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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief <b> SGGEVX 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 SGGEVX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sggevx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sggevx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sggevx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, 00022 * ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, 00023 * IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, 00024 * RCONDV, WORK, LWORK, IWORK, BWORK, INFO ) 00025 * 00026 * .. Scalar Arguments .. 00027 * CHARACTER BALANC, JOBVL, JOBVR, SENSE 00028 * INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N 00029 * REAL ABNRM, BBNRM 00030 * .. 00031 * .. Array Arguments .. 00032 * LOGICAL BWORK( * ) 00033 * INTEGER IWORK( * ) 00034 * REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), 00035 * $ B( LDB, * ), BETA( * ), LSCALE( * ), 00036 * $ RCONDE( * ), RCONDV( * ), RSCALE( * ), 00037 * $ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) 00038 * .. 00039 * 00040 * 00041 *> \par Purpose: 00042 * ============= 00043 *> 00044 *> \verbatim 00045 *> 00046 *> SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B) 00047 *> the generalized eigenvalues, and optionally, the left and/or right 00048 *> generalized eigenvectors. 00049 *> 00050 *> Optionally also, it computes a balancing transformation to improve 00051 *> the conditioning of the eigenvalues and eigenvectors (ILO, IHI, 00052 *> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for 00053 *> the eigenvalues (RCONDE), and reciprocal condition numbers for the 00054 *> right eigenvectors (RCONDV). 00055 *> 00056 *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar 00057 *> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is 00058 *> singular. It is usually represented as the pair (alpha,beta), as 00059 *> there is a reasonable interpretation for beta=0, and even for both 00060 *> being zero. 00061 *> 00062 *> The right eigenvector v(j) corresponding to the eigenvalue lambda(j) 00063 *> of (A,B) satisfies 00064 *> 00065 *> A * v(j) = lambda(j) * B * v(j) . 00066 *> 00067 *> The left eigenvector u(j) corresponding to the eigenvalue lambda(j) 00068 *> of (A,B) satisfies 00069 *> 00070 *> u(j)**H * A = lambda(j) * u(j)**H * B. 00071 *> 00072 *> where u(j)**H is the conjugate-transpose of u(j). 00073 *> 00074 *> \endverbatim 00075 * 00076 * Arguments: 00077 * ========== 00078 * 00079 *> \param[in] BALANC 00080 *> \verbatim 00081 *> BALANC is CHARACTER*1 00082 *> Specifies the balance option to be performed. 00083 *> = 'N': do not diagonally scale or permute; 00084 *> = 'P': permute only; 00085 *> = 'S': scale only; 00086 *> = 'B': both permute and scale. 00087 *> Computed reciprocal condition numbers will be for the 00088 *> matrices after permuting and/or balancing. Permuting does 00089 *> not change condition numbers (in exact arithmetic), but 00090 *> balancing does. 00091 *> \endverbatim 00092 *> 00093 *> \param[in] JOBVL 00094 *> \verbatim 00095 *> JOBVL is CHARACTER*1 00096 *> = 'N': do not compute the left generalized eigenvectors; 00097 *> = 'V': compute the left generalized eigenvectors. 00098 *> \endverbatim 00099 *> 00100 *> \param[in] JOBVR 00101 *> \verbatim 00102 *> JOBVR is CHARACTER*1 00103 *> = 'N': do not compute the right generalized eigenvectors; 00104 *> = 'V': compute the right generalized eigenvectors. 00105 *> \endverbatim 00106 *> 00107 *> \param[in] SENSE 00108 *> \verbatim 00109 *> SENSE is CHARACTER*1 00110 *> Determines which reciprocal condition numbers are computed. 00111 *> = 'N': none are computed; 00112 *> = 'E': computed for eigenvalues only; 00113 *> = 'V': computed for eigenvectors only; 00114 *> = 'B': computed for eigenvalues and eigenvectors. 00115 *> \endverbatim 00116 *> 00117 *> \param[in] N 00118 *> \verbatim 00119 *> N is INTEGER 00120 *> The order of the matrices A, B, VL, and VR. N >= 0. 00121 *> \endverbatim 00122 *> 00123 *> \param[in,out] A 00124 *> \verbatim 00125 *> A is REAL array, dimension (LDA, N) 00126 *> On entry, the matrix A in the pair (A,B). 00127 *> On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' 00128 *> or both, then A contains the first part of the real Schur 00129 *> form of the "balanced" versions of the input A and B. 00130 *> \endverbatim 00131 *> 00132 *> \param[in] LDA 00133 *> \verbatim 00134 *> LDA is INTEGER 00135 *> The leading dimension of A. LDA >= max(1,N). 00136 *> \endverbatim 00137 *> 00138 *> \param[in,out] B 00139 *> \verbatim 00140 *> B is REAL array, dimension (LDB, N) 00141 *> On entry, the matrix B in the pair (A,B). 00142 *> On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' 00143 *> or both, then B contains the second part of the real Schur 00144 *> form of the "balanced" versions of the input A and B. 00145 *> \endverbatim 00146 *> 00147 *> \param[in] LDB 00148 *> \verbatim 00149 *> LDB is INTEGER 00150 *> The leading dimension of B. LDB >= max(1,N). 00151 *> \endverbatim 00152 *> 00153 *> \param[out] ALPHAR 00154 *> \verbatim 00155 *> ALPHAR is REAL array, dimension (N) 00156 *> \endverbatim 00157 *> 00158 *> \param[out] ALPHAI 00159 *> \verbatim 00160 *> ALPHAI is REAL array, dimension (N) 00161 *> \endverbatim 00162 *> 00163 *> \param[out] BETA 00164 *> \verbatim 00165 *> BETA is REAL array, dimension (N) 00166 *> On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will 00167 *> be the generalized eigenvalues. If ALPHAI(j) is zero, then 00168 *> the j-th eigenvalue is real; if positive, then the j-th and 00169 *> (j+1)-st eigenvalues are a complex conjugate pair, with 00170 *> ALPHAI(j+1) negative. 00171 *> 00172 *> Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) 00173 *> may easily over- or underflow, and BETA(j) may even be zero. 00174 *> Thus, the user should avoid naively computing the ratio 00175 *> ALPHA/BETA. However, ALPHAR and ALPHAI will be always less 00176 *> than and usually comparable with norm(A) in magnitude, and 00177 *> BETA always less than and usually comparable with norm(B). 00178 *> \endverbatim 00179 *> 00180 *> \param[out] VL 00181 *> \verbatim 00182 *> VL is REAL array, dimension (LDVL,N) 00183 *> If JOBVL = 'V', the left eigenvectors u(j) are stored one 00184 *> after another in the columns of VL, in the same order as 00185 *> their eigenvalues. If the j-th eigenvalue is real, then 00186 *> u(j) = VL(:,j), the j-th column of VL. If the j-th and 00187 *> (j+1)-th eigenvalues form a complex conjugate pair, then 00188 *> u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1). 00189 *> Each eigenvector will be scaled so the largest component have 00190 *> abs(real part) + abs(imag. part) = 1. 00191 *> Not referenced if JOBVL = 'N'. 00192 *> \endverbatim 00193 *> 00194 *> \param[in] LDVL 00195 *> \verbatim 00196 *> LDVL is INTEGER 00197 *> The leading dimension of the matrix VL. LDVL >= 1, and 00198 *> if JOBVL = 'V', LDVL >= N. 00199 *> \endverbatim 00200 *> 00201 *> \param[out] VR 00202 *> \verbatim 00203 *> VR is REAL array, dimension (LDVR,N) 00204 *> If JOBVR = 'V', the right eigenvectors v(j) are stored one 00205 *> after another in the columns of VR, in the same order as 00206 *> their eigenvalues. If the j-th eigenvalue is real, then 00207 *> v(j) = VR(:,j), the j-th column of VR. If the j-th and 00208 *> (j+1)-th eigenvalues form a complex conjugate pair, then 00209 *> v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). 00210 *> Each eigenvector will be scaled so the largest component have 00211 *> abs(real part) + abs(imag. part) = 1. 00212 *> Not referenced if JOBVR = 'N'. 00213 *> \endverbatim 00214 *> 00215 *> \param[in] LDVR 00216 *> \verbatim 00217 *> LDVR is INTEGER 00218 *> The leading dimension of the matrix VR. LDVR >= 1, and 00219 *> if JOBVR = 'V', LDVR >= N. 00220 *> \endverbatim 00221 *> 00222 *> \param[out] ILO 00223 *> \verbatim 00224 *> ILO is INTEGER 00225 *> \endverbatim 00226 *> 00227 *> \param[out] IHI 00228 *> \verbatim 00229 *> IHI is INTEGER 00230 *> ILO and IHI are integer values such that on exit 00231 *> A(i,j) = 0 and B(i,j) = 0 if i > j and 00232 *> j = 1,...,ILO-1 or i = IHI+1,...,N. 00233 *> If BALANC = 'N' or 'S', ILO = 1 and IHI = N. 00234 *> \endverbatim 00235 *> 00236 *> \param[out] LSCALE 00237 *> \verbatim 00238 *> LSCALE is REAL array, dimension (N) 00239 *> Details of the permutations and scaling factors applied 00240 *> to the left side of A and B. If PL(j) is the index of the 00241 *> row interchanged with row j, and DL(j) is the scaling 00242 *> factor applied to row j, then 00243 *> LSCALE(j) = PL(j) for j = 1,...,ILO-1 00244 *> = DL(j) for j = ILO,...,IHI 00245 *> = PL(j) for j = IHI+1,...,N. 00246 *> The order in which the interchanges are made is N to IHI+1, 00247 *> then 1 to ILO-1. 00248 *> \endverbatim 00249 *> 00250 *> \param[out] RSCALE 00251 *> \verbatim 00252 *> RSCALE is REAL array, dimension (N) 00253 *> Details of the permutations and scaling factors applied 00254 *> to the right side of A and B. If PR(j) is the index of the 00255 *> column interchanged with column j, and DR(j) is the scaling 00256 *> factor applied to column j, then 00257 *> RSCALE(j) = PR(j) for j = 1,...,ILO-1 00258 *> = DR(j) for j = ILO,...,IHI 00259 *> = PR(j) for j = IHI+1,...,N 00260 *> The order in which the interchanges are made is N to IHI+1, 00261 *> then 1 to ILO-1. 00262 *> \endverbatim 00263 *> 00264 *> \param[out] ABNRM 00265 *> \verbatim 00266 *> ABNRM is REAL 00267 *> The one-norm of the balanced matrix A. 00268 *> \endverbatim 00269 *> 00270 *> \param[out] BBNRM 00271 *> \verbatim 00272 *> BBNRM is REAL 00273 *> The one-norm of the balanced matrix B. 00274 *> \endverbatim 00275 *> 00276 *> \param[out] RCONDE 00277 *> \verbatim 00278 *> RCONDE is REAL array, dimension (N) 00279 *> If SENSE = 'E' or 'B', the reciprocal condition numbers of 00280 *> the eigenvalues, stored in consecutive elements of the array. 00281 *> For a complex conjugate pair of eigenvalues two consecutive 00282 *> elements of RCONDE are set to the same value. Thus RCONDE(j), 00283 *> RCONDV(j), and the j-th columns of VL and VR all correspond 00284 *> to the j-th eigenpair. 00285 *> If SENSE = 'N' or 'V', RCONDE is not referenced. 00286 *> \endverbatim 00287 *> 00288 *> \param[out] RCONDV 00289 *> \verbatim 00290 *> RCONDV is REAL array, dimension (N) 00291 *> If SENSE = 'V' or 'B', the estimated reciprocal condition 00292 *> numbers of the eigenvectors, stored in consecutive elements 00293 *> of the array. For a complex eigenvector two consecutive 00294 *> elements of RCONDV are set to the same value. If the 00295 *> eigenvalues cannot be reordered to compute RCONDV(j), 00296 *> RCONDV(j) is set to 0; this can only occur when the true 00297 *> value would be very small anyway. 00298 *> If SENSE = 'N' or 'E', RCONDV is not referenced. 00299 *> \endverbatim 00300 *> 00301 *> \param[out] WORK 00302 *> \verbatim 00303 *> WORK is REAL array, dimension (MAX(1,LWORK)) 00304 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00305 *> \endverbatim 00306 *> 00307 *> \param[in] LWORK 00308 *> \verbatim 00309 *> LWORK is INTEGER 00310 *> The dimension of the array WORK. LWORK >= max(1,2*N). 00311 *> If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', 00312 *> LWORK >= max(1,6*N). 00313 *> If SENSE = 'E', LWORK >= max(1,10*N). 00314 *> If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16. 00315 *> 00316 *> If LWORK = -1, then a workspace query is assumed; the routine 00317 *> only calculates the optimal size of the WORK array, returns 00318 *> this value as the first entry of the WORK array, and no error 00319 *> message related to LWORK is issued by XERBLA. 00320 *> \endverbatim 00321 *> 00322 *> \param[out] IWORK 00323 *> \verbatim 00324 *> IWORK is INTEGER array, dimension (N+6) 00325 *> If SENSE = 'E', IWORK is not referenced. 00326 *> \endverbatim 00327 *> 00328 *> \param[out] BWORK 00329 *> \verbatim 00330 *> BWORK is LOGICAL array, dimension (N) 00331 *> If SENSE = 'N', BWORK is not referenced. 00332 *> \endverbatim 00333 *> 00334 *> \param[out] INFO 00335 *> \verbatim 00336 *> INFO is INTEGER 00337 *> = 0: successful exit 00338 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00339 *> = 1,...,N: 00340 *> The QZ iteration failed. No eigenvectors have been 00341 *> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) 00342 *> should be correct for j=INFO+1,...,N. 00343 *> > N: =N+1: other than QZ iteration failed in SHGEQZ. 00344 *> =N+2: error return from STGEVC. 00345 *> \endverbatim 00346 * 00347 * Authors: 00348 * ======== 00349 * 00350 *> \author Univ. of Tennessee 00351 *> \author Univ. of California Berkeley 00352 *> \author Univ. of Colorado Denver 00353 *> \author NAG Ltd. 00354 * 00355 *> \date April 2012 00356 * 00357 *> \ingroup realGEeigen 00358 * 00359 *> \par Further Details: 00360 * ===================== 00361 *> 00362 *> \verbatim 00363 *> 00364 *> Balancing a matrix pair (A,B) includes, first, permuting rows and 00365 *> columns to isolate eigenvalues, second, applying diagonal similarity 00366 *> transformation to the rows and columns to make the rows and columns 00367 *> as close in norm as possible. The computed reciprocal condition 00368 *> numbers correspond to the balanced matrix. Permuting rows and columns 00369 *> will not change the condition numbers (in exact arithmetic) but 00370 *> diagonal scaling will. For further explanation of balancing, see 00371 *> section 4.11.1.2 of LAPACK Users' Guide. 00372 *> 00373 *> An approximate error bound on the chordal distance between the i-th 00374 *> computed generalized eigenvalue w and the corresponding exact 00375 *> eigenvalue lambda is 00376 *> 00377 *> chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) 00378 *> 00379 *> An approximate error bound for the angle between the i-th computed 00380 *> eigenvector VL(i) or VR(i) is given by 00381 *> 00382 *> EPS * norm(ABNRM, BBNRM) / DIF(i). 00383 *> 00384 *> For further explanation of the reciprocal condition numbers RCONDE 00385 *> and RCONDV, see section 4.11 of LAPACK User's Guide. 00386 *> \endverbatim 00387 *> 00388 * ===================================================================== 00389 SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, 00390 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, 00391 $ IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, 00392 $ RCONDV, WORK, LWORK, IWORK, BWORK, INFO ) 00393 * 00394 * -- LAPACK driver routine (version 3.4.1) -- 00395 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00396 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00397 * April 2012 00398 * 00399 * .. Scalar Arguments .. 00400 CHARACTER BALANC, JOBVL, JOBVR, SENSE 00401 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N 00402 REAL ABNRM, BBNRM 00403 * .. 00404 * .. Array Arguments .. 00405 LOGICAL BWORK( * ) 00406 INTEGER IWORK( * ) 00407 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), 00408 $ B( LDB, * ), BETA( * ), LSCALE( * ), 00409 $ RCONDE( * ), RCONDV( * ), RSCALE( * ), 00410 $ VL( LDVL, * ), VR( LDVR, * ), WORK( * ) 00411 * .. 00412 * 00413 * ===================================================================== 00414 * 00415 * .. Parameters .. 00416 REAL ZERO, ONE 00417 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00418 * .. 00419 * .. Local Scalars .. 00420 LOGICAL ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL, 00421 $ PAIR, WANTSB, WANTSE, WANTSN, WANTSV 00422 CHARACTER CHTEMP 00423 INTEGER I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS, 00424 $ ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK, 00425 $ MINWRK, MM 00426 REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, 00427 $ SMLNUM, TEMP 00428 * .. 00429 * .. Local Arrays .. 00430 LOGICAL LDUMMA( 1 ) 00431 * .. 00432 * .. External Subroutines .. 00433 EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLABAD, 00434 $ SLACPY, SLASCL, SLASET, SORGQR, SORMQR, STGEVC, 00435 $ STGSNA, XERBLA 00436 * .. 00437 * .. External Functions .. 00438 LOGICAL LSAME 00439 INTEGER ILAENV 00440 REAL SLAMCH, SLANGE 00441 EXTERNAL LSAME, ILAENV, SLAMCH, SLANGE 00442 * .. 00443 * .. Intrinsic Functions .. 00444 INTRINSIC ABS, MAX, SQRT 00445 * .. 00446 * .. Executable Statements .. 00447 * 00448 * Decode the input arguments 00449 * 00450 IF( LSAME( JOBVL, 'N' ) ) THEN 00451 IJOBVL = 1 00452 ILVL = .FALSE. 00453 ELSE IF( LSAME( JOBVL, 'V' ) ) THEN 00454 IJOBVL = 2 00455 ILVL = .TRUE. 00456 ELSE 00457 IJOBVL = -1 00458 ILVL = .FALSE. 00459 END IF 00460 * 00461 IF( LSAME( JOBVR, 'N' ) ) THEN 00462 IJOBVR = 1 00463 ILVR = .FALSE. 00464 ELSE IF( LSAME( JOBVR, 'V' ) ) THEN 00465 IJOBVR = 2 00466 ILVR = .TRUE. 00467 ELSE 00468 IJOBVR = -1 00469 ILVR = .FALSE. 00470 END IF 00471 ILV = ILVL .OR. ILVR 00472 * 00473 NOSCL = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' ) 00474 WANTSN = LSAME( SENSE, 'N' ) 00475 WANTSE = LSAME( SENSE, 'E' ) 00476 WANTSV = LSAME( SENSE, 'V' ) 00477 WANTSB = LSAME( SENSE, 'B' ) 00478 * 00479 * Test the input arguments 00480 * 00481 INFO = 0 00482 LQUERY = ( LWORK.EQ.-1 ) 00483 IF( .NOT.( NOSCL .OR. LSAME( BALANC, 'S' ) .OR. 00484 $ LSAME( BALANC, 'B' ) ) ) THEN 00485 INFO = -1 00486 ELSE IF( IJOBVL.LE.0 ) THEN 00487 INFO = -2 00488 ELSE IF( IJOBVR.LE.0 ) THEN 00489 INFO = -3 00490 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) ) 00491 $ THEN 00492 INFO = -4 00493 ELSE IF( N.LT.0 ) THEN 00494 INFO = -5 00495 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00496 INFO = -7 00497 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN 00498 INFO = -9 00499 ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN 00500 INFO = -14 00501 ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN 00502 INFO = -16 00503 END IF 00504 * 00505 * Compute workspace 00506 * (Note: Comments in the code beginning "Workspace:" describe the 00507 * minimal amount of workspace needed at that point in the code, 00508 * as well as the preferred amount for good performance. 00509 * NB refers to the optimal block size for the immediately 00510 * following subroutine, as returned by ILAENV. The workspace is 00511 * computed assuming ILO = 1 and IHI = N, the worst case.) 00512 * 00513 IF( INFO.EQ.0 ) THEN 00514 IF( N.EQ.0 ) THEN 00515 MINWRK = 1 00516 MAXWRK = 1 00517 ELSE 00518 IF( NOSCL .AND. .NOT.ILV ) THEN 00519 MINWRK = 2*N 00520 ELSE 00521 MINWRK = 6*N 00522 END IF 00523 IF( WANTSE ) THEN 00524 MINWRK = 10*N 00525 ELSE IF( WANTSV .OR. WANTSB ) THEN 00526 MINWRK = 2*N*( N + 4 ) + 16 00527 END IF 00528 MAXWRK = MINWRK 00529 MAXWRK = MAX( MAXWRK, 00530 $ N + N*ILAENV( 1, 'SGEQRF', ' ', N, 1, N, 0 ) ) 00531 MAXWRK = MAX( MAXWRK, 00532 $ N + N*ILAENV( 1, 'SORMQR', ' ', N, 1, N, 0 ) ) 00533 IF( ILVL ) THEN 00534 MAXWRK = MAX( MAXWRK, N + 00535 $ N*ILAENV( 1, 'SORGQR', ' ', N, 1, N, 0 ) ) 00536 END IF 00537 END IF 00538 WORK( 1 ) = MAXWRK 00539 * 00540 IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN 00541 INFO = -26 00542 END IF 00543 END IF 00544 * 00545 IF( INFO.NE.0 ) THEN 00546 CALL XERBLA( 'SGGEVX', -INFO ) 00547 RETURN 00548 ELSE IF( LQUERY ) THEN 00549 RETURN 00550 END IF 00551 * 00552 * Quick return if possible 00553 * 00554 IF( N.EQ.0 ) 00555 $ RETURN 00556 * 00557 * 00558 * Get machine constants 00559 * 00560 EPS = SLAMCH( 'P' ) 00561 SMLNUM = SLAMCH( 'S' ) 00562 BIGNUM = ONE / SMLNUM 00563 CALL SLABAD( SMLNUM, BIGNUM ) 00564 SMLNUM = SQRT( SMLNUM ) / EPS 00565 BIGNUM = ONE / SMLNUM 00566 * 00567 * Scale A if max element outside range [SMLNUM,BIGNUM] 00568 * 00569 ANRM = SLANGE( 'M', N, N, A, LDA, WORK ) 00570 ILASCL = .FALSE. 00571 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN 00572 ANRMTO = SMLNUM 00573 ILASCL = .TRUE. 00574 ELSE IF( ANRM.GT.BIGNUM ) THEN 00575 ANRMTO = BIGNUM 00576 ILASCL = .TRUE. 00577 END IF 00578 IF( ILASCL ) 00579 $ CALL SLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) 00580 * 00581 * Scale B if max element outside range [SMLNUM,BIGNUM] 00582 * 00583 BNRM = SLANGE( 'M', N, N, B, LDB, WORK ) 00584 ILBSCL = .FALSE. 00585 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN 00586 BNRMTO = SMLNUM 00587 ILBSCL = .TRUE. 00588 ELSE IF( BNRM.GT.BIGNUM ) THEN 00589 BNRMTO = BIGNUM 00590 ILBSCL = .TRUE. 00591 END IF 00592 IF( ILBSCL ) 00593 $ CALL SLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) 00594 * 00595 * Permute and/or balance the matrix pair (A,B) 00596 * (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise) 00597 * 00598 CALL SGGBAL( BALANC, N, A, LDA, B, LDB, ILO, IHI, LSCALE, RSCALE, 00599 $ WORK, IERR ) 00600 * 00601 * Compute ABNRM and BBNRM 00602 * 00603 ABNRM = SLANGE( '1', N, N, A, LDA, WORK( 1 ) ) 00604 IF( ILASCL ) THEN 00605 WORK( 1 ) = ABNRM 00606 CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, 1, 1, WORK( 1 ), 1, 00607 $ IERR ) 00608 ABNRM = WORK( 1 ) 00609 END IF 00610 * 00611 BBNRM = SLANGE( '1', N, N, B, LDB, WORK( 1 ) ) 00612 IF( ILBSCL ) THEN 00613 WORK( 1 ) = BBNRM 00614 CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, 1, 1, WORK( 1 ), 1, 00615 $ IERR ) 00616 BBNRM = WORK( 1 ) 00617 END IF 00618 * 00619 * Reduce B to triangular form (QR decomposition of B) 00620 * (Workspace: need N, prefer N*NB ) 00621 * 00622 IROWS = IHI + 1 - ILO 00623 IF( ILV .OR. .NOT.WANTSN ) THEN 00624 ICOLS = N + 1 - ILO 00625 ELSE 00626 ICOLS = IROWS 00627 END IF 00628 ITAU = 1 00629 IWRK = ITAU + IROWS 00630 CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), 00631 $ WORK( IWRK ), LWORK+1-IWRK, IERR ) 00632 * 00633 * Apply the orthogonal transformation to A 00634 * (Workspace: need N, prefer N*NB) 00635 * 00636 CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, 00637 $ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), 00638 $ LWORK+1-IWRK, IERR ) 00639 * 00640 * Initialize VL and/or VR 00641 * (Workspace: need N, prefer N*NB) 00642 * 00643 IF( ILVL ) THEN 00644 CALL SLASET( 'Full', N, N, ZERO, ONE, VL, LDVL ) 00645 IF( IROWS.GT.1 ) THEN 00646 CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, 00647 $ VL( ILO+1, ILO ), LDVL ) 00648 END IF 00649 CALL SORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL, 00650 $ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) 00651 END IF 00652 * 00653 IF( ILVR ) 00654 $ CALL SLASET( 'Full', N, N, ZERO, ONE, VR, LDVR ) 00655 * 00656 * Reduce to generalized Hessenberg form 00657 * (Workspace: none needed) 00658 * 00659 IF( ILV .OR. .NOT.WANTSN ) THEN 00660 * 00661 * Eigenvectors requested -- work on whole matrix. 00662 * 00663 CALL SGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL, 00664 $ LDVL, VR, LDVR, IERR ) 00665 ELSE 00666 CALL SGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA, 00667 $ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IERR ) 00668 END IF 00669 * 00670 * Perform QZ algorithm (Compute eigenvalues, and optionally, the 00671 * Schur forms and Schur vectors) 00672 * (Workspace: need N) 00673 * 00674 IF( ILV .OR. .NOT.WANTSN ) THEN 00675 CHTEMP = 'S' 00676 ELSE 00677 CHTEMP = 'E' 00678 END IF 00679 * 00680 CALL SHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, 00681 $ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, 00682 $ LWORK, IERR ) 00683 IF( IERR.NE.0 ) THEN 00684 IF( IERR.GT.0 .AND. IERR.LE.N ) THEN 00685 INFO = IERR 00686 ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN 00687 INFO = IERR - N 00688 ELSE 00689 INFO = N + 1 00690 END IF 00691 GO TO 130 00692 END IF 00693 * 00694 * Compute Eigenvectors and estimate condition numbers if desired 00695 * (Workspace: STGEVC: need 6*N 00696 * STGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B', 00697 * need N otherwise ) 00698 * 00699 IF( ILV .OR. .NOT.WANTSN ) THEN 00700 IF( ILV ) THEN 00701 IF( ILVL ) THEN 00702 IF( ILVR ) THEN 00703 CHTEMP = 'B' 00704 ELSE 00705 CHTEMP = 'L' 00706 END IF 00707 ELSE 00708 CHTEMP = 'R' 00709 END IF 00710 * 00711 CALL STGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, 00712 $ LDVL, VR, LDVR, N, IN, WORK, IERR ) 00713 IF( IERR.NE.0 ) THEN 00714 INFO = N + 2 00715 GO TO 130 00716 END IF 00717 END IF 00718 * 00719 IF( .NOT.WANTSN ) THEN 00720 * 00721 * compute eigenvectors (STGEVC) and estimate condition 00722 * numbers (STGSNA). Note that the definition of the condition 00723 * number is not invariant under transformation (u,v) to 00724 * (Q*u, Z*v), where (u,v) are eigenvectors of the generalized 00725 * Schur form (S,T), Q and Z are orthogonal matrices. In order 00726 * to avoid using extra 2*N*N workspace, we have to recalculate 00727 * eigenvectors and estimate one condition numbers at a time. 00728 * 00729 PAIR = .FALSE. 00730 DO 20 I = 1, N 00731 * 00732 IF( PAIR ) THEN 00733 PAIR = .FALSE. 00734 GO TO 20 00735 END IF 00736 MM = 1 00737 IF( I.LT.N ) THEN 00738 IF( A( I+1, I ).NE.ZERO ) THEN 00739 PAIR = .TRUE. 00740 MM = 2 00741 END IF 00742 END IF 00743 * 00744 DO 10 J = 1, N 00745 BWORK( J ) = .FALSE. 00746 10 CONTINUE 00747 IF( MM.EQ.1 ) THEN 00748 BWORK( I ) = .TRUE. 00749 ELSE IF( MM.EQ.2 ) THEN 00750 BWORK( I ) = .TRUE. 00751 BWORK( I+1 ) = .TRUE. 00752 END IF 00753 * 00754 IWRK = MM*N + 1 00755 IWRK1 = IWRK + MM*N 00756 * 00757 * Compute a pair of left and right eigenvectors. 00758 * (compute workspace: need up to 4*N + 6*N) 00759 * 00760 IF( WANTSE .OR. WANTSB ) THEN 00761 CALL STGEVC( 'B', 'S', BWORK, N, A, LDA, B, LDB, 00762 $ WORK( 1 ), N, WORK( IWRK ), N, MM, M, 00763 $ WORK( IWRK1 ), IERR ) 00764 IF( IERR.NE.0 ) THEN 00765 INFO = N + 2 00766 GO TO 130 00767 END IF 00768 END IF 00769 * 00770 CALL STGSNA( SENSE, 'S', BWORK, N, A, LDA, B, LDB, 00771 $ WORK( 1 ), N, WORK( IWRK ), N, RCONDE( I ), 00772 $ RCONDV( I ), MM, M, WORK( IWRK1 ), 00773 $ LWORK-IWRK1+1, IWORK, IERR ) 00774 * 00775 20 CONTINUE 00776 END IF 00777 END IF 00778 * 00779 * Undo balancing on VL and VR and normalization 00780 * (Workspace: none needed) 00781 * 00782 IF( ILVL ) THEN 00783 CALL SGGBAK( BALANC, 'L', N, ILO, IHI, LSCALE, RSCALE, N, VL, 00784 $ LDVL, IERR ) 00785 * 00786 DO 70 JC = 1, N 00787 IF( ALPHAI( JC ).LT.ZERO ) 00788 $ GO TO 70 00789 TEMP = ZERO 00790 IF( ALPHAI( JC ).EQ.ZERO ) THEN 00791 DO 30 JR = 1, N 00792 TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) ) 00793 30 CONTINUE 00794 ELSE 00795 DO 40 JR = 1, N 00796 TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+ 00797 $ ABS( VL( JR, JC+1 ) ) ) 00798 40 CONTINUE 00799 END IF 00800 IF( TEMP.LT.SMLNUM ) 00801 $ GO TO 70 00802 TEMP = ONE / TEMP 00803 IF( ALPHAI( JC ).EQ.ZERO ) THEN 00804 DO 50 JR = 1, N 00805 VL( JR, JC ) = VL( JR, JC )*TEMP 00806 50 CONTINUE 00807 ELSE 00808 DO 60 JR = 1, N 00809 VL( JR, JC ) = VL( JR, JC )*TEMP 00810 VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP 00811 60 CONTINUE 00812 END IF 00813 70 CONTINUE 00814 END IF 00815 IF( ILVR ) THEN 00816 CALL SGGBAK( BALANC, 'R', N, ILO, IHI, LSCALE, RSCALE, N, VR, 00817 $ LDVR, IERR ) 00818 DO 120 JC = 1, N 00819 IF( ALPHAI( JC ).LT.ZERO ) 00820 $ GO TO 120 00821 TEMP = ZERO 00822 IF( ALPHAI( JC ).EQ.ZERO ) THEN 00823 DO 80 JR = 1, N 00824 TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) ) 00825 80 CONTINUE 00826 ELSE 00827 DO 90 JR = 1, N 00828 TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+ 00829 $ ABS( VR( JR, JC+1 ) ) ) 00830 90 CONTINUE 00831 END IF 00832 IF( TEMP.LT.SMLNUM ) 00833 $ GO TO 120 00834 TEMP = ONE / TEMP 00835 IF( ALPHAI( JC ).EQ.ZERO ) THEN 00836 DO 100 JR = 1, N 00837 VR( JR, JC ) = VR( JR, JC )*TEMP 00838 100 CONTINUE 00839 ELSE 00840 DO 110 JR = 1, N 00841 VR( JR, JC ) = VR( JR, JC )*TEMP 00842 VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP 00843 110 CONTINUE 00844 END IF 00845 120 CONTINUE 00846 END IF 00847 * 00848 * Undo scaling if necessary 00849 * 00850 130 CONTINUE 00851 * 00852 IF( ILASCL ) THEN 00853 CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N, IERR ) 00854 CALL SLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N, IERR ) 00855 END IF 00856 * 00857 IF( ILBSCL ) THEN 00858 CALL SLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) 00859 END IF 00860 * 00861 WORK( 1 ) = MAXWRK 00862 RETURN 00863 * 00864 * End of SGGEVX 00865 * 00866 END