Extremal and optimal properties of B-bases Collocation Matrices
Jorge Delgado, J. M. Peña
TL;DR
The paper investigates extremal and optimal properties of collocation matrices arising from normalized B-bases within spaces of functions. By exploiting corner-cutting factorizations of TP matrices and representing any NTP collocation matrix as $A=MK$ with $K$ a TP stochastic matrix, it proves that the minimal eigenvalue $\lambda_{\min}$ and minimal singular value $\sigma_{\min}$ of any NTP collocation matrix are bounded above by those of the normalized B-basis collocation matrix, which underscores the B-basis’s shape-preserving and conditioning advantages. It also shows that this extremal behavior extends to $\,\kappa_{\infty}$ conditioning, i.e., $\kappa_{\infty}(M)\le\kappa_{\infty}(A)$, while the maximal singular value does not enjoy a parallel bound. Numerical experiments with Bernstein, B-spline, rational Bernstein, Said-Ball, and DP bases corroborate the theory, illustrating the practical impact for stable, shape-preserving representations in approximation and CAGD contexts.
Abstract
Totally positive matrices are related with the shape preserving representations of a space of functions. The normalized B-basis of the space has optimal shape preserving properties. B-splines and rational Bernstein bases are examples of normalized B-bases. Some results on the optimal conditioning and on extremal properties of the minimal eigenvalue and singular value of the collocation matrices of normalized B-bases are proved. Numerical examples confirm the theoretical results and answer related questions.