Solution Methods and Perturbation Analysis for the Equality-Constrained Total Least Squares Problem over Reduced Biquaternions

TL;DR

This paper introduces a new framework for solving the reduced biquaternion equality-constrained total least squares problem, providing solution methods, stability analysis, and demonstrating superior performance over existing methods in noisy conditions.

Contribution

It develops a comprehensive theoretical and computational framework for RBTLSE, including existence conditions, solution algorithms, and sensitivity analysis, advancing the state of the art in biquaternion least squares problems.

Findings

The RBTLSE method outperforms the RBLSE method in noisy environments.

Derived explicit conditions for solution existence.

Provided tight bounds on solution sensitivity and error.

Abstract

This paper proposes a theoretical framework to address the reduced biquaternion equality-constrained total least squares (RBTLSE) problem. The objective is to find an approximate solution to the system $AX \approx B$, subject to linear constraints $CX = D$, while explicitly accounting for errors in both the coefficient matrix $A$ and the observation matrix $B$. We establish conditions under which real and complex solutions exist and develop corresponding solution methods using real and complex representations of reduced biquaternion matrices. To assess the sensitivity of the solutions to data perturbations, we derive relative normwise condition numbers and provide tight upper bounds on the relative forward errors. These results ensure the computational reliability of the proposed framework. Extensive numerical experiments validate the theoretical findings and demonstrate that the RBTLSE approach significantly outperforms the conventional reduced biquaternion equality-constrained least squares (RBLSE) method, particularly in noisy environments affecting both sides of the system.