On some iterative methods for solving nonlinear equations


Abstract views: 349


Let \(h(x):=f(x)+g(x)=0\), with \(f\) differentiable and \(g\) continuous, where \(f,g,h:X \rightarrow X\), \(X\) a Banach space. 

The Newton method cannot be applied for solving \(h(x)=0\), and we propose an iterative method for solving the nonlinear equation, by combining the Newton method (for the differentiable part) with the chord/secant method (for the nondifferentiable part): \[x_{k+1} = \big(f^\prime(x_k)+[x_{k-1},x_k;g]\big)^{-1}(f(x_k)+g(x_k)).\]

We show that the r-convergence order of the method is the same as of the chord/secant method.

We provide some numerical examples and compare different methods for a nonlinear system in \(\mathbb{R}^2\).


[Editor note: for a series of papers dealing with the notion of convergence orders, see the convergence orders category]


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How to Cite

Cătinaş, E. (1994). On some iterative methods for solving nonlinear equations. Rev. Anal. Numér. Théor. Approx., 23(1), 47–53. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art4