On some Steffensen-type iterative methods for a class of nonlinear equations
Let \(H(x):=F(x)+G(x)=0\), with \(F\) differentiable and \(G\) continuous, where \(F,G,H:X \rightarrow X\) are nonlinear operators and \(X\) is a Banach space. The Newton method cannot be applied for solving the nonlinear equation \(H(x)=0\), and we propose an iterative method for solving...
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| Format: | Article |
| Language: | English |
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Publishing House of the Romanian Academy
1995-08-01
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| Series: | Journal of Numerical Analysis and Approximation Theory |
| Online Access: | https://ictp.acad.ro/jnaat/journal/article/view/511 |
| Summary: |
Let \(H(x):=F(x)+G(x)=0\), with \(F\) differentiable and \(G\) continuous, where \(F,G,H:X \rightarrow X\) are nonlinear operators and \(X\) is a Banach space.
The Newton method cannot be applied for solving the nonlinear equation \(H(x)=0\), and we propose an iterative method for solving this equation by combining the Newton method with the Steffensen method: \[x_{k+1} = \big(F^\prime(x_k)+[x_k,\varphi(x_k);G]\big)^{-1}(F(x_k)+G(x_k)),\] where \(\varphi(x)=x-\lambda (F(x)+G(x))\), \(\lambda >0\) fixed.
The method is obtained by combining the Newton method for the differentiable part with the Steffensen method for the nondifferentiable part.
We show that the R-convergence order of this method is 2, the same as of the Newton method.
We provide some numerical examples and compare different methods for a nonlinear system in \(\mathbb{R}^2\).
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| ISSN: | 2457-6794 2501-059X |