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.
Du et al. (Sun,) studied this question.