DAMTP 2006/NA03 Fast evaluation of polyharmonic splines in three dimensions R.K. Beatson 1 , M.J.D. Powell and A.M. Tan 1 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 s(x)= p(x)+ N  j =1 d j |x - x j | 2ν -1 , x ∈R 3 , where ν is a positive integer, and p is a low degree polynomial. Splines s of this form are polyharmonic splines in R 3 and have been found to be very useful for providing solutions to scattered data interpolation problems in R 3 . As 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(N 2 ) 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. Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, England. April, 2006 1 Department of Mathematics and Statistics, University of Canterbury, Private Bag 4800, Christchurch 8020, New Zealand.