On the specific solutions of reduced biquaternion equality constrained least squares problem and their relative forward error bound
Sk. Safique Ahmad, Neha Bhadala
TL;DR
The paper addresses the RBLSE problem by transforming reduced biquaternion matrices into real and complex representations and solving resultant LSEs with QR-based methods. It provides explicit closed-form solutions for both real and complex cases, along with unitary/orthogonal-invariance properties and detailed perturbation analyses that yield forward-error bounds $U_{RL}$ and $U_{CL}$. Numerical experiments validate the high accuracy and show the real-solution algorithm is faster than the complex one, while confirming the reliability of the forward-error bounds under perturbations. The work advances numerical methods for reduced biquaternions and supports practical applications in robotics, signal and image processing, and related 3D/4D computational domains.
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.