Extended barycentric rational schemes for functions of singularities

• Published in: Calcolo Vol. 59; no. 4
• Main Authors: Kong, Desong, Xiang, Shuhuang, Li, Li, Li, Ran
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.11.2022; Springer Nature B.V
• Subjects: Chebyshev approximation; Interpolation; Mathematics; Mathematics and Statistics; Nodes; Numerical Analysis; Polynomials; Singularity (mathematics); Theory of Computation; Scaled transformations; 65R20; Algebraic singularity; 34A12; 41A20; Highly oscillatory integral; 45D05; Levin method; Barycentric rational interpolation
• ISSN: 0008-0624; 1126-5434
• DOI: 10.1007/s10092-022-00480-7

In this paper, we propose two kinds of extended barycentric rational schemes via (non-conformally) scaled transformations for approximating functions of singularities, which are bulit upon applying the barycentric interpolation formula of the second kind at two kinds of mapped nodes: (i) equispaced nodes and (ii) (shifted) Chebyshev nodes. While the weights in interpolation formula are selected inspired by the works of Berrut, Floater and Hormann. Ample numerical tests show that the extended barycentric rational schemes are efficient and can achieve higher convergence rates as the scaled parameter and the degree of the local approximation polynomial increase. Moreover, from the barycentric formula, it is easy to derive the difference matrices at these mapped nodes, which leads to an accurate Levin method for dealing with highly oscillatory integrals with the integrands of algebraic singularities. Numerical experiments are carried out to illustrate the effectiveness and accuracy of the proposed schemes.