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A modified Bairstow method for multiple zeros of a polynomial

Author: F. M. Carrano
Journal: Math. Comp. 27 (1973), 781-792
MSC: Primary 65H05
MathSciNet review: 0334492
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Abstract: A modification of Bairstow’s method to find multiple quadratic factors of a polynomial is presented. The nonlinear system of equations of the Bairstow method is replaced by high order partial derivatives of that system. The partials are computed by a repetition of the Bairstow recursion formulas. Numerical results demonstrate that the modified method converges in many cases where the Bairstow method fails due to the multiplicity of the quadratic factor. Rall [4] has described a generalization of Newton’s method for simultaneous nonlinear equations with multiple roots. This may be applied to solve the nonlinear Bairstow equations; however, it fails in some cases due to near-zero divisors. Examples are presented which illustrate the behavior of the author’s algorithm as well as the methods of Rall and Bairstow.

References [Enhancements On Off] (What's this?)

    L. Bairstow, Investigations Relating to the Stability of the Aeroplane, Reports and Memoranda #154, Advisory Committee for Aeronautics, October 1914, pp. 51-64. F. M. Carrano, A Generalized Bairstow Method for Multiple Zeros of a Polynomial, Ph.D. Dissertation, Syracuse University, Syracuse, N.Y., 1969.
  • Peter Henrici, Elements of numerical analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0166900
  • L. B. Rall, Convergence of the Newton process to multiple solutions, Numer. Math. 9 (1966), 23–37. MR 210316, DOI

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Keywords: Multiple zeros of a polynomial, root finding, Bairstow method, multiple quadratic factors of a polynomial
Article copyright: © Copyright 1973 American Mathematical Society