On Roots and Error Constants of Optimal Stability Polynomials

Optimal stability polynomials are polynomials whose stability region is as large as possible in a certain region, here the real-negative axis. We are interested in such polynomials wich in addition, obey a certain order condition. An important application of these polynomials is the construction of stabilized explicit Runge-Kutta methods (see 6] for a survey article and 4]). In this paper we will give some properties of the roots of these polynomials, and prove that their error constant is always positive. Furthermore, for a given order, the error constant decreases as the degree increases. AMS subject classiication: 65L20. 1 Introduction. Let R p s (z) be the optimal stability polynomial on the real axis of order p 1 and degree s, i.e.