A Root-finding Algorithm which uses the first few terms of the Taylor Series in the vicinity of a
suspected Root to zero in on the root. The Taylor Series of a function about the point is
given by

(1) |

(2) |

(3) |

Unfortunately, this procedure can be unstable near a horizontal Asymptote or a Local Minimum. However, with a
good initial choice of the Root's position, the algorithm can by applied iteratively to obtain

(4) |

The error
after the st iteration is given by

(5) |

But

(6) | |||

(7) |

so

(8) |

and (5) becomes

(9) |

A Fractal is obtained by applying Newton's method to finding a Root of (Mandelbrot 1983, Gleick 1988,
Peitgen and Saupe 1988, Press *et al. *1992, Dickau 1997). Iterating for a starting point gives

(10) |

Coloring the Basin of Attraction (the set of initial points which converge to the same Root) for each Root a different color then gives the above plots, corresponding to , 3, 4, and 5.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 18, 1972.

Acton, F. S. Ch. 2 in *Numerical Methods That Work.* Washington, DC: Math. Assoc. Amer., 1990.

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 963-964, 1985.

Dickau, R. M. ``Basins of Attraction for Using Newton's Method in the Complex Plane.'' http://forum.swarthmore.edu/advanced/robertd/newtons.html.

Dickau, R. M. ``Variations on Newton's Method.'' http://forum.swarthmore.edu/advanced/robertd/newnewton.html.

Dickau, R. M. ``Compilation of Iterative and List Operations.'' *Mathematica J.* **7**, 14-15, 1997.

Gleick, J. *Chaos: Making a New Science*. New York: Penguin Books, plate 6 (following pp. 114) and p. 220, 1988.

Householder, A. S. *Principles of Numerical Analysis.*ew York: McGraw-Hill, pp. 135-138, 1953.

Mandelbrot, B. B. *The Fractal Geometry of Nature.* San Francisco, CA: W. H. Freeman, 1983.

Ortega, J. M. and Rheinboldt, W. C. *Iterative Solution of Nonlinear Equations in Several Variables.*
New York: Academic Press, 1970.

Peitgen, H.-O. and Saupe, D. *The Science of Fractal Images.* New York: Springer-Verlag, 1988.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Newton-Raphson Method Using Derivatives''
and ``Newton-Raphson Methods for Nonlinear Systems of Equations.'' §9.4 and 9.6 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.*
Cambridge, England: Cambridge University Press, pp. 355-362 and 372-375, 1992.

Ralston, A. and Rabinowitz, P. §8.4 in *A First Course in Numerical Analysis, 2nd ed.* New York: McGraw-Hill,
1978.

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1999-05-25