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Dynamical Systems : Newton's method

At school we had this theme to make a short essay (in FRENCH) on it. For the first year I chose Newton's method.

What is it?

Newton's Method is a numerical technique for finding zeros of a function.
It uses the formula :

xn+1 = xn - f(xn) / f ' (xn)

to calculate a new approximate zero xn+1 given an old approximation xn. Starting with some initial guess for the zero, we can repeatedly apply this equation to generate a sequence of better and better approximations to the zero.

If you want to know more about it :

If you are interested in the entire essay (2 Mo in Word format (.doc)), please send an e-mail asking me to send it to you. Please mention your name, surname and your residential address : click here. These informations won't be known by somebody else than me, it's just to control how this document is downloaded. Thanks for the interest you bring to my work...

Two very simple examples :

With Newton's method, you can create beautiful pictures which are also known as Fractals.

1. (z-1).(z+1).(z-i).(z+i) :

2. (z-1)^2.(z+1)^2 :

Some more WEB references :

David E. Joyce 's pages on Newton basins

FractInt is a freeware fractal generator and in particular Newton Basin generator.

David Sumner and his on-line calculator; Another version using Newton's method to approximate real roots of real
polynomials.

Doug Ingalls 's page on Newton's method and self-similarity .

Bill Motyl and his tutorial on Newton's method (illustrates the method to find square roots with a JavaScript applet).

Matrix Operating System: a program to calculate with matrixes

What is a matrix ?

A matrix is a simple mathematical object, which really looks like a table.

Examples :

[1 2 4]
[8 3 2] is a 2 (rows)* 3(columns) matrix

    [4    8    2]
    [2    3    1]   is a 3*3 matrix
    [1    5    2]

Presentation of the program : MatriX Operating System (MX-OS v1.5)

This program has been realised with my friend Bronöel Yves as our computer sciences homework in 1997, on Turbo Pascal 7. It takes us quite a long time to make this program as complete as possible, and we are really happy to present it to you on the World Wide Web.
MX-OS is able to do nearly all basic operations (+, *, reverse, Det, Rank...) on real matrixes, using only lists (!) to code matrixes.
Why the use of lists instead of tables ?
The reason is quite simple, the main theme of our lessons was Lists, and this homework was made to illustrate this theme of course.
If :
1. you are interested in the executable program, click here : I want to download MX-OS
2. you want all the commented sources (in french), click here : The sources, now! Feel free to