Rational functions with maximal radius of absolute monotonicity

Lajos Loczi, David I. Ketcheson

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)159-205
Number of pages47
JournalLMS Journal of Computation and Mathematics
Volume17
Issue number1
DOIs
StatePublished - May 19 2014

Fingerprint Dive into the research topics of 'Rational functions with maximal radius of absolute monotonicity'. Together they form a unique fingerprint.

Cite this