3 research outputs found
Geometry of Polynomials and Root-Finding via Path-Lifting
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 where the form a decreasing sequence of
real numbers and 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 depends only on (where is
the radius of convergence of the branch of taking to a root
) and the logarithm of the angle between 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 , but only on the geometry of the
critical values.
Furthermore, for any polynomial 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 . The
average of over all polynomials with roots in the
unit disk is . This algorithm readily generalizes to
finding all roots of a polynomial (without deflation); doing so increases the
complexity by a factor of at most .Comment: 44 pages, 12 figure
Computing the Newtonian Graph
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)
A polynomial f\in \complex[z] defines a vector field on \complex. Certain degenerate curves of flow in give the edges of the Newtonian graph, as defined by \cite{Sma85}. These give a relation between the roots of and , much similar to the linear order, when 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)