Summary
We introduce a new three-stage process for calculating the zeros of a polynomial with complex coefficients. The algorithm is similar in spirit to the two stage algorithms studied by Traub in a series of papers. We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order. To obtain additional insight we recast the problem and algorithm into matrix form. The third stage is inverse iteration with the companion matrix, followed by generalized Rayleigh iteration.
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Bauer, F. L.: Beiträge zur Entwicklung numerischer Verfahren für programmgesteuerte Rechenanlagen. II. Direkte Faktorisierung eines Polynoms. Bayer. Akad. Wiss. Math.-Nat. Kl. S.-B., 163–203 (1956).
—— Samelson, K.: Polynomkerne und Iterationsverfahren. Math. Z.67, 93–93 (1957).
Jenkins, M. A.: Three-stage variable-shift iterations for the solution of polynomial equations with a posteriori error bounds for the zeros. Stanford Dissertation, 1969
— — — A three-stage algorithm for real polynomials using quadratic iteration. To appear in SIAM Journal of Numerical Analysis.
—, —: An algorithm for an automatic general polynomial solver. Constructive aspects of the fundamental theorem of algebra, edited by Dejon and Henrici, p. 151–180. New York: Wiley-Interscience 1969
Marden, M.: The geometry of the zeros of a polynomial in a complex variable. Providence, Rhode Island: Amer. Math. Soc. 1949
Ostrowski, A. M.: On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. IV. Generalized Rayleigh quotient for nonlinear elementary divisors. Arch. Rat. Mech. Anal.3, 341–347 (1959).
Traub, J. F.: A class of globally convergent iteration functions for the solution of polynomial equations. Math. Comp.20, 113–138 (1966).
—— Proof of global convergence of an iterative method for calculating complex zeros of a polynomial. Notices Amer. Math. Soc.13, 117 (1966).
——: The calculation of zeros of polynomials and analytic functions. Proceedings of Symposia in Applied Mathematics, volume 19, Mathematical aspects of computer science, p. 138–152. Providence, Rhode Island: Amer. Math. Soc., 1967.
Wielandt, H.: Bestimmung höherer Eigenwerte durch gebrochene Iteration. Ber. der Aerodynamischen Versuchsanstalt Göttingen, B 44/J/37. 1944.
Wilkinson, J. H.: Rounding errors in algebraic processes, chapter 2. Englewood Cliffs, New Jersey: Prentice-Hall 1963
——: The algebraic eigenvalue problem. Oxford: Clarendon Press 1965
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This work was supported in part by the National Science Foundation and the Office of Naval Research.
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Jenkins, M.A., Traub, J.F. A three-stage variable-shift iteration for polynomial zeros and its relation to generalized rayleigh iteration. Numer. Math. 14, 252–263 (1970). https://doi.org/10.1007/BF02163334
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DOI: https://doi.org/10.1007/BF02163334