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<p>In this paper, we consider iterative solution of a specific class of indefinite block three-by-three double saddle point problems, which have attracted much attention recently. A generalized successive overrelaxation (GSOR) iteration method is proposed and reliable convergence regions are established by leveraging a recently obtained result concerning the roots of real cubic polynomial being less than one in modulus. The selection of quasi-optimal iteration parameters to minimize the convergence rate of the iteration method is also explored under specific conditions. Furthermore, spectral properties of the GSOR preconditioned matrices are thoroughly analyzed, yielding clear bounds on the eigenvalue distributions. The effectiveness of the proposed GSOR method and its corresponding preconditioned GMRES variant is demonstrated through numerical experiments.</p>

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On generalized successive overrelaxation method for a class of block three-by-three double saddle point problems

  • 期刊论文

  • 发表日期：

    2026年2月5日

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  • Zhao-Zheng Liang,
  • Xiao-Kang Tan

摘要

In this paper, we consider iterative solution of a specific class of indefinite block three-by-three double saddle point problems, which have attracted much attention recently. A generalized successive overrelaxation (GSOR) iteration method is proposed and reliable convergence regions are established by leveraging a recently obtained result concerning the roots of real cubic polynomial being less than one in modulus. The selection of quasi-optimal iteration parameters to minimize the convergence rate of the iteration method is also explored under specific conditions. Furthermore, spectral properties of the GSOR preconditioned matrices are thoroughly analyzed, yielding clear bounds on the eigenvalue distributions. The effectiveness of the proposed GSOR method and its corresponding preconditioned GMRES variant is demonstrated through numerical experiments.