Newton's method for multiple roots

doi:10.1016/0097-8493(94)90097-3

Computers & Graphics

Volume 18, Issue 2, March–April 1994, Pages 227-229

https://doi.org/10.1016/0097-8493(94)90097-3 Get rights and content

Abstract

We investigate the basins of attraction in the complex plane of Newton's method for finding multiple roots and illustrate what happens as two simple roots coalesce to form a double root.

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Cayley's Problem and Julia Sets
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A well known result of J. Hubbard, D. Schleicher and S. Sutherland (see [27]) shows that if f is a complex polynomial of degree d, then there is a finite set $S_{d}$ depending only on d such that, given any root α of f, there exists at least one point in $S_{d}$ converging under iterations of $N_{f}$ to α. Their proof depends heavily on the simply connectedness of the immediate basins of attraction of Newton's method. We show that for all order $σ \geq 2$ , there exists a complex polynomial f such that the Julia set of König's method for multiple roots applied to it is disconnected. Consequently, our result establishes restrictions for extending the main result in [27] to higher order root-finding methods. As far as we know, there are no pictures of disconnected Julia sets for root finding algorithms applied to polynomials. Here we give a proof and provide pictures that illustrate such disconnectedness. We also show that the Fatou set of König's method for multiple roots converges to the Voronoi diagram under order of convergence growth, in the Hausdorff complementary metric.
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Both discrete and continuous structures, with frequency and/or temperature dependent viscoelastic elements, gives rise to a nonlinear eigenvalue problem. An accurate computation of eigenvalues (natural frequencies) and eigenvectors (mode shapes) is essential for control, design sensitivities, and optimization studies. In this paper, a pth order approximation of a general nonlinear eigenvalue problem is formulated. A numerical approach to simultaneously compute the eigenvalues and associated left and right eigenvectors is presented. This method can be used for both discrete and continuous systems with viscoelastic elements. Numerical examples are presented here to demonstrate its effectiveness and for the validation purposes.
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M and J sets from Newton's transformation of the transcendental mapping F(z) = e(<sup>z(w)+c</sup>) with vcps
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Newton's transformation f_w(z)=z−1/(wz^w−1) containing only one complex parameter w (w≠0 or 1) is constructed from the transcendental mapping F(z)=e^{z^w+c}. Although the number of critical points of f_w(z) is countably infinite, a method based on the Valid Critical Point Set, $vcps ={z_{k} ∈C −π< arg (z_{k})⩽π, f′_{w} (z_{k})=0, k∈Z}$ , is discussed for the generation of the generalized Mandelbrot set M of f_w(z). The petal fragments, the multi-period fragments, and the classical “Mandelbrot set” fragments are found in M. The dynamical characteristics of f_w(z) for different values of w are analyzed. The relationship between the parameter w in a classical “Mandelbrot set” fragment and the structure of the corresponding filled-in Julia set is investigated. A large number of novel images of filled-in Julia sets are generated based on the rate of convergence of orbits.
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Chaos and graphicsNewton's method for multiple roots

Abstract

Newton's method and complex dynamical systems

Acta Appl. Math.

Cayley's Problem and Julia Sets

Math. Intell.

Chaos and graphics
Newton's method for multiple roots