New Iterative Methods for Solving Nonlinear Equations and Their Basins of Attraction

The purpose of this paper is to propose new modified Newton’s method for solving nonlinear equations and free from second derivative. Convergence results show that the order of convergence is four. Several numerical examples are given to illustrate that the new iterative algorithms are effective.In the end, we present the basins of attraction to observe the fractal behavior and dynamical aspects of the proposed algorithms.

A note on machine method for root extraction

The present investigation deals with the critical study of the works of Lancaster and Traub, who have developed $n$th root extraction methods of a real number. It is found that their developed methods are equivalent and the particular cases of Halley's and Householder's methods. Again the methods presented by them are easily obtained from simple modifications of Newton's method, which is the extension of Heron's square root iteration formula. Further, the rate of convergency of their reported methods are studied.

A Low Distortion Mesh Parameterization Mapping Method Based on Proxy Function and Combined Newton

To address the issues of low efficiencies and serious mapping distortions in current mesh parameterization methods, we present a low distortion mesh parameterization mapping method based on proxy function and combined Newton’s method in this paper. First, the proposed method calculates visual blind areas and distortion prone areas of a 3D mesh model, and generates a model slit. Afterwards, the method performs the Tutte mapping on the cut three-dimensional mesh model, measures the mapping distortion of the model, and outputs a distortion metric function and distortion values. Finally, the method sets iteration parameters, establishes a reference mesh, and finds the optimal coordinate points to get a convergent mesh model. When calculating mapping distortions, Dirichlet energy function is used to measure the isometric mapping distortion, and MIPS energy function is used to measure the conformal mapping distortion. To find the minimum value of the mapping distortion metric function, we use an optimal solution method combining proxy functions and combined Newton’s method. The experimental data show that the proposed method has high execution efficiency, fast descending speed of mapping distortion energy and stable optimal value convergence quality. When a texture mapping is performed, the texture is evenly colored, close laid and uniformly lined, which meets the standards in practical applications.

A class of Newton maps with Julia sets of Lebesgue measure zero

Let $$g(z)=\int _0^zp(t)\exp (q(t))\,dt+c$$ g ( z ) = ∫ 0 z p ( t ) exp ( q ( t ) ) d t + c where p, q are polynomials and $$c\in {\mathbb {C}}$$ c ∈ C , and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of $$g''$$ g ′ ′ the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that $$f^n(z)$$ f n ( z ) converges to zeros of g almost everywhere in $${\mathbb {C}}$$ C if this is the case for each zero of $$g''$$ g ′ ′ that is not a zero of g or $$g'$$ g ′ . In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.