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<p>We propose an inner product free iterative method called GP-CMRH for solving block two-by-two nonsymmetric linear systems. GP-CMRH relies on a new simultaneous Hessenberg process that reduces two rectangular matrices to upper Hessenberg form simultaneously, without employing inner products. Compared with GPMR [SIAM J. Matrix Anal. Appl., 44 (2023), pp. 293–311], GP-CMRH requires less computational cost per iteration and may be more suitable for high-performance computing and low or mixed precision arithmetic due to its inner product free property. Our numerical experiments demonstrate that GP-CMRH and GPMR exhibit comparable convergence behavior (with GP-CMRH requiring slightly more iterations), yet GP-CMRH consumes less computational time in most cases. GP-CMRH significantly outperforms GMRES and CMRH in terms of convergence rate and runtime efficiency.</p>

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GP-CMRH: an inner product free iterative method for block two-by-two nonsymmetric linear systems

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Kui Du,
  • Jia-Jun Fan

摘要

We propose an inner product free iterative method called GP-CMRH for solving block two-by-two nonsymmetric linear systems. GP-CMRH relies on a new simultaneous Hessenberg process that reduces two rectangular matrices to upper Hessenberg form simultaneously, without employing inner products. Compared with GPMR [SIAM J. Matrix Anal. Appl., 44 (2023), pp. 293–311], GP-CMRH requires less computational cost per iteration and may be more suitable for high-performance computing and low or mixed precision arithmetic due to its inner product free property. Our numerical experiments demonstrate that GP-CMRH and GPMR exhibit comparable convergence behavior (with GP-CMRH requiring slightly more iterations), yet GP-CMRH consumes less computational time in most cases. GP-CMRH significantly outperforms GMRES and CMRH in terms of convergence rate and runtime efficiency.