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<p>Conjugate Gradient (CG) methods are widely recognized for their effectiveness in solving large-scale nonlinear systems of equations, primarily due to their reliance on efficient vector operations. However, the global convergence analysis of CG methods remains a challenging issue. Motivated by the broad range of applications involving symmetric nonlinear equations, this study develops two optimal strategies for the modified Riavie-Mamat-Ismail-Leong (RMIL) CG method. The first strategy is constructed by minimizing an appropriate measure function, while the second combines the modified RMIL direction with the classical quasi-Newton direction. To further enhance robustness, the resulting CG parameters are incorporated with the Li and Fukushima approximate gradient, leading to the formulation of a new CG-type algorithm for large-scale systems of symmetric nonlinear equations. The global convergence of the proposed algorithms is established under standard assumptions. Numerical experiments on benchmark problems demonstrate that the proposed methods exhibit superior efficiency and robustness compared to several existing approaches for symmetric nonlinear equations.</p>

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Globally convergent RMIL-Type conjugate gradient methods with optimal parameter strategies for large-scale symmetric nonlinear equations

  • 期刊论文

  • 发表日期：

    2026年5月9日

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  • Lawal Muhammad,
  • Mohammad Hamarsheh,
  • Sulaiman M. Ibrahim

摘要

Conjugate Gradient (CG) methods are widely recognized for their effectiveness in solving large-scale nonlinear systems of equations, primarily due to their reliance on efficient vector operations. However, the global convergence analysis of CG methods remains a challenging issue. Motivated by the broad range of applications involving symmetric nonlinear equations, this study develops two optimal strategies for the modified Riavie-Mamat-Ismail-Leong (RMIL) CG method. The first strategy is constructed by minimizing an appropriate measure function, while the second combines the modified RMIL direction with the classical quasi-Newton direction. To further enhance robustness, the resulting CG parameters are incorporated with the Li and Fukushima approximate gradient, leading to the formulation of a new CG-type algorithm for large-scale systems of symmetric nonlinear equations. The global convergence of the proposed algorithms is established under standard assumptions. Numerical experiments on benchmark problems demonstrate that the proposed methods exhibit superior efficiency and robustness compared to several existing approaches for symmetric nonlinear equations.