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

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Abstract

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.

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Data availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

Notes

  1. In order to obtain the solution of the QLSE problem, denote

    $$f(x,\lambda)=(Bx-d)^{H}(Bx-d)+\lambda(Ax-b)^{H}(Ax-b),$$

    in which λ is Lagrange multiplier. When \(\lambda \rightarrow +\infty \), the fixed point x of f(x,λ) tend to the solution of the QLSE problem. Notice that

    $$0=\frac{\partial f}{\partial x}=2B^{H}(Bx-d)+2\lambda A^{H}(Ax-b),$$

    and therefore BH(d − Bx) + λAH(b − Ax) = 0. Denote rB = d − Bx,v = λ(b − Ax), and then AHv + BHrB = 0, in which rB is the residual vector, and v is a vector of Lagrange multiplier.

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Acknowledgements

The authors would like to thank the handling editor Prof. Claude Brezinski and two anonymous reviewers for their constructive comments and suggestions, which greatly improve the presentation of this paper.

Funding

This paper is supported by the Natural Science Foundation of Shandong Province of China (No. ZR2020MA053), and the Scientific Research Foundation of Liaocheng University (No. 318011921).

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  1. College of Mathematical Science, Liaocheng University, Liaocheng, 252000, People’s Republic of China

    Fengxia Zhang & Jianli Zhao

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  1. Fengxia Zhang
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Correspondence to Fengxia Zhang.

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Zhang, F., Zhao, J. A real structure-preserving algorithm based on the quaternion QR decomposition for the quaternion equality constrained least squares problem. Numer Algor 91, 1815–1827 (2022). https://doi.org/10.1007/s11075-022-01323-w

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  • DOI: https://doi.org/10.1007/s11075-022-01323-w

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