TY - JOUR
T1 - A gradient descent method for solving a system of nonlinear equations
AU - Hao, Wenrui
N1 - Funding Information:
This work is supported by the National Science Foundation (NSF) (Grant No. DMS-1818769 ).
Publisher Copyright:
© 2020 Elsevier Ltd
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2021/2
Y1 - 2021/2
N2 - This paper develops a gradient descent (GD) method for solving a system of nonlinear equations with an explicit formulation. We theoretically prove that the GD method has linear convergence in general and, under certain conditions, is equivalent to Newton's method locally with quadratic convergence. A stochastic version of the gradient descent is also proposed for solving large-scale systems of nonlinear equations. Finally, several benchmark numerical examples are used to demonstrate the feasibility and efficiency compared to Newton's method.
AB - This paper develops a gradient descent (GD) method for solving a system of nonlinear equations with an explicit formulation. We theoretically prove that the GD method has linear convergence in general and, under certain conditions, is equivalent to Newton's method locally with quadratic convergence. A stochastic version of the gradient descent is also proposed for solving large-scale systems of nonlinear equations. Finally, several benchmark numerical examples are used to demonstrate the feasibility and efficiency compared to Newton's method.
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U2 - 10.1016/j.aml.2020.106739
DO - 10.1016/j.aml.2020.106739
M3 - Article
AN - SCOPUS:85090286609
VL - 112
JO - Applied Mathematics Letters
JF - Applied Mathematics Letters
SN - 0893-9659
M1 - 106739
ER -