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Lapack documentation
    Purpose
    =======

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

    Optionally also, it 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).


    Arguments
    =========

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

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

    SENSE   (input) 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       (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 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 real Schur
            form of the "balanced" versions of the input A and B.

    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 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 real Schur
            form of the "balanced" versions of the input A and B.

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

    ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
    ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
    BETA    (output) DOUBLE PRECISION array, dimension (N)
            On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
            be the generalized eigenvalues.  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) negative.

            Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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      (output) DOUBLE PRECISION array, dimension (LDVL,N)
            If JOBVL = 'V', the left eigenvectors u(j) are stored one
            after another in the columns of VL, in the same order as
            their eigenvalues. If the j-th eigenvalue is real, then
            u(j) = VL(:,j), the j-th column of VL. If the j-th and
            (j+1)-th 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 will be scaled so the largest component have
            abs(real part) + abs(imag. part) = 1.
            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 v(j) are stored one
            after another in the columns of VR, in the same order as
            their eigenvalues. If the j-th eigenvalue is real, then
            v(j) = VR(:,j), the j-th column of VR. If the j-th and
            (j+1)-th eigenvalues form a complex conjugate pair, then
            v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
            Each eigenvector will be scaled so the largest component have
            abs(real part) + abs(imag. part) = 1.
            Not referenced if JOBVR = 'N'.

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

    ILO     (output) INTEGER
    IHI     (output) 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  (output) DOUBLE PRECISION 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  (output) DOUBLE PRECISION 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   (output) DOUBLE PRECISION
            The one-norm of the balanced matrix A.

    BBNRM   (output) DOUBLE PRECISION
            The one-norm of the balanced matrix B.

    RCONDE  (output) DOUBLE PRECISION array, dimension (N)
            If SENSE = 'E' or 'B', the reciprocal condition numbers of
            the eigenvalues, stored in consecutive elements of the array.
            For a complex conjugate pair of eigenvalues two consecutive
            elements of RCONDE are set to the same value. Thus RCONDE(j),
            RCONDV(j), and the j-th columns of VL and VR all correspond
            to the j-th eigenpair.
            If SENSE = 'N or 'V', RCONDE is not referenced.

    RCONDV  (output) DOUBLE PRECISION array, dimension (N)
            If SENSE = 'V' or 'B', the estimated reciprocal condition
            numbers of the eigenvectors, stored in consecutive elements
            of the array. For a complex eigenvector two consecutive
            elements of RCONDV are set to the same value. 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    (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(1,2*N).
            If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
            LWORK >= max(1,6*N).
            If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
            If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.

            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.

    IWORK   (workspace) INTEGER array, dimension (N+6)
            If SENSE = 'E', IWORK is not referenced.

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

    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:  =N+1: other than QZ iteration failed in DHGEQZ.
                  =N+2: error return from DTGEVC.

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

    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.

(alphaReal, alphaImag, beta, lEigenVectors, rEigenVectors, info) = dggevx(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
lEigenVectors	Type: Real[size(A, 1),size(A, 1)] Description: left eigenvectors of matrix A
rEigenVectors	Type: Real[size(A, 1),size(A, 1)] Description: right eigenvectors of matrix A
info	Type: Integer