The Ehrlich-Aberth method is a root-finding algorithm that simultaneously approximates all roots of a polynomial (Ehrlich 1967, Aberth 1973). For a polynomial of polynomial degree and a vector of pairwise distinct approximations , its simultaneous update is

where every right-hand side uses the old vector and the denominators must be nonzero. This is the simultaneous, or Jacobi, variant. A cyclic variant uses each newly computed coordinate in subsequent updates of the same step.

Grabbel et al. (2026) proved that the simultaneous variant has an attracting periodic orbit of period 4 for an open set of polynomials of every polynomial degree . An open set of starting vectors converges to such an orbit instead of to a vector of roots. Thus this variant is not generally convergent, where general convergence means convergence to a root vector for an open set that is dense of starting vectors for an open set that is dense of polynomials. The authors report interval arithmetic certification and an independent verification of the attracting cycle. The result does not establish the corresponding failure for the cyclic variant. Grabbel et al. (2026) report that Claude assisted with search and certification code and with designing and analyzing mathematical experiments.

See also

Newton's Method, Periodic Orbit, Root-Finding Algorithm

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References

Aberth, O. "Iteration Methods for Finding All Zeros of a Polynomial Simultaneously." Math. Comput. 27, 339-344, 1973. https://doi.org/10.1090/S0025-5718-1973-0329236-7.Ehrlich, L. W. "A Modified Newton Method for Polynomials." Commun. ACM 10, 107-108, 1967. https://doi.org/10.1145/363067.363115.Grabbel, T.; Srkalovic, N.; and Schleicher, D. "The Ehrlich-Aberth Root Finder Is Not Generally Convergent." 1 Nov 2026. https://arxiv.org/abs/2611.02143v1.

Cite this as:

Weisstein, Eric W. "Ehrlich-Aberth Method." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Ehrlich-AberthMethod.html

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