Meshless weighting coefficients for arbitrary nodes: The efficient computation to machine precision using hyper-dual numbers

A computationally efficient algorithm to calculate the weighting coefficients required to evaluate derivatives for arbitrary multi-dimensional distributions of points is presented. The iterative algorithm guarantees IEEE 754 64-bit precision (at least 15 significant decimal digits) for the weighting coefficients. Convergence acceleration is achieved through the use of a Taylor series of up to third order, and hyper-dual numbers to obtain the derivatives required for the Taylor series. The method is applied as part of a finite point solution for three test examples, a Poisson equation, creeping flow around a cylinder, and heat conduction in a triangular annulus. The open source FORTRAN-90 implementation has been optimised for random distributions of points in 1 to 3 dimensions.