TY - JOUR
AU - Cătinaş, Emil
PY - 1995/08/01
Y2 - 2024/07/20
TI - On some Steffensen-type iterative methods for a class of nonlinear equations
JF - Rev. Anal. Numér. Théor. Approx.
JA - Rev. Anal. Numér. Théor. Approx.
VL - 24
IS - 1
SE - Articles
DO -
UR - https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art4
SP - 37-43
AB - <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>
ER -