Abstract
A three-step iterative method with fifth-order convergence as a new modification of Newton’s method was presented. This method is for finding multiple roots of nonlinear equation with unknown multiplicity m whose multiplicity m is the highest multiplicity. Its order of convergence is analyzed and proved. Results for some numerical examples show the efficiency of the new method.
Highlights
This paper addresses the problem of multiple roots x∗ of nonlinear equation f (x) = 0 with unknown multiplicity m whose multiplicity m is the highest multiplicity, where f : [a, b] ⊂ R → R is a nonlinearAlgorithms 2015, 8 differential function on [a, b]
From different initial iterative values, they can be convergent to the same iterative solution whose multiplicity is the highest multiplicity of nonlinear equation
For the special case which the multiplicity of the roots of nonlinear equation is a single multiplicity, we present the analysis result for comparison with previous methods
Summary
If the multiplicity m is not known explicitly, Traub [16] utilized a simple transformation F (x) = f (x)/f 0 (x) instead of f (x) for computing a multiple root of f (x) = 0. In this case, the aim of solving a multiple root is reduced to that of solving a simple root of the transformed equation f (x) = 0, and any iterative method can be used to preserve the original convergence order. In order to avoid the evaluations of these derivatives with the multiplicity m unknown, for multiple roots, King [17] proposed the secant f (x)
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