Estimates for the Spectral Condition Number of Cardinal B-Spline Collocation Matrices (Long version)

TL;DR

This paper investigates the spectral condition number of cardinal B-spline collocation matrices, providing numerical evidence supporting the de Boor conjecture for uniform knot sequences, despite known counterexamples for nonuniform meshes.

Contribution

The paper offers new numerical insights into the spectral condition number of cardinal B-spline matrices, supporting the conjecture for uniform knots.

Findings

Numerical tests suggest the conjecture holds for uniform knots.

Counterexamples exist for nonuniform knot meshes.

Spectral condition number appears bounded independently of knot sequence for uniform cases.

Abstract

The famous de Boor conjecture states that the condition of the polynomial B-spline collocation matrix at the knot averages is bounded independently of the knot sequence, i.e., depends only on the spline degree. For highly nonuniform knot meshes, like geometric meshes, the conjecture is known to be false. As an effort towards finding an answer for uniform meshes, we investigate the spectral condition number of cardinal B-spline collocation matrices. Numerical testing strongly suggests that the conjecture is true for cardinal B-splines.