Abstract
Recently Smale has obtained probabilistic estimates of the cost of computing a zero of a polynomial using a global version of Newton's method. Roughly speaking, his result says that, with the exception of a set of polynomials where the method fails or is very slow, the cost grows as a polynomial in the degree. He also asked whether similar results hold for PL homotopy methods.
This paper gives such a result for a special algorithm of the PL homotopy type devised by Kuhn. Its main result asserts that the cost of computing some zero of a polynomial of degreen to an accuracy of ε (measured by the number of evaluations of the polynomial) grows no faster than O(n 3 log2(n/ε)). This is a worst case analysis and holds for all polynomials without exception.
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References
M. Kojima, H. Nishino and N. Arima, “A PL homotopy for finding all the roots of a polynomial”,Mathematical Programming 16 (1979) 37–62.
H. Kuhn, “A new proof of the fundamental theorem of algebra”,Mathematical Programming Study 1 (1974) 148–158.
H. Kuhn, “Finding roots of polynomials by pivoting”, in: S. Karamardian, ed.,Fixed points: Algorithms and applications (Academic Press, New York, 1977), pp. 11–40.
S. Smale, “The fundamental theorem of algebra and complexity theory,Bulletin of the American Mathematical Society 4 (1981) 1–36.
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This work was supported, in part, by National Science Foundation Grant MCS79-10027 and, in part, by a fellowship of the Guggenheim Foundation.
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Kuhn, H.W., Wang, Z. & Xu, S. On the cost of computing roots of polynomials. Mathematical Programming 28, 156–163 (1984). https://doi.org/10.1007/BF02612355
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DOI: https://doi.org/10.1007/BF02612355