Inverse power and Durand-Kerner iterations for univariate polynomial root-finding

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

Real Polynomial Root-Finding by Means of Matrix and Polynomial Iterations

Univariate polynomial root-finding is a classical subject, still 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. The subject of devising efficient real root-finders has...

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...

Real root refinements for univariate polynomial equations

Real root finding of polynomial equations is a basic problem in computer algebra. This task is usually divided into two parts: isolation and refinement. In this paper, we propose two algorithms LZ1 and LZ2 to refine real roots of univariate polynomial equations. Our algorithms combine Newton’s method and the secant method to bound the unique solution in an interval of a monotonic convex isolati...

The Amended DSeSC Power Method for Polynomial Root-Finding

Cardinal’s matrix version of the Sebastiao e Silva polynomial root-finder rapidly approximates the roots as the eigenvalues of the associated Frobenius matrix. We preserve rapid convergence to the roots but amend the algorithm to allow input polynomials with multiple roots and root clusters. As in Cardinal’s algorithm, we repeatedly square the Frobenius matrix in nearly linear arithmetic time p...