Multipurpose modified iterative solver for nonlinear equations

A MODIFIED STEFFENSEN'S METHOD WITH MEMORY FOR NONLINEAR EQUATIONS

In this note, we propose a modification of Steffensen's method with some free parameters. These parameters are then be used for further acceleration via the concept of with memorization. In this way, we derive a fast Steffensen-type method with memory for solving nonlinear equations. Numerical results are also given to support the underlying theory of the article.  

An Iterative Solver in the Presence and Absence of Multiplicity for Nonlinear Equations

We develop a high-order fixed point type method to approximate a multiple root. By using three functional evaluations per full cycle, a new class of fourth-order methods for this purpose is suggested and established. The methods from the class require the knowledge of the multiplicity. We also present a method in the absence of multiplicity for nonlinear equations. In order to attest the effici...

Iterative Methods for Linear and Nonlinear Equations

Iterative Methods for Nonlinear Elliptic Equations

‖f(u)− f(v)‖−1 ≤ L‖u− v‖−1. We drop the dependence of f on x to emphasis the nonlinearity in u. Let Vh ⊂ H 0 (Ω) be the linear finite element space associated to a quasi-uniform mesh Th. The nonlinear Garlerkin method is: find uh ∈ Vh such that (2) Auh = f(uh) in Vh, where A : H 0 (Ω)→ H−1(Ω) is the elliptic operator defined by −∆. We shall assume the existence and locally uniqueness of the sol...

NITSOL: A Newton Iterative Solver for Nonlinear Systems

We introduce a well-developed Newton iterative (truncated Newton) algorithm for solving large-scale nonlinear systems. The framework is an inexact Newton method globalized by backtracking. Trial steps are obtained using one of several Krylov subspace methods. The algorithm is implemented in a Fortran solver called NITSOL that is robust yet easy to use and provides a number of useful options and...