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dgeevx examples
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Man Page Details
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NAME
       DGEEVX - 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 eigenval-
       ues 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 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, 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.  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) DOUBLE PRECISION 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 real Schur
               form of the balanced version of the input matrix A.

       LDA     (input) INTEGER
               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 eigen-
               values.  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 conju-
               gate 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     (output) INTEGER
               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 inter-
               changed 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)   DOUBLE   PRECISION   array,   dimension
       (MAX(1,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 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  WR and WI contain
               eigenvalues which have converged.

