Radial basis function approximation methods with extended precision floating point arithmetic

Radial basis function (RBF) methods that employ infinitely differentiable basis functions featuring a shape parameter are theoretically spectrally accurate methods for scattered data interpolation and for solving partial differential equations. It is also theoretically known that RBF methods are most accurate when the linear systems associated with the methods are extremely ill-conditioned. This often prevents the RBF methods from realizing spectral accuracy in applications. In this work we examine how extended precision floating point arithmetic can be used to improve the accuracy of RBF methods in an efficient manner. RBF methods using extended precision are compared to algorithms that evaluate RBF methods by bypassing the solution of the ill-conditioned linear systems.