Computation of Eigenvalues Associated with Functional Differential Equations

Abstract

This paper describes a method for computing the eigenvalues associated with systems of linear retarded functional differential equations (RFDE’s). The method finds the eigenvalues directly from a certain characteristic equation which is automatically determined from system matrices. The eigenvalues contained in some bounded region around the origin are approximately computed by a combinatorial algorithm suggested earlier by H. Kuhn [15] for approximations of zeros of ordinary polynomials. The eigenvalues of large modulus, which are distributed in some curvilinear strips, are computed from some asymptotic formulas obtained directly from the parameters of the characteristic equation. To verify that all the eigenvalues have been found, we use a highly reliable procedure proposed by Carpentier and Dos Santos, which evaluates the number of zeros of an analytic function in a given region. Numerical results are presented for several examples and compared with those obtained by a method based on finite-dimensional approximations of delay equations.