A Generalized Multivariable Newton Method

03/27/2021
by   Regina S. Burachik, et al.
0

It is well known that the Newton method may not converge when the initial guess does not belong to a specific quadratic convergence region. We propose a family of new variants of the Newton method with the potential advantage of having a larger convergence region as well as more desirable properties near a solution. We prove quadratic convergence of the new family, and provide specific bounds for the asymptotic error constant. We illustrate the advantages of the new methods by means of test problems, including two and six variable polynomial systems, as well as a challenging signal processing example. We present a numerical experimental methodology which uses a large number of randomized initial guesses for a number of methods from the new family, in turn providing advice as to which of the methods employed is preferable to use in a particular search domain.

READ FULL TEXT

page 16

page 18

page 21

page 26

research
11/13/2019

Benchmarking results for the Newton-Anderson method

This paper primarily presents numerical results for the Anderson acceler...
research
12/03/2019

Stochastic Newton and Cubic Newton Methods with Simple Local Linear-Quadratic Rates

We present two new remarkably simple stochastic second-order methods for...
research
09/12/2022

Backtracking New Q-Newton's method: a good algorithm for optimization and solving systems of equations

In this paper, by combining the algorithm New Q-Newton's method - develo...
research
07/06/2023

Convergence Properties of Newton's Method for Globally Optimal Free Flight Trajectory Optimization

The algorithmic efficiency of Newton-based methods for Free Flight Traje...
research
04/09/2014

The Secant-Newton Map is Optimal Among Contracting n^th Degree Maps for n^th Root Computation

Consider the problem: given a real number x and an error bound ϵ, find a...
research
11/27/2019

On the choice of initial guesses for the Newton-Raphson algorithm

The initialization of equation-based differential-algebraic system model...
research
01/29/2019

The projected Newton-Kleinman method for the algebraic Riccati equation

The numerical solution of the algebraic Riccati equation is a challengin...

Please sign up or login with your details

Forgot password? Click here to reset