Newton Basins

Technical details

When do you stop iterating?

There's one sticky technical detail in the algorithm as used in this Newton Generation program, namely: how close do you have to be to a root to conclude that you're in that root's basin? For this program, we know what the roots are, so we can use that information. If you're using Newton's method to find the roots, then, of course, you wouldn't be able to use that extra information. But we're not, instead we just want images of the basins.

So part of the algorithm in this program is to determine the radius r_i of a circle around each root z_i so that within that circle iterated values of Newton's method continually get closer to that root z_i. I used the distance d_ij to the nearest other root z_j to determine a safe radius r_i:

r_i = d_ij / (2n/m_i -1)

where m_i is the multiplicity (typically 1) of the root z_i.

Up to the Introduction.
Back to the Generation form.
On to some References in books and on the web.

These pages are located at http://aleph0.clarku.edu/~djoyce/newton/newton.html