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Lapack documentation
    Purpose
    =======

    DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
    eigenvalues and, optionally, the left and/or right eigenvectors.

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

    The right eigenvector v(j) of A satisfies
                     A * v(j) = lambda(j) * v(j)
    where lambda(j) is its eigenvalue.
    The left eigenvector u(j) of A satisfies
                  u(j)**H * A = lambda(j) * u(j)**H
    where u(j)**H denotes the conjugate transpose of u(j).

    The computed eigenvectors are normalized to have Euclidean norm
    equal to 1 and largest component real.

    Balancing a matrix means permuting the rows and columns to make it
    more nearly upper triangular, and applying a diagonal similarity
    transformation D * A * D**(-1), where D is a diagonal matrix, to
    make its rows and columns closer in norm and the condition numbers
    of its eigenvalues and eigenvectors smaller.  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 4.10.2 of the LAPACK
    Users' Guide.

    Arguments
    =========

    BALANC  (input) CHARACTER*1
            Indicates how the input matrix should be diagonally scaled
            and/or permuted to improve the conditioning of its
            eigenvalues.
            = 'N': Do not diagonally scale or permute;
            = 'P': Perform permutations to make the matrix more nearly
                   upper triangular. Do not diagonally scale;
            = 'S': Diagonally scale the matrix, i.e. replace A by
                   D*A*D**(-1), where D is a diagonal matrix chosen
                   to make the rows and columns of A more equal in
                   norm. Do not permute;
            = 'B': Both diagonally scale and permute A.

            Computed reciprocal condition numbers will be for the matrix
            after balancing and/or permuting. Permuting does not change
            condition numbers (in exact arithmetic), but balancing does.

    JOBVL   (input) CHARACTER*1
            = 'N': left eigenvectors of A are not computed;
            = 'V': left eigenvectors of A are computed.
            If SENSE = 'E' or 'B', JOBVL must = 'V'.

    JOBVR   (input) CHARACTER*1
            = 'N': right eigenvectors of A are not computed;
            = 'V': right eigenvectors of A are computed.
            If SENSE = 'E' or 'B', JOBVR must = 'V'.

    SENSE   (input) CHARACTER*1
            Determines which reciprocal condition numbers are computed.
            = 'N': None are computed;
            = 'E': Computed for eigenvalues only;
            = 'V': Computed for right eigenvectors only;
            = 'B': Computed for eigenvalues and right eigenvectors.

            If SENSE = 'E' or 'B', both left and right eigenvectors
            must also be computed (JOBVL = 'V' and JOBVR = 'V').

    N       (input) INTEGER
            The order of the matrix A. N >= 0.

    A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
            On entry, the N-by-N matrix A.
            On exit, A has been overwritten.  If JOBVL = 'V' or
            JOBVR = 'V', A contains the real Schur form of the balanced
            version of the input matrix A.

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

    WR      (output) DOUBLE PRECISION array, dimension (N)
    WI      (output) DOUBLE PRECISION array, dimension (N)
            WR and WI contain the real and imaginary parts,
            respectively, of the computed eigenvalues.  Complex
            conjugate pairs of eigenvalues will appear consecutively
            with the eigenvalue having the positive imaginary part
            first.

    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 JOBVL = 'N', VL is not referenced.
            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)-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).

    LDVL    (input) INTEGER
            The leading dimension of the array VL.  LDVL >= 1; 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 JOBVR = 'N', VR is not referenced.
            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)-st 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).

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

    ILO     (output) INTEGER
    IHI     (output) INTEGER
            ILO and IHI are integer values determined when A was
            balanced.  The balanced A(i,j) = 0 if I > J and
            J = 1,...,ILO-1 or I = IHI+1,...,N.

    SCALE   (output) DOUBLE PRECISION array, dimension (N)
            Details of the permutations and scaling factors applied
            when balancing A.  If P(j) is the index of the row and column
            interchanged with row and column j, and D(j) is the scaling
            factor applied to row and column j, then
            SCALE(J) = P(J),    for J = 1,...,ILO-1
                     = D(J),    for J = ILO,...,IHI
                     = P(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 (the maximum
            of the sum of absolute values of elements of any column).

    RCONDE  (output) DOUBLE PRECISION array, dimension (N)
            RCONDE(j) is the reciprocal condition number of the j-th
            eigenvalue.

    RCONDV  (output) DOUBLE PRECISION array, dimension (N)
            RCONDV(j) is the reciprocal condition number of the j-th
            right eigenvector.

    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.   If SENSE = 'N' or 'E',
            LWORK >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V',
            LWORK >= 3*N.  If SENSE = 'V' or 'B', LWORK >= N*(N+6).
            For good performance, LWORK must generally be larger.

            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 (2*N-2)
            If SENSE = 'N' or 'E', not referenced.

    INFO    (output) INTEGER
            = 0:  successful exit
            < 0:  if INFO = -i, the i-th argument had an illegal value.
            > 0:  if INFO = i, the QR algorithm failed to compute all the
                  eigenvalues, and no eigenvectors or condition numbers
                  have been computed; elements 1:ILO-1 and i+1:N of WR
                  and WI contain eigenvalues which have converged.