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arXiv · 1703.05847

Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

Marvin Randig·Dierk Schleicher·Robin Stoll
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Abstract

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$ and $O(d\ln^3 d)$ (measuring complexity in terms of number of Newton iterations or computing time). All computations were performed successfully on standard desktop computers built between 2007 and 2012.

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Marvin Randig, Dierk Schleicher, Robin Stoll. 2017-03-16. Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees. https://doi.org/10.1016/j.cam.2023.115427

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