From a dynamical point of view applied to complex polynomials, we study a number of root–finding iterative methods. We consider Newton’s method, Newton’s method for multiple roots, Jarratt’s method, the super–Halley method, the convex as well as the double convex acceleration of Whittaker’s method, the methods of Chebyshev, Stirling, and Steffensen, among others. Since all of the iterative root...

Root finding is the most time-consuming stage of McEliece cryptosystem decryption. The best method to find the zeroes of a polynomial for cryptographic parameters is the Berlekamp Trace Algorithm (BTA). Our idea is to mix BTA with ad-hoc methods proposed by Zinoviev. We obtain a significant gain in terms of time complexity for finding roots and so we decrease McEliece decryption time. This pape...

We analyze a path-lifting algorithm for finding an approximate zero of a complex polynomial, and show that for any polynomial with distinct roots in the unit disk, the average number of iterates this algorithm requires is universally bounded by a constant times the log of the condition number. In particular, this bound is independent of the degree d of the polynomial. The average is taken over ...

Computing roots of scalar polynomials as the eigenvalues of Frobenius companion matrices using backward stable eigenvalue algorithms is a classical approach. The introduction of new families of companion matrices allows for the use of other matrices in the root-finding problem. In this paper, we analyze the backward stability of polynomial root-finding algorithms via Fiedler companion matrices....

Polynomial root finding is a classical and well studied problem in mathematics with practical applications in many areas. Various relevant questions can be reformulated into the task of finding all common roots of a given system of polynomials in several variables, the so-called algebraic variety or zero set. Traditionally, root finding algorithms are implemented in exact arithmetics over algeb...