The following fractal images were created by applying the Newton-Raphson Method to various real-valued polynomials. Each pixel represents a point in the complex plane used by the Newton-Raphson Method as an initial approximation. The color of the pixel indicates which of the roots the point converged to, while the intensity is proportional to the number of iterations required to approximate that root to within a specified tolerance (.0001). Brighter pixels converged to the root more quickly than darker pixels, while white pixels did not converge at all after a specified number of iterations (usually 20 or 30), or encountered a derivative evaluation that was very close to zero.

The C++ program that generated these images was derived from Algorithm 2.3 (Newton-Raphson, pp. 57-58) and Algorithm 2.7 (Horner's, pp. 84-85) in Richard L. Burden and J. Douglas Fairies' *Numerical Analysis* (5th edition).

All images are 400 x 400 pixel GIFs. The real axis runs horizontally through the center of each image, while the imaginary axis runs vertically through the image (though not necessarily through the center).

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`x`^{3}+ 9`x`^{2}- 18`x`+ 6 (Laguerre polynomial #3) [16.4K] `x`^{4}- 16`x`^{3}+ 72`x`^{2}- 96`x`+ 24 (Laguerre polynomial #4) [18.1K]- -
`x`^{5}+ 25`x`^{4}- 200`x`^{3}+ 600`x`^{2}- 600`x`+ 120 (Laguerre polynomial #5) [21.9K] `x`^{3}-`x`- 1 (Exercise 3, Section 2.5) [22.4K]- 600
`x`^{4}- 550`x`^{3}+ 200`x`^{2}- 20`x`- 1 (Exercise 14, Section 2.6) [21.9K] `x`^{5}- 1 [47.7K]- 3
`x`^{3}+ 2`x`^{2}+`x`[16.4K]

©1996 Bryan Krofchok

Last updated: 28 January 2012