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__Rational functions with maximal radius of absolute monotonicity__

L. Lóczi and D.I. Ketcheson

**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, and we
prove earlier conjectured optimal radii in some families with 2 or 3 parameters. 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.