A new fast root-finder for black box polynomials

Abstract

Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our black box root-finder involves no coefficients and works for a black box polynomial, defined by an oracle (that is, black box subroutine) for its evaluation. Such root-finders have various benefits, e.g., are particularly efficient where a polynomial can be evaluated fast, say, is the sum of a small number of shifted monomials ${(x-c)}^a$. With incorporation of a fast algorithm by the first author for compression of a disc on the complex plane without losing roots, our root-finder approximates all d complex zeros of a dth degree polynomial $p\left(x\right)$ (aka roots of equation $p\left(x\right)=0$) by using near-optimal Las Vegas expected number of bit-operations,¹ that is, the root-finder is expected to run almost as fast as one accesses the coefficients with a precision required for the solution within a prescribed error bound. The only other known near-optimal polynomial root-finder was presented by the first author at ACM STOC 1995. It is quite involved and has never been implemented, while already in its initial implementation our new root-finder competed with user's choice package of root-finding subroutines MPSolve, according to extensive numerical experiments with standard test polynomials. Furthermore we readily extend our black box root-finder to approximation of the eigenvalues of a matrix in record expected bit operation time, while the root-finder of STOC 1995, using the coefficients of $p\left(x\right)$, does not support such extension.