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Numerical Computation of Polynomial Zeros by Means of Aberth's Method

D. Bini
Journal / Anthology

Numerical Algorithms
Year: 1996
Volume: 13
Page range: 179-200

An algorithm for computing polynomial zeros, based on Aberth's method, is presented. The starting approximations are chosen by means of a suitable application of Rouchés theorem. More precisely, an integer q > 1 and a set of annuli Ai, i=1,...,q, in the complex plane, are determined together with the number ki of zeros of the polynomial contained in each annulus Ai. As starting approximations we choose ki complex numbers lying on a suitable circle contained in the annulus Ai, for i=1,...,q. The computation of Newton's correction is performed in such a way that overflow situations are removed. A suitable stop condition, based on a rigorous backward rounding error analysis, guarantees that the computed approximations are the exact zeros of a "nearby" polynomial. This implies the backward stability of out algorithm. We provide a Fortran 77 implementation of the algorithm which is robust against overflow and allows us to deal with polynomials of an degree, not necessarily monic, whose zeros and coefficients are representable as floating point numbers. In all the tests performed with more than 1000 polynomials having degrees from 10 up to 25,600 and randomly generated coefficients, the Fortran 77 implementation of out algorithm computed conditioning theorems to all the zeros within the relative precision allowed by the classical conditioning theorems with 11.1 average iterations. In the worst case the number of iterations needed has been at most 17. Comparisons with available public domain software and with the algorithm PA16AD of Harwell re performed and show the effectiveness of out approach. A multiprecision implementation in Mathematica is presented together with the results of the numerical tests performed.

*Applied Mathematics > Numerical Methods
*Mathematics > Algebra > Polynomials