Efficient Construction, Update and Downdate Of The Coefficients Of Interpolants Based On Polynomials Satisfying A Three-Term Recurrence Relation

TL;DR

This paper introduces two algorithms for efficiently constructing, updating, and downdating polynomial interpolant coefficients based on polynomials satisfying a three-term recurrence, improving stability and efficiency.

Contribution

The paper presents two novel algorithms for polynomial interpolation coefficients: one for incremental updates and another for stable, efficient computation over multiple nodes.

Findings

Incremental algorithm allows updating coefficients with node changes.

Decomposition algorithm offers higher numerical stability.

Second algorithm is more efficient for multiple interpolations.

Abstract

In this paper, we consider methods to compute the coefficients of interpolants relative to a basis of polynomials satisfying a three-term recurrence relation. Two new algorithms are presented: the first constructs the coefficients of the interpolation incrementally and can be used to update the coefficients whenever a nodes is added to or removed from the interpolation. The second algorithm, which constructs the interpolation coefficients by decomposing the Vandermonde-like matrix iteratively, can not be used to update or downdate an interpolation, yet is more numerically stable than the first algorithm and is more efficient when the coefficients of multiple interpolations are to be computed over the same set of nodes.