On modified Newton–DGPMHSS method for solving nonlinear systems with complex symmetric Jacobian matrices

Chen, Minhong; Wu, Qingbiao

doi:10.1016/j.camwa.2018.04.003

Cited by 13 publications

(4 citation statements)

References 25 publications

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“…Kanwar et al [20] presented a new method to compute the multiple roots of problems. In [21], a method for solving large dissipative nonlinear systems with a symmetric Jacobian matrix was presented.…”

Section: Introductionmentioning

confidence: 99%

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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In this work, first, a family of fourth‐order methods is proposed to solve nonlinear equations. The methods satisfy the Kung‐Traub optimality conjecture. By developing the methods into memory methods, their efficiency indices are increased. Then, the methods are extended to the multi‐step methods for finding the solutions to systems of problems. The formula for the order of convergence of the multi‐step iterative methods is , where is the step number of the methods. It is clear that computing the Jacobian matrix derivative evaluation and its inversion are expensive; therefore, we compute them only once in every cycle of the methods. The important feature of these multi‐step methods is their high‐efficiency index. Numerical examples that confirm the theoretical results are performed. In applications, some nonlinear problems related to the numerical approximation of fractional differential equations (FDEs) are constructed and solved by the proposed methods.

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Section: Introductionmentioning

confidence: 99%

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Erfanifar,

Hajarian,

Sayevand

2023

Math Methods in App Sciences

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“…Furthermore, the multi-step modified Newton-HSS method (Li and Guo 2017a, b), which includes the Newton-HSS method and the modified Newton-HSS method as special cases, outperforms the modified Newton-HSS method. To our knowledge, there have been a succession of theses showing the efficiency of such kind of Newton-HSS based methods, see Zhong et al (2015); Chen and Wu (2018); Xie et al (2019) for more examples.…”

Section: Introductionmentioning

confidence: 99%

Two new effective iteration methods for nonlinear systems with complex symmetric Jacobian matrices

Zhang

Chen

et al. 2021

Comp. Appl. Math.

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In this paper, we mainly discuss the iterative methods for solving nonlinear systems with complex symmetric Jacobian matrices. By applying an FPAE iteration (a fixed-point iteration adding asymptotical error) as the inner iteration of the Newton method and modified Newton method, we get the so–called Newton-FPAE method and modified Newton-FPAE method. The local and semi-local convergence properties under Lipschitz condition are analyzed. Finally, some numerical examples are given to expound the feasibility and validity of the two new methods by comparing them with some other iterative methods.

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“…Furthermore, in order to increase the efficiency of algorithm, we optimized the outer iteration and then we propose modified Newton-PGSS method to solve the saddle problems. Because there was no Newton method to solve the saddle point system problem, we compare the Newton-PGSS method with the traditional methods, for example, the Newton-RHSS method [31,36,37]. e organization of the paper is as follows.…”

Section: Introductionmentioning

confidence: 99%

Newton-PGSS and Its Improvement Method for Solving Nonlinear Systems with Saddle Point Jacobian Matrices

Xiao

Zhang

2021

Journal of Mathematics

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The preconditioned generalized shift-splitting (PGSS) iteration method is unconditionally convergent for solving saddle point problems with nonsymmetric coefficient matrices. By making use of the PGSS iteration as the inner solver for the Newton method, we establish a class of Newton-PGSS method for solving large sparse nonlinear system with nonsymmetric Jacobian matrices about saddle point problems. For the new presented method, we give the local convergence analysis and semilocal convergence analysis under Hölder condition, which is weaker than Lipschitz condition. In order to further raise the efficiency of the algorithm, we improve the method to obtain the modified Newton-PGSS and prove its local convergence. Furthermore, we compare our new methods with the Newton-RHSS method, which is a considerable method for solving large sparse nonlinear system with saddle point nonsymmetric Jacobian matrix, and the numerical results show the efficiency of our new method.

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On modified Newton–DGPMHSS method for solving nonlinear systems with complex symmetric Jacobian matrices

Cited by 13 publications

References 25 publications

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

A family of iterative methods to solve nonlinear problems with applications in fractional differential equations

Two new effective iteration methods for nonlinear systems with complex symmetric Jacobian matrices

Newton-PGSS and Its Improvement Method for Solving Nonlinear Systems with Saddle Point Jacobian Matrices

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