Abstract
This paper is dedicated to the study of continuous Newton’s method, which is a generic differential equation whose associated flow tends to the zeros of a given polynomial. Firstly, we analyze some numerical features related to the root-finding methods obtained after applying different numerical methods for solving initial value problems. The relationship between the step size and the order of convergence is particularly considered. We have analyzed both the cases of a constant and non-constant step size in the procedure of integration. We show that working with a non-constant step, the well-known Chebyshev-Halley family of iterative methods for solving nonlinear scalar equations is obtained.
Highlights
We can find the origins of continuous Newton’s method in the seminal paper of Neuberger [1]
In the case of continuous Newton’s method, the function f (t, z(t)) that appears in differential equation Equation (4) is p(z(t))
The situation does not improve if we consider higher order numerical methods for the initial value problem Equation (2)
Summary
We can find the origins of continuous Newton’s method in the seminal paper of Neuberger [1] It appears in the study of the basins of attraction related with the relaxed Newton’s method for solving a complex equation p(z) = 0 zn+1 = zn − h p(zn ). The choice of the appropriate branch of the cube root, defined by the rays θ = π/3, θ = π and θ = −π/3, determines the ternary division of the complex plane that can be seen in the three graphics of Figure 1 According to these authors, the fractal boundaries in the basins of attraction of the roots can be originated by numerical errors inherent to the discretization of initial value problem Equation (2). The efficiency of the iterative processes obtained in this way is analyzed
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