Summary
Suppose all zeros of a polynomialp but one are known to lie in specified circular regions, and the value of the logarithmic derivativep′p −1 is known at a point. What can be said about the location of the remaining zero? This question is answered in the present paper, as well as its generalization where several zeros are missing and the values of some derivatives of the logarithmic derivative are known. A connection with a classical result due to Laguerre is established, and an application to the problem of locating zeros of certain transcendental functions is given. The results are used to construct (i) a version of Newton's method with error bounds, (ii) a cubically convergent algorithm for the simultaneous approximation of all zeros of a polynomial. The algorithms and their theoretical foundation make use of circular arithmetic, an extension, based on the theory of Moebius transformations, of interval arithmetic from the real line to the extended complex plane.
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Gargantini, I., Henrici, P. Circular arithmetic and the determination of polynomial zeros. Numer. Math. 18, 305–320 (1971). https://doi.org/10.1007/BF01404681
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DOI: https://doi.org/10.1007/BF01404681