KAUST DepartmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Numerical Mathematics Group
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AbstractWe 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.
CitationLajos Lóczi and David I. Ketcheson (2014). Rational functions with maximal radius of absolute monotonicity. LMS Journal of Computation and Mathematics, 17, pp 159-205. doi:10.1112/S1461157013000326.
SponsorsThis publication is based on work supported by Award No. FIC/2010/05 – 2000000231, made by King Abdullah University of Science and Technology (KAUST).
PublisherOxford University Press (OUP)