Mladen Rogina

Sharp error bounds for interpolating splines in tension

Abstract Sharp error bounds for interpolating splines in tension with variable tension parameters are considered. Error bounds given are the best possible ones in the limit case of the tension parameter p, that is for linear (p → ∞) and cubic (p → 0) spline. Error bounds sharper than those previously published are also developed for derivatives.

A collocation method for singularly perturbed two-point boundary value problems with splines in tension

An error bound for the collocation method by spline in tension is developed for a nonlinear boundary value problemay″+by′+cy=f(·,y),y(0)=y0,y(1)=y1. Sharp error bounds for the interpolating splines in tension are used in conjunction with recently obtained formulae for B-splines, to develop an error bound depending on the tension parameters and net spacing. For singularly perturbed boundary value problems (|a|=ε≪1), the representation of the error motivates a choice of tension parameters which makes the convergence of the collocation method problem at least linear. The B-representation of the spline in tension is also used in the actual computations. Some numerical experiments are given to illustrate the theory.

Variable degree polynomial splines are Chebyshev splines

Variable degree polynomial (VDP) splines have recently proved themselves as a valuable tool in obtaining shape preserving approximations. However, some usual properties which one would expect of a spline space in order to be useful in geometric modeling, do not follow easily from their definition. This includes total positivity (TP) and variation diminishing, but also constructive algorithms based on knot insertion. We consider variable degree polynomial splines of order

Basis of splines associated with some singular differential operators

k\geqslant 2

Non-uniform Exponential Tension Splines

spanned by

Quadratic convergence of approximations by CCC-Schoenberg operators

\{ 1,x,\ldots x^{k-3},(x-x_i)^{m_i-1},(x_{i+1}-x)^{n_i-1} \}

Original article: Singularly perturbed advection-diffusion-reaction problems: Comparison of operator-fitted methods

on each subinterval

On some inequalities for convex functions of higher order II

[x_i,x_{i+1}\rangle\subset [0,1]

Weighted Integrals of Polynomial Splines

, i = 0,1, ...l. Most of the paper deals with non-polynomial case mi,ni ∈ [4, ∞ ), and polynomial splines known as VDP–splines are the special case when mi, ni are integers. We describe VDP–splines as being piecewisely spanned by a Canonical Complete Chebyshev system of functions whose measure vector is determined by positive rational functions p(x), q(x). These functions are such that variable degree splines belong piecewisely to the kernel of the differential operator

On Calculating with Lower Order Chebyshev Splines

\frac{d}{dx} p \frac{d}{dx} q \frac{d^{k-2}} {dx^{k-2}}