I added two new fractals to Arthur. They are both calculated with Newton’s method, that is:
z -> z – f(z) / f’(z)
Newton invented this formula to estimate roots of functions. The initial z is an estimation of the root. The calculated z is closer to the root that is a better estimation. If we calculate the formula iteratively, z converges to one of the roots of f(z).
My newton fractals are colored according to the number of necessary iterations to get in a predefined distance with z from one of the roots.
The newton001 fractal is using the f(z) = z^3 – 1 polynomial.
f’(z) = 3z^2
z -> z – f(z) / f’(z) = z – (z^3 – 1) / 3z^2 = (2z^3 + 1) / 3z^2
The roots of f(z) are: 1, -0.5 + 0.86603i and -0.5 – 0.86603i
Here is an image of the fractal:
The newton002 fractal is using the f(z) = z^4 – 1 polynomial
f’(z) = 4z^3
z -> z – f(z) / f’(z) = z – (z^4 – 1) / 4z^3 = (3z^4 + 1) / 4z^3
The roots of f(z) are: 1, i, -1 and -i
Here is an image of the fractal:
You can get the newest sources of Arthur from here: fractal016
