"Newton's Method" is an iterative "successive approximation" procedure for locating the zeros of a function, F(z, Cj), where z is a complex variable, z=(x, y), and the Cj coefficients ( j=0, 1, 2, ...) are real numbers.   If the function is a polynomial of degree "n", there are n zeros, or "roots", at z=r1, r2, ..., rn which are either real or occur in conjurgate complex pairs ( for real coefficients ).

The general procedure is to start with an initial guess, z0; compute the function at z0 ( it won't be zero ); compute its slope; and jump along this slope to a zero at z1.   This won't be a zero of the function either, of course, but it should be closer than z0 to a root.

Use this z1 as a new starting point, and repeat this process to get z2.   Continue these iterations until the value of the function becomes acceptably small, and stop.   Store the iteration count for your starting point, z0, and the location of the root it approached.

Repeat this process for a grid of z-points in the X-Y plane.   You now know which root each of these points approached, and the iteration count ( i. e., how long it took to get there ).   This information can be converted to a colored map as before.

A simple function that illustrates all the features of these images is the quadratic polynomial with C0 = -1, C1 = 0, and C2 = +1:

A thin, but very complex, fractal boundary region separates these basins from each other and from the outer region extending to the point at infinity.   This fractal boundary looks something like a chain, with larger links at intervals.   Points in the outer region, and most points from within the "chain" will reach one of the roots, but not smoothly.

        Zoom View of Link with Mini-M             Deep Zoom on a Mini-M Island

It is remarkable that the boundary region between root basins contains miniature versions of the Mandelbrot set, as shown above.   The mathematical procedure used to generate these images is quite different from that used to create the Mset.

Two examples of Newton images that have been "processed" by filters to modify them are shown below.   Such techniques are sometimes used to "improve" images that have large regions with little structure.

   
Collision Point Newton's Web

The left image has threads added which are perpendicular to the color contours.   A mosaic filter was used on the righthand image.