Third-order iterative methods under Kantorovich conditions

Abstract

We study a class of third-order iterative methods for nonlinear equations on Banach spaces. A characterization of the convergence under Kantorovich type conditions and optimal estimates of the error are found. Though, in general, these methods are not very extended due to their computational costs, we will show some examples in which they are competitive and even cheaper than other simpler methods. We center our analysis in both, analytic and computational, aspects.

On two high‐order families of frozen Newton‐type methods

“…The classical third-order methods use second-order Fréchet derivatives. [10][11][12] These evaluations are very time consuming, and for this reason, these methods are hardly used in practice.…”

Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus

“…A generalized norm is defined to be an operator from a linear space into a partially order Banach space (as will be elaborated in Section 2). Earlier studies such as [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] for Newton's method have shown that a more precise convergence analysis is obtained when compared with the real norm theory. However, the main assumption is that the operator involved is Fréchet differentiable.…”

“…Furthermore, the advantages (i)-(iii) are obtained under the same or less computational cost. Notice that in the recent elegant work by Adly et al, [1] Newton's method has also been generalized to other important directions for solving inclusions and set-valued approximations. In the classical Banach space setting though these results that rely on non smooth analysis and metric regularity do not provide sufficient convergence criteria in the local as well as semilocal convergence case that are verifiable using Lipschitz-type constants as we utilize in the present study.…”

Expanding the Applicability of a Third Order Newton-Type Method Free of Bilinear Operators

“…Under some regularity assumptions, the methods are at least second order convergent. The classical third order schemes, such as Halley or Chebyshev methods, evaluate second order Fréchet derivatives [1][2][3]. These evaluations are very time-consuming for systems of equations.…”