Newton Basins for a cubic

This particular cubic polynomial is the simplest one such that Newton's method doesn't nearly always work. You'll notice that there's a black lozenge in the middle of the image. If your initial estimate of a root begins in that black region, then Newton's method won't settle down on a root of the polynomial.

Try changing the values of the roots very slightly. You'll find that the lozenge changes into very interesting shapes. You can, of course, click on the image to expand a portion of the image.


x in [-1.00000000,1.00000000] ;
y in [-1.00000000,1.00000000].

Click at a point in the image to magnify it by a factor of

This is an image of the Newton Basins for a polynomial with the 3 roots

x1 = y1 = m1 =
x2 = y2 = m2 =
x3 = y3 = m3 =

The image is pixels wide by pixels high.

Newtons method is applied up to iterations per pixel.

Each basin is colored a single color or in shades.

Load the image on its own page
Up to the introduction
Back to the examples


David E. Joyce