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2015, Applied Mathematics and Computation
An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to o...
A nonlinear equation f(x)=0 is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme xn+1=xn−f(xn)/[a+bf(xn)], which requires one evaluation of the given function per iteration. A local convergence analysis is adopted to determine the optimal values of a and b. Moreover, upon combining the fractional iterative scheme to the generalized quadrature methods, the fourth-order optimal iterative schemes are derived. The finite differences based on three data are used to estimate the optimal values of a and b. We recast the Newton iterative method to two types of derivative-free iterative schemes by using the finite difference technique. A three-point generalized Hermite interpolation technique is developed, which includes the weight functions with certain constraints. Inserting the derived interpolation formulas into the triple Newton method, the eighth-order optimal ite...
2014 •
2008 •
2020 •
We propose a novel family of seventh-order iterative methods for computing multiple zeros of a nonlinear function. The algorithm consists of three steps, of which the first two are the steps of recently developed Liu–Zhou fourth-order method, whereas the third step is based on a Newton-like step. The efficiency index of the proposed scheme is 1.627, which is better than the efficiency index 1.587 of the basic Liu–Zhou fourth-order method. In this sense, the proposed iteration is the modification over the Liu–Zhou iteration. Theoretical results are fully studied including the main theorem of local convergence analysis. Moreover, convergence domains are also assessed using the graphical tool, namely, basins of attraction which are symmetrical through the fractal like boundaries. Accuracy and computational efficiency are demonstrated by implementing the algorithms on different numerical problems. Comparison of numerical experiments shows that the new methods have an edge over the exist...
Journal of Numerical Analysis and Approximation Theory
Convergence analysis for the two-step Newton method of order fourWe provide a tighter than before convergence analysis for the two-step Newton method of order four using recurrent functions. Numerical examples are also provided in this study.
International Journal of Computer Mathematics
An optimal eighth-order class of three-step weighted Newton's methods and their dynamics behind the purely imaginary extraneous fixed points2017 •
Journal of Optimization Theory and Applications
A Fast and Simple Modification of Newton’s Method Avoiding Saddle Points2023 •
Journal of Computational and Applied Mathematics
A modified Newton method with cubic convergence: the multivariate case2004 •
Mathematical Modelling and Analysis
On a Class of Efficient Higher Order Newton-Like Methods2019 •
Based on a two-step Newton-like scheme, we propose a three-step scheme of convergence order p+2 (p >=3) for solving systems of nonlinear equations. Furthermore, on the basis of this scheme a generalized k+2-step scheme with increasing convergence order p+2k is presented. Local convergence analysis including radius of convergence and uniqueness results of the methods is presented. Computational efficiency in the general form is discussed. Theoretical results are verified through numerical experimentation. Finally, the performance is demonstrated by the application of the methods on some nonlinear systems of equations.
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Sustainability
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Soil Biology and Biochemistry
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