One of the most important themes of this book is the implementation of radial basis function (interpolation) methods. Therefore, after four chapters on the theory of radial basis functions which we have investigated so far, we now turn to some more practical aspects. Concretely, in this chapter, we will focus on the numerical solution of the interpolation problems we considered here, i.e. the computation of the interpolation coefficients. In practice, interpolation methods such as radial basis functions are often required for approximations with very large numbers of data sites ξ, and this is where the numerical solution of the resulting linear systems becomes nontrivial in the face of rounding and other errors. Moreover, storage can also become a significant problem if |Ξ| is very large, even with the most modern workstations which often have gigabytes of main memory.

Several researchers have reported that the method provides high quality solutions to the scattered data interpolation problem. The adoption of the method in wider applications, e.g. in engineering and finance, where the number of data points is large, was hindered by the high computational cost, however, that is associated with the numerical solution of the interpolation equations and the evaluation of the resulting approximant.