Numerical Analysis of equivariant evolution equations
10-007 Wolf-Jürgen Beyn.
An integral method for solving nonlinear eigenvalue problems
We propose a numerical method for
computing all eigenvalues (and the corresponding eigenvectors) of a
nonlinear holomorphic eigenvalue problem that lie within a given
contour in the complex plane. The method uses complex integrals of the
resolvent operator, applied to at least k column vectors, where k is
the number of eigenvalues inside the contour. The theorem of Keldysh
is employed to show that the original nonlinear eigenvalue problem
reduces to a linear eigenvalue problem of dimension k. No initial
approximations of eigenvalues and eigenvectors are needed. The method
is particularly suitable for moderately large eigenvalue problems
where k is much smaller than the matrix dimension. We also give an
extension of the method to the case where k is larger than the matrix
dimension. The quadrature errors caused by the trapezoid sum are
discussed for the case of analytic closed contours. Using well known
techniques it is shown that the error decays exponentially with an
exponent given by the product of the number of quadrature points and
the minimal distance of the eigenvalues to the contour.
