sgeev(3F)

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

SYNOPSIS
     SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR,
                       WORK, LWORK, INFO )

         CHARACTER     JOBVL, JOBVR

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

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

PURPOSE
     SGEEV computes for an N-by-N real nonsymmetric matrix A, the eigenvalues
     and, optionally, the left and/or right eigenvectors.

     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.


ARGUMENTS
     JOBVL   (input) CHARACTER*1
             = 'N': left eigenvectors of A are not computed;
             = 'V': left eigenvectors of A are computed.

     JOBVR   (input) CHARACTER*1
             = 'N': right eigenvectors of A are not computed;
             = 'V': right eigenvectors of A are computed.

     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.

     LDA     (input) INTEGER
             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, respectively, of the computed
             eigenvalues.  Complex conjugate pairs of eigenvalues appear
             consecutively 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; if JOBVR =
             'V', LDVR >= N.

     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.  LWORK >= max(1,3*N), and if
             JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  For good performance,
             LWORK must generally be larger.

     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 have been computed; elements
             i+1:N of WR and WI contain eigenvalues which have converged.