Error analysis for piecewise quadratic curve fitting algorithms

doi:10.1016/0167-8396(86)90038-5

Computer Aided Geometric Design

Volume 3, Issue 3, November 1986, Pages 205-215

https://doi.org/10.1016/0167-8396(86)90038-5 Get rights and content

Abstract

We analyze the convergence of piecewise quadratic curve fitting algorithms which preserve geometric properties of the data such as monotonicity and convexity. In the process, we suggest two new algorithms which improve the order of convergence of existing algorithms.

References (6)

H. Akima
A new method of interpolation and smooth curve fitting based on local procedures
J. ACM
(1970)
C. de Boor
A Practical Guide to Splines
F.N. Fritsch et al.
Monotone piecewise cubic interpolation
SIAM J. Numer. Anal.
(1980)

There are more references available in the full text version of this article.

Cited by (24)

A shape-preserving spline interpolation for sampling designs from inverse distributions
2023, Results in Applied Mathematics
Given a strictly monotone cumulative function $F : [a, b] ⟶ [c, d]$ , with $a, b, c, d \in R$ , $a < b$ and $c < d$ , such that $F ([a, b]) = [c, d]$ . The possibility of using the spline interpolation procedure, to approximate the inverse of $F$ , is a natural way in some applications, and it leads to schemes that are stable and feasible to implement. We prove that, with a shape-preserving quadratic Hermite interpolation scheme based on quadratic B-splines, we can preserve the monotonicity of the inverse of $F$ while maintaining a third-order estimate of the interpolation error. To illustrate the effectiveness of this approach, we provide some examples of application, such as the inversion of cumulative distribution functions (arc-length and normal cumulative distributions) and the computation of the Lambert W-function.
Error estimation of a kind of rational spline
2007, Journal of Computational and Applied Mathematics
This paper deals with the approximation properties of a kind of rational spline with linear denominator when the function being interpolated is $C^{3}$ in an interpolating interval. Error estimate expressions of interpolating functions are derived, convergence is established, the optimal error coefficient, $c_{i}$ , is proved to be symmetric about the parameters of the rational interpolation and it is bounded. Finally, the precise jump measurements of the second derivatives of the interpolating function at the knots are given.
Data visualization using rational spline interpolation
2006, Journal of Computational and Applied Mathematics
A new C<sup>2</sup> rational interpolation based on function values and constrained control of the interpolant curves
2005, Applied Mathematics and Computation
In this paper a new method is developed to create a high-order smoothness interpolation using values of the function being interpolated. This is a kind of rational cubic interpolation with quadratic denominator. This rational spline not only belongs to C² in the interpolating interval, but could also be used to constrain the shape of the interpolant curve such as to force it to be in the given region, all because of the selectable parameters in the rational spline itself. The more important achievement mathematically of this method is that the uniqueness of the interpolating function for the given data would be replaced by the uniqueness of the interpolating curve for the given data and the selected parameters.
Circular Bernstein-Bézier spline approximation with knot removal
2003, Journal of Computational and Applied Mathematics
Constrained control and approximation properties of a rational interpolating curve
2003, Information Sciences
This paper deals with the convexity control and the strain energy control of interpolating curves using a rational cubic spline with linear denominator. The sufficient and necessary conditions for controlling the interpolating curve to be convex or concave are derived. When the function being interpolated is f(t)∈C⁽³⁾[t₀,t_n], the error estimation of the interpolating function and the boundedness of the optimal error coefficient and its double symmetry with regard to parameters are obtained.

View all citing articles on Scopus

^∗: This author was supported by NSF grant DMS 8320562.

View full text

Error analysis for piecewise quadratic curve fitting algorithms

Abstract

A new method of interpolation and smooth curve fitting based on local procedures

J. ACM

A Practical Guide to Splines

Monotone piecewise cubic interpolation

SIAM J. Numer. Anal.