LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zhgeqz.f
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00001 *> \brief \b ZHGEQZ
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download ZHGEQZ + 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/zhgeqz.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZHGEQZ( 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 *       DOUBLE PRECISION   RWORK( * )
00031 *       COMPLEX*16         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 *> ZHGEQZ 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 ZGGHRD.
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 ZGGHRD 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*16 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*16 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*16 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*16 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*16 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*16 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*16 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 DOUBLE PRECISION 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 complex16GEcomputational
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 ZHGEQZ( 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       DOUBLE PRECISION   RWORK( * )
00298       COMPLEX*16         ALPHA( * ), BETA( * ), H( LDH, * ),
00299      $                   Q( LDQ, * ), T( LDT, * ), WORK( * ),
00300      $                   Z( LDZ, * )
00301 *     ..
00302 *
00303 *  =====================================================================
00304 *
00305 *     .. Parameters ..
00306       COMPLEX*16         CZERO, CONE
00307       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
00308      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
00309       DOUBLE PRECISION   ZERO, ONE
00310       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00311       DOUBLE PRECISION   HALF
00312       PARAMETER          ( HALF = 0.5D+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       DOUBLE PRECISION   ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
00320      $                   C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
00321       COMPLEX*16         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       DOUBLE PRECISION   DLAMCH, ZLANHS
00328       EXTERNAL           LSAME, DLAMCH, ZLANHS
00329 *     ..
00330 *     .. External Subroutines ..
00331       EXTERNAL           XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL
00332 *     ..
00333 *     .. Intrinsic Functions ..
00334       INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN,
00335      $                   SQRT
00336 *     ..
00337 *     .. Statement Functions ..
00338       DOUBLE PRECISION   ABS1
00339 *     ..
00340 *     .. Statement Function definitions ..
00341       ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
00342 *     ..
00343 *     .. Executable Statements ..
00344 *
00345 *     Decode JOB, COMPQ, COMPZ
00346 *
00347       IF( LSAME( JOB, 'E' ) ) THEN
00348          ILSCHR = .FALSE.
00349          ISCHUR = 1
00350       ELSE IF( LSAME( JOB, 'S' ) ) THEN
00351          ILSCHR = .TRUE.
00352          ISCHUR = 2
00353       ELSE
00354          ISCHUR = 0
00355       END IF
00356 *
00357       IF( LSAME( COMPQ, 'N' ) ) THEN
00358          ILQ = .FALSE.
00359          ICOMPQ = 1
00360       ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
00361          ILQ = .TRUE.
00362          ICOMPQ = 2
00363       ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
00364          ILQ = .TRUE.
00365          ICOMPQ = 3
00366       ELSE
00367          ICOMPQ = 0
00368       END IF
00369 *
00370       IF( LSAME( COMPZ, 'N' ) ) THEN
00371          ILZ = .FALSE.
00372          ICOMPZ = 1
00373       ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
00374          ILZ = .TRUE.
00375          ICOMPZ = 2
00376       ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
00377          ILZ = .TRUE.
00378          ICOMPZ = 3
00379       ELSE
00380          ICOMPZ = 0
00381       END IF
00382 *
00383 *     Check Argument Values
00384 *
00385       INFO = 0
00386       WORK( 1 ) = MAX( 1, N )
00387       LQUERY = ( LWORK.EQ.-1 )
00388       IF( ISCHUR.EQ.0 ) THEN
00389          INFO = -1
00390       ELSE IF( ICOMPQ.EQ.0 ) THEN
00391          INFO = -2
00392       ELSE IF( ICOMPZ.EQ.0 ) THEN
00393          INFO = -3
00394       ELSE IF( N.LT.0 ) THEN
00395          INFO = -4
00396       ELSE IF( ILO.LT.1 ) THEN
00397          INFO = -5
00398       ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
00399          INFO = -6
00400       ELSE IF( LDH.LT.N ) THEN
00401          INFO = -8
00402       ELSE IF( LDT.LT.N ) THEN
00403          INFO = -10
00404       ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
00405          INFO = -14
00406       ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
00407          INFO = -16
00408       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
00409          INFO = -18
00410       END IF
00411       IF( INFO.NE.0 ) THEN
00412          CALL XERBLA( 'ZHGEQZ', -INFO )
00413          RETURN
00414       ELSE IF( LQUERY ) THEN
00415          RETURN
00416       END IF
00417 *
00418 *     Quick return if possible
00419 *
00420 *     WORK( 1 ) = CMPLX( 1 )
00421       IF( N.LE.0 ) THEN
00422          WORK( 1 ) = DCMPLX( 1 )
00423          RETURN
00424       END IF
00425 *
00426 *     Initialize Q and Z
00427 *
00428       IF( ICOMPQ.EQ.3 )
00429      $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
00430       IF( ICOMPZ.EQ.3 )
00431      $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
00432 *
00433 *     Machine Constants
00434 *
00435       IN = IHI + 1 - ILO
00436       SAFMIN = DLAMCH( 'S' )
00437       ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
00438       ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
00439       BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
00440       ATOL = MAX( SAFMIN, ULP*ANORM )
00441       BTOL = MAX( SAFMIN, ULP*BNORM )
00442       ASCALE = ONE / MAX( SAFMIN, ANORM )
00443       BSCALE = ONE / MAX( SAFMIN, BNORM )
00444 *
00445 *
00446 *     Set Eigenvalues IHI+1:N
00447 *
00448       DO 10 J = IHI + 1, N
00449          ABSB = ABS( T( J, J ) )
00450          IF( ABSB.GT.SAFMIN ) THEN
00451             SIGNBC = DCONJG( T( J, J ) / ABSB )
00452             T( J, J ) = ABSB
00453             IF( ILSCHR ) THEN
00454                CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
00455                CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
00456             ELSE
00457                H( J, J ) = H( J, J )*SIGNBC
00458             END IF
00459             IF( ILZ )
00460      $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
00461          ELSE
00462             T( J, J ) = CZERO
00463          END IF
00464          ALPHA( J ) = H( J, J )
00465          BETA( J ) = T( J, J )
00466    10 CONTINUE
00467 *
00468 *     If IHI < ILO, skip QZ steps
00469 *
00470       IF( IHI.LT.ILO )
00471      $   GO TO 190
00472 *
00473 *     MAIN QZ ITERATION LOOP
00474 *
00475 *     Initialize dynamic indices
00476 *
00477 *     Eigenvalues ILAST+1:N have been found.
00478 *        Column operations modify rows IFRSTM:whatever
00479 *        Row operations modify columns whatever:ILASTM
00480 *
00481 *     If only eigenvalues are being computed, then
00482 *        IFRSTM is the row of the last splitting row above row ILAST;
00483 *        this is always at least ILO.
00484 *     IITER counts iterations since the last eigenvalue was found,
00485 *        to tell when to use an extraordinary shift.
00486 *     MAXIT is the maximum number of QZ sweeps allowed.
00487 *
00488       ILAST = IHI
00489       IF( ILSCHR ) THEN
00490          IFRSTM = 1
00491          ILASTM = N
00492       ELSE
00493          IFRSTM = ILO
00494          ILASTM = IHI
00495       END IF
00496       IITER = 0
00497       ESHIFT = CZERO
00498       MAXIT = 30*( IHI-ILO+1 )
00499 *
00500       DO 170 JITER = 1, MAXIT
00501 *
00502 *        Check for too many iterations.
00503 *
00504          IF( JITER.GT.MAXIT )
00505      $      GO TO 180
00506 *
00507 *        Split the matrix if possible.
00508 *
00509 *        Two tests:
00510 *           1: H(j,j-1)=0  or  j=ILO
00511 *           2: T(j,j)=0
00512 *
00513 *        Special case: j=ILAST
00514 *
00515          IF( ILAST.EQ.ILO ) THEN
00516             GO TO 60
00517          ELSE
00518             IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
00519                H( ILAST, ILAST-1 ) = CZERO
00520                GO TO 60
00521             END IF
00522          END IF
00523 *
00524          IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
00525             T( ILAST, ILAST ) = CZERO
00526             GO TO 50
00527          END IF
00528 *
00529 *        General case: j<ILAST
00530 *
00531          DO 40 J = ILAST - 1, ILO, -1
00532 *
00533 *           Test 1: for H(j,j-1)=0 or j=ILO
00534 *
00535             IF( J.EQ.ILO ) THEN
00536                ILAZRO = .TRUE.
00537             ELSE
00538                IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
00539                   H( J, J-1 ) = CZERO
00540                   ILAZRO = .TRUE.
00541                ELSE
00542                   ILAZRO = .FALSE.
00543                END IF
00544             END IF
00545 *
00546 *           Test 2: for T(j,j)=0
00547 *
00548             IF( ABS( T( J, J ) ).LT.BTOL ) THEN
00549                T( J, J ) = CZERO
00550 *
00551 *              Test 1a: Check for 2 consecutive small subdiagonals in A
00552 *
00553                ILAZR2 = .FALSE.
00554                IF( .NOT.ILAZRO ) THEN
00555                   IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
00556      $                J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
00557      $                ILAZR2 = .TRUE.
00558                END IF
00559 *
00560 *              If both tests pass (1 & 2), i.e., the leading diagonal
00561 *              element of B in the block is zero, split a 1x1 block off
00562 *              at the top. (I.e., at the J-th row/column) The leading
00563 *              diagonal element of the remainder can also be zero, so
00564 *              this may have to be done repeatedly.
00565 *
00566                IF( ILAZRO .OR. ILAZR2 ) THEN
00567                   DO 20 JCH = J, ILAST - 1
00568                      CTEMP = H( JCH, JCH )
00569                      CALL ZLARTG( CTEMP, H( JCH+1, JCH ), C, S,
00570      $                            H( JCH, JCH ) )
00571                      H( JCH+1, JCH ) = CZERO
00572                      CALL ZROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
00573      $                          H( JCH+1, JCH+1 ), LDH, C, S )
00574                      CALL ZROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
00575      $                          T( JCH+1, JCH+1 ), LDT, C, S )
00576                      IF( ILQ )
00577      $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
00578      $                             C, DCONJG( S ) )
00579                      IF( ILAZR2 )
00580      $                  H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
00581                      ILAZR2 = .FALSE.
00582                      IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
00583                         IF( JCH+1.GE.ILAST ) THEN
00584                            GO TO 60
00585                         ELSE
00586                            IFIRST = JCH + 1
00587                            GO TO 70
00588                         END IF
00589                      END IF
00590                      T( JCH+1, JCH+1 ) = CZERO
00591    20             CONTINUE
00592                   GO TO 50
00593                ELSE
00594 *
00595 *                 Only test 2 passed -- chase the zero to T(ILAST,ILAST)
00596 *                 Then process as in the case T(ILAST,ILAST)=0
00597 *
00598                   DO 30 JCH = J, ILAST - 1
00599                      CTEMP = T( JCH, JCH+1 )
00600                      CALL ZLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
00601      $                            T( JCH, JCH+1 ) )
00602                      T( JCH+1, JCH+1 ) = CZERO
00603                      IF( JCH.LT.ILASTM-1 )
00604      $                  CALL ZROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
00605      $                             T( JCH+1, JCH+2 ), LDT, C, S )
00606                      CALL ZROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
00607      $                          H( JCH+1, JCH-1 ), LDH, C, S )
00608                      IF( ILQ )
00609      $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
00610      $                             C, DCONJG( S ) )
00611                      CTEMP = H( JCH+1, JCH )
00612                      CALL ZLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
00613      $                            H( JCH+1, JCH ) )
00614                      H( JCH+1, JCH-1 ) = CZERO
00615                      CALL ZROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
00616      $                          H( IFRSTM, JCH-1 ), 1, C, S )
00617                      CALL ZROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
00618      $                          T( IFRSTM, JCH-1 ), 1, C, S )
00619                      IF( ILZ )
00620      $                  CALL ZROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
00621      $                             C, S )
00622    30             CONTINUE
00623                   GO TO 50
00624                END IF
00625             ELSE IF( ILAZRO ) THEN
00626 *
00627 *              Only test 1 passed -- work on J:ILAST
00628 *
00629                IFIRST = J
00630                GO TO 70
00631             END IF
00632 *
00633 *           Neither test passed -- try next J
00634 *
00635    40    CONTINUE
00636 *
00637 *        (Drop-through is "impossible")
00638 *
00639          INFO = 2*N + 1
00640          GO TO 210
00641 *
00642 *        T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
00643 *        1x1 block.
00644 *
00645    50    CONTINUE
00646          CTEMP = H( ILAST, ILAST )
00647          CALL ZLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
00648      $                H( ILAST, ILAST ) )
00649          H( ILAST, ILAST-1 ) = CZERO
00650          CALL ZROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
00651      $              H( IFRSTM, ILAST-1 ), 1, C, S )
00652          CALL ZROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
00653      $              T( IFRSTM, ILAST-1 ), 1, C, S )
00654          IF( ILZ )
00655      $      CALL ZROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
00656 *
00657 *        H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
00658 *
00659    60    CONTINUE
00660          ABSB = ABS( T( ILAST, ILAST ) )
00661          IF( ABSB.GT.SAFMIN ) THEN
00662             SIGNBC = DCONJG( T( ILAST, ILAST ) / ABSB )
00663             T( ILAST, ILAST ) = ABSB
00664             IF( ILSCHR ) THEN
00665                CALL ZSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
00666                CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
00667      $                     1 )
00668             ELSE
00669                H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC
00670             END IF
00671             IF( ILZ )
00672      $         CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
00673          ELSE
00674             T( ILAST, ILAST ) = CZERO
00675          END IF
00676          ALPHA( ILAST ) = H( ILAST, ILAST )
00677          BETA( ILAST ) = T( ILAST, ILAST )
00678 *
00679 *        Go to next block -- exit if finished.
00680 *
00681          ILAST = ILAST - 1
00682          IF( ILAST.LT.ILO )
00683      $      GO TO 190
00684 *
00685 *        Reset counters
00686 *
00687          IITER = 0
00688          ESHIFT = CZERO
00689          IF( .NOT.ILSCHR ) THEN
00690             ILASTM = ILAST
00691             IF( IFRSTM.GT.ILAST )
00692      $         IFRSTM = ILO
00693          END IF
00694          GO TO 160
00695 *
00696 *        QZ step
00697 *
00698 *        This iteration only involves rows/columns IFIRST:ILAST.  We
00699 *        assume IFIRST < ILAST, and that the diagonal of B is non-zero.
00700 *
00701    70    CONTINUE
00702          IITER = IITER + 1
00703          IF( .NOT.ILSCHR ) THEN
00704             IFRSTM = IFIRST
00705          END IF
00706 *
00707 *        Compute the Shift.
00708 *
00709 *        At this point, IFIRST < ILAST, and the diagonal elements of
00710 *        T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
00711 *        magnitude)
00712 *
00713          IF( ( IITER / 10 )*10.NE.IITER ) THEN
00714 *
00715 *           The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
00716 *           the bottom-right 2x2 block of A inv(B) which is nearest to
00717 *           the bottom-right element.
00718 *
00719 *           We factor B as U*D, where U has unit diagonals, and
00720 *           compute (A*inv(D))*inv(U).
00721 *
00722             U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
00723      $            ( BSCALE*T( ILAST, ILAST ) )
00724             AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
00725      $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
00726             AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
00727      $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
00728             AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
00729      $             ( BSCALE*T( ILAST, ILAST ) )
00730             AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
00731      $             ( BSCALE*T( ILAST, ILAST ) )
00732             ABI22 = AD22 - U12*AD21
00733 *
00734             T1 = HALF*( AD11+ABI22 )
00735             RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 )
00736             TEMP = DBLE( T1-ABI22 )*DBLE( RTDISC ) +
00737      $             DIMAG( T1-ABI22 )*DIMAG( RTDISC )
00738             IF( TEMP.LE.ZERO ) THEN
00739                SHIFT = T1 + RTDISC
00740             ELSE
00741                SHIFT = T1 - RTDISC
00742             END IF
00743          ELSE
00744 *
00745 *           Exceptional shift.  Chosen for no particularly good reason.
00746 *
00747             ESHIFT = ESHIFT + H(ILAST,ILAST-1)/T(ILAST-1,ILAST-1)
00748             SHIFT = ESHIFT
00749          END IF
00750 *
00751 *        Now check for two consecutive small subdiagonals.
00752 *
00753          DO 80 J = ILAST - 1, IFIRST + 1, -1
00754             ISTART = J
00755             CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
00756             TEMP = ABS1( CTEMP )
00757             TEMP2 = ASCALE*ABS1( H( J+1, J ) )
00758             TEMPR = MAX( TEMP, TEMP2 )
00759             IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
00760                TEMP = TEMP / TEMPR
00761                TEMP2 = TEMP2 / TEMPR
00762             END IF
00763             IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
00764      $         GO TO 90
00765    80    CONTINUE
00766 *
00767          ISTART = IFIRST
00768          CTEMP = ASCALE*H( IFIRST, IFIRST ) -
00769      $           SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
00770    90    CONTINUE
00771 *
00772 *        Do an implicit-shift QZ sweep.
00773 *
00774 *        Initial Q
00775 *
00776          CTEMP2 = ASCALE*H( ISTART+1, ISTART )
00777          CALL ZLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
00778 *
00779 *        Sweep
00780 *
00781          DO 150 J = ISTART, ILAST - 1
00782             IF( J.GT.ISTART ) THEN
00783                CTEMP = H( J, J-1 )
00784                CALL ZLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
00785                H( J+1, J-1 ) = CZERO
00786             END IF
00787 *
00788             DO 100 JC = J, ILASTM
00789                CTEMP = C*H( J, JC ) + S*H( J+1, JC )
00790                H( J+1, JC ) = -DCONJG( S )*H( J, JC ) + C*H( J+1, JC )
00791                H( J, JC ) = CTEMP
00792                CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
00793                T( J+1, JC ) = -DCONJG( S )*T( J, JC ) + C*T( J+1, JC )
00794                T( J, JC ) = CTEMP2
00795   100       CONTINUE
00796             IF( ILQ ) THEN
00797                DO 110 JR = 1, N
00798                   CTEMP = C*Q( JR, J ) + DCONJG( S )*Q( JR, J+1 )
00799                   Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
00800                   Q( JR, J ) = CTEMP
00801   110          CONTINUE
00802             END IF
00803 *
00804             CTEMP = T( J+1, J+1 )
00805             CALL ZLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
00806             T( J+1, J ) = CZERO
00807 *
00808             DO 120 JR = IFRSTM, MIN( J+2, ILAST )
00809                CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
00810                H( JR, J ) = -DCONJG( S )*H( JR, J+1 ) + C*H( JR, J )
00811                H( JR, J+1 ) = CTEMP
00812   120       CONTINUE
00813             DO 130 JR = IFRSTM, J
00814                CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
00815                T( JR, J ) = -DCONJG( S )*T( JR, J+1 ) + C*T( JR, J )
00816                T( JR, J+1 ) = CTEMP
00817   130       CONTINUE
00818             IF( ILZ ) THEN
00819                DO 140 JR = 1, N
00820                   CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
00821                   Z( JR, J ) = -DCONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
00822                   Z( JR, J+1 ) = CTEMP
00823   140          CONTINUE
00824             END IF
00825   150    CONTINUE
00826 *
00827   160    CONTINUE
00828 *
00829   170 CONTINUE
00830 *
00831 *     Drop-through = non-convergence
00832 *
00833   180 CONTINUE
00834       INFO = ILAST
00835       GO TO 210
00836 *
00837 *     Successful completion of all QZ steps
00838 *
00839   190 CONTINUE
00840 *
00841 *     Set Eigenvalues 1:ILO-1
00842 *
00843       DO 200 J = 1, ILO - 1
00844          ABSB = ABS( T( J, J ) )
00845          IF( ABSB.GT.SAFMIN ) THEN
00846             SIGNBC = DCONJG( T( J, J ) / ABSB )
00847             T( J, J ) = ABSB
00848             IF( ILSCHR ) THEN
00849                CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
00850                CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
00851             ELSE
00852                H( J, J ) = H( J, J )*SIGNBC
00853             END IF
00854             IF( ILZ )
00855      $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
00856          ELSE
00857             T( J, J ) = CZERO
00858          END IF
00859          ALPHA( J ) = H( J, J )
00860          BETA( J ) = T( J, J )
00861   200 CONTINUE
00862 *
00863 *     Normal Termination
00864 *
00865       INFO = 0
00866 *
00867 *     Exit (other than argument error) -- return optimal workspace size
00868 *
00869   210 CONTINUE
00870       WORK( 1 ) = DCMPLX( N )
00871       RETURN
00872 *
00873 *     End of ZHGEQZ
00874 *
00875       END
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