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**Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs.**
*(English)*
Zbl 1316.65053

The authors propose a multi-step iterative method for approximating nonsingular solutions of nonlinear systems in \({\mathbb R}^n\). Under sufficient differentiability conditions of the nonlinear function they establish high (sixth, eighth) orders of the method along the idea of a fixed point theorem. Although no second or higher order of Fréchet derivatives of the nonlinear mapping are involved in the proposed method, in each iteration step, multiple linear systems need to be solved to create intermediate approximations. The authors also discuss and compare computational efficiency indices of different methods. Numerical examples are used to illustrate the performance of their method. It would be also useful to compare the proposed method directly with the use of the Newton method in the corresponding multiple steps.

Reviewer: Zhen Mei (Toronto)

### MSC:

65H10 | Numerical computation of solutions to systems of equations |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65Y20 | Complexity and performance of numerical algorithms |

68Q25 | Analysis of algorithms and problem complexity |

### Keywords:

Newton method; nonlinear systems; multi-step iterative methods; nonsingular solution; convergence; nonlinear PDE### Software:

Mathematica
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\textit{M. Z. Ullah} et al., Numer. Algorithms 67, No. 1, 223--242 (2014; Zbl 1316.65053)

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### References:

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