Abstract
This work examines the computational complexity of a homotopy algorithm in approximating all roots of a complex polynomialf. It is shown that, probabilistically, monotonic convergence to each of the roots occurs after a determined number of steps. Moreover, in all subsequent steps, each rootz is approximated by a complex numberx, where ifx 0 =x, x j =x j−1 −f(x j−1)/f′(x j−1),j = 1, 2,⋯, then |x j −z| < (1/|x 0 −z|)|x j−1−z|2.
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Renegar, J. On the cost of approximating all roots of a complex polynomial. Mathematical Programming 32, 319–336 (1985). https://doi.org/10.1007/BF01582052
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DOI: https://doi.org/10.1007/BF01582052