An algorithmic approach to direct spline products: procedures and computational aspects
For researchers and practitioners using spline-based methods, this provides a more efficient and numerically stable alternative to implicit approaches for computing spline products.
The paper introduces an efficient algorithmic procedure for computing the product of splines in the B-spline basis, demonstrating that implicit methods like collocation can fail due to ill-conditioning while the direct formula remains robust. The proposed method achieves substantial reduction in computational cost through a factorization of terms.
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.