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# Mathematics > Algebraic Geometry

# Title: A more direct and better variant of New Q-Newton's method Backtracking for m equations in m variables

(Submitted on 14 Oct 2021 (v1), last revised 25 Oct 2021 (this version, v2))

Abstract: In this paper we apply the ideas of New Q-Newton's method directly to a system of equations, utilising the specialties of the cost function $f=||F||^2$, where $F=(f_1,\ldots ,f_m)$.

The first algorithm proposed here is a modification of Levenberg-Marquardt algorithm, where we prove some new results on global convergence and avoidance of saddle points.

The second algorithm proposed here is a modification of New Q-Newton's method Backtracking, where we use the operator $\nabla ^2f(x)+\delta ||F(x)||^{\tau}$ instead of $\nabla ^2f(x)+\delta ||\nabla f(x)||^{\tau}$. This new version is more suitable than New Q-Newton's method Backtracking itself, while currently has better avoidance of saddle points guarantee than Levenberg-Marquardt algorithms.

Also, a general scheme for second order methods for solving systems of equations is proposed. We will also discuss a way to avoid that the limit of the constructed sequence is a solution of $H(x)^{\intercal}F(x)=0$ but not of $F(x)=0$.

## Submission history

From: Tuyen Truong [view email]**[v1]**Thu, 14 Oct 2021 14:40:03 GMT (9kb)

**[v2]**Mon, 25 Oct 2021 14:22:24 GMT (13kb)

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