Abstract
We present another simple way of deriving several iterative methods for solving nonlinear equations numerically. The presented approach of deriving these methods is based on exponentially fitted osculating straight line. These methods are the modifications of Newton′s method. Also, we obtain well‐known methods as special cases, for example, Halley′s method, super‐Halley method, Ostrowski′s square‐root method, Chebyshev′s method, and so forth. Further, new classes of third‐order multipoint iterative methods free from a second‐order derivative are derived by semidiscrete modifications of cubically convergent iterative methods. Furthermore, a simple linear combination of two third‐order multipoint iterative methods is used for designing new optimal methods of order four.
Highlights
Various problems arising in mathematical and engineering science can be formulated in terms of nonlinear equation of the form f x 0.To solve 1.1, we can use iterative methods such as Newton’s method 1–19 and its variants, namely, Halley’s method 1–3, 5, 6, 8, 9, Euler’s method irrational Halley’s method 1, 3, Chebyshev’s method 1, 2, super-Halley method 2, 4 as Ostrowski’s square-root method 5, 6, and so forth, available in the literature.Among these iterative methods, Newton’s method is probably the best known and most widely used algorithm for solving such problems
We present another simple way of deriving several iterative methods for solving nonlinear equations numerically
The following stopping criteria are used for computer programs: i |xn 1 − xn|
Summary
Various problems arising in mathematical and engineering science can be formulated in terms of nonlinear equation of the form f x 0. To solve 1.1 , we can use iterative methods such as Newton’s method 1–19 and its variants, namely, Halley’s method 1–3, 5, 6, 8, 9 , Euler’s method irrational Halley’s method 1, 3 , Chebyshev’s method 1, 2 , super-Halley method 2, 4 as Ostrowski’s square-root method 5, 6 , and so forth, available in the literature. Newton’s method for multiple roots appears in the work of Schroder 19 , which is given as xn 1 xn − This method has a second-order convergence, including the case of multiple roots. It may be obtained by applying Newton’s method to the function uf x f x /f x , which has simple roots in each multiple root of f x. If we can obtain the error equation for any iterative method, the value of P is its order of convergence. E P 1/d, 1.9 where P is the order of the method
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