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


SYNOPSIS

       SUBROUTINE CGEEVX( 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

           REAL           ABNRM

           REAL           RCONDE(  *  ), RCONDV( * ), RWORK( * ),
                          SCALE( * )

           COMPLEX        A( LDA, * ), VL( LDVL, * ), VR( LDVR, *
                          ), W( * ), WORK( * )


PURPOSE

       CGEEVX  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  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, 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.  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) COMPLEX 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.

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

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

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

       VR      (output) COMPLEX 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  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)
               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) 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(j) is the reciprocal  condition  number  of
               the j-th right eigenvector.

       WORK    (workspace/output) COMPLEX 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 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) REAL array, dimension (2*N)

       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 W contain  eigenvalues  which
               have converged.


Man(1) output converted with man2html