A New Broyden rank reduction method to solve large systems of nonlinear equations

Ouarit Mostafa,Ali Souissi,Mohamed Ziani

doi:10.11121/ijocta.01.2019.00743

Abstract

We propose a modification of limited memory Broyden methods, called dynamical Broyden rank reduction method, to solve high dimensional systems of nonlinear equations. Based on a thresholding process of singular values, the proposed method determines a priori the rank of the reduced update matrix. It significantly reduces the number of singular values decomposition calls of the update matrix during the iterations. Local superlinear convergence of the method is proved and some numerical examples are displayed.

Highlights

Let us consider the problem of finding a solution of the system of nonlinear equationsF (x) = 0, F : Rn → Rn. (1)The mapping F is assumed to fulfill the following classical assumptions (CA): it is continuously differentiable in an open convex set D ⊂ Rn, there is an x∗ in D such that F (x∗) = 0, the Jacobian F ′ is Lipschitz continuous at x∗ and F ′(x∗) is nonsingular.Newton’s methodis well suited to solve the system (1) due to its local quadratic convergence
We propose a modification of limited memory Broyden methods, called dynamical Broyden rank reduction method, to solve high dimensional systems of nonlinear equations
We present numerical tests by applying the proposed method to some classical test functions from the literature and we present a comparison of this method with the classical Broyden rank reduction (BRR) method (q = p − 1)

Summary

Introduction

Let us consider the problem of finding a solution of the system of nonlinear equations. An alternative to Newton’s method is the Broyden’s quasiNewton method This method uses approximations to the Jacobian matrix at each iteration by performing rank-one updates, see [4]. It requires only one F -evaluation per iteration and achieves, under the classical hypotheses (CA), local superlinear convergence as shown in [5]. The BRR method, as presented in [7], does not give any idea how to fix a priori the rank of the matrix Q, only the smallest singular value is removed. In many cases there are more than one singular value that are close to zero and so they can be removed In this case, memory will be free to store more than one Broyden’s update.

The proposed method

Numerical results

Conclusion