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<p>Building upon the Petrov-Galerkin framework and the recent 2-D randomized Gauss-Seidel (D2RGS) and 2-D randomized extended Gauss-Seidel (D2REGS) methods proposed by Mustafa and Saha for linear least-squares problems, this paper introduces a new greedy two-dimensional Gauss-Seidel method (D2GGS). Unlike D2RGS and D2REGS, which employ uniform random selection for the two-dimensional search subspace basis vectors and avoid explicit pseudoinverse computation, the proposed D2GGS method utilizes a novel greedy sampling strategy. At each iteration, D2GGS deterministically selects the two column indices corresponding to the maximum and minimum values of a specific criterion. We further extend this approach to handle rank-deficient and underdetermined linear systems, developing the 2-D greedy extended Gauss-Seidel (D2GEGS) method. Convergence properties for both new methods are established. Numerical experiments demonstrate the efficiency of D2GGS and D2GEGS. Notably, for matrices with highly coherent columns, D2GGS significantly outperforms existing Gauss-Seidel-type methods.</p>

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A new greedy two-dimensional extended Gauss-Seidel method for least squares problems

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • Jianhua Zhang,
  • Shuling Dai,
  • Jing Zhao

摘要

Building upon the Petrov-Galerkin framework and the recent 2-D randomized Gauss-Seidel (D2RGS) and 2-D randomized extended Gauss-Seidel (D2REGS) methods proposed by Mustafa and Saha for linear least-squares problems, this paper introduces a new greedy two-dimensional Gauss-Seidel method (D2GGS). Unlike D2RGS and D2REGS, which employ uniform random selection for the two-dimensional search subspace basis vectors and avoid explicit pseudoinverse computation, the proposed D2GGS method utilizes a novel greedy sampling strategy. At each iteration, D2GGS deterministically selects the two column indices corresponding to the maximum and minimum values of a specific criterion. We further extend this approach to handle rank-deficient and underdetermined linear systems, developing the 2-D greedy extended Gauss-Seidel (D2GEGS) method. Convergence properties for both new methods are established. Numerical experiments demonstrate the efficiency of D2GGS and D2GEGS. Notably, for matrices with highly coherent columns, D2GGS significantly outperforms existing Gauss-Seidel-type methods.