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On a compensated Ehrlich-Aberth method for the accurate computation of all polynomial roots

Abstract : In this article, we use the complex compensated Horner's method to derive a compensated Ehrlich-Aberth method for the accurate computation of all roots of a polynomial. In particular, under suitable conditions, we prove that the limiting accuracy for the compensated Ehrlich-Aberth iterations is as accurate as if computed in twice the working precision and then rounded into the working precision. Moreover, we derive a running error bound for the complex compensated Horner's and use it to form robust stopping criteria for the compensated Ehrlich-Aberth iterations. Finally, extensive numerical experiments illustrate that the backward and forward errors of the root approximations computed via the compensated Ehrlich-Aberth method are similar to those obtained with a quadruple precision implementation of the Ehrlich-Aberth method with a significant speed-up in terms of the computation time.
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https://hal.archives-ouvertes.fr/hal-03335604
Contributor : Stef Graillat Connect in order to contact the contributor
Submitted on : Monday, September 6, 2021 - 1:21:30 PM
Last modification on : Wednesday, March 30, 2022 - 9:47:38 AM
Long-term archiving on: : Tuesday, December 7, 2021 - 6:42:21 PM

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Thomas Cameron, Stef Graillat. On a compensated Ehrlich-Aberth method for the accurate computation of all polynomial roots. Electronic Transactions on Numerical Analysis, Kent State University Library, 2022, 55, pp.401-423. ⟨10.1553/etna_vol55s401⟩. ⟨hal-03335604⟩

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