Numerical computation of polynomial zeros by means of Aberth's method
 Dario Andrea Bini
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Abstract
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 Rouché's theorem. More precisely, an integerq ≥ 1 and a set of annuliA _{i},i=1,...,q, in the complex plane, are determined together with the numberk _{i} of zeros of the polynomial contained in each annulusA _{i}. As starting approximations we choosek _{i} complex numbers lying on a suitable circle contained in the annulusA _{i}, fori=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 our algorithm. We provide a Fortran 77 implementation of the algorithm which is robust against overflow and allows us to deal with polynomials of any 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 our algorithm computed approximations 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 are performed and show the effectiveness of our approach. A multiprecision implementation in MATHEMATICA^{ ™ } is presented together with the results of the numerical tests performed.
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 Title
 Numerical computation of polynomial zeros by means of Aberth's method
 Journal

Numerical Algorithms
Volume 13, Issue 2 , pp 179200
 Cover Date
 19960201
 DOI
 10.1007/BF02207694
 Print ISSN
 10171398
 Online ISSN
 15729265
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 polynomial zeros
 Aberth's method
 numerical test
 starting approximations
 65H05
 65Y20
 Industry Sectors
 Authors

 Dario Andrea Bini ^{(1)}
 Author Affiliations

 1. Dipartimento di Matematica, Università di Pisa, via Buonarroti 2, I56127, Pisa, Italy