Total positivity and least squares problems in the Lagrange basis

TL;DR

This paper develops accurate algorithms for polynomial least squares fitting in the Lagrange basis by leveraging total positivity properties of Lagrange-Vandermonde matrices, enabling precise computations of inverses and projections.

Contribution

It introduces a fast bidiagonal decomposition algorithm for rectangular totally positive matrices, improving the accuracy of least squares solutions in polynomial fitting.

Findings

Algorithms demonstrate high accuracy in numerical experiments.

Efficient computation of Moore-Penrose inverse and projection matrices.

Enhanced stability in polynomial least squares fitting.

Abstract

The problem of polynomial least squares fitting in the standard Lagrange basis is addressed in this work. Although the matrices involved in the corresponding overdetermined linear systems are not totally positive, rectangular totally positive Lagrange-Vandermonde matrices are used to take advantage of total positivity in the construction of accurate algorithms to solve the considered problem. In particular, a fast and accurate algorithm to compute the bidiagonal decomposition of such rectangular totally positive matrices is crucial to solve the problem. This algorithm also allows the accurate computation of the Moore-Penrose inverse and the projection matrix of the collocation matrices involved in these problems. Numerical experiments showing the good behaviour of the proposed algorithms are included.