Polynomial least squares fitting in the Bernstein basis

TL;DR

This paper presents a new numerical method for polynomial least squares fitting using the Bernstein basis, involving bidiagonal and QR decompositions of Bernstein-Vandermonde matrices, with numerical experiments demonstrating its effectiveness.

Contribution

It introduces an efficient algorithm for polynomial least squares fitting in the Bernstein basis utilizing bidiagonal and QR decompositions of Bernstein-Vandermonde matrices.

Findings

The method effectively computes polynomial fits in the Bernstein basis.

Numerical experiments confirm the stability and accuracy of the approach.

The algorithm improves computational efficiency over traditional methods.

Abstract

The problem of polynomial regression in which the usual monomial basis is replaced by the Bernstein basis is considered. The coefficient matrix A of the overdetermined system to be solved in the least squares sense is then a rectangular Bernstein-Vandermonde matrix. In order to use the method based on the QR decomposition of A, the first stage consists of computing the bidiagonal decomposition of the coefficient matrix A. Starting from that bidiagonal decomposition, an algorithm for obtaining the QR decomposition of A is the applied. Finally, a triangular system is solved by using the bidiagonal decomposition of the R-factor of A. Some numerical experiments showing the behavior of this approach are included.