Fast solving of Weighted Pairing Least-Squares systems

TL;DR

This paper introduces a new weighted pairing least-squares (WPLS) method for data alignment, along with two fast algorithms, demonstrating superior speed, accuracy, and stability through computational experiments.

Contribution

It generalizes WLS to WPLS with a rectangular weight matrix and develops efficient solving methods, including a novel inverse-based approach.

Findings

The proposed method outperforms existing techniques in speed and accuracy.

The inverse-based solver is numerically stable and simple to implement.

Computational experiments validate the effectiveness of the new approach.

Abstract

This paper presents a generalization of the "weighted least-squares" (WLS), named "weighted pairing least-squares" (WPLS), which uses a rectangular weight matrix and is suitable for data alignment problems. Two fast solving methods, suitable for solving full rank systems as well as rank deficient systems, are studied. Computational experiments clearly show that the best method, in terms of speed, accuracy, and numerical stability, is based on a special {1, 2, 3}-inverse, whose computation reduces to a very simple generalization of the usual "Cholesky factorization-backward substitution" method for solving linear systems.