Approximation / Ellipse

The best uniform approximation algorithm with degree two. (2016)

A uniform quadratic approximation of degree 2 is created in explicit parametric form to represent elliptical arcs. The error function is identical to that of the Chebyshev polynomial of degree 4 and equioscillates five times with an approximation order of four. In this paper we provide the approximation method, show it is efficient, its error bound to be accurate and demonstrate that it satisfies properties of the best uniform approximation. Keywords: B´ezier curves; quadratic best uniform approximation; elliptical arc; high accuracy; approximation order; equioscillation.

AIP Conference Proceedings 1863, 060003 (2017); https://doi.org/10.1063/1.4992217

 

Approximation / Polynomials

Cubic Quadrature Formula with Fifth Degree of Accuracy (2019)

Given constants a, b and a function f, our aim is to approximate the Integral of f over an interval [a, b] by a quadrature formula Q(f). Such a formula to approximate the integral has degree of accuracy n if it is exact for all polynomials of degree n. There are two approaches to approximate the integral. The first is the interpolatory Newton-Cotes formula, and the second is the Gaussian quadrature method. A cubic Newton-Cotes formula has degree of accuracy three, while a cubic Gaussian quadrature formula has degree of accuracy five. Both approaches are limited to functions and can not be used to approximate the area enclosed by parametric curves.