Topic
Steffensen's method
About: Steffensen's method is a research topic. Over the lifetime, 545 publications have been published within this topic receiving 11649 citations.
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01 Jan 1987
TL;DR: This chapter discusses how to get the Newton Step with Gaussian Elimination software and some of the methods used to achieve this goal.
Abstract: Preface How to Get the Software 1 Introduction 2 Finding the Newton Step with Gaussian Elimination 3 Newton-Krylov Methods 4 Broyden's Method Bibliography Index
1,002 citations
TL;DR: It is shown that the order of convergence of the new method is three, and computed results support this theory.
813 citations
TL;DR: A new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables, which appears to have significant practical potential for large-scale problems.
Abstract: We propose a new algorithm, a reflective Newton method, for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables. This method applies to a general (indefinite) quadratic function, for which a local minimizer subject to bounds is required, and is particularly suitable for the large-scale problem. Our new method exhibits strong convergence properties, global and quadratic convergence, and appears to have significant practical potential. Strictly feasible points are generated. Experimental results on moderately large and sparse problems support the claim of practicality for large-scale problems.
640 citations
TL;DR: The primary goal is to introduce and analyze new inexact Newton methods, but consideration is also given to “globally convergence” features designed to improve convergence from arbitrary starting points.
Abstract: Inexact Newton methods for finding a zero of $F:\mathbf{R}^n \to \mathbf{R}^n $ are variations of Newton’s method in which each step only approximately satisfies the linear Newton equation but still reduces the norm of the local linear model of F. Here, inexact Newton methods are formulated that incorporate features designed to improve convergence from arbitrary starting points. For each method, a basic global convergence result is established to the effect that, under reasonable assumptions, if a sequence of iterates has a limit point at which $F^\prime $ is invertible, then that limit point is a solution and the sequence converges to it. When appropriate, it is shown that initial inexact Newton steps are taken near the solution, and so the convergence can ultimately be made as fast as desired, up to the rate of Newton’s method, by forcing the initial linear residuals to be appropriately small. The primary goal is to introduce and analyze new inexact Newton methods, but consideration is also given to “gl...
447 citations