We present a local convergence analysis of an eighth order three step method in order to approximate a locally unique solution of nonlinear equation in a Banach space setting. In an earlier study by Sharma and Arora (2015), the order of convergence was shown using Taylor series expansions and hypotheses up to the fourth order derivative or even higher of the function involved which restrict the...

Oversampled functions can be evaluated using generalized sinc-series and filter functions connected with these series. A standard filter function is given by exp((ξ2 − 1)−1). We show that the Fourier transform of this filter posseses the convergence order O(exp(−√x)), improving an estimation given in [10]. Moreover, we define a family of filter functions with convergence order O(x · exp(−xσ)) w...

Article history: Available online 30 July 2011

The method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self-adjoint operator. We extend the method to normal operators and find optimal convergence rates for eigenvalues and eigenspaces. The convergence to eigenspaces is new, while the convergence rate for eigenvalues improves on the previous estimate by an order of magnitude.

In this paper we control the first moment of the initial approximations and obtain the order of convergence and the asymptotic profile of a general solution by two explicit “canonical” approximations: a diffusive N-wave and a diffusion wave solution. The order of convergence of both approximations is O(t1/(2r)−3/2) in Lr norm, 1 ≤ r ≤ ∞, as t → ∞, which is faster than the well-known classical c...

A family of eighth-order iterative methods for solution of nonlinear equations is presented. We propose an optimal three-step method with eight-order convergence for finding the simple roots of nonlinear equations by Hermite interpolation method. Per iteration of this method requires two evaluations of the function and two evaluations of its first derivative, which implies that the efficiency i...