3 research outputs found

    Geometry of Polynomials and Root-Finding via Path-Lifting

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    Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate zeros, similar to those studied by Smale, Shub, Kim, and others. Given any polynomial, this simple algorithm always converges to a root, except on a finite set of initial points lying on a circle of a given radius. Specifically, the algorithm we analyze consists of iterating zf(z)tkf(z0)f(z)z - \frac{f(z)-t_kf(z_0)}{f'(z)} where the tkt_k form a decreasing sequence of real numbers and z0z_0 is chosen on a circle containing all the roots. We show that the number of iterates required to locate an approximate zero of a polynomial ff depends only on logf(z0)/ρζ\log|f(z_0)/\rho_\zeta| (where ρζ\rho_\zeta is the radius of convergence of the branch of f1f^{-1} taking 00 to a root ζ\zeta) and the logarithm of the angle between f(z0)f(z_0) and certain critical values. Previous complexity results for related algorithms depend linearly on the reciprocals of these angles. Note that the complexity of the algorithm does not depend directly on the degree of ff, but only on the geometry of the critical values. Furthermore, for any polynomial ff with distinct roots, the average number of steps required over all starting points taken on a circle containing all the roots is bounded by a constant times the average of log(1/ρζ)\log(1/\rho_\zeta). The average of log(1/ρζ)\log(1/\rho_\zeta) over all polynomials ff with dd roots in the unit disk is O(d){\mathcal{O}}({d}). This algorithm readily generalizes to finding all roots of a polynomial (without deflation); doing so increases the complexity by a factor of at most dd.Comment: 44 pages, 12 figure

    Computing the Newtonian Graph

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    AbstractIn his study of Newton's root approximation method, Smale (1985) defined the Newtonian graph of a complex univariate polynomialf. The vertices of this graph are the roots offandf′and the edges are the degenerate curves of flow of the Newtonian vector fieldNf(z) = −f(z)/f′(z). The embedded edges of this graph form the boundaries of root basins in Newton's root approximation method. The graph defines a treelike relation on the roots offandf′, similar to the linear order whenfhas only real roots.We give an efficient algebraic algorithm based on cell decomposition to compute the Newtonian graph. The resulting structure can be used to query whether two points in C are in the same basin. This suggests a modified version of Newton's method in which one can test whether a step has crossed a basin boundary. We show that this modified version does not necessarily converge to a root.Stefánsson (1995) has recently extended this algorithm to handle rational and algebraic functions without a significant increase in complexity. He has shown that the Newtonian graph tesselates the associated Riemann surface and can be used in conjunction with Euler's formula to give anNCalgorithm to calculate the genus of an algebraic curve

    Computing the Newtonian Graph (Extended abstract)

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    A polynomial f\in \complex[z] defines a vector field Nf(z)=f(z)/f(z)N_f(z) = -f(z)/f'(z) on \complex. Certain degenerate curves of flow in NfN_f give the edges of the Newtonian graph, as defined by \cite{Sma85}. These give a relation between the roots of ff and ff', much similar to the linear order, when ff has real roots only. We give a purely algebraic algorithm to compute the Newtonian graph and the basins of attraction in the Newtonian field. The resulting structure can be used to query whether two points in \complex are within the same basin of attraction. This gives us an algebraic approach to root-finding using Newton's method. This method extends to rational functions and more generally to any functions on \complex whose flow is algebraic over \complex(e)
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