Newton’s method is one of the most powerful and well-known numerical methods for solving a root-finding problem f(x) = 0. The algorithm of Newton’s method is as following:
float Newton (float p0, float Tolerance) //initial approximation p0
{
float p;
while(1)
{
p=p0-f(p0)/f’(p0);
if (fabs(p-p0)<Tolerance)
return p;
p0=p; //update p0
}
}
In this project,
let f (x)=Z4-1 and Tolerance = 0.001. Let z=x+iy in which x is the real and y the imaginary parts be a point in the range W=[-2 £ x £ 2, -2 £ y £ 2].
Now consider your screen in the VGA mode 640 x 480 to map to W.
In this case, z = x+iy becomes a pixel in the range 640 x 480.
The Z4-1=0 has four root: -1, 1, i, -i.
We can color each pixel in following way:
(1) If z=x+iy converge the root –1 by Newton’s method, we color the corresponding pixel by Red.
(2) If z=x+iy converge the root 1 by Newton’s method, we color the corresponding pixel by Green.
(3) If z=x+iy converge the root i by Newton’s method, we color the corresponding pixel by Blue.
(4) If z=x+iy converge the root -i by Newton’s method, we color the corresponding pixel by Black.
Of course, you can choose another four colors. Therefore, this project is that
you make a picture which is in 640x480 VGA mode such that each pixel corresponding to a point in W is colored by converging which root by Newton’s method.
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