ZGEEVX(3)      LAPACK routine of NEC Numeric Library Collection      ZGEEVX(3)



NAME
       ZGEEVX

SYNOPSIS
       SUBROUTINE ZGEEVX (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL,
           VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK,
           RWORK, INFO)



PURPOSE
            ZGEEVX computes for an N-by-N complex 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)
                     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, ie. 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)
                     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     (input)
                     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     (input)
                     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         (input)
                     N is INTEGER
                     The order of the matrix A. N >= 0.

           A         (input/output)
                     A is COMPLEX*16 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 Schur form of the balanced
                     version of the matrix A.

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

           W         (output)
                     W is COMPLEX*16 array, dimension (N)
                     W contains the computed eigenvalues.

           VL        (output)
                     VL is COMPLEX*16 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.
                     u(j) = VL(:,j), the j-th column of VL.

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

           VR        (output)
                     VR is COMPLEX*16 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.
                     v(j) = VR(:,j), the j-th column of VR.

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

           ILO       (output)
                     ILO is INTEGER

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

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

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     LWORK is INTEGER
                     The dimension of the array WORK.  If SENSE = 'N' or 'E',
                     LWORK >= max(1,2*N), and if SENSE = 'V' or 'B',
                     LWORK >= N*N+2*N.
                     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.

           RWORK     (output)
                     RWORK is DOUBLE PRECISION array, dimension (2*N)

           INFO      (output)
                     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 W
                           contain eigenvalues which have converged.



LAPACK routine                  31 October 2017                      ZGEEVX(3)