Computational aspects of data fitting with a new multivariate spline

Abstract

The multivariate splines considered are piecewise polynomials of total degree s, with continuous derivatives of order s-1. The piecewise domains consist of the polyhedra obtained by partitioning En with any k hyperplanes. For nondegenerate partitions, the splines have data fitting power which is greater than a single polynomial and less than the standard tensor product splines. An especially simple canonical form represents these splines. This representation, although numerically ill-conditioned, can be effectively used with standard software on problems with 1≤s≤3, 1≤n≤3, 1≤k≤8. Fixed partition problems can be solved with IBM's Scientific Subroutine Package programs for min-max or least squares fitting. Variable partitions can be handled with Marquardt's method, modified to avoid redundant placement of partitions.