Fast linear barycentric rational interpolation for singular functions via scaled transformations

Abstract

In this paper, applied strictly monotonic increasing scaled maps, a kind of wellconditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as x α for α ∈ (0, 1) and log(x). It just takes O(N ) flops and can achieve fast convergence rates with the choice the scaled parameter, where N is the maximum degree of the denominator and numerator. The construction of the rational interpolant couples rational polynomials in the barycentric form of second kind with the transformed Jacobi-Gauss-Lobatto points. Numerical experiments are considered which illustrate the accuracy and efficiency of the algorithms. The convergence of the rational interpolation is also considered.