@article{Cătinaş_1994, title={On some iterative methods for solving nonlinear equations}, volume={23}, url={https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art4}, abstractNote={<p>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. </p> <p>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): \[x_{k+1} = \big(f^\prime(x_k)+[x_{k-1},x_k;g]\big)^{-1}(f(x_k)+g(x_k)).\]</p> <p>We show that the <a href="https://ictp.acad.ro/how-many-steps-still-left-to-x/">r-convergence order</a> of the method is the same as of the chord/secant method.</p> <p>We provide some numerical examples and compare different methods for a nonlinear system in \(\mathbb{R}^2\).</p> <p> </p> <p>[Editor note: for a series of papers dealing with the notion of convergence orders, see the <a href="https://ictp.acad.ro/category/numerical-analysis/convergence-orders/">convergence orders</a> category]</p>}, number={1}, journal={Rev. Anal. Numér. Théor. Approx.}, author={Cătinaş, Emil}, year={1994}, month={Aug.}, pages={47–53} }