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The Largest Roots of the Mandelbrot Polynomials

  • Robert M. Corless
  • Piers W. Lawrence
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 50)

Abstract

This paper gives some details of the experimental discovery and partial proof of a simple asymptotic development for the largest magnitude roots of the Mandelbrot polynomials defined by p 0(z) = 1 and \(p_{n+1}(z) = zp_{n}^{2}(z) + 1\).

Key words

Asymptotic expansion Eigenvalues Mandelbrot polynomials Periodic points Polynomial zeros 

Mathematics Subject Classifications (2010)

Primary 37F10 Secondary 37F45 41A60 

Notes

Acknowledgements

The discussion with Neil Calkin, during the course of the JonFest DownUnder conference, was very helpful. One might even say that had those discussions not taken place, RMC would not have been bitten (by the problem). Peter Borwein pointed out the work of Aczél on functional differential equations. This research work was partially supported by the Natural Sciences and Engineering Research Council of Canada, the University of Western Ontario (and in particular by the Department of Applied Mathematics), the Australian National University (special thanks to the Coffee Gang, Joe Gani, Mike Osborne, David Heath, Ken Brewer, and the other regulars, for their interest), and by the University of Newcastle.

References

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    Aczél, J.: On history, applications and theory of functional equations. In: Functional Equations: History, Applications and Theory. D. Reidel, Dordrecht (1984)Google Scholar
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    Corless, R.M., Fillion, N.: A graduate introduction to numerical methods. Springer, to appear (2013)Google Scholar
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    Lawrence, P.W., Corless, R.M.: Stability of rootfinding for barycentric Lagrange interpolants. (submitted)Google Scholar
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    Lawrence, P.W., Corless, R.M.: Numerical stability of barycentric Hermite root-finding. In: Proceedings of the 4th International Workshop on Symbolic-Numeric Computation, pp. 147–148. San Jose, USA (2011)Google Scholar
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    Lawrence, P.W., Corless, R.M., Jeffrey, D.J.: Mandelbrot polynomials and matrices (2012) (in preparation)Google Scholar
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    Sloane, N.J.A.: An on-line version of the encyclopedia of integer sequences. Electron. J. Combin. 1, 1–5 (1994). http://oeis.org/ Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Applied MathematicsThe University of Western OntarioLondonCanada

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