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 been long and intensively studied. Nevertheless, we propose some novel ideas and techniques and obtain dramatic acceleration of the known numerical algorithms. In order to achieve our progress we exploit the correlation between the computations with matrices and polynomials, randomized matrix computations, and complex plane geometry, extend the techniques of the matrix sign iterations, and use the structure of the companion matrix of the input polynomial. The results of our extensive numerical tests with benchmark polynomials and random matrices are quite encouraging. In particular in these tests we have consistently computed accurate approximations of the real roots of benchmark polynomials of degree up to 1024 by using the IEEE standard double precision. Moreover the number of iterations required for convergence of our algorithms grew very slowly (if at all) as we increased the degree of the univariate input polynomials and the dimension of the input matrices from 64 to 1024.