Abstract
Aberth's method for finding the roots of a polynomial was shown to be robust. However, complex arithmetic is needed in this method even if the polynomial is real, because it starts with complex initial approximations. A novel method is proposed for real polynomials that does not require any complex arithmetic within iterations. It is based on the observation that Aberth's method is a systematic use of Newton's method. The analogous technique is then applied to Bairstow's procedure in the proposed method. As a result, the method needs half the computations per iteration than Aberth's method. Numerical experiments showed that the new method exhibited a competitive overall performance for the test polynomials.
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Luk, W.S. Finding roots of a real polynomial simultaneously by means of Bairstow's method. Bit Numer Math 36, 302–308 (1996). https://doi.org/10.1007/BF01731985
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DOI: https://doi.org/10.1007/BF01731985