Fast evaluation of radial basis functions: methods for two-dimensional polyharmonic splines

Received on 23 February 1995. Revised on 7 May 1996. This paper concerns the fast evaluation of radial basis functions.It describes the mathematics of a methos for splines of theform where p is a low-degree polynomial. Such functions are veryuseful for the interpolation of scattered data, but can be computationallyexpensive to use when N is large. The method described is ageneralization of the fast multipole method of Greengard andRokhlin for the potential case (m=0), and reduces the incrementalcost of a single extra evaluation from O(N) operations to O(1)operations. The paper develops the required series expansionsand uniqueness results. It pays particular attention to therate of convergence of the series approximations involved, obtainingimproved estimates which explain why numerical experiments revealfaster convergence than predicted by previous work for the potential(m=0) and thin-plate spline (m=1) cases.     

A Comparison Between Chebyshev and Ultraspherical Expansions

This paper examines the norms of the projections from C[–1,1] to P_M[–1, 1] which are obtained by truncating Chebyshevand ultraspherical expansions after n terms. The norms are evaluatednumerically for n = 1, 2, 3 , ... , 10 and –0.1 5 wherethe ultraspherical weight function is (1 – x²)^x–.In particular the data show that the norm of the Chebyshev projectionis not the smallest in this range for 1 n 10.     

Envelope Solutions for Implicit Ordinary Differential Equations

In recent papers the numerical solution of implicit ordinarydifferential equations of the form f(x, y(x), y'(x))=0 has beendiscussed. In this paper we address the problem of computingnumerically the so-called envelope solutions to these equations.In particular we suggest a numerical method for the solutionof this problem-one which is in spirit a predictor-correctormethod. We discuss the numerical difficulties encountered andgive some numerical examples.