Newton-Raphson's Method, picture generator by S. Kennedy


s_kennedy@lineone.net
http://website.lineone.net/~s_kennedy/index.htm


This is a simple programme to illustrate the choatic paths generated within the Newton-Raphson method of solving a small group of complex equations.

The method is a way of finding the roots (solutions) of an equation by starting with an initial guess which leads to a better guess which in turn leads to an even better guess and so on until the guesses converge to the root. The fascinating thing is that, because most equations have more than one solution, the solution that the method finds is dependant on the initial guess and in fact two initial guesses that are very close can lead to wildly different solutions. This programme looks at about 170000 initial guesses and draws an Argand diagram that is colour coded depending on which solution the guess leads to. 

The family of graphs that I have concentrated on is of the form 

		x^n - 1 = 0

where x is a complex number, and n is between 2 and 9.

There are n solutions to the equation with each solution lying on the unit circle around the origin.   




I was initially tempted into writing this program after reading the book Chaos by James Gleick (ISBN: 0 7474 0413 5) which has a colour plate showing the Argand diagram of Newton's method on x^4 - 1 = 0. I read the book while at college in 1990 and wrote a couple of very simple programs using AMOS on my Amiga computer to draw the Mandelbrot and Julia sets, Sierpinski's gasket and the Koch curve but never got around to doing this one until 1997/98 on a PC in Visual Basic. The program has always been just for fun and to prove to myself that I could do it but I have recently re-written it using Delphi Developer Version 2 and added some slight functionally to it (although I'm sure much more could be done if I get the time and inclination). 

I would definitely recommend James Gleick's Chaos to anyone interested in learning about the subject of Chaos and Fractals.

If you have any comments about the program (or if this readme contains any errors) please mail me at s_kennedy@lineone.net. My website contains slightly more information about the Newton-Raphson method if you are interested.


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Version history

V1.01 		02.06.01 (Delphi v2)
		Added co-ordinate buttons with increments of 0.01
		Added Start button (previously just clicked on main form)

V1.00		28.09.98 (Delphi v1)
		Re-written version of the programme I created using Visual Basic. 

Original	1997/98 (Visual Basic)
