Thrice Differentiable Affine Conic Spline Interpolation.

Abstract

Abstract : SWe present interpolating functions which have three orders of differentiability at each (convex) data point. These functions are defined as piecewise conics and are therefore guaranteed to be convex in the case of (strict) convex data. The modifier 'affine' refers to the fact that we make no use of Euclidean distance or angle in the discussion. We also include a discussion of numerical differentiation using conics. The nodal derivatives for the conic splines satisfy a locally quadrivariate quadratic System solved by Newton iteration-each iteration involving the solution of a pentadiagonal linear system. Initial values for Newton iteration are obtained by the aforementioned conic numerical differentiation. A discussion of numerical quadrature based on conic splines is also included, as well as a discussion of what we refer to as 'sketched' interpolation, which makes use of the mathematical machinery behind conic differentiation and local C conic spline. Sketched interpolation is more generally applicable than global C conic splines are, as well as being computationally simpler, more flexible, and smoother in a local pointwise sense This apparent increase of Smoothness beyond C is obtained through a process of re-sketching during the construction of the interpolant. Sketched interpolants reproduce conics with or without re-sketching. This is to say that if the discrete data comes from a conic. all the points of the sketched interpolant will lie on that conic.