On the specific solutions of reduced biquaternion equality constrained least squares problem and their relative forward error bound

TL;DR

This paper develops algebraic techniques and algorithms for solving the reduced biquaternion equality constrained least squares problem, analyzes their stability, and validates their effectiveness through numerical examples, with implications for 3D and 4D applications.

Contribution

It introduces new algebraic methods and algorithms for solving the RBLSE problem, along with a perturbation analysis and error bounds, advancing the stability and accuracy of solutions.

Findings

Algorithms effectively solve RBLSE problem

Perturbation analysis provides reliable error bounds

Numerical examples confirm accuracy and stability

Abstract

This study focuses on addressing the challenge of solving the reduced biquaternion equality constrained least squares (RBLSE) problem. We develop algebraic techniques to derive real and complex solutions for the RBLSE problem by utilizing the real and complex forms of reduced biquaternion matrices. Furthermore, we propose algorithms and provide a detailed analysis of their computational complexity for finding special solutions to the RBLSE problem. A perturbation analysis is conducted, establishing an upper bound for the relative forward error of these solutions. This analysis ensures the accuracy and stability of the solutions in the presence of data perturbations, which is crucial for practical applications where errors arising from input inaccuracies can cause deviations between computed and true solutions. Numerical examples are presented to validate the proposed algorithms, demonstrate their effectiveness, and verify the accuracy of the established upper bound for the relative forward errors. These findings lay the groundwork for exploring applications in 3D and 4D algebra such as robotics, signal, and image processing, expanding their impact on practical and emerging domains.