# A global convergent derivative-free method for solving a system of non-linear equations

A global convergent derivative-free method for solving a system of non-linear equations Finding all zeros of a system of m ∈ ℕ $m \in \mathbb {N}$ real non-linear equations in n ∈ ℕ $n \in \mathbb {N}$ variables often arises in engineering problems. Using Newtons’ iterative method is one way to solve the problem; however, the convergence order is at most two, it depends on the starting point, there must be as many equations as variables and the function F, which defines the system of nonlinear equations F(x)=0 must be at least continuously differentiable. In other words, finding all zeros under weaker conditions is in general an impossible task. In this paper, we present a global convergent derivative-free method that is capable to calculate all zeros using an appropriate Schauder base. The component functions of F are only assumed to be Lipschitz-continuous. Therefore, our method outperforms the classical counterparts. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Numerical Algorithms Springer Journals

# A global convergent derivative-free method for solving a system of non-linear equations

, Volume 76 (1) – Dec 10, 2016
16 pages

/lp/springer_journal/a-global-convergent-derivative-free-method-for-solving-a-system-of-non-wYsD2uP61B
Publisher
Springer Journals
Subject
Computer Science; Numeric Computing; Algorithms; Algebra; Theory of Computation; Numerical Analysis
ISSN
1017-1398
eISSN
1572-9265
D.O.I.
10.1007/s11075-016-0246-0
Publisher site
See Article on Publisher Site

### Abstract

Finding all zeros of a system of m ∈ ℕ $m \in \mathbb {N}$ real non-linear equations in n ∈ ℕ $n \in \mathbb {N}$ variables often arises in engineering problems. Using Newtons’ iterative method is one way to solve the problem; however, the convergence order is at most two, it depends on the starting point, there must be as many equations as variables and the function F, which defines the system of nonlinear equations F(x)=0 must be at least continuously differentiable. In other words, finding all zeros under weaker conditions is in general an impossible task. In this paper, we present a global convergent derivative-free method that is capable to calculate all zeros using an appropriate Schauder base. The component functions of F are only assumed to be Lipschitz-continuous. Therefore, our method outperforms the classical counterparts.

### Journal

Numerical AlgorithmsSpringer Journals

Published: Dec 10, 2016

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