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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Quasi-nonexpansivity and two classical methods for solving nonlinear equations

Author: St. Măruşter
Journal: Proc. Amer. Math. Soc. 62 (1977), 119-123
MSC: Primary 65H05
MathSciNet review: 0455354
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Abstract: Let F: $ {{\mathbf{R}}_n} \to {{\mathbf{R}}_n}$ be a vector-valued function and let $ J(x)$ denote the corresponding Jacobi matrix. The main result states that the functions $ x - {J^{ - 1}}(x) \cdot F(x)$ and $ x - \lambda {J^T}(x)\cdot F(x)$, where $ \lambda $ is a certain positive number, are quasi-nonexpansive. This property is used for establishing the convergence of the Newton and the gradient methods in a finite-dimensional space.

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Keywords: Quasi-nonexpansive mapping, nonlinear equation, Newton method, gradient method
Article copyright: © Copyright 1977 American Mathematical Society