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NAME

        ZGEEVX  -  compute  for  an  N-by-N  complex nonsymmetric matrix A, the
        eigenvalues and, optionally, the left and/or right eigenvectors
 

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 )
 
            CHARACTER      BALANC, JOBVL, JOBVR, SENSE
 
            INTEGER        IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
 
            DOUBLE         PRECISION ABNRM
 
            DOUBLE         PRECISION RCONDE( * ), RCONDV(  *  ),  RWORK(  *  ),
                           SCALE( * )
 
            COMPLEX*16     A(  LDA,  * ), VL( LDVL, * ), VR( LDVR, * ), W( * ),
                           WORK( * )
 

PURPOSE

        ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the eigen‐
        values and, optionally, the left and/or right eigenvectors.  Optionally
        also, it computes a balancing transformation to improve the  condition‐
        ing  of  the eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM),
        reciprocal condition numbers for the eigenvalues (RCONDE), and recipro‐
        cal 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 transforma‐
        tion 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, 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) 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) COMPLEX*16 array, dimension (LDA,N)
                On  entry,  the N-by-N matrix A.  On exit, A has been overwrit‐
                ten.  If JOBVL = ’V’ or JOBVR = ’V’, A contains the Schur  form
                of the balanced version of the matrix A.
 
        LDA     (input) INTEGER
                The leading dimension of the array A.  LDA >= max(1,N).
 
        W       (output) COMPLEX*16 array, dimension (N)
                W contains the computed eigenvalues.
 
        VL      (output) 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 eigen‐
                values.  If JOBVL = ’N’, VL is not referenced.  u(j) = VL(:,j),
                the j-th column of VL.
 
        LDVL    (input) INTEGER
                The leading dimension of the array VL.  LDVL >= 1; if  JOBVL  =
                ’V’, LDVL >= N.
 
        VR      (output) 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) INTEGER
                The leading dimension of the array VR.  LDVR >= 1; if  JOBVR  =
                ’V’, LDVR >= N.
 
                ILO,IHI  (output) INTEGER ILO and IHI are integer values deter‐
                mined 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 eigen‐
                value.
 
        RCONDV  (output) DOUBLE PRECISION array, dimension (N)
                RCONDV(j)  is the reciprocal condition number of the j-th right
                eigenvector.
 
        WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
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  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   (workspace) DOUBLE PRECISION array, dimension (2*N)
 
        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 W contain eigenvalues
                which have converged.
 

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