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Generalized Newton Frozen Jacobian Multi-Step Iterative Methods GMN_p,n for Solving a System of Nonlinear Equations

Eman Salem Al-Aidarous, Fayyaz Ahmad, Dalal Adnan Al-Maturi, Dina Abdullah Alrehaili

Abstract



Some generalized Newton multi-step iterative methods GMNp;m for solving a system of nonlinear equations are constructed. Generalized Newton multi-step iterative methods depend on parameters and consist of two parts, namely the base method and the multi-step part. In the base method we evaluate the Jacobian at the initial guess and then freeze it to solve system of linear equations in the multi-step part. Direct inversion of the Jacobian is an expansion operation and hence we utilize LU-factorization of the frozen Jacobian for moderately large system of linear equations. In the multi-step part we only solve lower and upper triangular systems that makes the computational process economical. The GMNp;m
involve parameters and we are interested to find the parameters for maximizing the convergence order of iterative method. In the present article we explore some iterative method with p = 5 and convergence of order six for the base method. Each multi-step part adds one in the convergence order of the base method and hence the convergence order of GMN5;m(in this article) is 6 + m. The validity and numerical accuracy of the solution of the
system of nonlinear equations are presented via numerical simulations.

Keywords


Multi-step Newton iterative methods, Systems of nonlinear equations, Ordinary differential equations, Partial differential equations

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