A new iterative method to find the root of a nonlinear scalar function f is proposed. The method is based on a suitable Taylor polynomial model of order n around the current point xk and involves at each iteration the solution of a linear system of dimension n. It is shown that the coefficient matrix of the linear system is nonsingular if and only if the first derivative of f at xk is not null. Moreover, it is proved that the method is locally convergent with order of convergence at least n + 1. Finally, an easily implementable scheme is provided and some numerical results are reported
Germani, A., Manes, C., Palumbo, P., Sciandrone, M. (2006). A higher order method for the solution of nonlinear scalar equations. JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS, 131(3), 347-364 [10.1007/s10957-006-9154-0].
A higher order method for the solution of nonlinear scalar equations
PALUMBO, PASQUALE;
2006
Abstract
A new iterative method to find the root of a nonlinear scalar function f is proposed. The method is based on a suitable Taylor polynomial model of order n around the current point xk and involves at each iteration the solution of a linear system of dimension n. It is shown that the coefficient matrix of the linear system is nonsingular if and only if the first derivative of f at xk is not null. Moreover, it is proved that the method is locally convergent with order of convergence at least n + 1. Finally, an easily implementable scheme is provided and some numerical results are reported
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