
Complexity Analysis of Root Clustering for a Complex Polynomial
Let F(z) be an arbitrary complex polynomial. We introduce the local root...
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Efficiently Computing Real Roots of Sparse Polynomials
We propose an efficient algorithm to compute the real roots of a sparse ...
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Implementation of a NearOptimal Complex Root Clustering Algorithm
We describe Ccluster, a software for computing natural ϵclusters of com...
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Root Radii and Subdivision for Polynomial RootFinding
The recent subdivision algorithms for univariate polynomial Complex Root...
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A Globally Convergent Newton Method for Polynomials
Newton's method for polynomial root finding is one of mathematics' most ...
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The SecantNewton Map is Optimal Among Contracting n^th Degree Maps for n^th Root Computation
Consider the problem: given a real number x and an error bound ϵ, find a...
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Nearly Optimal Hybrid Polynomial Rootfinders
Univariate polynomial rootfinding has been studied for four millennia a...
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A NearOptimal Subdivision Algorithm for Complex Root Isolation based on the Pellet Test and Newton Iteration
We describe a subdivision algorithm for isolating the complex roots of a polynomial F∈C[x]. Given an oracle that provides approximations of each of the coefficients of F to any absolute error bound and given an arbitrary square B in the complex plane containing only simple roots of F, our algorithm returns disjoint isolating disks for the roots of F in B. Our complexity analysis bounds the absolute error to which the coefficients of F have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square B, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is nearoptimal. For the benchmark problem, namely, to isolate all the roots of a polynomial of degree n with integer coefficients of bit size less than τ, our algorithm needs Õ(n^3+n^2τ) bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divideandconquer techniques such as Schönhage's splitting circle technique. Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a "softtest" to count the number of roots in a disk. Using Schröder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divideconquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is selfcontained: we provide pseudocode for all subroutines used by our algorithm.
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