@article{Cătinaş_1995, title={On some Steffensen-type iterative methods for a class of nonlinear equations}, volume={24}, url={https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art4}, abstractNote={<p> </p> <p>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. </p> <p>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.</p> <p>The method is obtained by combining the Newton method for the differentiable part with the Steffensen method for the nondifferentiable part.</p> <p>We show that the <a href="https://ictp.acad.ro/a-survey-on-the-high-convergence-orders-and-computational-convergence-orders-of-sequences/">R-convergence order</a> of this method is 2, the same as of the Newton method.</p> <p>We provide some numerical examples and compare different methods for a nonlinear system in \(\mathbb{R}^2\).</p>}, number={1}, journal={Rev. Anal. Numér. Théor. Approx.}, author={Cătinaş, Emil}, year={1995}, month={Aug.}, pages={37-43} }