TY - JOUR

T1 - A global convergent derivative-free method for solving a system of non-linear equations

AU - Wörz, Sascha

AU - Bernhardt, Heinz

N1 - Publisher Copyright:
© 2016, Springer Science+Business Media New York.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - Finding all zeros of a system of m∈ ℕ real non-linear equations in n∈ 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.

AB - Finding all zeros of a system of m∈ ℕ real non-linear equations in n∈ 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.

KW - Calculation of zeros

KW - Derivative-free method

KW - Non-linear equations

UR - http://www.scopus.com/inward/record.url?scp=85003944814&partnerID=8YFLogxK

U2 - 10.1007/s11075-016-0246-0

DO - 10.1007/s11075-016-0246-0

M3 - Article

AN - SCOPUS:85003944814

SN - 1017-1398

VL - 76

SP - 109

EP - 124

JO - Numerical Algorithms

JF - Numerical Algorithms

IS - 1

ER -