2025-07-25 –, Main Room 5
Approximation of functions is an enabling technology for scientific computing. The Julia ecosystem has excellent options for polynomial-based approximation methods. New algorithms for approximation by ratios of polynomials have sparked increasing interest in computational rational approximation. The RationalFunctionApproximation.jl package supplies the fastest known versions of these methods for approximation of functions over an interval or any connected domain in the complex plane.
Approximation of functions is an enabling technology for scientific computing. The Julia ecosystem has excellent options for approximation in packages such as Interpolations for piecewise polynomials and ApproxFun for global polynomials.
Approximation by rational functions is perhaps the most fruitful known nonlinear approximation method, with theoretical and constructive interest stretching back centuries. Unlike polynomial approximation, rational approximation is not constrained in principle by an accuracy–stability tradeoff. It is also well established that the poles of rational interpolants can give root-exponential convergence even to functions with branch cuts.
Recent developments of the AAA and greedy Thiele algorithms have sparked renewed interest in computational rational approximation. The RationalFunctionApproximation package supplies the fastest known versions of these methods and the only arbitrary-precision implementations. Combined with the ComplexRegions package, we can produce compact and accurate representations of a huge variety of functions over intervals or other domains in the complex plane.
Toby Driscoll is a Unidel Chaired Professor of Mathematical Sciences at the University of Delaware. He is author of numerous publications in computational and applied mathematics, mathematical software packages in MATLAB and Julia, and five books, including the online textbook Fundamentals of Numerical Computation at fncbook.com.