Algebraic technique for mixed least squares and total least squares problem in the reduced biquaternion algebra

Abstract

This paper presents the reduced biquaternion mixed least squares and total least squares (RBMTLS) method for solving an overdetermined system (Formula presented.) in the reduced biquaternion algebra. The RBMTLS method is suitable when matrix B and a few columns of matrix A contain errors. By examining real representations of reduced biquaternion matrices, we investigate the conditions for the existence and uniqueness of the real RBMTLS solution and derive an explicit expression for the real RBMTLS solution. The proposed technique covers two special cases: the reduced biquaternion total least squares (RBTLS) method and the reduced biquaternion least squares (RBLS) method. Furthermore, the developed method is used to find the best approximate solution to (Formula presented.) over a complex field. Perturbation analysis of the real RBMTLS, RBTLS, and RBLS solutions is also studied. Lastly, numerical examples are presented to illustrate our findings. © 2024 Informa UK Limited, trading as Taylor & Francis Group.