# A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations

@article{Liao2022ASG, title={A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations}, author={Zeyu Liao and Ken Hayami and Keiichi Morikuni and Junfeng Yin}, journal={Japan Journal of Industrial and Applied Mathematics}, year={2022} }

Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized…

## One Citation

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- Computer Science, MathematicsArXiv
- 2022

This work derived the necessary and suﬃcient conditions for GMRES to determine a least squares solution of inconsistent and consistent range symmetric systems assuming exact arithmetic except for the computation of the elements of the Hessenberg matrix.

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