TY - JOUR

T1 - Rational functions with maximal radius of absolute monotonicity

AU - Loczi, Lajos

AU - Ketcheson, David I.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): FIC/2010/05 – 2000000231
Acknowledgements: This publication is based on work supported by Award No. FIC/2010/05 – 2000000231, made by King Abdullah University of Science and Technology (KAUST).

PY - 2014/5/19

Y1 - 2014/5/19

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/551012

UR - http://www.journals.cambridge.org/abstract_S1461157013000326

UR - http://www.scopus.com/inward/record.url?scp=84924649979&partnerID=8YFLogxK

U2 - 10.1112/S1461157013000326

DO - 10.1112/S1461157013000326

M3 - Article

VL - 17

SP - 159

EP - 205

JO - LMS Journal of Computation and Mathematics

JF - LMS Journal of Computation and Mathematics

SN - 1461-1570

IS - 1

ER -