dhgeqz.f

          SUBROUTINE DHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
         $                   ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
         $                   LWORK, 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   ALPHAI( * ), ALPHAR( * ), BETA( * ),
         $                   H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
         $                   WORK( * ), Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
    *  where H is an upper Hessenberg matrix and T is upper triangular,
    *  using the double-shift QZ method.
    *  Matrix pairs of this type are produced by the reduction to
    *  generalized upper Hessenberg form of a real matrix pair (A,B):
    *
    *     A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
    *
    *  as computed by DGGHRD.
    *
    *  If JOB='S', then the Hessenberg-triangular pair (H,T) is
    *  also reduced to generalized Schur form,
    *
    *     H = Q*S*Z**T,  T = Q*P*Z**T,
    *
    *  where Q and Z are orthogonal matrices, P is an upper triangular
    *  matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
    *  diagonal blocks.
    *
    *  The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
    *  (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
    *  eigenvalues.
    *
    *  Additionally, the 2-by-2 upper triangular diagonal blocks of P
    *  corresponding to 2-by-2 blocks of S are reduced to positive diagonal
    *  form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
    *  P(j,j) > 0, and P(j+1,j+1) > 0.
    *
    *  Optionally, the orthogonal matrix Q from the generalized Schur
    *  factorization may be postmultiplied into an input matrix Q1, and the
    *  orthogonal matrix Z may be postmultiplied into an input matrix Z1.
    *  If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
    *  the matrix pair (A,B) to generalized upper Hessenberg form, then the
    *  output matrices Q1*Q and Z1*Z are the orthogonal factors from the
    *  generalized Schur factorization of (A,B):
    *
    *     A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
    *
    *  To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
    *  of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
    *  complex and beta real.
    *  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.
    *  Real eigenvalues 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': Compute 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 an orthogonal matrix Q1 on entry and
    *                 the product Q1*Q is returned.
    *
    *  COMPZ   (input) CHARACTER*1
    *          = 'N': Right Schur vectors (Z) are not computed;
    *          = 'I': Z is initialized to the unit matrix and the matrix Z
    *                 of right Schur vectors of (H,T) is returned;
    *          = 'V': Z must contain an orthogonal 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) DOUBLE PRECISION array, dimension (LDH, N)
    *          On entry, the N-by-N upper Hessenberg matrix H.
    *          On exit, if JOB = 'S', H contains the upper quasi-triangular
    *          matrix S from the generalized Schur factorization;
    *          2-by-2 diagonal blocks (corresponding to complex conjugate
    *          pairs of eigenvalues) are returned in standard form, with
    *          H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0.
    *          If JOB = 'E', the diagonal blocks of H match those 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) DOUBLE PRECISION 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;
    *          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
    *          are reduced to positive diagonal form, i.e., if H(j+1,j) is
    *          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and
    *          T(j+1,j+1) > 0.
    *          If JOB = 'E', the diagonal blocks of T match those of P, but
    *          the rest of T is unspecified.
    *
    *  LDT     (input) INTEGER
    *          The leading dimension of the array T.  LDT >= max( 1, N ).
    *
    *  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
    *          The real parts of each scalar alpha defining an eigenvalue
    *          of GNEP.
    *
    *  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
    *          The imaginary parts of each scalar alpha defining an
    *          eigenvalue of GNEP.
    *          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
    *          positive, then the j-th and (j+1)-st eigenvalues are a
    *          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
    *
    *  BETA    (output) DOUBLE PRECISION array, dimension (N)
    *          The scalars beta that define the eigenvalues of GNEP.
    *          Together, the quantities alpha = (ALPHAR(j),ALPHAI(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) DOUBLE PRECISION array, dimension (LDQ, N)
    *          On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
    *          the reduction of (A,B) to generalized Hessenberg form.
    *          On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
    *          vectors of (H,T), and if COMPZ = 'V', the orthogonal 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) DOUBLE PRECISION array, dimension (LDZ, N)
    *          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
    *          the reduction of (A,B) to generalized Hessenberg form.
    *          On exit, if COMPZ = 'I', the orthogonal matrix of
    *          right Schur vectors of (H,T), and if COMPZ = 'V', the
    *          orthogonal 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) DOUBLE PRECISION 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.
    *
    *  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 ALPHAR(i), ALPHAI(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 ALPHAR(i), ALPHAI(i), and
    *                     BETA(i), i=INFO-N+1,...,N should be correct.
    *
    *  Further Details
    *  ===============
    *
    *  Iteration counters:
    *
    *  JITER  -- counts iterations.
    *  IITER  -- counts iterations run since ILAST was last
    *            changed.  This is therefore reset only when a 1-by-1 or
    *            2-by-2 block deflates off the bottom.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
    *    $                     SAFETY = 1.0E+0 )
          DOUBLE PRECISION   HALF, ZERO, ONE, SAFETY
          PARAMETER          ( HALF = 0.5D+0, ZERO = 0.0D+0, ONE = 1.0D+0,
         $                   SAFETY = 1.0D+2 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            ILAZR2, ILAZRO, ILPIVT, ILQ, ILSCHR, ILZ,
         $                   LQUERY
          INTEGER            ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
         $                   ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
         $                   JR, MAXIT
          DOUBLE PRECISION   A11, A12, A1I, A1R, A21, A22, A2I, A2R, AD11,
         $                   AD11L, AD12, AD12L, AD21, AD21L, AD22, AD22L,
         $                   AD32L, AN, ANORM, ASCALE, ATOL, B11, B1A, B1I,
         $                   B1R, B22, B2A, B2I, B2R, BN, BNORM, BSCALE,
         $                   BTOL, C, C11I, C11R, C12, C21, C22I, C22R, CL,
         $                   CQ, CR, CZ, ESHIFT, S, S1, S1INV, S2, SAFMAX,
         $                   SAFMIN, SCALE, SL, SQI, SQR, SR, SZI, SZR, T1,
         $                   TAU, TEMP, TEMP2, TEMPI, TEMPR, U1, U12, U12L,
         $                   U2, ULP, VS, W11, W12, W21, W22, WABS, WI, WR,
         $                   WR2
    *     ..
    *     .. Local Arrays ..
          DOUBLE PRECISION   V( 3 )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DLAMCH, DLANHS, DLAPY2, DLAPY3
          EXTERNAL           LSAME, DLAMCH, DLANHS, DLAPY2, DLAPY3
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLAG2, DLARFG, DLARTG, DLASET, DLASV2, DROT,
         $                   XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, MAX, MIN, SQRT
    *     ..
    *     .. 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 = -15
          ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
             INFO = -17
          ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
             INFO = -19
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DHGEQZ', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.LE.0 ) THEN
             WORK( 1 ) = DBLE( 1 )
             RETURN
          END IF
    *
    *     Initialize Q and Z
    *
          IF( ICOMPQ.EQ.3 )
         $   CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
          IF( ICOMPZ.EQ.3 )
         $   CALL DLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
    *
    *     Machine Constants
    *
          IN = IHI + 1 - ILO
          SAFMIN = DLAMCH( 'S' )
          SAFMAX = ONE / SAFMIN
          ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
          ANORM = DLANHS( 'F', IN, H( ILO, ILO ), LDH, WORK )
          BNORM = DLANHS( 'F', IN, T( ILO, ILO ), LDT, WORK )
          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 30 J = IHI + 1, N
             IF( T( J, J ).LT.ZERO ) THEN
                IF( ILSCHR ) THEN
                   DO 10 JR = 1, J
                      H( JR, J ) = -H( JR, J )
                      T( JR, J ) = -T( JR, J )
       10          CONTINUE
                ELSE
                   H( J, J ) = -H( J, J )
                   T( J, J ) = -T( J, J )
                END IF
                IF( ILZ ) THEN
                   DO 20 JR = 1, N
                      Z( JR, J ) = -Z( JR, J )
       20          CONTINUE
                END IF
             END IF
             ALPHAR( J ) = H( J, J )
             ALPHAI( J ) = ZERO
             BETA( J ) = T( J, J )
       30 CONTINUE
    *
    *     If IHI < ILO, skip QZ steps
    *
          IF( IHI.LT.ILO )
         $   GO TO 380
    *
    *     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 = ZERO
          MAXIT = 30*( IHI-ILO+1 )
    *
          DO 360 JITER = 1, MAXIT
    *
    *        Split the matrix if possible.
    *
    *        Two tests:
    *           1: H(j,j-1)=0  or  j=ILO
    *           2: T(j,j)=0
    *
             IF( ILAST.EQ.ILO ) THEN
    *
    *           Special case: j=ILAST
    *
                GO TO 80
             ELSE
                IF( ABS( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
                   H( ILAST, ILAST-1 ) = ZERO
                   GO TO 80
                END IF
             END IF
    *
             IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
                T( ILAST, ILAST ) = ZERO
                GO TO 70
             END IF
    *
    *        General case: j<ILAST
    *
             DO 60 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( ABS( H( J, J-1 ) ).LE.ATOL ) THEN
                      H( J, J-1 ) = ZERO
                      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 ) = ZERO
    *
    *              Test 1a: Check for 2 consecutive small subdiagonals in A
    *
                   ILAZR2 = .FALSE.
                   IF( .NOT.ILAZRO ) THEN
                      TEMP = ABS( H( J, J-1 ) )
                      TEMP2 = ABS( H( J, J ) )
                      TEMPR = MAX( TEMP, TEMP2 )
                      IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
                         TEMP = TEMP / TEMPR
                         TEMP2 = TEMP2 / TEMPR
                      END IF
                      IF( TEMP*( ASCALE*ABS( H( J+1, J ) ) ).LE.TEMP2*
         $                ( 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 40 JCH = J, ILAST - 1
                         TEMP = H( JCH, JCH )
                         CALL DLARTG( TEMP, H( JCH+1, JCH ), C, S,
         $                            H( JCH, JCH ) )
                         H( JCH+1, JCH ) = ZERO
                         CALL DROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
         $                          H( JCH+1, JCH+1 ), LDH, C, S )
                         CALL DROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
         $                          T( JCH+1, JCH+1 ), LDT, C, S )
                         IF( ILQ )
         $                  CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
         $                             C, S )
                         IF( ILAZR2 )
         $                  H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
                         ILAZR2 = .FALSE.
                         IF( ABS( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
                            IF( JCH+1.GE.ILAST ) THEN
                               GO TO 80
                            ELSE
                               IFIRST = JCH + 1
                               GO TO 110
                            END IF
                         END IF
                         T( JCH+1, JCH+1 ) = ZERO
       40             CONTINUE
                      GO TO 70
                   ELSE
    *
    *                 Only test 2 passed -- chase the zero to T(ILAST,ILAST)
    *                 Then process as in the case T(ILAST,ILAST)=0
    *
                      DO 50 JCH = J, ILAST - 1
                         TEMP = T( JCH, JCH+1 )
                         CALL DLARTG( TEMP, T( JCH+1, JCH+1 ), C, S,
         $                            T( JCH, JCH+1 ) )
                         T( JCH+1, JCH+1 ) = ZERO
                         IF( JCH.LT.ILASTM-1 )
         $                  CALL DROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
         $                             T( JCH+1, JCH+2 ), LDT, C, S )
                         CALL DROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
         $                          H( JCH+1, JCH-1 ), LDH, C, S )
                         IF( ILQ )
         $                  CALL DROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
         $                             C, S )
                         TEMP = H( JCH+1, JCH )
                         CALL DLARTG( TEMP, H( JCH+1, JCH-1 ), C, S,
         $                            H( JCH+1, JCH ) )
                         H( JCH+1, JCH-1 ) = ZERO
                         CALL DROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
         $                          H( IFRSTM, JCH-1 ), 1, C, S )
                         CALL DROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
         $                          T( IFRSTM, JCH-1 ), 1, C, S )
                         IF( ILZ )
         $                  CALL DROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
         $                             C, S )
       50             CONTINUE
                      GO TO 70
                   END IF
                ELSE IF( ILAZRO ) THEN
    *
    *              Only test 1 passed -- work on J:ILAST
    *
                   IFIRST = J
                   GO TO 110
                END IF
    *
    *           Neither test passed -- try next J
    *
       60    CONTINUE
    *
    *        (Drop-through is "impossible")
    *
             INFO = N + 1
             GO TO 420
    *
    *        T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
    *        1x1 block.
    *
       70    CONTINUE
             TEMP = H( ILAST, ILAST )
             CALL DLARTG( TEMP, H( ILAST, ILAST-1 ), C, S,
         $                H( ILAST, ILAST ) )
             H( ILAST, ILAST-1 ) = ZERO
             CALL DROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
         $              H( IFRSTM, ILAST-1 ), 1, C, S )
             CALL DROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
         $              T( IFRSTM, ILAST-1 ), 1, C, S )
             IF( ILZ )
         $      CALL DROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
    *
    *        H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHAR, ALPHAI,
    *                              and BETA
    *
       80    CONTINUE
             IF( T( ILAST, ILAST ).LT.ZERO ) THEN
                IF( ILSCHR ) THEN
                   DO 90 J = IFRSTM, ILAST
                      H( J, ILAST ) = -H( J, ILAST )
                      T( J, ILAST ) = -T( J, ILAST )
       90          CONTINUE
                ELSE
                   H( ILAST, ILAST ) = -H( ILAST, ILAST )
                   T( ILAST, ILAST ) = -T( ILAST, ILAST )
                END IF
                IF( ILZ ) THEN
                   DO 100 J = 1, N
                      Z( J, ILAST ) = -Z( J, ILAST )
      100          CONTINUE
                END IF
             END IF
             ALPHAR( ILAST ) = H( ILAST, ILAST )
             ALPHAI( ILAST ) = ZERO
             BETA( ILAST ) = T( ILAST, ILAST )
    *
    *        Go to next block -- exit if finished.
    *
             ILAST = ILAST - 1
             IF( ILAST.LT.ILO )
         $      GO TO 380
    *
    *        Reset counters
    *
             IITER = 0
             ESHIFT = ZERO
             IF( .NOT.ILSCHR ) THEN
                ILASTM = ILAST
                IF( IFRSTM.GT.ILAST )
         $         IFRSTM = ILO
             END IF
             GO TO 350
    *
    *        QZ step
    *
    *        This iteration only involves rows/columns IFIRST:ILAST. We
    *        assume IFIRST < ILAST, and that the diagonal of B is non-zero.
    *
      110    CONTINUE
             IITER = IITER + 1
             IF( .NOT.ILSCHR ) THEN
                IFRSTM = IFIRST
             END IF
    *
    *        Compute single shifts.
    *
    *        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.EQ.IITER ) THEN
    *
    *           Exceptional shift.  Chosen for no particularly good reason.
    *           (Single shift only.)
    *
                IF( ( DBLE( MAXIT )*SAFMIN )*ABS( H( ILAST-1, ILAST ) ).LT.
         $          ABS( T( ILAST-1, ILAST-1 ) ) ) THEN
                   ESHIFT = ESHIFT + H( ILAST-1, ILAST ) /
         $                  T( ILAST-1, ILAST-1 )
                ELSE
                   ESHIFT = ESHIFT + ONE / ( SAFMIN*DBLE( MAXIT ) )
                END IF
                S1 = ONE
                WR = ESHIFT
    *
             ELSE
    *
    *           Shifts based on the generalized eigenvalues of the
    *           bottom-right 2x2 block of A and B. The first eigenvalue
    *           returned by DLAG2 is the Wilkinson shift (AEP p.512),
    *
                CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
         $                  T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
         $                  S2, WR, WR2, WI )
    *
                TEMP = MAX( S1, SAFMIN*MAX( ONE, ABS( WR ), ABS( WI ) ) )
                IF( WI.NE.ZERO )
         $         GO TO 200
             END IF
    *
    *        Fiddle with shift to avoid overflow
    *
             TEMP = MIN( ASCALE, ONE )*( HALF*SAFMAX )
             IF( S1.GT.TEMP ) THEN
                SCALE = TEMP / S1
             ELSE
                SCALE = ONE
             END IF
    *
             TEMP = MIN( BSCALE, ONE )*( HALF*SAFMAX )
             IF( ABS( WR ).GT.TEMP )
         $      SCALE = MIN( SCALE, TEMP / ABS( WR ) )
             S1 = SCALE*S1
             WR = SCALE*WR
    *
    *        Now check for two consecutive small subdiagonals.
    *
             DO 120 J = ILAST - 1, IFIRST + 1, -1
                ISTART = J
                TEMP = ABS( S1*H( J, J-1 ) )
                TEMP2 = ABS( S1*H( J, J )-WR*T( J, J ) )
                TEMPR = MAX( TEMP, TEMP2 )
                IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
                   TEMP = TEMP / TEMPR
                   TEMP2 = TEMP2 / TEMPR
                END IF
                IF( ABS( ( ASCALE*H( J+1, J ) )*TEMP ).LE.( ASCALE*ATOL )*
         $          TEMP2 )GO TO 130
      120    CONTINUE
    *
             ISTART = IFIRST
      130    CONTINUE
    *
    *        Do an implicit single-shift QZ sweep.
    *
    *        Initial Q
    *
             TEMP = S1*H( ISTART, ISTART ) - WR*T( ISTART, ISTART )
             TEMP2 = S1*H( ISTART+1, ISTART )
             CALL DLARTG( TEMP, TEMP2, C, S, TEMPR )
    *
    *        Sweep
    *
             DO 190 J = ISTART, ILAST - 1
                IF( J.GT.ISTART ) THEN
                   TEMP = H( J, J-1 )
                   CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
                   H( J+1, J-1 ) = ZERO
                END IF
    *
                DO 140 JC = J, ILASTM
                   TEMP = C*H( J, JC ) + S*H( J+1, JC )
                   H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
                   H( J, JC ) = TEMP
                   TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
                   T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
                   T( J, JC ) = TEMP2
      140       CONTINUE
                IF( ILQ ) THEN
                   DO 150 JR = 1, N
                      TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
                      Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
                      Q( JR, J ) = TEMP
      150          CONTINUE
                END IF
    *
                TEMP = T( J+1, J+1 )
                CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
                T( J+1, J ) = ZERO
    *
                DO 160 JR = IFRSTM, MIN( J+2, ILAST )
                   TEMP = C*H( JR, J+1 ) + S*H( JR, J )
                   H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
                   H( JR, J+1 ) = TEMP
      160       CONTINUE
                DO 170 JR = IFRSTM, J
                   TEMP = C*T( JR, J+1 ) + S*T( JR, J )
                   T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
                   T( JR, J+1 ) = TEMP
      170       CONTINUE
                IF( ILZ ) THEN
                   DO 180 JR = 1, N
                      TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
                      Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
                      Z( JR, J+1 ) = TEMP
      180          CONTINUE
                END IF
      190    CONTINUE
    *
             GO TO 350
    *
    *        Use Francis double-shift
    *
    *        Note: the Francis double-shift should work with real shifts,
    *              but only if the block is at least 3x3.
    *              This code may break if this point is reached with
    *              a 2x2 block with real eigenvalues.
    *
      200    CONTINUE
             IF( IFIRST+1.EQ.ILAST ) THEN
    *
    *           Special case -- 2x2 block with complex eigenvectors
    *
    *           Step 1: Standardize, that is, rotate so that
    *
    *                       ( B11  0  )
    *                   B = (         )  with B11 non-negative.
    *                       (  0  B22 )
    *
                CALL DLASV2( T( ILAST-1, ILAST-1 ), T( ILAST-1, ILAST ),
         $                   T( ILAST, ILAST ), B22, B11, SR, CR, SL, CL )
    *
                IF( B11.LT.ZERO ) THEN
                   CR = -CR
                   SR = -SR
                   B11 = -B11
                   B22 = -B22
                END IF
    *
                CALL DROT( ILASTM+1-IFIRST, H( ILAST-1, ILAST-1 ), LDH,
         $                 H( ILAST, ILAST-1 ), LDH, CL, SL )
                CALL DROT( ILAST+1-IFRSTM, H( IFRSTM, ILAST-1 ), 1,
         $                 H( IFRSTM, ILAST ), 1, CR, SR )
    *
                IF( ILAST.LT.ILASTM )
         $         CALL DROT( ILASTM-ILAST, T( ILAST-1, ILAST+1 ), LDT,
         $                    T( ILAST, ILAST+1 ), LDH, CL, SL )
                IF( IFRSTM.LT.ILAST-1 )
         $         CALL DROT( IFIRST-IFRSTM, T( IFRSTM, ILAST-1 ), 1,
         $                    T( IFRSTM, ILAST ), 1, CR, SR )
    *
                IF( ILQ )
         $         CALL DROT( N, Q( 1, ILAST-1 ), 1, Q( 1, ILAST ), 1, CL,
         $                    SL )
                IF( ILZ )
         $         CALL DROT( N, Z( 1, ILAST-1 ), 1, Z( 1, ILAST ), 1, CR,
         $                    SR )
    *
                T( ILAST-1, ILAST-1 ) = B11
                T( ILAST-1, ILAST ) = ZERO
                T( ILAST, ILAST-1 ) = ZERO
                T( ILAST, ILAST ) = B22
    *
    *           If B22 is negative, negate column ILAST
    *
                IF( B22.LT.ZERO ) THEN
                   DO 210 J = IFRSTM, ILAST
                      H( J, ILAST ) = -H( J, ILAST )
                      T( J, ILAST ) = -T( J, ILAST )
      210          CONTINUE
    *
                   IF( ILZ ) THEN
                      DO 220 J = 1, N
                         Z( J, ILAST ) = -Z( J, ILAST )
      220             CONTINUE
                   END IF
                END IF
    *
    *           Step 2: Compute ALPHAR, ALPHAI, and BETA (see refs.)
    *
    *           Recompute shift
    *
                CALL DLAG2( H( ILAST-1, ILAST-1 ), LDH,
         $                  T( ILAST-1, ILAST-1 ), LDT, SAFMIN*SAFETY, S1,
         $                  TEMP, WR, TEMP2, WI )
    *
    *           If standardization has perturbed the shift onto real line,
    *           do another (real single-shift) QR step.
    *
                IF( WI.EQ.ZERO )
         $         GO TO 350
                S1INV = ONE / S1
    *
    *           Do EISPACK (QZVAL) computation of alpha and beta
    *
                A11 = H( ILAST-1, ILAST-1 )
                A21 = H( ILAST, ILAST-1 )
                A12 = H( ILAST-1, ILAST )
                A22 = H( ILAST, ILAST )
    *
    *           Compute complex Givens rotation on right
    *           (Assume some element of C = (sA - wB) > unfl )
    *                            __
    *           (sA - wB) ( CZ   -SZ )
    *                     ( SZ    CZ )
    *
                C11R = S1*A11 - WR*B11
                C11I = -WI*B11
                C12 = S1*A12
                C21 = S1*A21
                C22R = S1*A22 - WR*B22
                C22I = -WI*B22
    *
                IF( ABS( C11R )+ABS( C11I )+ABS( C12 ).GT.ABS( C21 )+
         $          ABS( C22R )+ABS( C22I ) ) THEN
                   T1 = DLAPY3( C12, C11R, C11I )
                   CZ = C12 / T1
                   SZR = -C11R / T1
                   SZI = -C11I / T1
                ELSE
                   CZ = DLAPY2( C22R, C22I )
                   IF( CZ.LE.SAFMIN ) THEN
                      CZ = ZERO
                      SZR = ONE
                      SZI = ZERO
                   ELSE
                      TEMPR = C22R / CZ
                      TEMPI = C22I / CZ
                      T1 = DLAPY2( CZ, C21 )
                      CZ = CZ / T1
                      SZR = -C21*TEMPR / T1
                      SZI = C21*TEMPI / T1
                   END IF
                END IF
    *
    *           Compute Givens rotation on left
    *
    *           (  CQ   SQ )
    *           (  __      )  A or B
    *           ( -SQ   CQ )
    *
                AN = ABS( A11 ) + ABS( A12 ) + ABS( A21 ) + ABS( A22 )
                BN = ABS( B11 ) + ABS( B22 )
                WABS = ABS( WR ) + ABS( WI )
                IF( S1*AN.GT.WABS*BN ) THEN
                   CQ = CZ*B11
                   SQR = SZR*B22
                   SQI = -SZI*B22
                ELSE
                   A1R = CZ*A11 + SZR*A12
                   A1I = SZI*A12
                   A2R = CZ*A21 + SZR*A22
                   A2I = SZI*A22
                   CQ = DLAPY2( A1R, A1I )
                   IF( CQ.LE.SAFMIN ) THEN
                      CQ = ZERO
                      SQR = ONE
                      SQI = ZERO
                   ELSE
                      TEMPR = A1R / CQ
                      TEMPI = A1I / CQ
                      SQR = TEMPR*A2R + TEMPI*A2I
                      SQI = TEMPI*A2R - TEMPR*A2I
                   END IF
                END IF
                T1 = DLAPY3( CQ, SQR, SQI )
                CQ = CQ / T1
                SQR = SQR / T1
                SQI = SQI / T1
    *
    *           Compute diagonal elements of QBZ
    *
                TEMPR = SQR*SZR - SQI*SZI
                TEMPI = SQR*SZI + SQI*SZR
                B1R = CQ*CZ*B11 + TEMPR*B22
                B1I = TEMPI*B22
                B1A = DLAPY2( B1R, B1I )
                B2R = CQ*CZ*B22 + TEMPR*B11
                B2I = -TEMPI*B11
                B2A = DLAPY2( B2R, B2I )
    *
    *           Normalize so beta > 0, and Im( alpha1 ) > 0
    *
                BETA( ILAST-1 ) = B1A
                BETA( ILAST ) = B2A
                ALPHAR( ILAST-1 ) = ( WR*B1A )*S1INV
                ALPHAI( ILAST-1 ) = ( WI*B1A )*S1INV
                ALPHAR( ILAST ) = ( WR*B2A )*S1INV
                ALPHAI( ILAST ) = -( WI*B2A )*S1INV
    *
    *           Step 3: Go to next block -- exit if finished.
    *
                ILAST = IFIRST - 1
                IF( ILAST.LT.ILO )
         $         GO TO 380
    *
    *           Reset counters
    *
                IITER = 0
                ESHIFT = ZERO
                IF( .NOT.ILSCHR ) THEN
                   ILASTM = ILAST
                   IF( IFRSTM.GT.ILAST )
         $            IFRSTM = ILO
                END IF
                GO TO 350
             ELSE
    *
    *           Usual case: 3x3 or larger block, using Francis implicit
    *                       double-shift
    *
    *                                    2
    *           Eigenvalue equation is  w  - c w + d = 0,
    *
    *                                         -1 2        -1
    *           so compute 1st column of  (A B  )  - c A B   + d
    *           using the formula in QZIT (from EISPACK)
    *
    *           We assume that the block is at least 3x3
    *
                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 ) )
                U12 = T( ILAST-1, ILAST ) / T( ILAST, ILAST )
                AD11L = ( ASCALE*H( IFIRST, IFIRST ) ) /
         $              ( BSCALE*T( IFIRST, IFIRST ) )
                AD21L = ( ASCALE*H( IFIRST+1, IFIRST ) ) /
         $              ( BSCALE*T( IFIRST, IFIRST ) )
                AD12L = ( ASCALE*H( IFIRST, IFIRST+1 ) ) /
         $              ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
                AD22L = ( ASCALE*H( IFIRST+1, IFIRST+1 ) ) /
         $              ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
                AD32L = ( ASCALE*H( IFIRST+2, IFIRST+1 ) ) /
         $              ( BSCALE*T( IFIRST+1, IFIRST+1 ) )
                U12L = T( IFIRST, IFIRST+1 ) / T( IFIRST+1, IFIRST+1 )
    *
                V( 1 ) = ( AD11-AD11L )*( AD22-AD11L ) - AD12*AD21 +
         $               AD21*U12*AD11L + ( AD12L-AD11L*U12L )*AD21L
                V( 2 ) = ( ( AD22L-AD11L )-AD21L*U12L-( AD11-AD11L )-
         $               ( AD22-AD11L )+AD21*U12 )*AD21L
                V( 3 ) = AD32L*AD21L
    *
                ISTART = IFIRST
    *
                CALL DLARFG( 3, V( 1 ), V( 2 ), 1, TAU )
                V( 1 ) = ONE
    *
    *           Sweep
    *
                DO 290 J = ISTART, ILAST - 2
    *
    *              All but last elements: use 3x3 Householder transforms.
    *
    *              Zero (j-1)st column of A
    *
                   IF( J.GT.ISTART ) THEN
                      V( 1 ) = H( J, J-1 )
                      V( 2 ) = H( J+1, J-1 )
                      V( 3 ) = H( J+2, J-1 )
    *
                      CALL DLARFG( 3, H( J, J-1 ), V( 2 ), 1, TAU )
                      V( 1 ) = ONE
                      H( J+1, J-1 ) = ZERO
                      H( J+2, J-1 ) = ZERO
                   END IF
    *
                   DO 230 JC = J, ILASTM
                      TEMP = TAU*( H( J, JC )+V( 2 )*H( J+1, JC )+V( 3 )*
         $                   H( J+2, JC ) )
                      H( J, JC ) = H( J, JC ) - TEMP
                      H( J+1, JC ) = H( J+1, JC ) - TEMP*V( 2 )
                      H( J+2, JC ) = H( J+2, JC ) - TEMP*V( 3 )
                      TEMP2 = TAU*( T( J, JC )+V( 2 )*T( J+1, JC )+V( 3 )*
         $                    T( J+2, JC ) )
                      T( J, JC ) = T( J, JC ) - TEMP2
                      T( J+1, JC ) = T( J+1, JC ) - TEMP2*V( 2 )
                      T( J+2, JC ) = T( J+2, JC ) - TEMP2*V( 3 )
      230          CONTINUE
                   IF( ILQ ) THEN
                      DO 240 JR = 1, N
                         TEMP = TAU*( Q( JR, J )+V( 2 )*Q( JR, J+1 )+V( 3 )*
         $                      Q( JR, J+2 ) )
                         Q( JR, J ) = Q( JR, J ) - TEMP
                         Q( JR, J+1 ) = Q( JR, J+1 ) - TEMP*V( 2 )
                         Q( JR, J+2 ) = Q( JR, J+2 ) - TEMP*V( 3 )
      240             CONTINUE
                   END IF
    *
    *              Zero j-th column of B (see DLAGBC for details)
    *
    *              Swap rows to pivot
    *
                   ILPIVT = .FALSE.
                   TEMP = MAX( ABS( T( J+1, J+1 ) ), ABS( T( J+1, J+2 ) ) )
                   TEMP2 = MAX( ABS( T( J+2, J+1 ) ), ABS( T( J+2, J+2 ) ) )
                   IF( MAX( TEMP, TEMP2 ).LT.SAFMIN ) THEN
                      SCALE = ZERO
                      U1 = ONE
                      U2 = ZERO
                      GO TO 250
                   ELSE IF( TEMP.GE.TEMP2 ) THEN
                      W11 = T( J+1, J+1 )
                      W21 = T( J+2, J+1 )
                      W12 = T( J+1, J+2 )
                      W22 = T( J+2, J+2 )
                      U1 = T( J+1, J )
                      U2 = T( J+2, J )
                   ELSE
                      W21 = T( J+1, J+1 )
                      W11 = T( J+2, J+1 )
                      W22 = T( J+1, J+2 )
                      W12 = T( J+2, J+2 )
                      U2 = T( J+1, J )
                      U1 = T( J+2, J )
                   END IF
    *
    *              Swap columns if nec.
    *
                   IF( ABS( W12 ).GT.ABS( W11 ) ) THEN
                      ILPIVT = .TRUE.
                      TEMP = W12
                      TEMP2 = W22
                      W12 = W11
                      W22 = W21
                      W11 = TEMP
                      W21 = TEMP2
                   END IF
    *
    *              LU-factor
    *
                   TEMP = W21 / W11
                   U2 = U2 - TEMP*U1
                   W22 = W22 - TEMP*W12
                   W21 = ZERO
    *
    *              Compute SCALE
    *
                   SCALE = ONE
                   IF( ABS( W22 ).LT.SAFMIN ) THEN
                      SCALE = ZERO
                      U2 = ONE
                      U1 = -W12 / W11
                      GO TO 250
                   END IF
                   IF( ABS( W22 ).LT.ABS( U2 ) )
         $            SCALE = ABS( W22 / U2 )
                   IF( ABS( W11 ).LT.ABS( U1 ) )
         $            SCALE = MIN( SCALE, ABS( W11 / U1 ) )
    *
    *              Solve
    *
                   U2 = ( SCALE*U2 ) / W22
                   U1 = ( SCALE*U1-W12*U2 ) / W11
    *
      250          CONTINUE
                   IF( ILPIVT ) THEN
                      TEMP = U2
                      U2 = U1
                      U1 = TEMP
                   END IF
    *
    *              Compute Householder Vector
    *
                   T1 = SQRT( SCALE**2+U1**2+U2**2 )
                   TAU = ONE + SCALE / T1
                   VS = -ONE / ( SCALE+T1 )
                   V( 1 ) = ONE
                   V( 2 ) = VS*U1
                   V( 3 ) = VS*U2
    *
    *              Apply transformations from the right.
    *
                   DO 260 JR = IFRSTM, MIN( J+3, ILAST )
                      TEMP = TAU*( H( JR, J )+V( 2 )*H( JR, J+1 )+V( 3 )*
         $                   H( JR, J+2 ) )
                      H( JR, J ) = H( JR, J ) - TEMP
                      H( JR, J+1 ) = H( JR, J+1 ) - TEMP*V( 2 )
                      H( JR, J+2 ) = H( JR, J+2 ) - TEMP*V( 3 )
      260          CONTINUE
                   DO 270 JR = IFRSTM, J + 2
                      TEMP = TAU*( T( JR, J )+V( 2 )*T( JR, J+1 )+V( 3 )*
         $                   T( JR, J+2 ) )
                      T( JR, J ) = T( JR, J ) - TEMP
                      T( JR, J+1 ) = T( JR, J+1 ) - TEMP*V( 2 )
                      T( JR, J+2 ) = T( JR, J+2 ) - TEMP*V( 3 )
      270          CONTINUE
                   IF( ILZ ) THEN
                      DO 280 JR = 1, N
                         TEMP = TAU*( Z( JR, J )+V( 2 )*Z( JR, J+1 )+V( 3 )*
         $                      Z( JR, J+2 ) )
                         Z( JR, J ) = Z( JR, J ) - TEMP
                         Z( JR, J+1 ) = Z( JR, J+1 ) - TEMP*V( 2 )
                         Z( JR, J+2 ) = Z( JR, J+2 ) - TEMP*V( 3 )
      280             CONTINUE
                   END IF
                   T( J+1, J ) = ZERO
                   T( J+2, J ) = ZERO
      290       CONTINUE
    *
    *           Last elements: Use Givens rotations
    *
    *           Rotations from the left
    *
                J = ILAST - 1
                TEMP = H( J, J-1 )
                CALL DLARTG( TEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
                H( J+1, J-1 ) = ZERO
    *
                DO 300 JC = J, ILASTM
                   TEMP = C*H( J, JC ) + S*H( J+1, JC )
                   H( J+1, JC ) = -S*H( J, JC ) + C*H( J+1, JC )
                   H( J, JC ) = TEMP
                   TEMP2 = C*T( J, JC ) + S*T( J+1, JC )
                   T( J+1, JC ) = -S*T( J, JC ) + C*T( J+1, JC )
                   T( J, JC ) = TEMP2
      300       CONTINUE
                IF( ILQ ) THEN
                   DO 310 JR = 1, N
                      TEMP = C*Q( JR, J ) + S*Q( JR, J+1 )
                      Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
                      Q( JR, J ) = TEMP
      310          CONTINUE
                END IF
    *
    *           Rotations from the right.
    *
                TEMP = T( J+1, J+1 )
                CALL DLARTG( TEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
                T( J+1, J ) = ZERO
    *
                DO 320 JR = IFRSTM, ILAST
                   TEMP = C*H( JR, J+1 ) + S*H( JR, J )
                   H( JR, J ) = -S*H( JR, J+1 ) + C*H( JR, J )
                   H( JR, J+1 ) = TEMP
      320       CONTINUE
                DO 330 JR = IFRSTM, ILAST - 1
                   TEMP = C*T( JR, J+1 ) + S*T( JR, J )
                   T( JR, J ) = -S*T( JR, J+1 ) + C*T( JR, J )
                   T( JR, J+1 ) = TEMP
      330       CONTINUE
                IF( ILZ ) THEN
                   DO 340 JR = 1, N
                      TEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
                      Z( JR, J ) = -S*Z( JR, J+1 ) + C*Z( JR, J )
                      Z( JR, J+1 ) = TEMP
      340          CONTINUE
                END IF
    *
    *           End of Double-Shift code
    *
             END IF
    *
             GO TO 350
    *
    *        End of iteration loop
    *
      350    CONTINUE
      360 CONTINUE
    *
    *     Drop-through = non-convergence
    *
          INFO = ILAST
          GO TO 420
    *
    *     Successful completion of all QZ steps
    *
      380 CONTINUE
    *
    *     Set Eigenvalues 1:ILO-1
    *
          DO 410 J = 1, ILO - 1
             IF( T( J, J ).LT.ZERO ) THEN
                IF( ILSCHR ) THEN
                   DO 390 JR = 1, J
                      H( JR, J ) = -H( JR, J )
                      T( JR, J ) = -T( JR, J )
      390          CONTINUE
                ELSE
                   H( J, J ) = -H( J, J )
                   T( J, J ) = -T( J, J )
                END IF
                IF( ILZ ) THEN
                   DO 400 JR = 1, N
                      Z( JR, J ) = -Z( JR, J )
      400          CONTINUE
                END IF
             END IF
             ALPHAR( J ) = H( J, J )
             ALPHAI( J ) = ZERO
             BETA( J ) = T( J, J )
      410 CONTINUE
    *
    *     Normal Termination
    *
          INFO = 0
    *
    *     Exit (other than argument error) -- return optimal workspace size
    *
      420 CONTINUE
          WORK( 1 ) = DBLE( N )
          RETURN
    *
    *     End of DHGEQZ
    *
          END