CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
 (A,B) the generalized eigenvalues, and optionally, the left and/or
 right generalized eigenvectors.

 Optionally, it also computes a balancing transformation to improve
 the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
 LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
 the eigenvalues (RCONDE), and reciprocal condition numbers for the
 right eigenvectors (RCONDV).

 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
 singular. It is usually represented as the pair (alpha,beta), as
 there is a reasonable interpretation for beta=0, and even for both
 being zero.

 The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  A * v(j) = lambda(j) * B * v(j) .
 The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
 of (A,B) satisfies
                  u(j)**H * A  = lambda(j) * u(j)**H * B.
 where u(j)**H is the conjugate-transpose of u(j).

          BALANC is CHARACTER*1
          Specifies the balance option to be performed:
          = 'N':  do not diagonally scale or permute;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale.
          Computed reciprocal condition numbers will be for the
          matrices after permuting and/or balancing. Permuting does
          not change condition numbers (in exact arithmetic), but
          balancing does.

          JOBVL is CHARACTER*1
          = 'N':  do not compute the left generalized eigenvectors;
          = 'V':  compute the left generalized eigenvectors.

          JOBVR is CHARACTER*1
          = 'N':  do not compute the right generalized eigenvectors;
          = 'V':  compute the right generalized eigenvectors.

          SENSE is CHARACTER*1
          Determines which reciprocal condition numbers are computed.
          = 'N': none are computed;
          = 'E': computed for eigenvalues only;
          = 'V': computed for eigenvectors only;
          = 'B': computed for eigenvalues and eigenvectors.

          N is INTEGER
          The order of the matrices A, B, VL, and VR.  N >= 0.

          A is COMPLEX array, dimension (LDA, N)
          On entry, the matrix A in the pair (A,B).
          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then A contains the first part of the complex Schur
          form of the "balanced" versions of the input A and B.

          LDA is INTEGER
          The leading dimension of A.  LDA >= max(1,N).

          B is COMPLEX array, dimension (LDB, N)
          On entry, the matrix B in the pair (A,B).
          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
          or both, then B contains the second part of the complex
          Schur form of the "balanced" versions of the input A and B.

          LDB is INTEGER
          The leading dimension of B.  LDB >= max(1,N).

          ALPHA is COMPLEX array, dimension (N)

          BETA is COMPLEX array, dimension (N)
          On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized
          eigenvalues.

          Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
          underflow, and BETA(j) may even be zero.  Thus, the user
          should avoid naively computing the ratio ALPHA/BETA.
          However, ALPHA will be always less than and usually
          comparable with norm(A) in magnitude, and BETA always less
          than and usually comparable with norm(B).

          VL is COMPLEX array, dimension (LDVL,N)
          If JOBVL = 'V', the left generalized eigenvectors u(j) are
          stored one after another in the columns of VL, in the same
          order as their eigenvalues.
          Each eigenvector will be scaled so the largest component
          will have abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVL = 'N'.

          LDVL is INTEGER
          The leading dimension of the matrix VL. LDVL >= 1, and
          if JOBVL = 'V', LDVL >= N.

          VR is COMPLEX array, dimension (LDVR,N)
          If JOBVR = 'V', the right generalized eigenvectors v(j) are
          stored one after another in the columns of VR, in the same
          order as their eigenvalues.
          Each eigenvector will be scaled so the largest component
          will have abs(real part) + abs(imag. part) = 1.
          Not referenced if JOBVR = 'N'.

          LDVR is INTEGER
          The leading dimension of the matrix VR. LDVR >= 1, and
          if JOBVR = 'V', LDVR >= N.

          ILO is INTEGER

          IHI is INTEGER
          ILO and IHI are integer values such that on exit
          A(i,j) = 0 and B(i,j) = 0 if i > j and
          j = 1,...,ILO-1 or i = IHI+1,...,N.
          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

          LSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B.  If PL(j) is the index of the
          row interchanged with row j, and DL(j) is the scaling
          factor applied to row j, then
            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
                      = DL(j)  for j = ILO,...,IHI
                      = PL(j)  for j = IHI+1,...,N.
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

          RSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B.  If PR(j) is the index of the
          column interchanged with column j, and DR(j) is the scaling
          factor applied to column j, then
            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
                      = DR(j)  for j = ILO,...,IHI
                      = PR(j)  for j = IHI+1,...,N
          The order in which the interchanges are made is N to IHI+1,
          then 1 to ILO-1.

          ABNRM is REAL
          The one-norm of the balanced matrix A.

          BBNRM is REAL
          The one-norm of the balanced matrix B.

          RCONDE is REAL array, dimension (N)
          If SENSE = 'E' or 'B', the reciprocal condition numbers of
          the eigenvalues, stored in consecutive elements of the array.
          If SENSE = 'N' or 'V', RCONDE is not referenced.

          RCONDV is REAL array, dimension (N)
          If SENSE = 'V' or 'B', the estimated reciprocal condition
          numbers of the eigenvectors, stored in consecutive elements
          of the array. If the eigenvalues cannot be reordered to
          compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
          when the true value would be very small anyway. 
          If SENSE = 'N' or 'E', RCONDV is not referenced.

          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

          LWORK is INTEGER
          The dimension of the array WORK. LWORK >= max(1,2*N).
          If SENSE = 'E', LWORK >= max(1,4*N).
          If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*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 is REAL array, dimension (lrwork)
          lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B',
          and at least max(1,2*N) otherwise.
          Real workspace.

          IWORK is INTEGER array, dimension (N+2)
          If SENSE = 'E', IWORK is not referenced.

          BWORK is LOGICAL array, dimension (N)
          If SENSE = 'N', BWORK is not referenced.

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  No eigenvectors have been
                calculated, but ALPHA(j) and BETA(j) should be correct
                for j=INFO+1,...,N.
          > N:  =N+1: other than QZ iteration failed in CHGEQZ.
                =N+2: error return from CTGEVC.

  Balancing a matrix pair (A,B) includes, first, permuting rows and
  columns to isolate eigenvalues, second, applying diagonal similarity
  transformation to the rows and columns to make the rows and columns
  as close in norm as possible. The computed reciprocal condition
  numbers correspond to the balanced matrix. Permuting rows and columns
  will not change the condition numbers (in exact arithmetic) but
  diagonal scaling will.  For further explanation of balancing, see
  section 4.11.1.2 of LAPACK Users' Guide.

  An approximate error bound on the chordal distance between the i-th
  computed generalized eigenvalue w and the corresponding exact
  eigenvalue lambda is

       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

  An approximate error bound for the angle between the i-th computed
  eigenvector VL(i) or VR(i) is given by

       EPS * norm(ABNRM, BBNRM) / DIF(i).

  For further explanation of the reciprocal condition numbers RCONDE
  and RCONDV, see section 4.11 of LAPACK User's Guide.