zhgeqz.f

          SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
         $                   ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
         $                   RWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          COMPQ, COMPZ, JOB
          INTEGER            IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   RWORK( * )
          COMPLEX*16         ALPHA( * ), BETA( * ), H( LDH, * ),
         $                   Q( LDQ, * ), T( LDT, * ), WORK( * ),
         $                   Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
    *  where H is an upper Hessenberg matrix and T is upper triangular,
    *  using the single-shift QZ method.
    *  Matrix pairs of this type are produced by the reduction to
    *  generalized upper Hessenberg form of a complex matrix pair (A,B):
    *
    *     A = Q1*H*Z1**H,  B = Q1*T*Z1**H,
    *
    *  as computed by ZGGHRD.
    *
    *  If JOB='S', then the Hessenberg-triangular pair (H,T) is
    *  also reduced to generalized Schur form,
    *
    *     H = Q*S*Z**H,  T = Q*P*Z**H,
    *
    *  where Q and Z are unitary matrices and S and P are upper triangular.
    *
    *  Optionally, the unitary matrix Q from the generalized Schur
    *  factorization may be postmultiplied into an input matrix Q1, and the
    *  unitary matrix Z may be postmultiplied into an input matrix Z1.
    *  If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
    *  the matrix pair (A,B) to generalized Hessenberg form, then the output
    *  matrices Q1*Q and Z1*Z are the unitary factors from the generalized
    *  Schur factorization of (A,B):
    *
    *     A = (Q1*Q)*S*(Z1*Z)**H,  B = (Q1*Q)*P*(Z1*Z)**H.
    *
    *  To avoid overflow, eigenvalues of the matrix pair (H,T)
    *  (equivalently, of (A,B)) are computed as a pair of complex values
    *  (alpha,beta).  If beta is nonzero, lambda = alpha / beta is an
    *  eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
    *     A*x = lambda*B*x
    *  and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
    *  alternate form of the GNEP
    *     mu*A*y = B*y.
    *  The values of alpha and beta for the i-th eigenvalue can be read
    *  directly from the generalized Schur form:  alpha = S(i,i),
    *  beta = P(i,i).
    *
    *  Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
    *       Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
    *       pp. 241--256.
    *
    *  Arguments
    *  =========
    *
    *  JOB     (input) CHARACTER*1
    *          = 'E': Compute eigenvalues only;
    *          = 'S': Computer eigenvalues and the Schur form.
    *
    *  COMPQ   (input) CHARACTER*1
    *          = 'N': Left Schur vectors (Q) are not computed;
    *          = 'I': Q is initialized to the unit matrix and the matrix Q
    *                 of left Schur vectors of (H,T) is returned;
    *          = 'V': Q must contain a unitary matrix Q1 on entry and
    *                 the product Q1*Q is returned.
    *
    *  COMPZ   (input) CHARACTER*1
    *          = 'N': Right Schur vectors (Z) are not computed;
    *          = 'I': Q is initialized to the unit matrix and the matrix Z
    *                 of right Schur vectors of (H,T) is returned;
    *          = 'V': Z must contain a unitary matrix Z1 on entry and
    *                 the product Z1*Z is returned.
    *
    *  N       (input) INTEGER
    *          The order of the matrices H, T, Q, and Z.  N >= 0.
    *
    *  ILO     (input) INTEGER
    *  IHI     (input) INTEGER
    *          ILO and IHI mark the rows and columns of H which are in
    *          Hessenberg form.  It is assumed that A is already upper
    *          triangular in rows and columns 1:ILO-1 and IHI+1:N.
    *          If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
    *
    *  H       (input/output) COMPLEX*16 array, dimension (LDH, N)
    *          On entry, the N-by-N upper Hessenberg matrix H.
    *          On exit, if JOB = 'S', H contains the upper triangular
    *          matrix S from the generalized Schur factorization.
    *          If JOB = 'E', the diagonal of H matches that of S, but
    *          the rest of H is unspecified.
    *
    *  LDH     (input) INTEGER
    *          The leading dimension of the array H.  LDH >= max( 1, N ).
    *
    *  T       (input/output) COMPLEX*16 array, dimension (LDT, N)
    *          On entry, the N-by-N upper triangular matrix T.
    *          On exit, if JOB = 'S', T contains the upper triangular
    *          matrix P from the generalized Schur factorization.
    *          If JOB = 'E', the diagonal of T matches that of P, but
    *          the rest of T is unspecified.
    *
    *  LDT     (input) INTEGER
    *          The leading dimension of the array T.  LDT >= max( 1, N ).
    *
    *  ALPHA   (output) COMPLEX*16 array, dimension (N)
    *          The complex scalars alpha that define the eigenvalues of
    *          GNEP.  ALPHA(i) = S(i,i) in the generalized Schur
    *          factorization.
    *
    *  BETA    (output) COMPLEX*16 array, dimension (N)
    *          The real non-negative scalars beta that define the
    *          eigenvalues of GNEP.  BETA(i) = P(i,i) in the generalized
    *          Schur factorization.
    *
    *          Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
    *          represent the j-th eigenvalue of the matrix pair (A,B), in
    *          one of the forms lambda = alpha/beta or mu = beta/alpha.
    *          Since either lambda or mu may overflow, they should not,
    *          in general, be computed.
    *
    *  Q       (input/output) COMPLEX*16 array, dimension (LDQ, N)
    *          On entry, if COMPZ = 'V', the unitary matrix Q1 used in the
    *          reduction of (A,B) to generalized Hessenberg form.
    *          On exit, if COMPZ = 'I', the unitary matrix of left Schur
    *          vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
    *          left Schur vectors of (A,B).
    *          Not referenced if COMPZ = 'N'.
    *
    *  LDQ     (input) INTEGER
    *          The leading dimension of the array Q.  LDQ >= 1.
    *          If COMPQ='V' or 'I', then LDQ >= N.
    *
    *  Z       (input/output) COMPLEX*16 array, dimension (LDZ, N)
    *          On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
    *          reduction of (A,B) to generalized Hessenberg form.
    *          On exit, if COMPZ = 'I', the unitary matrix of right Schur
    *          vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
    *          right Schur vectors of (A,B).
    *          Not referenced if COMPZ = 'N'.
    *
    *  LDZ     (input) INTEGER
    *          The leading dimension of the array Z.  LDZ >= 1.
    *          If COMPZ='V' or 'I', then LDZ >= N.
    *
    *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
    *          On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK.  LWORK >= max(1,N).
    *
    *          If LWORK = -1, then a workspace query is assumed; the routine
    *          only calculates the optimal size of the WORK array, returns
    *          this value as the first entry of the WORK array, and no error
    *          message related to LWORK is issued by XERBLA.
    *
    *  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
    *
    *  INFO    (output) INTEGER
    *          = 0: successful exit
    *          < 0: if INFO = -i, the i-th argument had an illegal value
    *          = 1,...,N: the QZ iteration did not converge.  (H,T) is not
    *                     in Schur form, but ALPHA(i) and BETA(i),
    *                     i=INFO+1,...,N should be correct.
    *          = N+1,...,2*N: the shift calculation failed.  (H,T) is not
    *                     in Schur form, but ALPHA(i) and BETA(i),
    *                     i=INFO-N+1,...,N should be correct.
    *
    *  Further Details
    *  ===============
    *
    *  We assume that complex ABS works as long as its value is less than
    *  overflow.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          COMPLEX*16         CZERO, CONE
          PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
         $                   CONE = ( 1.0D+0, 0.0D+0 ) )
          DOUBLE PRECISION   ZERO, ONE
          PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
          DOUBLE PRECISION   HALF
          PARAMETER          ( HALF = 0.5D+0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
          INTEGER            ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
         $                   ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
         $                   JR, MAXIT
          DOUBLE PRECISION   ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
         $                   C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
          COMPLEX*16         ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
         $                   CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1,
         $                   U12, X
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH, ZLANHS
          EXTERNAL           LSAME, DLAMCH, ZLANHS
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN,
         $                   SQRT
    *     ..
    *     .. Statement Functions ..
          DOUBLE PRECISION   ABS1
    *     ..
    *     .. Statement Function definitions ..
          ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
    *     ..
    *     .. Executable Statements ..
    *
    *     Decode JOB, COMPQ, COMPZ
    *
          IF( LSAME( JOB, 'E' ) ) THEN
             ILSCHR = .FALSE.
             ISCHUR = 1
          ELSE IF( LSAME( JOB, 'S' ) ) THEN
             ILSCHR = .TRUE.
             ISCHUR = 2
          ELSE
             ISCHUR = 0
          END IF
    *
          IF( LSAME( COMPQ, 'N' ) ) THEN
             ILQ = .FALSE.
             ICOMPQ = 1
          ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
             ILQ = .TRUE.
             ICOMPQ = 2
          ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
             ILQ = .TRUE.
             ICOMPQ = 3
          ELSE
             ICOMPQ = 0
          END IF
    *
          IF( LSAME( COMPZ, 'N' ) ) THEN
             ILZ = .FALSE.
             ICOMPZ = 1
          ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
             ILZ = .TRUE.
             ICOMPZ = 2
          ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
             ILZ = .TRUE.
             ICOMPZ = 3
          ELSE
             ICOMPZ = 0
          END IF
    *
    *     Check Argument Values
    *
          INFO = 0
          WORK( 1 ) = MAX( 1, N )
          LQUERY = ( LWORK.EQ.-1 )
          IF( ISCHUR.EQ.0 ) THEN
             INFO = -1
          ELSE IF( ICOMPQ.EQ.0 ) THEN
             INFO = -2
          ELSE IF( ICOMPZ.EQ.0 ) THEN
             INFO = -3
          ELSE IF( N.LT.0 ) THEN
             INFO = -4
          ELSE IF( ILO.LT.1 ) THEN
             INFO = -5
          ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
             INFO = -6
          ELSE IF( LDH.LT.N ) THEN
             INFO = -8
          ELSE IF( LDT.LT.N ) THEN
             INFO = -10
          ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
             INFO = -14
          ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
             INFO = -16
          ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
             INFO = -18
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'ZHGEQZ', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
    *     WORK( 1 ) = CMPLX( 1 )
          IF( N.LE.0 ) THEN
             WORK( 1 ) = DCMPLX( 1 )
             RETURN
          END IF
    *
    *     Initialize Q and Z
    *
          IF( ICOMPQ.EQ.3 )
         $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
          IF( ICOMPZ.EQ.3 )
         $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
    *
    *     Machine Constants
    *
          IN = IHI + 1 - ILO
          SAFMIN = DLAMCH( 'S' )
          ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
          ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
          BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
          ATOL = MAX( SAFMIN, ULP*ANORM )
          BTOL = MAX( SAFMIN, ULP*BNORM )
          ASCALE = ONE / MAX( SAFMIN, ANORM )
          BSCALE = ONE / MAX( SAFMIN, BNORM )
    *
    *
    *     Set Eigenvalues IHI+1:N
    *
          DO 10 J = IHI + 1, N
             ABSB = ABS( T( J, J ) )
             IF( ABSB.GT.SAFMIN ) THEN
                SIGNBC = DCONJG( T( J, J ) / ABSB )
                T( J, J ) = ABSB
                IF( ILSCHR ) THEN
                   CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
                   CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
                ELSE
                   H( J, J ) = H( J, J )*SIGNBC
                END IF
                IF( ILZ )
         $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
             ELSE
                T( J, J ) = CZERO
             END IF
             ALPHA( J ) = H( J, J )
             BETA( J ) = T( J, J )
       10 CONTINUE
    *
    *     If IHI < ILO, skip QZ steps
    *
          IF( IHI.LT.ILO )
         $   GO TO 190
    *
    *     MAIN QZ ITERATION LOOP
    *
    *     Initialize dynamic indices
    *
    *     Eigenvalues ILAST+1:N have been found.
    *        Column operations modify rows IFRSTM:whatever
    *        Row operations modify columns whatever:ILASTM
    *
    *     If only eigenvalues are being computed, then
    *        IFRSTM is the row of the last splitting row above row ILAST;
    *        this is always at least ILO.
    *     IITER counts iterations since the last eigenvalue was found,
    *        to tell when to use an extraordinary shift.
    *     MAXIT is the maximum number of QZ sweeps allowed.
    *
          ILAST = IHI
          IF( ILSCHR ) THEN
             IFRSTM = 1
             ILASTM = N
          ELSE
             IFRSTM = ILO
             ILASTM = IHI
          END IF
          IITER = 0
          ESHIFT = CZERO
          MAXIT = 30*( IHI-ILO+1 )
    *
          DO 170 JITER = 1, MAXIT
    *
    *        Check for too many iterations.
    *
             IF( JITER.GT.MAXIT )
         $      GO TO 180
    *
    *        Split the matrix if possible.
    *
    *        Two tests:
    *           1: H(j,j-1)=0  or  j=ILO
    *           2: T(j,j)=0
    *
    *        Special case: j=ILAST
    *
             IF( ILAST.EQ.ILO ) THEN
                GO TO 60
             ELSE
                IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
                   H( ILAST, ILAST-1 ) = CZERO
                   GO TO 60
                END IF
             END IF
    *
             IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
                T( ILAST, ILAST ) = CZERO
                GO TO 50
             END IF
    *
    *        General case: j<ILAST
    *
             DO 40 J = ILAST - 1, ILO, -1
    *
    *           Test 1: for H(j,j-1)=0 or j=ILO
    *
                IF( J.EQ.ILO ) THEN
                   ILAZRO = .TRUE.
                ELSE
                   IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
                      H( J, J-1 ) = CZERO
                      ILAZRO = .TRUE.
                   ELSE
                      ILAZRO = .FALSE.
                   END IF
                END IF
    *
    *           Test 2: for T(j,j)=0
    *
                IF( ABS( T( J, J ) ).LT.BTOL ) THEN
                   T( J, J ) = CZERO
    *
    *              Test 1a: Check for 2 consecutive small subdiagonals in A
    *
                   ILAZR2 = .FALSE.
                   IF( .NOT.ILAZRO ) THEN
                      IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
         $                J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
         $                ILAZR2 = .TRUE.
                   END IF
    *
    *              If both tests pass (1 & 2), i.e., the leading diagonal
    *              element of B in the block is zero, split a 1x1 block off
    *              at the top. (I.e., at the J-th row/column) The leading
    *              diagonal element of the remainder can also be zero, so
    *              this may have to be done repeatedly.
    *
                   IF( ILAZRO .OR. ILAZR2 ) THEN
                      DO 20 JCH = J, ILAST - 1
                         CTEMP = H( JCH, JCH )
                         CALL ZLARTG( CTEMP, H( JCH+1, JCH ), C, S,
         $                            H( JCH, JCH ) )
                         H( JCH+1, JCH ) = CZERO
                         CALL ZROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
         $                          H( JCH+1, JCH+1 ), LDH, C, S )
                         CALL ZROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
         $                          T( JCH+1, JCH+1 ), LDT, C, S )
                         IF( ILQ )
         $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
         $                             C, DCONJG( S ) )
                         IF( ILAZR2 )
         $                  H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
                         ILAZR2 = .FALSE.
                         IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
                            IF( JCH+1.GE.ILAST ) THEN
                               GO TO 60
                            ELSE
                               IFIRST = JCH + 1
                               GO TO 70
                            END IF
                         END IF
                         T( JCH+1, JCH+1 ) = CZERO
       20             CONTINUE
                      GO TO 50
                   ELSE
    *
    *                 Only test 2 passed -- chase the zero to T(ILAST,ILAST)
    *                 Then process as in the case T(ILAST,ILAST)=0
    *
                      DO 30 JCH = J, ILAST - 1
                         CTEMP = T( JCH, JCH+1 )
                         CALL ZLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
         $                            T( JCH, JCH+1 ) )
                         T( JCH+1, JCH+1 ) = CZERO
                         IF( JCH.LT.ILASTM-1 )
         $                  CALL ZROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
         $                             T( JCH+1, JCH+2 ), LDT, C, S )
                         CALL ZROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
         $                          H( JCH+1, JCH-1 ), LDH, C, S )
                         IF( ILQ )
         $                  CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
         $                             C, DCONJG( S ) )
                         CTEMP = H( JCH+1, JCH )
                         CALL ZLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
         $                            H( JCH+1, JCH ) )
                         H( JCH+1, JCH-1 ) = CZERO
                         CALL ZROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
         $                          H( IFRSTM, JCH-1 ), 1, C, S )
                         CALL ZROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
         $                          T( IFRSTM, JCH-1 ), 1, C, S )
                         IF( ILZ )
         $                  CALL ZROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
         $                             C, S )
       30             CONTINUE
                      GO TO 50
                   END IF
                ELSE IF( ILAZRO ) THEN
    *
    *              Only test 1 passed -- work on J:ILAST
    *
                   IFIRST = J
                   GO TO 70
                END IF
    *
    *           Neither test passed -- try next J
    *
       40    CONTINUE
    *
    *        (Drop-through is "impossible")
    *
             INFO = 2*N + 1
             GO TO 210
    *
    *        T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
    *        1x1 block.
    *
       50    CONTINUE
             CTEMP = H( ILAST, ILAST )
             CALL ZLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
         $                H( ILAST, ILAST ) )
             H( ILAST, ILAST-1 ) = CZERO
             CALL ZROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
         $              H( IFRSTM, ILAST-1 ), 1, C, S )
             CALL ZROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
         $              T( IFRSTM, ILAST-1 ), 1, C, S )
             IF( ILZ )
         $      CALL ZROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
    *
    *        H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
    *
       60    CONTINUE
             ABSB = ABS( T( ILAST, ILAST ) )
             IF( ABSB.GT.SAFMIN ) THEN
                SIGNBC = DCONJG( T( ILAST, ILAST ) / ABSB )
                T( ILAST, ILAST ) = ABSB
                IF( ILSCHR ) THEN
                   CALL ZSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
                   CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
         $                     1 )
                ELSE
                   H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC
                END IF
                IF( ILZ )
         $         CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
             ELSE
                T( ILAST, ILAST ) = CZERO
             END IF
             ALPHA( ILAST ) = H( ILAST, ILAST )
             BETA( ILAST ) = T( ILAST, ILAST )
    *
    *        Go to next block -- exit if finished.
    *
             ILAST = ILAST - 1
             IF( ILAST.LT.ILO )
         $      GO TO 190
    *
    *        Reset counters
    *
             IITER = 0
             ESHIFT = CZERO
             IF( .NOT.ILSCHR ) THEN
                ILASTM = ILAST
                IF( IFRSTM.GT.ILAST )
         $         IFRSTM = ILO
             END IF
             GO TO 160
    *
    *        QZ step
    *
    *        This iteration only involves rows/columns IFIRST:ILAST.  We
    *        assume IFIRST < ILAST, and that the diagonal of B is non-zero.
    *
       70    CONTINUE
             IITER = IITER + 1
             IF( .NOT.ILSCHR ) THEN
                IFRSTM = IFIRST
             END IF
    *
    *        Compute the Shift.
    *
    *        At this point, IFIRST < ILAST, and the diagonal elements of
    *        T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
    *        magnitude)
    *
             IF( ( IITER / 10 )*10.NE.IITER ) THEN
    *
    *           The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
    *           the bottom-right 2x2 block of A inv(B) which is nearest to
    *           the bottom-right element.
    *
    *           We factor B as U*D, where U has unit diagonals, and
    *           compute (A*inv(D))*inv(U).
    *
                U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
         $            ( BSCALE*T( ILAST, ILAST ) )
                AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
         $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
                AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
         $             ( BSCALE*T( ILAST-1, ILAST-1 ) )
                AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
         $             ( BSCALE*T( ILAST, ILAST ) )
                AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
         $             ( BSCALE*T( ILAST, ILAST ) )
                ABI22 = AD22 - U12*AD21
    *
                T1 = HALF*( AD11+ABI22 )
                RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 )
                TEMP = DBLE( T1-ABI22 )*DBLE( RTDISC ) +
         $             DIMAG( T1-ABI22 )*DIMAG( RTDISC )
                IF( TEMP.LE.ZERO ) THEN
                   SHIFT = T1 + RTDISC
                ELSE
                   SHIFT = T1 - RTDISC
                END IF
             ELSE
    *
    *           Exceptional shift.  Chosen for no particularly good reason.
    *
                ESHIFT = ESHIFT + DCONJG( ( ASCALE*H( ILAST-1, ILAST ) ) /
         $               ( BSCALE*T( ILAST-1, ILAST-1 ) ) )
                SHIFT = ESHIFT
             END IF
    *
    *        Now check for two consecutive small subdiagonals.
    *
             DO 80 J = ILAST - 1, IFIRST + 1, -1
                ISTART = J
                CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
                TEMP = ABS1( CTEMP )
                TEMP2 = ASCALE*ABS1( H( J+1, J ) )
                TEMPR = MAX( TEMP, TEMP2 )
                IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
                   TEMP = TEMP / TEMPR
                   TEMP2 = TEMP2 / TEMPR
                END IF
                IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
         $         GO TO 90
       80    CONTINUE
    *
             ISTART = IFIRST
             CTEMP = ASCALE*H( IFIRST, IFIRST ) -
         $           SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
       90    CONTINUE
    *
    *        Do an implicit-shift QZ sweep.
    *
    *        Initial Q
    *
             CTEMP2 = ASCALE*H( ISTART+1, ISTART )
             CALL ZLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
    *
    *        Sweep
    *
             DO 150 J = ISTART, ILAST - 1
                IF( J.GT.ISTART ) THEN
                   CTEMP = H( J, J-1 )
                   CALL ZLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
                   H( J+1, J-1 ) = CZERO
                END IF
    *
                DO 100 JC = J, ILASTM
                   CTEMP = C*H( J, JC ) + S*H( J+1, JC )
                   H( J+1, JC ) = -DCONJG( S )*H( J, JC ) + C*H( J+1, JC )
                   H( J, JC ) = CTEMP
                   CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
                   T( J+1, JC ) = -DCONJG( S )*T( J, JC ) + C*T( J+1, JC )
                   T( J, JC ) = CTEMP2
      100       CONTINUE
                IF( ILQ ) THEN
                   DO 110 JR = 1, N
                      CTEMP = C*Q( JR, J ) + DCONJG( S )*Q( JR, J+1 )
                      Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
                      Q( JR, J ) = CTEMP
      110          CONTINUE
                END IF
    *
                CTEMP = T( J+1, J+1 )
                CALL ZLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
                T( J+1, J ) = CZERO
    *
                DO 120 JR = IFRSTM, MIN( J+2, ILAST )
                   CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
                   H( JR, J ) = -DCONJG( S )*H( JR, J+1 ) + C*H( JR, J )
                   H( JR, J+1 ) = CTEMP
      120       CONTINUE
                DO 130 JR = IFRSTM, J
                   CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
                   T( JR, J ) = -DCONJG( S )*T( JR, J+1 ) + C*T( JR, J )
                   T( JR, J+1 ) = CTEMP
      130       CONTINUE
                IF( ILZ ) THEN
                   DO 140 JR = 1, N
                      CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
                      Z( JR, J ) = -DCONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
                      Z( JR, J+1 ) = CTEMP
      140          CONTINUE
                END IF
      150    CONTINUE
    *
      160    CONTINUE
    *
      170 CONTINUE
    *
    *     Drop-through = non-convergence
    *
      180 CONTINUE
          INFO = ILAST
          GO TO 210
    *
    *     Successful completion of all QZ steps
    *
      190 CONTINUE
    *
    *     Set Eigenvalues 1:ILO-1
    *
          DO 200 J = 1, ILO - 1
             ABSB = ABS( T( J, J ) )
             IF( ABSB.GT.SAFMIN ) THEN
                SIGNBC = DCONJG( T( J, J ) / ABSB )
                T( J, J ) = ABSB
                IF( ILSCHR ) THEN
                   CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
                   CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
                ELSE
                   H( J, J ) = H( J, J )*SIGNBC
                END IF
                IF( ILZ )
         $         CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
             ELSE
                T( J, J ) = CZERO
             END IF
             ALPHA( J ) = H( J, J )
             BETA( J ) = T( J, J )
      200 CONTINUE
    *
    *     Normal Termination
    *
          INFO = 0
    *
    *     Exit (other than argument error) -- return optimal workspace size
    *
      210 CONTINUE
          WORK( 1 ) = DCMPLX( N )
          RETURN
    *
    *     End of ZHGEQZ
    *
          END