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Lapack documentation
    Purpose
    =======

    This routine is deprecated and has been replaced by routine DGGEV.

    DGEGV computes the eigenvalues and, optionally, the left and/or right
    eigenvectors of a real matrix pair (A,B).
    Given two square matrices A and B,
    the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
    eigenvalues lambda and corresponding (non-zero) eigenvectors x such
    that

       A*x = lambda*B*x.

    An alternate form is to find the eigenvalues mu and corresponding
    eigenvectors y such that

       mu*A*y = B*y.

    These two forms are equivalent with mu = 1/lambda and x = y if
    neither lambda nor mu is zero.  In order to deal with the case that
    lambda or mu is zero or small, two values alpha and beta are returned
    for each eigenvalue, such that lambda = alpha/beta and
    mu = beta/alpha.

    The vectors x and y in the above equations are right eigenvectors of
    the matrix pair (A,B).  Vectors u and v satisfying

       u**H*A = lambda*u**H*B  or  mu*v**H*A = v**H*B

    are left eigenvectors of (A,B).

    Note: this routine performs "full balancing" on A and B -- see
    "Further Details", below.

    Arguments
    =========

    JOBVL   (input) CHARACTER*1
            = 'N':  do not compute the left generalized eigenvectors;
            = 'V':  compute the left generalized eigenvectors (returned
                    in VL).

    JOBVR   (input) CHARACTER*1
            = 'N':  do not compute the right generalized eigenvectors;
            = 'V':  compute the right generalized eigenvectors (returned
                    in VR).

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

    A       (input/output) DOUBLE PRECISION array, dimension (LDA, N)
            On entry, the matrix A.
            If JOBVL = 'V' or JOBVR = 'V', then on exit A
            contains the real Schur form of A from the generalized Schur
            factorization of the pair (A,B) after balancing.
            If no eigenvectors were computed, then only the diagonal
            blocks from the Schur form will be correct.  See DGGHRD and
            DHGEQZ for details.

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

    B       (input/output) DOUBLE PRECISION array, dimension (LDB, N)
            On entry, the matrix B.
            If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
            upper triangular matrix obtained from B in the generalized
            Schur factorization of the pair (A,B) after balancing.
            If no eigenvectors were computed, then only those elements of
            B corresponding to the diagonal blocks from the Schur form of
            A will be correct.  See DGGHRD and DHGEQZ for details.

    LDB     (input) INTEGER
            The leading dimension of B.  LDB >= 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.

    VL      (output) DOUBLE PRECISION array, dimension (LDVL,N)
            If JOBVL = 'V', the left eigenvectors u(j) are stored
            in the columns of VL, in the same order as their eigenvalues.
            If the j-th eigenvalue is real, then u(j) = VL(:,j).
            If the j-th and (j+1)-st eigenvalues form a complex conjugate
            pair, then
               u(j) = VL(:,j) + i*VL(:,j+1)
            and
              u(j+1) = VL(:,j) - i*VL(:,j+1).

            Each eigenvector is scaled so that its largest component has
            abs(real part) + abs(imag. part) = 1, except for eigenvectors
            corresponding to an eigenvalue with alpha = beta = 0, which
            are set to zero.
            Not referenced if JOBVL = 'N'.

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

    VR      (output) DOUBLE PRECISION array, dimension (LDVR,N)
            If JOBVR = 'V', the right eigenvectors x(j) are stored
            in the columns of VR, in the same order as their eigenvalues.
            If the j-th eigenvalue is real, then x(j) = VR(:,j).
            If the j-th and (j+1)-st eigenvalues form a complex conjugate
            pair, then
              x(j) = VR(:,j) + i*VR(:,j+1)
            and
              x(j+1) = VR(:,j) - i*VR(:,j+1).

            Each eigenvector is scaled so that its largest component has
            abs(real part) + abs(imag. part) = 1, except for eigenvalues
            corresponding to an eigenvalue with alpha = beta = 0, which
            are set to zero.
            Not referenced if JOBVR = 'N'.

    LDVR    (input) INTEGER
            The leading dimension of the matrix VR. LDVR >= 1, and
            if JOBVR = 'V', LDVR >= 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(1,8*N).
            For good performance, LWORK must generally be larger.
            To compute the optimal value of LWORK, call ILAENV to get
            blocksizes (for DGEQRF, DORMQR, and DORGQR.)  Then compute:
            NB  -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
            The optimal LWORK is:
                2*N + MAX( 6*N, N*(NB+1) ).

            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 failed.  No eigenvectors have been
                  calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
                  should be correct for j=INFO+1,...,N.
            > N:  errors that usually indicate LAPACK problems:
                  =N+1: error return from DGGBAL
                  =N+2: error return from DGEQRF
                  =N+3: error return from DORMQR
                  =N+4: error return from DORGQR
                  =N+5: error return from DGGHRD
                  =N+6: error return from DHGEQZ (other than failed
                                                  iteration)
                  =N+7: error return from DTGEVC
                  =N+8: error return from DGGBAK (computing VL)
                  =N+9: error return from DGGBAK (computing VR)
                  =N+10: error return from DLASCL (various calls)

    Further Details
    ===============

    Balancing
    ---------

    This driver calls DGGBAL to both permute and scale rows and columns
    of A and B.  The permutations PL and PR are chosen so that PL*A*PR
    and PL*B*R will be upper triangular except for the diagonal blocks
    A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
    possible.  The diagonal scaling matrices DL and DR are chosen so
    that the pair  DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
    one (except for the elements that start out zero.)

    After the eigenvalues and eigenvectors of the balanced matrices
    have been computed, DGGBAK transforms the eigenvectors back to what
    they would have been (in perfect arithmetic) if they had not been
    balanced.

    Contents of A and B on Exit
    -------- -- - --- - -- ----

    If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
    both), then on exit the arrays A and B will contain the real Schur
    form[*] of the "balanced" versions of A and B.  If no eigenvectors
    are computed, then only the diagonal blocks will be correct.

    [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
        by Golub & van Loan, pub. by Johns Hopkins U. Press.

(alphaReal, alphaImag, beta, info) = dgegv(A, B)

A	Type: Real[:,size(A, 1)]
B	Type: Real[size(A, 1),size(A, 1)]

alphaReal	Type: Real[size(A, 1)] Description: Real part of alpha (eigenvalue=(alphaReal+i*alphaImag)/beta)
alphaImag	Type: Real[size(A, 1)] Description: Imaginary part of alpha
beta	Type: Real[size(A, 1)] Description: Denominator of eigenvalue
info	Type: Integer