Wendland Functions - A C++ Code to Compute Them

Abstract

Abstract:In this paper we present a code in C++ to compute Wendland functions for arbitrary smoothness parameters. Wendland functions are compactly supported Radial Basis Functions that are used for interpolation of data or solving Partial Differential Equations with mesh-free collocation. For the computations of Lyapunov functions using Wendland functions their derivatives are also needed so we include this in the code. Wendland functions with a few fixed smoothness parameters are included in some C++ libraries, but for the general case the only code freely available was implemented in MAPLE taking advantage of the computer algebra system. The aim of this contribution is to allow scientists to use Wendland functions in their C++ code without having to implement them themselves. The computed Wendland functions are polynomials and their coefficients are computed and stored in a vector, which allows for efficient computation of their values using the Horner scheme.

Eigenpairs for the Analysis of Complete Lyapunov Functions

“…Hence, we take k ≥ 2 to assure that the Hessian of v is well defined. We use the C++ tool from [16] to generate ψ 0 � ψ and ψ i (r): � (1/r)(d/dr)ψ i−1 (r), i � 1, 2, 3, in optimised form for evaluation.…”