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      SUBROUTINE <a name="DHGEQZ.1"></a><a href="dhgeqz.f.html#DHGEQZ.1">DHGEQZ</a>( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
     $                   ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
     $                   LWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      CHARACTER          COMPQ, COMPZ, JOB
      INTEGER            IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ),
     $                   H( LDH, * ), Q( LDQ, * ), T( LDT, * ),
     $                   WORK( * ), Z( LDZ, * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DHGEQZ.22"></a><a href="dhgeqz.f.html#DHGEQZ.1">DHGEQZ</a> computes the eigenvalues of a real matrix pair (H,T),
</span><span class="comment">*</span><span class="comment">  where H is an upper Hessenberg matrix and T is upper triangular,
</span><span class="comment">*</span><span class="comment">  using the double-shift QZ method.
</span><span class="comment">*</span><span class="comment">  Matrix pairs of this type are produced by the reduction to
</span><span class="comment">*</span><span class="comment">  generalized upper Hessenberg form of a real matrix pair (A,B):
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  as computed by <a name="DGGHRD.30"></a><a href="dgghrd.f.html#DGGHRD.1">DGGHRD</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If JOB='S', then the Hessenberg-triangular pair (H,T) is
</span><span class="comment">*</span><span class="comment">  also reduced to generalized Schur form,
</span><span class="comment">*</span><span class="comment">  
</span><span class="comment">*</span><span class="comment">     H = Q*S*Z**T,  T = Q*P*Z**T,
</span><span class="comment">*</span><span class="comment">  
</span><span class="comment">*</span><span class="comment">  where Q and Z are orthogonal matrices, P is an upper triangular
</span><span class="comment">*</span><span class="comment">  matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
</span><span class="comment">*</span><span class="comment">  diagonal blocks.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
</span><span class="comment">*</span><span class="comment">  (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
</span><span class="comment">*</span><span class="comment">  eigenvalues.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Additionally, the 2-by-2 upper triangular diagonal blocks of P
</span><span class="comment">*</span><span class="comment">  corresponding to 2-by-2 blocks of S are reduced to positive diagonal
</span><span class="comment">*</span><span class="comment">  form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
</span><span class="comment">*</span><span class="comment">  P(j,j) &gt; 0, and P(j+1,j+1) &gt; 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Optionally, the orthogonal matrix Q from the generalized Schur
</span><span class="comment">*</span><span class="comment">  factorization may be postmultiplied into an input matrix Q1, and the
</span><span class="comment">*</span><span class="comment">  orthogonal matrix Z may be postmultiplied into an input matrix Z1.
</span><span class="comment">*</span><span class="comment">  If Q1 and Z1 are the orthogonal matrices from <a name="DGGHRD.53"></a><a href="dgghrd.f.html#DGGHRD.1">DGGHRD</a> that reduced
</span><span class="comment">*</span><span class="comment">  the matrix pair (A,B) to generalized upper Hessenberg form, then the
</span><span class="comment">*</span><span class="comment">  output matrices Q1*Q and Z1*Z are the orthogonal factors from the
</span><span class="comment">*</span><span class="comment">  generalized Schur factorization of (A,B):
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
</span><span class="comment">*</span><span class="comment">  
</span><span class="comment">*</span><span class="comment">  To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
</span><span class="comment">*</span><span class="comment">  of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
</span><span class="comment">*</span><span class="comment">  complex and beta real.
</span><span class="comment">*</span><span class="comment">  If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
</span><span class="comment">*</span><span class="comment">  generalized nonsymmetric eigenvalue problem (GNEP)
</span><span class="comment">*</span><span class="comment">     A*x = lambda*B*x
</span><span class="comment">*</span><span class="comment">  and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
</span><span class="comment">*</span><span class="comment">  alternate form of the GNEP
</span><span class="comment">*</span><span class="comment">     mu*A*y = B*y.
</span><span class="comment">*</span><span class="comment">  Real eigenvalues can be read directly from the generalized Schur
</span><span class="comment">*</span><span class="comment">  form: 
</span><span class="comment">*</span><span class="comment">    alpha = S(i,i), beta = P(i,i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Ref: C.B. Moler &amp; G.W. Stewart, &quot;An Algorithm for Generalized Matrix
</span><span class="comment">*</span><span class="comment">       Eigenvalue Problems&quot;, SIAM J. Numer. Anal., 10(1973),
</span><span class="comment">*</span><span class="comment">       pp. 241--256.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  JOB     (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'E': Compute eigenvalues only;
</span><span class="comment">*</span><span class="comment">          = 'S': Compute eigenvalues and the Schur form. 
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  COMPQ   (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'N': Left Schur vectors (Q) are not computed;
</span><span class="comment">*</span><span class="comment">          = 'I': Q is initialized to the unit matrix and the matrix Q
</span><span class="comment">*</span><span class="comment">                 of left Schur vectors of (H,T) is returned;
</span><span class="comment">*</span><span class="comment">          = 'V': Q must contain an orthogonal matrix Q1 on entry and
</span><span class="comment">*</span><span class="comment">                 the product Q1*Q is returned.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  COMPZ   (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          = 'N': Right Schur vectors (Z) are not computed;
</span><span class="comment">*</span><span class="comment">          = 'I': Z is initialized to the unit matrix and the matrix Z
</span><span class="comment">*</span><span class="comment">                 of right Schur vectors of (H,T) is returned;
</span><span class="comment">*</span><span class="comment">          = 'V': Z must contain an orthogonal matrix Z1 on entry and
</span><span class="comment">*</span><span class="comment">                 the product Z1*Z is returned.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrices H, T, Q, and Z.  N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ILO     (input) INTEGER
</span><span class="comment">*</span><span class="comment">  IHI     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          ILO and IHI mark the rows and columns of H which are in
</span><span class="comment">*</span><span class="comment">          Hessenberg form.  It is assumed that A is already upper
</span><span class="comment">*</span><span class="comment">          triangular in rows and columns 1:ILO-1 and IHI+1:N.
</span><span class="comment">*</span><span class="comment">          If N &gt; 0, 1 &lt;= ILO &lt;= IHI &lt;= N; if N = 0, ILO=1 and IHI=0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  H       (input/output) DOUBLE PRECISION array, dimension (LDH, N)
</span><span class="comment">*</span><span class="comment">          On entry, the N-by-N upper Hessenberg matrix H.
</span><span class="comment">*</span><span class="comment">          On exit, if JOB = 'S', H contains the upper quasi-triangular
</span><span class="comment">*</span><span class="comment">          matrix S from the generalized Schur factorization;
</span><span class="comment">*</span><span class="comment">          2-by-2 diagonal blocks (corresponding to complex conjugate
</span><span class="comment">*</span><span class="comment">          pairs of eigenvalues) are returned in standard form, with
</span><span class="comment">*</span><span class="comment">          H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) &lt; 0.
</span><span class="comment">*</span><span class="comment">          If JOB = 'E', the diagonal blocks of H match those of S, but
</span><span class="comment">*</span><span class="comment">          the rest of H is unspecified.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDH     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array H.  LDH &gt;= max( 1, N ).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  T       (input/output) DOUBLE PRECISION array, dimension (LDT, N)
</span><span class="comment">*</span><span class="comment">          On entry, the N-by-N upper triangular matrix T.
</span><span class="comment">*</span><span class="comment">          On exit, if JOB = 'S', T contains the upper triangular
</span><span class="comment">*</span><span class="comment">          matrix P from the generalized Schur factorization;
</span><span class="comment">*</span><span class="comment">          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
</span><span class="comment">*</span><span class="comment">          are reduced to positive diagonal form, i.e., if H(j+1,j) is
</span><span class="comment">*</span><span class="comment">          non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) &gt; 0, and
</span><span class="comment">*</span><span class="comment">          T(j+1,j+1) &gt; 0.
</span><span class="comment">*</span><span class="comment">          If JOB = 'E', the diagonal blocks of T match those of P, but
</span><span class="comment">*</span><span class="comment">          the rest of T is unspecified.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDT     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array T.  LDT &gt;= max( 1, N ).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The real parts of each scalar alpha defining an eigenvalue
</span><span class="comment">*</span><span class="comment">          of GNEP.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The imaginary parts of each scalar alpha defining an
</span><span class="comment">*</span><span class="comment">          eigenvalue of GNEP.
</span><span class="comment">*</span><span class="comment">          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
</span><span class="comment">*</span><span class="comment">          positive, then the j-th and (j+1)-st eigenvalues are a
</span><span class="comment">*</span><span class="comment">          complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  BETA    (output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The scalars beta that define the eigenvalues of GNEP.
</span><span class="comment">*</span><span class="comment">          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
</span><span class="comment">*</span><span class="comment">          beta = BETA(j) represent the j-th eigenvalue of the matrix
</span><span class="comment">*</span><span class="comment">          pair (A,B), in one of the forms lambda = alpha/beta or
</span><span class="comment">*</span><span class="comment">          mu = beta/alpha.  Since either lambda or mu may overflow,
</span><span class="comment">*</span><span class="comment">          they should not, in general, be computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
</span><span class="comment">*</span><span class="comment">          On entry, if COMPZ = 'V', the orthogonal matrix Q1 used in
</span><span class="comment">*</span><span class="comment">          the reduction of (A,B) to generalized Hessenberg form.
</span><span class="comment">*</span><span class="comment">          On exit, if COMPZ = 'I', the orthogonal matrix of left Schur
</span><span class="comment">*</span><span class="comment">          vectors of (H,T), and if COMPZ = 'V', the orthogonal matrix
</span><span class="comment">*</span><span class="comment">          of left Schur vectors of (A,B).
</span><span class="comment">*</span><span class="comment">          Not referenced if COMPZ = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDQ     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array Q.  LDQ &gt;= 1.
</span><span class="comment">*</span><span class="comment">          If COMPQ='V' or 'I', then LDQ &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Z       (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
</span><span class="comment">*</span><span class="comment">          On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
</span><span class="comment">*</span><span class="comment">          the reduction of (A,B) to generalized Hessenberg form.
</span><span class="comment">*</span><span class="comment">          On exit, if COMPZ = 'I', the orthogonal matrix of
</span><span class="comment">*</span><span class="comment">          right Schur vectors of (H,T), and if COMPZ = 'V', the
</span><span class="comment">*</span><span class="comment">          orthogonal matrix of right Schur vectors of (A,B).
</span><span class="comment">*</span><span class="comment">          Not referenced if COMPZ = 'N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDZ     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array Z.  LDZ &gt;= 1.
</span><span class="comment">*</span><span class="comment">          If COMPZ='V' or 'I', then LDZ &gt;= N.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment">          On exit, if INFO &gt;= 0, WORK(1) returns the optimal LWORK.
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