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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b CHGEQZ 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download CHGEQZ + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chgeqz.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chgeqz.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chgeqz.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, 00022 * ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, 00023 * RWORK, INFO ) 00024 * 00025 * .. Scalar Arguments .. 00026 * CHARACTER COMPQ, COMPZ, JOB 00027 * INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N 00028 * .. 00029 * .. Array Arguments .. 00030 * REAL RWORK( * ) 00031 * COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ), 00032 * $ Q( LDQ, * ), T( LDT, * ), WORK( * ), 00033 * $ Z( LDZ, * ) 00034 * .. 00035 * 00036 * 00037 *> \par Purpose: 00038 * ============= 00039 *> 00040 *> \verbatim 00041 *> 00042 *> CHGEQZ computes the eigenvalues of a complex matrix pair (H,T), 00043 *> where H is an upper Hessenberg matrix and T is upper triangular, 00044 *> using the single-shift QZ method. 00045 *> Matrix pairs of this type are produced by the reduction to 00046 *> generalized upper Hessenberg form of a complex matrix pair (A,B): 00047 *> 00048 *> A = Q1*H*Z1**H, B = Q1*T*Z1**H, 00049 *> 00050 *> as computed by CGGHRD. 00051 *> 00052 *> If JOB='S', then the Hessenberg-triangular pair (H,T) is 00053 *> also reduced to generalized Schur form, 00054 *> 00055 *> H = Q*S*Z**H, T = Q*P*Z**H, 00056 *> 00057 *> where Q and Z are unitary matrices and S and P are upper triangular. 00058 *> 00059 *> Optionally, the unitary matrix Q from the generalized Schur 00060 *> factorization may be postmultiplied into an input matrix Q1, and the 00061 *> unitary matrix Z may be postmultiplied into an input matrix Z1. 00062 *> If Q1 and Z1 are the unitary matrices from CGGHRD that reduced 00063 *> the matrix pair (A,B) to generalized Hessenberg form, then the output 00064 *> matrices Q1*Q and Z1*Z are the unitary factors from the generalized 00065 *> Schur factorization of (A,B): 00066 *> 00067 *> A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H. 00068 *> 00069 *> To avoid overflow, eigenvalues of the matrix pair (H,T) 00070 *> (equivalently, of (A,B)) are computed as a pair of complex values 00071 *> (alpha,beta). If beta is nonzero, lambda = alpha / beta is an 00072 *> eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP) 00073 *> A*x = lambda*B*x 00074 *> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the 00075 *> alternate form of the GNEP 00076 *> mu*A*y = B*y. 00077 *> The values of alpha and beta for the i-th eigenvalue can be read 00078 *> directly from the generalized Schur form: alpha = S(i,i), 00079 *> beta = P(i,i). 00080 *> 00081 *> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix 00082 *> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973), 00083 *> pp. 241--256. 00084 *> \endverbatim 00085 * 00086 * Arguments: 00087 * ========== 00088 * 00089 *> \param[in] JOB 00090 *> \verbatim 00091 *> JOB is CHARACTER*1 00092 *> = 'E': Compute eigenvalues only; 00093 *> = 'S': Computer eigenvalues and the Schur form. 00094 *> \endverbatim 00095 *> 00096 *> \param[in] COMPQ 00097 *> \verbatim 00098 *> COMPQ is CHARACTER*1 00099 *> = 'N': Left Schur vectors (Q) are not computed; 00100 *> = 'I': Q is initialized to the unit matrix and the matrix Q 00101 *> of left Schur vectors of (H,T) is returned; 00102 *> = 'V': Q must contain a unitary matrix Q1 on entry and 00103 *> the product Q1*Q is returned. 00104 *> \endverbatim 00105 *> 00106 *> \param[in] COMPZ 00107 *> \verbatim 00108 *> COMPZ is CHARACTER*1 00109 *> = 'N': Right Schur vectors (Z) are not computed; 00110 *> = 'I': Q is initialized to the unit matrix and the matrix Z 00111 *> of right Schur vectors of (H,T) is returned; 00112 *> = 'V': Z must contain a unitary matrix Z1 on entry and 00113 *> the product Z1*Z is returned. 00114 *> \endverbatim 00115 *> 00116 *> \param[in] N 00117 *> \verbatim 00118 *> N is INTEGER 00119 *> The order of the matrices H, T, Q, and Z. N >= 0. 00120 *> \endverbatim 00121 *> 00122 *> \param[in] ILO 00123 *> \verbatim 00124 *> ILO is INTEGER 00125 *> \endverbatim 00126 *> 00127 *> \param[in] IHI 00128 *> \verbatim 00129 *> IHI is INTEGER 00130 *> ILO and IHI mark the rows and columns of H which are in 00131 *> Hessenberg form. It is assumed that A is already upper 00132 *> triangular in rows and columns 1:ILO-1 and IHI+1:N. 00133 *> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0. 00134 *> \endverbatim 00135 *> 00136 *> \param[in,out] H 00137 *> \verbatim 00138 *> H is COMPLEX array, dimension (LDH, N) 00139 *> On entry, the N-by-N upper Hessenberg matrix H. 00140 *> On exit, if JOB = 'S', H contains the upper triangular 00141 *> matrix S from the generalized Schur factorization. 00142 *> If JOB = 'E', the diagonal of H matches that of S, but 00143 *> the rest of H is unspecified. 00144 *> \endverbatim 00145 *> 00146 *> \param[in] LDH 00147 *> \verbatim 00148 *> LDH is INTEGER 00149 *> The leading dimension of the array H. LDH >= max( 1, N ). 00150 *> \endverbatim 00151 *> 00152 *> \param[in,out] T 00153 *> \verbatim 00154 *> T is COMPLEX array, dimension (LDT, N) 00155 *> On entry, the N-by-N upper triangular matrix T. 00156 *> On exit, if JOB = 'S', T contains the upper triangular 00157 *> matrix P from the generalized Schur factorization. 00158 *> If JOB = 'E', the diagonal of T matches that of P, but 00159 *> the rest of T is unspecified. 00160 *> \endverbatim 00161 *> 00162 *> \param[in] LDT 00163 *> \verbatim 00164 *> LDT is INTEGER 00165 *> The leading dimension of the array T. LDT >= max( 1, N ). 00166 *> \endverbatim 00167 *> 00168 *> \param[out] ALPHA 00169 *> \verbatim 00170 *> ALPHA is COMPLEX array, dimension (N) 00171 *> The complex scalars alpha that define the eigenvalues of 00172 *> GNEP. ALPHA(i) = S(i,i) in the generalized Schur 00173 *> factorization. 00174 *> \endverbatim 00175 *> 00176 *> \param[out] BETA 00177 *> \verbatim 00178 *> BETA is COMPLEX array, dimension (N) 00179 *> The real non-negative scalars beta that define the 00180 *> eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized 00181 *> Schur factorization. 00182 *> 00183 *> Together, the quantities alpha = ALPHA(j) and beta = BETA(j) 00184 *> represent the j-th eigenvalue of the matrix pair (A,B), in 00185 *> one of the forms lambda = alpha/beta or mu = beta/alpha. 00186 *> Since either lambda or mu may overflow, they should not, 00187 *> in general, be computed. 00188 *> \endverbatim 00189 *> 00190 *> \param[in,out] Q 00191 *> \verbatim 00192 *> Q is COMPLEX array, dimension (LDQ, N) 00193 *> On entry, if COMPZ = 'V', the unitary matrix Q1 used in the 00194 *> reduction of (A,B) to generalized Hessenberg form. 00195 *> On exit, if COMPZ = 'I', the unitary matrix of left Schur 00196 *> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of 00197 *> left Schur vectors of (A,B). 00198 *> Not referenced if COMPZ = 'N'. 00199 *> \endverbatim 00200 *> 00201 *> \param[in] LDQ 00202 *> \verbatim 00203 *> LDQ is INTEGER 00204 *> The leading dimension of the array Q. LDQ >= 1. 00205 *> If COMPQ='V' or 'I', then LDQ >= N. 00206 *> \endverbatim 00207 *> 00208 *> \param[in,out] Z 00209 *> \verbatim 00210 *> Z is COMPLEX array, dimension (LDZ, N) 00211 *> On entry, if COMPZ = 'V', the unitary matrix Z1 used in the 00212 *> reduction of (A,B) to generalized Hessenberg form. 00213 *> On exit, if COMPZ = 'I', the unitary matrix of right Schur 00214 *> vectors of (H,T), and if COMPZ = 'V', the unitary matrix of 00215 *> right Schur vectors of (A,B). 00216 *> Not referenced if COMPZ = 'N'. 00217 *> \endverbatim 00218 *> 00219 *> \param[in] LDZ 00220 *> \verbatim 00221 *> LDZ is INTEGER 00222 *> The leading dimension of the array Z. LDZ >= 1. 00223 *> If COMPZ='V' or 'I', then LDZ >= N. 00224 *> \endverbatim 00225 *> 00226 *> \param[out] WORK 00227 *> \verbatim 00228 *> WORK is COMPLEX array, dimension (MAX(1,LWORK)) 00229 *> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. 00230 *> \endverbatim 00231 *> 00232 *> \param[in] LWORK 00233 *> \verbatim 00234 *> LWORK is INTEGER 00235 *> The dimension of the array WORK. LWORK >= max(1,N). 00236 *> 00237 *> If LWORK = -1, then a workspace query is assumed; the routine 00238 *> only calculates the optimal size of the WORK array, returns 00239 *> this value as the first entry of the WORK array, and no error 00240 *> message related to LWORK is issued by XERBLA. 00241 *> \endverbatim 00242 *> 00243 *> \param[out] RWORK 00244 *> \verbatim 00245 *> RWORK is REAL array, dimension (N) 00246 *> \endverbatim 00247 *> 00248 *> \param[out] INFO 00249 *> \verbatim 00250 *> INFO is INTEGER 00251 *> = 0: successful exit 00252 *> < 0: if INFO = -i, the i-th argument had an illegal value 00253 *> = 1,...,N: the QZ iteration did not converge. (H,T) is not 00254 *> in Schur form, but ALPHA(i) and BETA(i), 00255 *> i=INFO+1,...,N should be correct. 00256 *> = N+1,...,2*N: the shift calculation failed. (H,T) is not 00257 *> in Schur form, but ALPHA(i) and BETA(i), 00258 *> i=INFO-N+1,...,N should be correct. 00259 *> \endverbatim 00260 * 00261 * Authors: 00262 * ======== 00263 * 00264 *> \author Univ. of Tennessee 00265 *> \author Univ. of California Berkeley 00266 *> \author Univ. of Colorado Denver 00267 *> \author NAG Ltd. 00268 * 00269 *> \date April 2012 00270 * 00271 *> \ingroup complexGEcomputational 00272 * 00273 *> \par Further Details: 00274 * ===================== 00275 *> 00276 *> \verbatim 00277 *> 00278 *> We assume that complex ABS works as long as its value is less than 00279 *> overflow. 00280 *> \endverbatim 00281 *> 00282 * ===================================================================== 00283 SUBROUTINE CHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT, 00284 $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, 00285 $ RWORK, INFO ) 00286 * 00287 * -- LAPACK computational routine (version 3.4.1) -- 00288 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00289 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00290 * April 2012 00291 * 00292 * .. Scalar Arguments .. 00293 CHARACTER COMPQ, COMPZ, JOB 00294 INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N 00295 * .. 00296 * .. Array Arguments .. 00297 REAL RWORK( * ) 00298 COMPLEX ALPHA( * ), BETA( * ), H( LDH, * ), 00299 $ Q( LDQ, * ), T( LDT, * ), WORK( * ), 00300 $ Z( LDZ, * ) 00301 * .. 00302 * 00303 * ===================================================================== 00304 * 00305 * .. Parameters .. 00306 COMPLEX CZERO, CONE 00307 PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ), 00308 $ CONE = ( 1.0E+0, 0.0E+0 ) ) 00309 REAL ZERO, ONE 00310 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00311 REAL HALF 00312 PARAMETER ( HALF = 0.5E+0 ) 00313 * .. 00314 * .. Local Scalars .. 00315 LOGICAL ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY 00316 INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST, 00317 $ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER, 00318 $ JR, MAXIT 00319 REAL ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL, 00320 $ C, SAFMIN, TEMP, TEMP2, TEMPR, ULP 00321 COMPLEX ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2, 00322 $ CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1, 00323 $ U12, X 00324 * .. 00325 * .. External Functions .. 00326 LOGICAL LSAME 00327 REAL CLANHS, SLAMCH 00328 EXTERNAL LSAME, CLANHS, SLAMCH 00329 * .. 00330 * .. External Subroutines .. 00331 EXTERNAL CLARTG, CLASET, CROT, CSCAL, XERBLA 00332 * .. 00333 * .. Intrinsic Functions .. 00334 INTRINSIC ABS, AIMAG, CMPLX, CONJG, MAX, MIN, REAL, SQRT 00335 * .. 00336 * .. Statement Functions .. 00337 REAL ABS1 00338 * .. 00339 * .. Statement Function definitions .. 00340 ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) ) 00341 * .. 00342 * .. Executable Statements .. 00343 * 00344 * Decode JOB, COMPQ, COMPZ 00345 * 00346 IF( LSAME( JOB, 'E' ) ) THEN 00347 ILSCHR = .FALSE. 00348 ISCHUR = 1 00349 ELSE IF( LSAME( JOB, 'S' ) ) THEN 00350 ILSCHR = .TRUE. 00351 ISCHUR = 2 00352 ELSE 00353 ISCHUR = 0 00354 END IF 00355 * 00356 IF( LSAME( COMPQ, 'N' ) ) THEN 00357 ILQ = .FALSE. 00358 ICOMPQ = 1 00359 ELSE IF( LSAME( COMPQ, 'V' ) ) THEN 00360 ILQ = .TRUE. 00361 ICOMPQ = 2 00362 ELSE IF( LSAME( COMPQ, 'I' ) ) THEN 00363 ILQ = .TRUE. 00364 ICOMPQ = 3 00365 ELSE 00366 ICOMPQ = 0 00367 END IF 00368 * 00369 IF( LSAME( COMPZ, 'N' ) ) THEN 00370 ILZ = .FALSE. 00371 ICOMPZ = 1 00372 ELSE IF( LSAME( COMPZ, 'V' ) ) THEN 00373 ILZ = .TRUE. 00374 ICOMPZ = 2 00375 ELSE IF( LSAME( COMPZ, 'I' ) ) THEN 00376 ILZ = .TRUE. 00377 ICOMPZ = 3 00378 ELSE 00379 ICOMPZ = 0 00380 END IF 00381 * 00382 * Check Argument Values 00383 * 00384 INFO = 0 00385 WORK( 1 ) = MAX( 1, N ) 00386 LQUERY = ( LWORK.EQ.-1 ) 00387 IF( ISCHUR.EQ.0 ) THEN 00388 INFO = -1 00389 ELSE IF( ICOMPQ.EQ.0 ) THEN 00390 INFO = -2 00391 ELSE IF( ICOMPZ.EQ.0 ) THEN 00392 INFO = -3 00393 ELSE IF( N.LT.0 ) THEN 00394 INFO = -4 00395 ELSE IF( ILO.LT.1 ) THEN 00396 INFO = -5 00397 ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN 00398 INFO = -6 00399 ELSE IF( LDH.LT.N ) THEN 00400 INFO = -8 00401 ELSE IF( LDT.LT.N ) THEN 00402 INFO = -10 00403 ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN 00404 INFO = -14 00405 ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN 00406 INFO = -16 00407 ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN 00408 INFO = -18 00409 END IF 00410 IF( INFO.NE.0 ) THEN 00411 CALL XERBLA( 'CHGEQZ', -INFO ) 00412 RETURN 00413 ELSE IF( LQUERY ) THEN 00414 RETURN 00415 END IF 00416 * 00417 * Quick return if possible 00418 * 00419 * WORK( 1 ) = CMPLX( 1 ) 00420 IF( N.LE.0 ) THEN 00421 WORK( 1 ) = CMPLX( 1 ) 00422 RETURN 00423 END IF 00424 * 00425 * Initialize Q and Z 00426 * 00427 IF( ICOMPQ.EQ.3 ) 00428 $ CALL CLASET( 'Full', N, N, CZERO, CONE, Q, LDQ ) 00429 IF( ICOMPZ.EQ.3 ) 00430 $ CALL CLASET( 'Full', N, N, CZERO, CONE, Z, LDZ ) 00431 * 00432 * Machine Constants 00433 * 00434 IN = IHI + 1 - ILO 00435 SAFMIN = SLAMCH( 'S' ) 00436 ULP = SLAMCH( 'E' )*SLAMCH( 'B' ) 00437 ANORM = CLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK ) 00438 BNORM = CLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK ) 00439 ATOL = MAX( SAFMIN, ULP*ANORM ) 00440 BTOL = MAX( SAFMIN, ULP*BNORM ) 00441 ASCALE = ONE / MAX( SAFMIN, ANORM ) 00442 BSCALE = ONE / MAX( SAFMIN, BNORM ) 00443 * 00444 * 00445 * Set Eigenvalues IHI+1:N 00446 * 00447 DO 10 J = IHI + 1, N 00448 ABSB = ABS( T( J, J ) ) 00449 IF( ABSB.GT.SAFMIN ) THEN 00450 SIGNBC = CONJG( T( J, J ) / ABSB ) 00451 T( J, J ) = ABSB 00452 IF( ILSCHR ) THEN 00453 CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 ) 00454 CALL CSCAL( J, SIGNBC, H( 1, J ), 1 ) 00455 ELSE 00456 H( J, J ) = H( J, J )*SIGNBC 00457 END IF 00458 IF( ILZ ) 00459 $ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 ) 00460 ELSE 00461 T( J, J ) = CZERO 00462 END IF 00463 ALPHA( J ) = H( J, J ) 00464 BETA( J ) = T( J, J ) 00465 10 CONTINUE 00466 * 00467 * If IHI < ILO, skip QZ steps 00468 * 00469 IF( IHI.LT.ILO ) 00470 $ GO TO 190 00471 * 00472 * MAIN QZ ITERATION LOOP 00473 * 00474 * Initialize dynamic indices 00475 * 00476 * Eigenvalues ILAST+1:N have been found. 00477 * Column operations modify rows IFRSTM:whatever 00478 * Row operations modify columns whatever:ILASTM 00479 * 00480 * If only eigenvalues are being computed, then 00481 * IFRSTM is the row of the last splitting row above row ILAST; 00482 * this is always at least ILO. 00483 * IITER counts iterations since the last eigenvalue was found, 00484 * to tell when to use an extraordinary shift. 00485 * MAXIT is the maximum number of QZ sweeps allowed. 00486 * 00487 ILAST = IHI 00488 IF( ILSCHR ) THEN 00489 IFRSTM = 1 00490 ILASTM = N 00491 ELSE 00492 IFRSTM = ILO 00493 ILASTM = IHI 00494 END IF 00495 IITER = 0 00496 ESHIFT = CZERO 00497 MAXIT = 30*( IHI-ILO+1 ) 00498 * 00499 DO 170 JITER = 1, MAXIT 00500 * 00501 * Check for too many iterations. 00502 * 00503 IF( JITER.GT.MAXIT ) 00504 $ GO TO 180 00505 * 00506 * Split the matrix if possible. 00507 * 00508 * Two tests: 00509 * 1: H(j,j-1)=0 or j=ILO 00510 * 2: T(j,j)=0 00511 * 00512 * Special case: j=ILAST 00513 * 00514 IF( ILAST.EQ.ILO ) THEN 00515 GO TO 60 00516 ELSE 00517 IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN 00518 H( ILAST, ILAST-1 ) = CZERO 00519 GO TO 60 00520 END IF 00521 END IF 00522 * 00523 IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN 00524 T( ILAST, ILAST ) = CZERO 00525 GO TO 50 00526 END IF 00527 * 00528 * General case: j<ILAST 00529 * 00530 DO 40 J = ILAST - 1, ILO, -1 00531 * 00532 * Test 1: for H(j,j-1)=0 or j=ILO 00533 * 00534 IF( J.EQ.ILO ) THEN 00535 ILAZRO = .TRUE. 00536 ELSE 00537 IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN 00538 H( J, J-1 ) = CZERO 00539 ILAZRO = .TRUE. 00540 ELSE 00541 ILAZRO = .FALSE. 00542 END IF 00543 END IF 00544 * 00545 * Test 2: for T(j,j)=0 00546 * 00547 IF( ABS( T( J, J ) ).LT.BTOL ) THEN 00548 T( J, J ) = CZERO 00549 * 00550 * Test 1a: Check for 2 consecutive small subdiagonals in A 00551 * 00552 ILAZR2 = .FALSE. 00553 IF( .NOT.ILAZRO ) THEN 00554 IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1, 00555 $ J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) ) 00556 $ ILAZR2 = .TRUE. 00557 END IF 00558 * 00559 * If both tests pass (1 & 2), i.e., the leading diagonal 00560 * element of B in the block is zero, split a 1x1 block off 00561 * at the top. (I.e., at the J-th row/column) The leading 00562 * diagonal element of the remainder can also be zero, so 00563 * this may have to be done repeatedly. 00564 * 00565 IF( ILAZRO .OR. ILAZR2 ) THEN 00566 DO 20 JCH = J, ILAST - 1 00567 CTEMP = H( JCH, JCH ) 00568 CALL CLARTG( CTEMP, H( JCH+1, JCH ), C, S, 00569 $ H( JCH, JCH ) ) 00570 H( JCH+1, JCH ) = CZERO 00571 CALL CROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH, 00572 $ H( JCH+1, JCH+1 ), LDH, C, S ) 00573 CALL CROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT, 00574 $ T( JCH+1, JCH+1 ), LDT, C, S ) 00575 IF( ILQ ) 00576 $ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1, 00577 $ C, CONJG( S ) ) 00578 IF( ILAZR2 ) 00579 $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C 00580 ILAZR2 = .FALSE. 00581 IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN 00582 IF( JCH+1.GE.ILAST ) THEN 00583 GO TO 60 00584 ELSE 00585 IFIRST = JCH + 1 00586 GO TO 70 00587 END IF 00588 END IF 00589 T( JCH+1, JCH+1 ) = CZERO 00590 20 CONTINUE 00591 GO TO 50 00592 ELSE 00593 * 00594 * Only test 2 passed -- chase the zero to T(ILAST,ILAST) 00595 * Then process as in the case T(ILAST,ILAST)=0 00596 * 00597 DO 30 JCH = J, ILAST - 1 00598 CTEMP = T( JCH, JCH+1 ) 00599 CALL CLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S, 00600 $ T( JCH, JCH+1 ) ) 00601 T( JCH+1, JCH+1 ) = CZERO 00602 IF( JCH.LT.ILASTM-1 ) 00603 $ CALL CROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT, 00604 $ T( JCH+1, JCH+2 ), LDT, C, S ) 00605 CALL CROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH, 00606 $ H( JCH+1, JCH-1 ), LDH, C, S ) 00607 IF( ILQ ) 00608 $ CALL CROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1, 00609 $ C, CONJG( S ) ) 00610 CTEMP = H( JCH+1, JCH ) 00611 CALL CLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S, 00612 $ H( JCH+1, JCH ) ) 00613 H( JCH+1, JCH-1 ) = CZERO 00614 CALL CROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1, 00615 $ H( IFRSTM, JCH-1 ), 1, C, S ) 00616 CALL CROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1, 00617 $ T( IFRSTM, JCH-1 ), 1, C, S ) 00618 IF( ILZ ) 00619 $ CALL CROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1, 00620 $ C, S ) 00621 30 CONTINUE 00622 GO TO 50 00623 END IF 00624 ELSE IF( ILAZRO ) THEN 00625 * 00626 * Only test 1 passed -- work on J:ILAST 00627 * 00628 IFIRST = J 00629 GO TO 70 00630 END IF 00631 * 00632 * Neither test passed -- try next J 00633 * 00634 40 CONTINUE 00635 * 00636 * (Drop-through is "impossible") 00637 * 00638 INFO = 2*N + 1 00639 GO TO 210 00640 * 00641 * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a 00642 * 1x1 block. 00643 * 00644 50 CONTINUE 00645 CTEMP = H( ILAST, ILAST ) 00646 CALL CLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S, 00647 $ H( ILAST, ILAST ) ) 00648 H( ILAST, ILAST-1 ) = CZERO 00649 CALL CROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1, 00650 $ H( IFRSTM, ILAST-1 ), 1, C, S ) 00651 CALL CROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1, 00652 $ T( IFRSTM, ILAST-1 ), 1, C, S ) 00653 IF( ILZ ) 00654 $ CALL CROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S ) 00655 * 00656 * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA 00657 * 00658 60 CONTINUE 00659 ABSB = ABS( T( ILAST, ILAST ) ) 00660 IF( ABSB.GT.SAFMIN ) THEN 00661 SIGNBC = CONJG( T( ILAST, ILAST ) / ABSB ) 00662 T( ILAST, ILAST ) = ABSB 00663 IF( ILSCHR ) THEN 00664 CALL CSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 ) 00665 CALL CSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ), 00666 $ 1 ) 00667 ELSE 00668 H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC 00669 END IF 00670 IF( ILZ ) 00671 $ CALL CSCAL( N, SIGNBC, Z( 1, ILAST ), 1 ) 00672 ELSE 00673 T( ILAST, ILAST ) = CZERO 00674 END IF 00675 ALPHA( ILAST ) = H( ILAST, ILAST ) 00676 BETA( ILAST ) = T( ILAST, ILAST ) 00677 * 00678 * Go to next block -- exit if finished. 00679 * 00680 ILAST = ILAST - 1 00681 IF( ILAST.LT.ILO ) 00682 $ GO TO 190 00683 * 00684 * Reset counters 00685 * 00686 IITER = 0 00687 ESHIFT = CZERO 00688 IF( .NOT.ILSCHR ) THEN 00689 ILASTM = ILAST 00690 IF( IFRSTM.GT.ILAST ) 00691 $ IFRSTM = ILO 00692 END IF 00693 GO TO 160 00694 * 00695 * QZ step 00696 * 00697 * This iteration only involves rows/columns IFIRST:ILAST. We 00698 * assume IFIRST < ILAST, and that the diagonal of B is non-zero. 00699 * 00700 70 CONTINUE 00701 IITER = IITER + 1 00702 IF( .NOT.ILSCHR ) THEN 00703 IFRSTM = IFIRST 00704 END IF 00705 * 00706 * Compute the Shift. 00707 * 00708 * At this point, IFIRST < ILAST, and the diagonal elements of 00709 * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in 00710 * magnitude) 00711 * 00712 IF( ( IITER / 10 )*10.NE.IITER ) THEN 00713 * 00714 * The Wilkinson shift (AEP p.512), i.e., the eigenvalue of 00715 * the bottom-right 2x2 block of A inv(B) which is nearest to 00716 * the bottom-right element. 00717 * 00718 * We factor B as U*D, where U has unit diagonals, and 00719 * compute (A*inv(D))*inv(U). 00720 * 00721 U12 = ( BSCALE*T( ILAST-1, ILAST ) ) / 00722 $ ( BSCALE*T( ILAST, ILAST ) ) 00723 AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) / 00724 $ ( BSCALE*T( ILAST-1, ILAST-1 ) ) 00725 AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) / 00726 $ ( BSCALE*T( ILAST-1, ILAST-1 ) ) 00727 AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) / 00728 $ ( BSCALE*T( ILAST, ILAST ) ) 00729 AD22 = ( ASCALE*H( ILAST, ILAST ) ) / 00730 $ ( BSCALE*T( ILAST, ILAST ) ) 00731 ABI22 = AD22 - U12*AD21 00732 * 00733 T1 = HALF*( AD11+ABI22 ) 00734 RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 ) 00735 TEMP = REAL( T1-ABI22 )*REAL( RTDISC ) + 00736 $ AIMAG( T1-ABI22 )*AIMAG( RTDISC ) 00737 IF( TEMP.LE.ZERO ) THEN 00738 SHIFT = T1 + RTDISC 00739 ELSE 00740 SHIFT = T1 - RTDISC 00741 END IF 00742 ELSE 00743 * 00744 * Exceptional shift. Chosen for no particularly good reason. 00745 * 00746 ESHIFT = ESHIFT + H(ILAST,ILAST-1)/T(ILAST-1,ILAST-1) 00747 SHIFT = ESHIFT 00748 END IF 00749 * 00750 * Now check for two consecutive small subdiagonals. 00751 * 00752 DO 80 J = ILAST - 1, IFIRST + 1, -1 00753 ISTART = J 00754 CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) ) 00755 TEMP = ABS1( CTEMP ) 00756 TEMP2 = ASCALE*ABS1( H( J+1, J ) ) 00757 TEMPR = MAX( TEMP, TEMP2 ) 00758 IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN 00759 TEMP = TEMP / TEMPR 00760 TEMP2 = TEMP2 / TEMPR 00761 END IF 00762 IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL ) 00763 $ GO TO 90 00764 80 CONTINUE 00765 * 00766 ISTART = IFIRST 00767 CTEMP = ASCALE*H( IFIRST, IFIRST ) - 00768 $ SHIFT*( BSCALE*T( IFIRST, IFIRST ) ) 00769 90 CONTINUE 00770 * 00771 * Do an implicit-shift QZ sweep. 00772 * 00773 * Initial Q 00774 * 00775 CTEMP2 = ASCALE*H( ISTART+1, ISTART ) 00776 CALL CLARTG( CTEMP, CTEMP2, C, S, CTEMP3 ) 00777 * 00778 * Sweep 00779 * 00780 DO 150 J = ISTART, ILAST - 1 00781 IF( J.GT.ISTART ) THEN 00782 CTEMP = H( J, J-1 ) 00783 CALL CLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) ) 00784 H( J+1, J-1 ) = CZERO 00785 END IF 00786 * 00787 DO 100 JC = J, ILASTM 00788 CTEMP = C*H( J, JC ) + S*H( J+1, JC ) 00789 H( J+1, JC ) = -CONJG( S )*H( J, JC ) + C*H( J+1, JC ) 00790 H( J, JC ) = CTEMP 00791 CTEMP2 = C*T( J, JC ) + S*T( J+1, JC ) 00792 T( J+1, JC ) = -CONJG( S )*T( J, JC ) + C*T( J+1, JC ) 00793 T( J, JC ) = CTEMP2 00794 100 CONTINUE 00795 IF( ILQ ) THEN 00796 DO 110 JR = 1, N 00797 CTEMP = C*Q( JR, J ) + CONJG( S )*Q( JR, J+1 ) 00798 Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 ) 00799 Q( JR, J ) = CTEMP 00800 110 CONTINUE 00801 END IF 00802 * 00803 CTEMP = T( J+1, J+1 ) 00804 CALL CLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) ) 00805 T( J+1, J ) = CZERO 00806 * 00807 DO 120 JR = IFRSTM, MIN( J+2, ILAST ) 00808 CTEMP = C*H( JR, J+1 ) + S*H( JR, J ) 00809 H( JR, J ) = -CONJG( S )*H( JR, J+1 ) + C*H( JR, J ) 00810 H( JR, J+1 ) = CTEMP 00811 120 CONTINUE 00812 DO 130 JR = IFRSTM, J 00813 CTEMP = C*T( JR, J+1 ) + S*T( JR, J ) 00814 T( JR, J ) = -CONJG( S )*T( JR, J+1 ) + C*T( JR, J ) 00815 T( JR, J+1 ) = CTEMP 00816 130 CONTINUE 00817 IF( ILZ ) THEN 00818 DO 140 JR = 1, N 00819 CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J ) 00820 Z( JR, J ) = -CONJG( S )*Z( JR, J+1 ) + C*Z( JR, J ) 00821 Z( JR, J+1 ) = CTEMP 00822 140 CONTINUE 00823 END IF 00824 150 CONTINUE 00825 * 00826 160 CONTINUE 00827 * 00828 170 CONTINUE 00829 * 00830 * Drop-through = non-convergence 00831 * 00832 180 CONTINUE 00833 INFO = ILAST 00834 GO TO 210 00835 * 00836 * Successful completion of all QZ steps 00837 * 00838 190 CONTINUE 00839 * 00840 * Set Eigenvalues 1:ILO-1 00841 * 00842 DO 200 J = 1, ILO - 1 00843 ABSB = ABS( T( J, J ) ) 00844 IF( ABSB.GT.SAFMIN ) THEN 00845 SIGNBC = CONJG( T( J, J ) / ABSB ) 00846 T( J, J ) = ABSB 00847 IF( ILSCHR ) THEN 00848 CALL CSCAL( J-1, SIGNBC, T( 1, J ), 1 ) 00849 CALL CSCAL( J, SIGNBC, H( 1, J ), 1 ) 00850 ELSE 00851 H( J, J ) = H( J, J )*SIGNBC 00852 END IF 00853 IF( ILZ ) 00854 $ CALL CSCAL( N, SIGNBC, Z( 1, J ), 1 ) 00855 ELSE 00856 T( J, J ) = CZERO 00857 END IF 00858 ALPHA( J ) = H( J, J ) 00859 BETA( J ) = T( J, J ) 00860 200 CONTINUE 00861 * 00862 * Normal Termination 00863 * 00864 INFO = 0 00865 * 00866 * Exit (other than argument error) -- return optimal workspace size 00867 * 00868 210 CONTINUE 00869 WORK( 1 ) = CMPLX( N ) 00870 RETURN 00871 * 00872 * End of CHGEQZ 00873 * 00874 END