one-step numerical method for solving Ordinary Differential Equations (ODE). When this method is applied to 

an initial value problem, it requires the computation of the dynamical  coefficients, spectral coeffients, 

eigenvalues  l_N,k , and residues R_N,k. The computation of the dynamical coefficients and especially spectral 

coefficients for an arbitrary ODE system is rather difficult, which is explained in [15]. In [16] and [1] some 

techniques are developed to calculate these coefficients for the systems with polynomial right hand side 

functions.In this study, the order of convergence of Evolution Operator Method is obtained. Also the stability 

characteristics of the method are given for some special values of approximation order.A recursive formulatioin 

of the dynamical and spectral coefficients for ODE systems whose right hand-side functions are not only 

polynomials but in the form of products of polynomials and exponential functions is developed. A new algorithm 

for the calculation of  l_N,k and R_N,k is presented. This new algorithm is compared with the previously published 

algorithms. The numerical applications of the algorithms presented here and previously published ones are given 

for the calculation of the dynamical coefficients, spectral coefficients, l_N,k  and  R_N,k. For testing the algorithms 

mentioned above, the Lotka-Volterra system is used. 

Key words: Evolution Operator Method, Numerical methods for ODE systems, Order of convergence, 

Stability.