Resolving some conjectures about the radius of absolute monotonicity of rational functions

Témavezető:	Lóczi Lajos
	ELTE Informatikai Kar
email:	LLoczi@inf.elte.hu

Projekt leírás

The radius of absolute monotonicity of polynomials and rational functions plays an important role in the analysis of positivity, monotonicity, and contractivity properties of numerical methods for ordinary differential equations (ODEs). In particular, it governs the largest allowable time-step size for discretizations preserving these qualitative properties. It is therefore natural to consider the problem of finding a function that achieves the maximal radius of absolute monotonicity within a given function class. In [1], we studied the absolute monotonicity of rational functions that correspond to the stability functions of certain implicit or singly diagonally implicit Runge–Kutta methods. We disproved a 28-year-old open conjecture for a certain class of rational functions, and we also determined the exact optimal values of these radii in many other classes of rational functions by using Wolfram Mathematica. We formulated many new conjectures. To answer these questions, one needs to solve highly non-linear optimization problems with a large number of parameters. These theoretical results would guide us in the construction of optimal Runge–Kutta methods, corresponding to more efficient numerical integration schemes for ODEs in terms of the discretization step size. In these studies, we may also explore utilizing the Hungarian supercomputer Komondor located in Debrecen.

Előfeltételek

Familiarity with, or willingness to explore, symbolic computing, e.g., Wolfram Mathematica.

Hivatkozások

[1] Rational functions with maximal radius of absolute monotonicity, LMS J. Comp. Math., https://doi.org/10.1112/S1461157013000326