New Progress in Classic Area: Polynomial Root-squaring and Root-finding
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The DLG root-squaring iterations by Dandelin 1826, Lobachevsky 1834, and Graeffe 1837 have been the main approach to root-finding for a univariate polynomial p(x) in the 19th century and beyond, but not so nowadays because of severe numerical instability of these iterations. We circumvent this problem based on simple but novel links of the iterations at first to the evaluation of high order derivatives of the ratio p'(x)/p(x) at x=0 and then further to computing the power sums of the reciprocals of the roots of p(x). By expressing the power sums as Cauchy integrals over a sufficiently large circle on a complex plane, we devise fast and numerically stable algorithms for the DLG iterations and for their more recent extension to Fiedler-Gemignani's iterations. Our algorithms can be applied to a black box polynomial p(x) – given by a black box for its evaluation rather than by its coefficients, which enables important computational benefits, including efficient recursive as well as concurrent approximation of a selected set of the zeros of p(x) or even all of the zeros.
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