Cubic spline functions revisited

TL;DR

This paper introduces a new cubic spline, the Q-spline, with an improved asymptotic error bound, and compares it to traditional not-a-knot splines, showing potential advantages in accuracy and conditioning.

Contribution

The paper derives a fourth order asymptotically optimal error bound for the Q-spline and proposes a modified not-a-knot spline with better conditioning and comparable or lower interpolation error.

Findings

Q-spline has a stronger error bound than traditional cubic splines.

Modified not-a-knot spline improves conditioning and reduces error.

Numerical examples confirm the effectiveness of the proposed spline methods.

Abstract

In this paper a fourth order asymptotically optimal error bound for a new cubic interpolating spline function, denoted by Q-spline, is derived for the case that only function values at given points are used but not any derivative information. The bound seems to be stronger than earlier error bounds for cubic spline interpolation in such setting such as the not-a-knot spline. A brief analysis of the conditioning of the end conditions of cubic spline interpolation leads to a modification of the not-a-knot spline, and some numerical examples suggest that the interpolation error of this revised not-a-knot spline generally is comparable to the near optimal Q-spline and lower than for the not-a-knot spline when the mesh size is small.