ELSE IF( N.LT.0 ) THEN 00208 INFO = -3 00209 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00210 INFO = -5 00211 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00212 INFO = -7 00213 ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN 00214 INFO = -12 00215 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN 00216 INFO = -14 00217 ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN 00218 INFO = -16 00219 END IF 00220 * 00221 IF( INFO.EQ.0 ) THEN 00222 NB1 = ILAENV( 1, 'SGEQRF', ' ', N, N, -1, -1 ) 00223 NB2 = ILAENV( 1, 'SORMQR', ' ', N, N, N, -1 ) 00224 NB3 = ILAENV( 1, 'SORGQR', ' ', N, N, N, -1 ) 00225 NB = MAX( NB1, NB2, NB3 ) 00226 LOPT = 2*N+N*(NB+1) 00227 WORK( 1 ) = LOPT 00228 END IF 00229 * 00230 IF( INFO.NE.0 ) THEN 00231 CALL XERBLA( 'SGEGS ', -INFO ) 00232 RETURN 00233 ELSE IF( LQUERY ) THEN 00234 RETURN 00235 END IF 00236 * 00237 * Quick return if possible 00238 * 00239 IF( N.EQ.0 ) 00240 $ RETURN 00241 * 00242 * Get machine constants 00243 * 00244 EPS = SLAMCH( 'E' )*SLAMCH( 'B' ) 00245 SAFMIN = SLAMCH( 'S' ) 00246 SMLNUM = N*SAFMIN / EPS 00247 BIGNUM = ONE / SMLNUM 00248 * 00249 * Scale A if max element outside range [SMLNUM,BIGNUM] 00250 * 00251 ANRM = SLANGE( 'M', N, N, A, LDA, WORK ) 00252 ILASCL = .FALSE. 00253 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 00254 ANRMTO = SMLNUM 00255 ILASCL = .TRUE. 00256 ELSE IF( ANRM.GT.BIGNUM ) THEN 00257 ANRMTO = BIGNUM 00258 ILASCL = .TRUE. 00259 END IF 00260 * 00261 IF( ILASCL ) THEN 00262 CALL SLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO ) 00263 IF( IINFO.NE.0 ) THEN 00264 INFO = N + 9 00265 RETURN 00266 END IF 00267 END IF 00268 * 00269 * Scale B if max element outside range [SMLNUM,BIGNUM] 00270 * 00271 BNRM = SLANGE( 'M', N, N, B, LDB, WORK ) 00272 ILBSCL = .FALSE. 00273 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN 00274 BNRMTO = SMLNUM 00275 ILBSCL = .TRUE. 00276 ELSE IF( BNRM.GT.BIGNUM ) THEN 00277 BNRMTO = BIGNUM 00278 ILBSCL = .TRUE. 00279 END IF 00280 * 00281 IF( ILBSCL ) THEN 00282 CALL SLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO ) 00283 IF( IINFO.NE.0 ) THEN 00284 INFO = N + 9 00285 RETURN 00286 END IF 00287 END IF 00288 * 00289 * Permute the matrix to make it more nearly triangular 00290 * Workspace layout: (2*N words -- "work..." not actually used) 00291 * left_permutation, right_permutation, work... 00292 * 00293 ILEFT = 1 00294 IRIGHT = N + 1 00295 IWORK = IRIGHT + N 00296 CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ), 00297 $ WORK( IRIGHT ), WORK( IWORK ), IINFO ) 00298 IF( IINFO.NE.0 ) THEN 00299 INFO = N + 1 00300 GO TO 10 00301 END IF 00302 * 00303 * Reduce B to triangular form, and initialize VSL and/or VSR 00304 * Workspace layout: ("work..." must have at least N words) 00305 * left_permutation, right_permutation, tau, work... 00306 * 00307 IROWS = IHI + 1 - ILO 00308 ICOLS = N + 1 - ILO 00309 ITAU = IWORK 00310 IWORK = ITAU + IROWS 00311 CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), 00312 $ WORK( IWORK ), LWORK+1-IWORK, IINFO ) 00313 IF( IINFO.GE.0 ) 00314 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00315 IF( IINFO.NE.0 ) THEN 00316 INFO = N + 2 00317 GO TO 10 00318 END IF 00319 * 00320 CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, 00321 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ), 00322 $ LWORK+1-IWORK, IINFO ) 00323 IF( IINFO.GE.0 ) 00324 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00325 IF( IINFO.NE.0 ) THEN 00326 INFO = N + 3 00327 GO TO 10 00328 END IF 00329 * 00330 IF( ILVSL ) THEN 00331 CALL SLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL ) 00332 CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, 00333 $ VSL( ILO+1, ILO ), LDVSL ) 00334 CALL SORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, 00335 $ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK, 00336 $ IINFO ) 00337 IF( IINFO.GE.0 ) 00338 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00339 IF( IINFO.NE.0 ) THEN 00340 INFO = N + 4 00341 GO TO 10 00342 END IF 00343 END IF 00344 * 00345 IF( ILVSR ) 00346 $ CALL SLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR ) 00347 * 00348 * Reduce to generalized Hessenberg form 00349 * 00350 CALL SGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL, 00351 $ LDVSL, VSR, LDVSR, IINFO ) 00352 IF( IINFO.NE.0 ) THEN 00353 INFO = N + 5 00354 GO TO 10 00355 END IF 00356 * 00357 * Perform QZ algorithm, computing Schur vectors if desired 00358 * Workspace layout: ("work..." must have at least 1 word) 00359 * left_permutation, right_permutation, work... 00360 * 00361 IWORK = ITAU 00362 CALL SHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, 00363 $ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, 00364 $ WORK( IWORK ), LWORK+1-IWORK, IINFO ) 00365 IF( IINFO.GE.0 ) 00366 $ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 ) 00367 IF( IINFO.NE.0 ) THEN 00368 IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN 00369 INFO = IINFO 00370 ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN 00371 INFO = IINFO - N 00372 ELSE 00373 INFO = N + 6 00374 END IF 00375 GO TO 10 00376 END IF 00377 * 00378 * Apply permutation to VSL and VSR 00379 * 00380 IF( ILVSL ) THEN 00381 CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ), 00382 $ WORK( IRIGHT ), N, VSL, LDVSL, IINFO ) 00383 IF( IINFO.NE.0 ) THEN 00384 INFO = N + 7 00385 GO TO 10 00386 END IF 00387 END IF 00388 IF( ILVSR ) THEN 00389 CALL SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ), 00390 $ WORK( IRIGHT ), N, VSR, LDVSR, IINFO ) 00391 IF( IINFO.NE.0 ) THEN 00392 INFO = N + 8 00393 GO TO 10 00394 END IF 00395 END IF 00396 * 00397 * Undo scaling 00398 * 00399 IF( ILASCL ) THEN 00400 CALL SLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO ) 00401 IF( IINFO.NE.0 ) THEN 00402 INFO = N + 9 00403 RETURN 00404 END IF 00405 CALL SLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N, 00406 $ IINFO ) 00407 IF( IINFO.NE.0 ) THEN 00408 INFO = N + 9 00409 RETURN 00410 END IF 00411 CALL SLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N, 00412 $ IINFO ) 00413 IF( IINFO.NE.0 ) THEN 00414 INFO = N + 9 00415 RETURN 00416 END IF 00417 END IF 00418 * 00419 IF( ILBSCL ) THEN 00420 CALL SLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO ) 00421 IF( IINFO.NE.0 ) THEN 00422 INFO = N + 9 00423 RETURN 00424 END IF 00425 CALL SLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO ) 00426 IF( IINFO.NE.0 ) THEN 00427 INFO = N + 9 00428 RETURN 00429 END IF 00430 END IF 00431 * 00432 10 CONTINUE 00433 WORK( 1 ) = LWKOPT 00434 * 00435 RETURN 00436 * 00437 * End of SGEGS 00438 * 00439 END
1.7.3 
