When reading the current threads about root finding methods and Newton fractals,

https://fractalforums.org/fractal-mathematics-and-new-theories/28/finding-all-roots-of-a-polynomial-with-mathematical-guarantee/2959 , and

https://fractalforums.org/fractal-mathematics-and-new-theories/28/newton-hines-fractals/2737I remembered that I wanted to write something about own experiences on variations an mutations on Newton style fractals.

Since I have now a bit more time than usual I would like to start a loose series about several variations in that area.

My approach contains much less mathematics than for instance the above linked threads. It is more "try it, and look what happens".

In several cases one may even doubt a matematical meaningfulness.

But anyway - as long as the resulting pictures are nice enough it doesn't matter to me ;-)

In fact the root of these variations goes back to times where calculation was still expensive.

The thought was:

Take a simple method that shows fast results, and then extend it systematically.

After such extension it may be possible to vary a parameter, or to change a power of the polynomial - without disturbing the complete picture (which is usually the problem with mutating simple formulas).

You will find extensions that contain for instance applications of the Newton methods on functions with multiple variables, some use of n-dimensional numbers, or matrices, or cases in which the Newton method is applied to the result of a Newton method, applied to the result of...

There is one hint which may look a bit mathematically though:

To explain what I have done I will use a specific notation.

When I write

\( x \leftarrow N[N[f(x,y) , x] - x , y] \)

then I mean:

1. Take a function with two variables ( \( f(x,y) \) ) , apply the Newton method with respect to x when it comes to the differentiation ( \( N[... , x] \) )

2. Subtract x from the resulting formula

3. Take the function you have received in 2. , and apply the Newton method with respect to y when it comes to the differentiation ( \( N[... , y] \) )

4. Iterate the resulting formula (only x in this case) and look whether you get a fractal.

I will add formulas and parameters created in UF5.

I will attach those files to each of my entries. Each will contain the params/formulas from the previous entries. This is because I had difficulties to replace attachments in existing entries.

Please do not hesitate to add your own examples, or other variations. - Have fun :-)

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/slightly-extended-newton-method-fractals/3067/