Newton's iteration is a fundamental tool for root-finding and numerical solutions of systems of equations. The iteration rapidly refines the initial approximation to the exact root, and in general the convergence is quadratic. Since the method requires finding the function value and its derivative at each iteration, in some cases, it may not converge. This is because the value of the derivative gets close to zero. In this paper, we introduce a modified and stable algorithm of Newton's iteration method that addresses this issue, reduces computational cost, and improves efficiency. In addition. we analyze the convergence properties of the modified method to demonstrate its effectiveness.
The Gauss–Newton–Broyden method is proposed and investigated for solving a nonlinear least squares problem with operator decomposition. This method is obtained from the Gauss–Newton method by replacing the Jacobian matrix of a nonlinear operator with the sum of the derivative of the differentiable part of the operator...
A new direct accelerated Newton method for minimizing convex functions with Lipschitz continuous Hessian that achieves the global convergence rate of $O(1/k^3)$ in terms of the functional residual, the first second-order method for this problem class attaining this rate.
This paper considers the local convergence for solving general nonlinear equations. We establish randomized quasi-Gauss--Newton methods based on the approximation of the Gram matrix, addressing both the underdetermined and the overdetermined settings. For the underdetermined case, we show that our methods achieve local...
This paper is concerned with the linear approximation method (i.e. the iterative method in which a sequence of vectors is generated by solving certain linearized subproblems) for solving the variational inequality. The global convergent iterative process is proposed by applying the continuation method, and the related...
Zhao-Yong You, Si-Quan Zhu· Journal of Computational Mat...· 1 citation
We study the quasi-Newton method by using set-valued approximations for solving generalized equations without smoothness assumptions. The set-valued approximations appear naturally when dealing with nonsmooth problems, or even in smooth cases, data in almost concrete applications are not exact. We present a generalizat...
Samir Adly, Ngai van Huynh· Journal of Convex Analysis· 6 citations
In this paper, we propose and analyze a new one-parameter family of two-step iterative methods without requiring higher derivatives for solving nonlinear equations. This family is constructed based on a Newton-type method utilizing the Heron mean, a predictor-corrector technique, and an appropriate choice of deriv...
Noori Y. Abdul-Hassan, Ali Hasan Ali, F. Ghanim· Iraqi Journal of Science· 0 citations
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