Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems

Summary

This paper deals with polynomial approximations ø(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument, denoted by −R(ø), at which ø is absolutely monotonic. For given integersp≧1,m≧1 we determine the maximum ofR(ø) when ø varies over the class of all polynomials of a degree ≦m with$\phi \left( x \right) = \exp \left( x \right) + \mathcal{O}\left( {x^{p + 1} } \right)\left( {for x \to 0} \right)$ (forx→0).