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


SYNOPSIS

       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 )

           CHARACTER      BALANC, JOBVL, JOBVR, SENSE

           INTEGER        IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK,
                          N

           DOUBLE         PRECISION ABNRM

           INTEGER        IWORK( * )

           DOUBLE         PRECISION  A(  LDA,  *  ), RCONDE( * ),
                          RCONDV( * ), SCALE( * ), VL( LDVL, * ),
                          VR(  LDVR, * ), WI( * ), WORK( * ), WR(
                          * )


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 eigen­
       values 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 diag­
       onal 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

       Users' Guide.


ARGUMENTS

       BALANC  (input) CHARACTER*1
               Indicates  how  the  input matrix should be diago­
               nally scaled and/or permuted to improve the condi­
               tioning  of its eigenvalues.  = 'N': Do not diago­
               nally 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. Per­
               muting 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  eigen­
               vectors.

               If  SENSE = 'E' or 'B', both left and right eigen­
               vectors 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.

               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 posi­
               tive 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,IHI  (output)  INTEGER ILO and IHI are integer
               values determined when A was balanced.   The  bal­
               anced 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  col­
               umn  j,  and D(j) is the scaling factor applied to

               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 (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 ille­
               gal value.
               > 0:  if INFO = i, the QR algorithm failed to com­
               pute  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.


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