One-parameter Family Of Methods Research Articles

A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space

The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in Rm spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects.

ON A FAMILY OF NEWTON-LIKE METHODS FOR BILATERAL APPROXIMATION OF THE EIGENVALUES OF NONLINEAR SPECTRAL PROBLEMS

Methods and algorithms that form a one-parameter family of methods of bilateral approximations to the eigenvalues of nonlinear spectral problems are constructed. Their convergence is proved. Numerical results are given.

Certain improvements of Chebyshev–Halley methods with accelerated fourth-order convergence

In this paper, we present a new one-parameter family of methods which improves the classical Chebyshev–Halley methods with cubic convergence. The convergence analysis shows that each method of the family is quartically convergent and per iteration it requires one evaluation of the given function and two of its first derivative. The well-known Jarratt fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performance of the presented methods.