# Extremal and optimal properties of B-bases Collocation Matrices

## TL;DR

This paper investigates the extremal and optimal properties of collocation matrices derived from B-bases, highlighting their shape-preserving qualities and providing theoretical and numerical insights into their eigenvalues and conditioning.

## Contribution

It presents new theoretical results on the extremal eigenvalue and singular value properties of B-base collocation matrices, supported by numerical examples.

## Key findings

- Minimal eigenvalue and singular value bounds are established.
- Numerical experiments confirm theoretical extremal properties.
- Normalized B-bases exhibit optimal shape-preserving and conditioning characteristics.

## 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.