Fast Approximation of Polynomial Zeros and Matrix Eigenvalues

TL;DR

This paper introduces simpler, randomized root-finding algorithms that approximate polynomial zeros and matrix eigenvalues efficiently, outperforming existing methods and applicable even when only black box evaluations are available.

Contribution

The paper presents nearly optimal, randomized subdivision algorithms for polynomial root-finding that are simpler to implement and extend to eigenvalue problems, surpassing previous methods in efficiency and applicability.

Findings

New root-finders outperform MPSolve on standard tests.

Algorithms work with black box polynomial evaluations.

Applicable to fast-evaluated polynomials like Mandelbrot polynomials.

Abstract

We approximate the d complex zeros of a univariate polynomial p(x) of a degree d or those zeros that lie in a fixed region of interest on the complex plane such as a disc or a square. Our divide and conquer algorithm of STOC 1995 supports solution of this problem in optimal Boolean time (up to a poly-logarithmic factor), that is, runs nearly as fast as one can access the coefficients of p with the precision necessary to support required accuracy of the output. That record complexity has not been matched by any other algorithm yet, but our root-finder of 1995 is quite involved and has never been implemented. We present alternative nearly optimal root-finders based on our novel variants of the classical subdivision iterations. Unlike our predecessor of 1995, we require randomization of Las Vegas type, allowing us to detect any output error at a dominated computational cost, but our new root-finders are much simpler to implement than their predecessor of 1995. According to the results of extensive test with standard test polynomials for their preliminary version, which incorporates only a part of our novel techniques, the new root-finders compete and for a large class of inputs significantly supersedes the package of root-finding subroutines MPSolve, which for decades has been user's choice package. Unlike our predecessor of 1995 and all known fast algorithms for the cited tasks of polynomial root-finding, our new algorithms can be also applied to a polynomial given by a black box oracle for its evaluation rather than by its coefficients. This makes our root-finders particularly efficient for polynomials p(x) that can be evaluated fast such as the Mandelbrot polynomials or those given by the sum of a small number of shifted monomials. Our algorithm can be readily extended to fast approximation of the eigenvalues of a matrix or a matrix polynomial.