Rational functions with maximal radius of absolute monotonicity

TL;DR

This paper investigates the maximum radius of absolute monotonicity for rational functions used in implicit Runge-Kutta methods, providing exact algebraic results and disproving previous conjectures.

Contribution

It constructs a rational function with p=2 and R>2s, disproving a prior conjecture, and determines optimal radii for various families of functions using algebraic methods.

Findings

Constructed a rational function with p=2 and R>2s.

Proved optimal radii for certain function families.

Validated the optimality of specific Runge-Kutta methods.

Abstract

We study the radius of absolute monotonicity R of rational functions with numerator and denominator of degree s that approximate the exponential function to order p. Such functions arise in the application of implicit s-stage, order p Runge-Kutta methods for initial value problems and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with p=2 and R>2s, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with 2 or 3 parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge-Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.