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 a 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(x) (aka roots of equation p(x)=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(x), does not support such extension.
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