A real structure-preserving algorithm based on the quaternion QR decomposition for the quaternion equality constrained least squares problem

• Published in: Numerical algorithms Vol. 91; no. 4; pp. 1815 - 1827
• Main Authors: Zhang, Fengxia, Zhao, Jianli
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2022; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computer Science; Decomposition; Floating point arithmetic; Least squares method; Numeric Computing; Numerical Analysis; Original Paper; Quaternions; Representations; Theory of Computation; Real representation matrix; Quaternion equality constrained least squares problem; Quaternion QR decomposition; Real structure-preserving algorithm
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-022-01323-w

In this paper, we consider the computational problem of the quaternion equality constrained least squares problem. First, applying the quaternion QR decomposition and an equivalence problem of the quaternion equality constrained least squares problem, we obtain the expressions of the general solution and the minimal norm solution of the quaternion equality constrained least squares problem. And then by using the real representation matrices of quaternion matrices, the special structure of the real representation matrices and the real structure-preserving algorithm of the quaternion QR decomposition, we purpose a real structure-preserving algorithm for the minimal norm solution of the quaternion equality constrained least squares problem. The purposed algorithm only involves real algebraic operations, and the number of real floating-point operations and assignments has been minimized. Finally, we give two examples, which show that our purposed algorithm is more efficient and time-saving than direct computation in the quaternion Toolbox for Matlab.