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A Fractal Eigenvector

Profile image of Neil CalkinNeil Calkin

2021, arXiv (Cornell University)

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20 pages

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Abstract

The recursively-constructed family of Mandelbrot matrices Mn for n = 1, 2, . . . have nonnegative entries (indeed just 0 and 1, so each Mn can be called a binary matrix) and have eigenvalues whose negatives -λ = c give periodic orbits under the Mandelbrot iteration, namely z k = z 2 k-1 + c with z0 = 0, and are thus contained in the Mandelbrot set. By the Perron-Frobenius theorem, the matrices Mn have a dominant real positive eigenvalue, which we call ρn. This article examines the eigenvector belonging to that dominant eigenvalue and its fractal-like structure, and similarly examines (with less success) the dominant singular vectors of Mn from the singular value decomposition. This article seeks to explain a visual curiosity, namely that of Figure , where visible structures seem to repeat, slightly transformed, at smaller scales. But what do we mean by an "explanation?" What constitutes a mathematical explanation? By the way, those structures are not the result of rounding errors, in spite of our doing the computation only in standard hardware precision floating-point arithmetic. We will see a connection with the Mandelbrot set. After seeing the name Mandelbrot get involved, the reader might no longer be surprised that repeating transformed

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References (19)

  1. Bini, D. A., Fiorentino, G. (2000). Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms, 23(2): 127-173. doi.org/10.1023/a:1019199917103.
  2. Bini, D. A., Robol, L. (2014). Solving secular and polynomial equations: A multiprecision algorithm. J. Comput. Appl. Math, 272: 276-292. doi.org/10.1016/j.cam.2013.04.037.
  3. Boyd, J. P. (2014). Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Nu- merical Rootfinders, Perturbation Series, and Oracles. Philadelphia, PA: SIAM.
  4. Chan, E. Y. S. (2016). A comparison of solution methods for Mandelbrot-like polynomials. Master's thesis, Western University, London, Canada.
  5. Chan, E. Y. S., Corless, R. M., Gonzalez-Vega, L., Sendra, J. R., Sendra, J. (2019). Algebraic linearizations of matrix polynomials. Linear Algebra Appl, 563: 373-399. doi.org/doi.org/10.1016/j.laa.2018.10.028.
  6. Corless, R. M., Lawrence, P. W. (2013). The largest roots of the Mandelbrot polynomials. In: Bailey, D. H., Bauschke, H. H., Borwein, P., Garvan, F., Théra, M., Vanderwerff, J. D., Wolkowicz, H., eds., Computational and Analytical Mathematics. New York, NY: Springer, pp. 305-324.
  7. Davis, T. A. (2019). Algorithm 1000: Suitesparse:graphblas: Graph algorithms in the language of sparse linear algebra. ACM Trans. Math. Software, 45(4): 1-25. doi.org/10.1145/3322125.
  8. Dietzenbacher, E. (1988). Perturbations of matrices: A theorem on the Perron vector and its applications to input-output models. J. Economics, 48(4): 389-412. doi.org/10.1007/bf01227544.
  9. Eidelman, Y., Gohberg, I., Olshevsky, V. (2005). Eigenstructure of order-one-quasiseparable matrices. three-term and two-term recurrence relations. Linear Algebra Appl, 405: 1-40. doi.org/10.1016/j.laa.2005.02.039.
  10. Greenbaum, A., Li, R.-C., Overton, M. L. (2020). First-order perturbation theory for eigenvalues and eigenvectors. SIAM Rev, 62(2): 463-482. doi.org/10.1137/19m124784x.
  11. Hogben, L., ed. (2014). Handbook of Linear Algebra, 2nd ed. Boca Raton, FL: Taylor & Francis Group.
  12. MacCluer, C. R. (2000). The many proofs and applications of Perron's theorem. SIAM Rev, 42(3): 487- 498. doi.org/10.1137/s0036144599359449.
  13. Morton, P., Patel, P. (1994). The Galois theory of periodic points of polynomial maps. Proc. London Math. Soc. (3), 68(2): 225-263. doi.org/10.1112/plms/s3-68.2.225.
  14. OEIS Foundation Inc. (2021). The On-Line Encyclopedia of Integer Sequences. Published electronically at oeis.org.
  15. Poincaré, H. (1905). Science and Hypothesis. New York, NY: The Walter Scott Publishing Co., Ltd. en.wikisource.org/wiki/Science and Hypothesis.
  16. Schleicher, D. (2017). Internal addresses of the Mandelbrot set and Galois groups of polynomials. Arnold Math. J, 3(1): 1-35. doi.org/10.1007/s40598-016-0042-x.
  17. Shampine, L. F., Corless, R. M. (2000). Initial value problems for ODEs in problem solving environ- ments. J. Comput. Appl. Math, 125(1-2): 31-40. doi.org/10.1016/s0377-0427(00)00456-8.
  18. Tyaglov, M. (2017). Self-interlacing polynomials. Linear Algebra Appl, 535: 12-34. doi.org/10.1016/j.laa.2017.08.020.
  19. Vivaldi, F., Hatjispyros, S. (1992). Galois theory of periodic orbits of rational maps. Nonlinearity, 5(4): 961-978. doi.org/doi:10.1088/0951-7715/5/4/007.

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