Fast Evaluation of Radial Basis Functions: Methods for Generalized Multiquadrics in Rn

A generalised multiquadric radial basis function is a function of the form s(x) = ∑N i=1 diφ(|x − ti|), where φ(r) = ( r2 + τ2 )k/2 , x ∈ Rn, and k ∈ Z is odd. The direct evaluation of an N centre generalised multiquadric radial basis function at m points requires O(mN) flops, which is prohibitive when m and N are large. Similar considerations apparently rule out fitting an interpolating N centre generalised multiquadric to N data points by either direct or iterative solution of the associated system of linear equations in realistic problems. In this paper we will develop far field expansions, recurrence relations for efficient formation of the expansions, error estimates, and translation formulas, for generalised multiquadric radial basis functions in n-variables. These pieces are combined in a hierarchical fast evaluator requiring only O((m+N) logN | log |n+1) flops for evaluation of an N centre generalised multiquadric at m points. This flop count is significantly less than that of the direct method. Moreover, used to compute matrix-vector products, the fast evaluator provides a basis for fast iterative fitting strategies.