# The danger of combining block red–black ordering with modified incomplete factorizations and its remedy by perturbation or relaxation

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## Abstract

Modified incomplete LU/Cholesky factorizations without fill-ins are popular preconditioners for Krylov subspace methods, because they require no extra memory and have more potential of accelerating the convergence than simple ILU/IC preconditioners. For parallelizing preconditioners, the block red–black ordering is attractive due to its highly parallel nature and small number of synchronization points. Hence, their combination seems to produce powerful and parallelizable preconditioners. In fact, however, this combination can cause breakdown of the factorization due to the occurrence of zero pivots. We analyze this phenomenon and give necessary and sufficient conditions of zero pivots in the case of a regular grid. We also show both theoretically and experimentally that adding perturbation to the diagonal elements or relaxing the compensation of dropped fill-ins is useful to alleviate the problem. Numerical tests show that the resulting preconditioners are highly effective and are applicable for up to \(10^3\) level of parallelism.

## Keywords

Linear system Modified incomplete factorizations Block red–black ordering Parallel processing## Mathematics Subject Classification

65F08## Notes

### Acknowledgements

We are grateful to the anonymous reviewer, whose comments helped us to improve the quality of this paper. We thank Prof. Takeshi Iwashita of Hokkaido University for valuable comments on our work. The present study is supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (nos. 26286087, 15H02708, 15H02709, 16KT0016, 17H02828, 17K19966). The computational experiments in this paper were performed using the FX10 parallel computer at the Education Center on Computational Science and Engineering of Kobe University.

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