Abstract
This paper concerns the fast evaluation of radial basis functions. It describes the mathematics of hierarchical and fast multipole methods for fast evaluation of splines of the form N S (X)=p(X) +Σ d j |X - X j | ( 2V-1 , X ∈R 3 , j=1 where v is a positive integer and p is a low-degree polynomial. Splines s of this form are polyharmonic splines in Ρ 3 and have been found to be very useful for providing solutions to scattered data interpolation problems in Ρ 3 . As it is now well known, hierarchical methods reduce the incremental cost of a single extra evaluation from O(N) to O (log N) operations and reduce the cost of a matrix-vector product (evaluation of s at all the centres) from O(N2) to O(N log N) operations. We give appropriate far- and near-field expansions, together with error estimates, uniqueness theorems and translation formulae. A hierarchical code based on these formulae is detailed and some numerical results are given.
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