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NAME

       dgeevx.f -

SYNOPSIS

   Functions/Subroutines
       subroutine dgeevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR,
           ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO)
            DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
           for GE matrices

Function/Subroutine Documentation

   subroutine dgeevx (characterBALANC, characterJOBVL, characterJOBVR, characterSENSE, integerN,
       double precision, dimension( lda, * )A, integerLDA, double precision, dimension( * )WR,
       double precision, dimension( * )WI, double precision, dimension( ldvl, * )VL, integerLDVL,
       double precision, dimension( ldvr, * )VR, integerLDVR, integerILO, integerIHI, double
       precision, dimension( * )SCALE, double precisionABNRM, double precision, dimension( *
       )RCONDE, double precision, dimension( * )RCONDV, double precision, dimension( * )WORK,
       integerLWORK, integer, dimension( * )IWORK, integerINFO)
        DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices

       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.

       Parameters:
           BALANC

                     BALANC is 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

                     JOBVL is 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

                     JOBVR is 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

                     SENSE is 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

                     N is INTEGER
                     The order of the matrix A. N >= 0.

           A

                     A is 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

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

           WR

                     WR is DOUBLE PRECISION array, dimension (N)

           WI

                     WI is 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

                     VL is 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

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

           VR

                     VR is 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

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

           ILO

                     ILO is INTEGER

           IHI

                     IHI is 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

                     SCALE is 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

                     ABNRM is DOUBLE PRECISION
                     The one-norm of the balanced matrix (the maximum
                     of the sum of absolute values of elements of any column).

           RCONDE

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

           RCONDV

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

           WORK

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

           LWORK

                     LWORK is 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

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

           INFO

                     INFO is 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.

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

       Definition at line 302 of file dgeevx.f.

Author

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