# Modified Newton–SHSS method for a class of systems of nonlinear equations

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## Abstract

It has been proved that Newton–HSS method is efficient and robust for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. In this paper, by utilizing the single-step Hermitian and skew-Hermitian splitting (SHSS) iteration technique, which performs efficiently under certain conditions, as the inner solver of the modified Newton method, we propose a class of modified Newton–SHSS methods. Subsequently, the local and semilocal convergence properties of our method will be discussed under some reasonable assumptions. Furthermore, we introduce the modified Newton–SHSS method with a backtracking strategy and analyze its basic global convergence theorem. Finally, several typical instances are used to illustrate the advantages of our methods when the Hermitian part of the Jacobian matrices are dominant.

## Keywords

Nonlinear systems Single-step Hermitian and skew-Hermitian splitting Modified Newton method Convergence properties## Mathematics Subject Classification

65H10## Notes

### Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11771393, 11632015) and Zhejiang Natural Science Foundation (Grant No. LZ14A010002).

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