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


SYNOPSIS

       SUBROUTINE SGEEVX( 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

           REAL           ABNRM

           INTEGER        IWORK( * )

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


PURPOSE

       SGEEVX  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
       explanation of balancing, see section 4.10.2 of the LAPACK



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) REAL 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) REAL array, dimension (N)
               WI      (output) REAL array, dimension (N) WR  and
               WI  contain  the real and imaginary parts, respec­
               tively, of the computed eigenvalues.  Complex con­
               jugate  pairs  of eigenvalues will appear consecu­
               tively with the  eigenvalue  having  the  positive
               imaginary part first.

       VL      (output) REAL 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) REAL 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) REAL 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
               row and column j, then SCALE(J) = P(J),    for J =
               1,...,ILO-1  = D(J),    for J = ILO,...,IHI = P(J)

               interchanges  are  made  is  N to IHI+1, then 1 to
               ILO-1.

       ABNRM   (output) REAL
               The one-norm of the balanced matrix  (the  maximum
               of  the  sum of absolute values of elements of any
               column).

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

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

       WORK    (workspace/output) REAL 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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