An iterative method for solving the generalized total least squares problem

Document Type : Research Article

Authors

1 Department of Mathematics, College of Sciences, Shiraz University, Shiraz 7187919556, Iran & Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran

2 Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran

Abstract

This paper introduces a novel method for solving the generalized total least squares problem, an extension of the total least squares problem. The generalized total least squares problem emerges when solving overdetermined linear systems with the multiple right-hand sides $\mathbf{AX} \thickapprox \mathbf{B}$, where both the observation matrix $\mathbf{B}$ and the data matrix $\mathbf{A}$ contain errors. Our approach involves extending the Taylor series expansion to reformulate the generalized total least squares problem into a linear problem, allowing us to employ the tensor form of the generalized least squares algorithm for efficient computation. This technique streamlines the computational process and enhances solution accuracy. For a more detailed survey, we compare the proposed method for solving the generalized total least squares problem with one of the matrix format methods for the associated total least squares problem. Empirical results show that our method significantly improves computational efficiency and solution precision. Additionally, we demonstrate its practical application in the context of image blurring.

Keywords

Main Subjects

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Journal of Mathematical Modeling
Volume 13, Issue 3
July 2025
Pages 573-592

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