iterative scheme

This manuscript puts forward two new generalized families of Jarratt’s iterative schemes for deciding the solution of scalar and systems of non-linear equations. The schemes involve weight functions that are based on bi-variate rational approximation polynomial of degree two in both its numerator and denominator. The convergence study conducted on the schemes, indicated that they have convergence order (CO) four in scalar space and retain the same number of CO in vector space. The numerical experiments conducted on the schemes when used to decide the solutions of some real-life nonlinear models show that they are good challengers of some well-known and robust existing iterative schemes.

In this paper, introduced a new accelerated iterative algorithm in \((\lambda,\rho )\) -quasi firmly nonexpansive multi-valued mappings in modular function spaces and present some results for convergence to a fixed point in this mapping, we use faster convergence theorem to comparison our iteration with some other iterations and introduced numerical example. As an application, we have referred to previous work by other researchers.

In this work, we define an iterative scheme for a generalized spectral problem associated with two operators defined on a Banach space of infinite dimension. We show that under the norm convergence, the generalized approximated eigenvalues and eigenvectors converge to the exact eigenpairs. As a numerical application, we tackle a generalized eigenvalue problem associated with integral operators, where the accuracy and efficiency are illustrated in some numerical examples.

In this paper, we introduce a new type of modified generalized $\alpha$-nonexpansive mapping and establish some fixed point properties and demiclosedness principle for this class of mappings in the framework of  uniformly convex Banach spaces. We further  propose a new iterative method for approximating a common fixed point of two modified generalized $\alpha$-nonexpansive mappings and present some weak and strong convergence theorems for these mappings in uniformly convex Banach spaces. In addition, we apply our result to solve a  convex-constrained minimization problem, variational inequality and split feasibility problem and present some numerical experiments in infinite dimensional spaces to establish the applicability and efficiency of our proposed algorithm. The obtained results in this paper improve and extend   some related results in the literature.

The Householder iterative scheme (HIS) for determining solution of equations that are nonlinear have existed for over fifty decades and have enjoyed several modifications in literature. However, in most HIS modifications, they usually require function derivative evaluation in their implementation. Obtaining derivative of some functions is difficult and in some cases, it is not achievable.To circumvent this setback, the divided difference operator was utilised to approximate function derivatives that appear in the scheme. This resulted to the development of a new variant of the HIS with high precision and require no function derivative. The theoretical convergence of the new scheme was established using Taylor’s expansion approach. From the computational results obtained when the new scheme was tested on some non-linear problems in literature, it performed better than the Householder scheme.