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<p>Iterative methods play a crucial role in solving large and sparse linear systems, where direct methods become impractical due to computational constraints. Among these, Krylov subspace methods (KSMs) have been widely studied and applied for efficient iterative solutions. The s-step Krylov subspace techniques have gained attention for their ability to reduce communication overhead and improve computational performance, particularly in large-scale systems. However, these methods can introduce numerical instability, necessitating the use of regularization techniques. In this paper, we propose a modified variable s-step BiCGSTAB method, incorporating regularization parameters, to enhance the stability and convergence of shifted linear systems. The approach combines s-step iterations with techniques such as Tikhonov regularization and truncated singular value decomposition to mitigate error accumulation while maintaining computational efficiency. Comparative analysis demonstrates that the proposed method achieves improved accuracy and reduced computational cost compared to existing s-step Krylov subspace approaches. The findings contribute to the ongoing development of high-performance iterative solvers for large-scale linear algebra problems. The code can be downloaded from <a href="https://github.com/shahbaz1982/S-step">https://github.com/shahbaz1982/S-step</a>.</p>

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A new modified variable s-step BiCGSTAB method with regularization for solving shifted linear systems

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Shahbaz Ahmad,
  • Hifsa

摘要

Iterative methods play a crucial role in solving large and sparse linear systems, where direct methods become impractical due to computational constraints. Among these, Krylov subspace methods (KSMs) have been widely studied and applied for efficient iterative solutions. The s-step Krylov subspace techniques have gained attention for their ability to reduce communication overhead and improve computational performance, particularly in large-scale systems. However, these methods can introduce numerical instability, necessitating the use of regularization techniques. In this paper, we propose a modified variable s-step BiCGSTAB method, incorporating regularization parameters, to enhance the stability and convergence of shifted linear systems. The approach combines s-step iterations with techniques such as Tikhonov regularization and truncated singular value decomposition to mitigate error accumulation while maintaining computational efficiency. Comparative analysis demonstrates that the proposed method achieves improved accuracy and reduced computational cost compared to existing s-step Krylov subspace approaches. The findings contribute to the ongoing development of high-performance iterative solvers for large-scale linear algebra problems. The code can be downloaded from https://github.com/shahbaz1982/S-step.