This thesis studies two aspects of polynomial interpolation theory. The first part sets forth explicit formulas for the coefficients of polynomial interpolants to implicit functions. A formula for the higher-order derivatives of implicit functions appears as a limiting case of these formulas. The second part delves into certain configurations of points in space — generalized principal lattices — that are well suited as interpolation points. Applying the theory of real algebraic curves then allows the construction of many new examples of such configurations.
List of papers.
Paper I / Chapter 2
Georg Muntingh and Michael Floater:
Divided Differences of Univariate Implicit Functions
Math. Comp. 80 (2011), 2185-2195 First published in Mathematics of Computation in volume 80 (2011), published by the American Mathematical Society
Mathematics of Computation
Paper II / Chapter 3
Divided Differences of Multivariate Implicit Functions
BIT Numerical Mathematics (2012) Published Online 29 February 2012. The original publication is available at www.springerlink.com