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 Num. Meth. Prog., 2015, Volume 16, Issue 2, Pages 298–306 (Mi vmp541)

An approach for constructing one-point iterative methods for solving nonlinear equations of one variable

A. N. Gromov

Odintsovo university of Humanities

Abstract: An approach for constructing one-point iterative methods for solving nonlinear equations of one variable is proposed. This approach is based on the concept of a pole as a singular point and on using Cauchy's convergence criterion. It is shown that such an approach leads to new iterative processes of higher order with larger convergence domains compared to the known iterative methods. Convergence theorems are proved and convergence rate estimates are obtained. For polynomials having only real roots, the iterative process converges for any initial approximation to the sought root. Generally, in the case of real roots of transcendental equations, the convergence takes place when an initial approximation is chosen near the sought root.

Keywords: iterative processes, Newton's method, logarithmic derivative, simple pole, contracted mapping, third order method, singular point, transcendental equations.

Full text: PDF file (209 kB)
UDC: 519.6

Citation: A. N. Gromov, “An approach for constructing one-point iterative methods for solving nonlinear equations of one variable”, Num. Meth. Prog., 16:2 (2015), 298–306

Citation in format AMSBIB
\Bibitem{Gro15} \by A.~N.~Gromov \paper An approach for constructing one-point iterative methods for solving nonlinear equations of one variable \jour Num. Meth. Prog. \yr 2015 \vol 16 \issue 2 \pages 298--306 \mathnet{http://mi.mathnet.ru/vmp541}