An algorithm based on QSVD for the quaternion equality constrained least squares problem
- PDF / 522,466 Bytes
- 14 Pages / 439.642 x 666.49 pts Page_size
- 39 Downloads / 275 Views
An algorithm based on QSVD for the quaternion equality constrained least squares problem Yanzhen Zhang1 · Ying Li1 · Musheng Wei1 · Hong Zhao2 Received: 5 May 2020 / Accepted: 22 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Quaternion equality constrained least squares (QLSE) problems have attracted extensive attention in the field of mathematical physics due to its applicability as an extremely effective tool. However, the knowledge gap among numerous QLSE problems has not been settled now. In this paper, by using quaternion SVD (Q-SVD) and the equivalence of the QLSE problem and Karush-Kuhb-Tucker (KKT) equation, we obtain some equations about the matrices in the general solution of the QLSE problem. Using these equations, an equivalent form of the solution of the QLSE problem is obtained. Then, applying the special structure of real representation of quaternion, we propose a real structure-preserving algorithm based on Q-SVD. At last, we give numerical example, which illustrates the effectiveness of our algorithm. Keywords Quaternion equality constrained least squares (QLSE) problem · Quaternion SVD (Q-SVD) · Karush-Kuhb-Tucker (KKT) equation · Real representation · Real structure-preserving algorithm
1 Introduction In 1843, the British physicist W. R. Hamilton first proposed the concept of quaternion. Since then, a lot of achievements about quaternion have been made by researchers which came from the disciplines of mathematics and physics. For instance, the American physicist Adler published his monograph Quaternionic Quantum Mechanics and Quantum Fields in 1995, which vastly expanded the quaternionic
Ying Li
[email protected] 1
School of Mathematical Science, Liaocheng University, Liaocheng, 252000, Shandong, People’s Republic of China
2
School of Physics Science and Information Engineering, Liaocheng University, Liaocheng, 252000, Shandong, People’s Republic of China
Numerical Algorithms
generalization of standard complex quantum mechanics [1]. As an extremely effective research tool, the quaternion least squares problems, especially quaternion equality constrained least squares (QLSE) problem, have attracted extensive attention in many fields, such as mathematical physics, vibration theory, automatic control and optimal control theory and structure design [2–17]. However, the computation of the quaternion least squares problem is extremely complex and the knowledge gap among many quaternion least squares problem has not been settled now. Currently, the equality constrained least squares (LSE) problem has gained much attention in the real and complex number field and its applications are extensive [15–20]. And some conclusions from complex LSE problem can also be applied to solve QLSE problem. In [21–23], Jiang et al. studied the QLSE problem by means of complex representation of a quaternion matrix. Recently, we found that QLSE problem is equivalent to weighted quaternion least squares problem when the parameter τ → + ∝ by applying the special stru
Data Loading...
