Actual 3D newton fractals

[cKleinhuis]

first: [ latext ] can be used in postings

second: never use == on comparing floats or doubles, i think you run into a classical instability problem of floating point representation here

use a small delta bias to check if it is so close to zero as you like

just my five cents!

[Pauldelbrot]

What about this? Compute the gradient vector g at x, then compute x - gF(x)/|g|². That moves x in the direction opposite g and moves it a distance equal to F(x)/|g|.

In your example, (1,0.01) is now moved by (-1,-0.01) divided by the square root of 1.01, so almost right onto zero.

[David Makin]

One problem with using methods that converge faster/better than Newton's is (as already mentioned) that you lose any fractal detail - in fact I think I remember trying one myself (Householder's?) and got 3 perfect triangles with straight edges !!

Maybe the extension to general 2D using Jacobian methods can be extended to 3D ?

Edit: Of course the Jacobian does get a little messy using 3 dimensions instead of 2

See my "Multivariable Newton" formula in mmf3.ufm for Ultra Fractal which essentially gives the Newton for any system of 2 reals - actually it would of course be easy to convert to any system of 2 complex (Which of course gives us 4 dimensions to play with without too much difficulty)

Example:

Code:

GeneralNewton {
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}

[Kabuto]

What about this? Compute the gradient vector g at x, then compute x - gF(x)/|g|². That moves x in the direction opposite g and moves it a distance equal to F(x)/|g|.

In your example, (1,0.01) is now moved by (-1,-0.01) divided by the square root of 1.01, so almost right onto zero.

That's exactly the method this thread is about :-)

[Pauldelbrot]

Is it? Then there's a few extra absolute bars in your OP.