An algorithmic approach to direct spline products: procedures and computational aspects

TL;DR

This paper presents an efficient, robust algorithm for computing the product of splines in the B-spline basis, outperforming implicit methods in terms of stability and computational cost.

Contribution

The authors develop a novel algorithmic framework based on the Oslo Algorithm, significantly improving efficiency and robustness in spline product computations.

Findings

The direct formula remains stable where implicit methods fail due to ill-conditioning.

The proposed algorithm reduces computational cost substantially.

Numerical experiments confirm the efficiency and stability of the method.

Abstract

We introduce an efficient algorithmic procedure for implementing the direct formula that represents the product of splines in the B-spline basis. We first demonstrate the relevance of this direct approach through numerical evidence showing that implicit methods, such as collocation, may fail in some instances due to severe ill-conditioning of the associated system matrices, whereas the direct formula remains robust. We then recast the direct formula into an algorithmic framework based on the Oslo Algorithm and subsequently enhance it, through a factorization of the terms to be computed, to dramatically improve computational efficiency. Extensive numerical experiments illustrate the substantial reduction in computational cost achieved by the proposed method. Implementation aspects are also discussed to ensure numerical stability and applicability.