(1934-1999)
One of the more simpler concepts in chaos is Newton's method (frequently called the Newton method) for finding roots of equations. In an analysis by A. Cayley, he found that initial or starting values which were negative converged to -1 (the white pixel in the center of the small red dot on the left in the figure above) while intial or starting value which were positive converged to +1 (the white pixel inside the small red dot on the right). Values on the axis corresponding to x = 0 + ai, did not converge to either, and were considered as a poor choice of the initial value. With the advent of the digital computer our knowledge has expanded and it was realized that the concepts of chaos could be extended to the Newton method and are prevalent in this simple case.
The results presented here are the part of an ongoing investigation and as new results are determined this site will be updated.
The tutorials are adequate to explain the Newton Method and no effort will be made to present duplicate what can be readily found on the internet.
Click back or next to go through this site page by page or go directly to:
- Introduction to Newton's method
- The graph for square root of 1
- The graph for cube root of of 1
- The graph for fourth root of 1
- The graph for fifth root of 1
- Gallery of graphs for the above cases
- Gallery for the polynomial (x+1)x(x-1) = 0
- Two methods of calculating convergence.
- FORTRAN program for studying the equation.
- References to the Newton's method
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