Stability radius of polynomials occurring in the numerical solution of initial value problems (Q922644)

Let \(T_{mp}\) be the set of all polynomials \(\phi\) (x) with real coefficients and degree \(\leq m\) for which \(\phi (x)=\exp (x)+O(x^{p+1})\) (for \(x\to 0)\) holds. \(\phi\) occurs in many numerical methods for initial value problems of ordinary differential equations and for initial-boundary value problems of partial differential equations. Motivated by stability requirements, denote by r(\(\phi\)) the radius of the largest disk \(D(\rho)=\{z\in C:| z+\rho | \leq \rho \}\) contained in the stability region \(S(\rho)=\{z\in C:| \phi (z)| \leq 1\}\). Then some conclusions are given which deal with \(r_{mp}=\sup \{r(\phi):\phi \in T_{mp}\}\).