Fast Cauchy Sum Algorithms for Polynomial Zeros and Matrix Eigenvalues.

Given a black box oracle that evaluates a univariate polynomial p ( x ) of a degree d , we seek its zeros, aka the roots of the equation . At FOCS 2016, Louis and Vempala approximated within an absolutely largest zero of such a real-rooted polynomial at the cost of the evaluation of Newton’s ratio at points x and then extended this algorithm to approximation of an absolutely largest eigenvalue of a symmetric matrix at a record Boolean cost. By applying distinct approach and techniques we obtain much more general results at the same computational cost. Our use of Cauchy integrals and randomization is non-trivial and pioneering in this field. Somewhat surprisingly, the Boolean complexity of the accelerated versions of our algorithms in [ 25 , 26 ] reached below the known lower bounds on the Boolean complexity of polynomial root-finding.