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In this paper, we present an iterative three-point method with memory based on the family of Kings methods to solve nonlinear equations. This proposed method has eighth order convergence and costs only four function evaluations per iteration which supports the Kung-Traub conjecture on the optimal order of convergence. An acceleration of the convergence speed is achieved by an appropriate variation of a free parameter in each step. This self accelerator parameter is estimated using Newtons interpolation polynomial of fourth degree. The order of convergence is increased from 8 to 12 without any extra function evaluation. Consequently, this method, possesses a high computational efficiency. Finally, a numerical comparison of the proposed method with related methods shows its effectiveness and performance in high precision computations.
Spectral residual methods are powerful tools for solving nonlinear systems of equations without derivatives. In a recent paper, it was shown that an acceleration technique based on the Sequential Secant Method can greatly improve its efficiency and r
In this paper, we propose a $mu$-mode integrator for computing the solution of stiff evolution equations. The integrator is based on a d-dimensional splitting approach and uses exact (usually precomputed) one-dimensional matrix exponentials. We show
Differential algebraic Riccati equations are at the heart of many applications in control theory. They are time-depent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been u
In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is sho
We introduce a novel spatio-temporal discretization for nonlinear Fokker-Planck equations on the multi-dimensional unit cube. This discretization is based on two structural properties of these equations: the first is the representation as a gradient