LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
chgeqz.f
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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 
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00011 *> [TGZ]</a> 
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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
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