TR-2014011: Real Polynomial Root-Finding: New Advances

TR-2014007: Real Polynomial Root-Finding by Means of Matrix and Polynomial Iterations

Recently we proposed to extend the matrix sign classical iteration to the approximation of the real eigenvalues of a companion matrix of a polynomial and consequently to the approximation of its real roots. In our present paper we advance this approach further by combining it with the alternative square root iteration for polynomials and also show a variation using repeated squaring in polynomi...

TR-2006007: Root-Finding with Eigen-Solving

We survey and extend the recent progress in polynomial root-finding based on reducing the problem to eigen-solving for highly structured generalized companion matrices. In particular we cover the selection of the eigen-solvers and the matrices and the resulting benefits based on exploiting matrix structure. No good estimates for global convergence of the basic eigen-solvers have been proved, bu...

TR-2002020: Inverse Power and Durand-Kerner Iterations for Univariate Polynomial Root-Finding

New method for bounding the roots of a univariate polynomial

We present a new algorithm for computing upper bounds for the maximum positive real root of a univariate polynomial. The algorithm improves complexity and accuracy of current methods. These improvements do impact in the performance of methods for root isolation, which are the first step (and most expensive, in terms of computational effort) executed by current methods for computing the real roo...

Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving

Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial has no nonreal roots, but typically nonreal roots are much more numerous than the real ones. We dramatically accelerate the known algorithms in this case by...