| Newton's method for approximating the roots of a function |
| Newton's method becomes interesting for functions with multiple roots. Then a natural question is which initial guesses x0 will iterate to which root? For a given root, the collection of all such guesses is called the basin of attraction of that root. Here is a simple example. |
| We can reformulate Newton's method using graphical iteration. |
| With graphical iteration, finding the basins of attraction is straightforward. |
| For some functions, the basins of attraction are intricately interwound, so much that they possess the Wada property. |
| Finally, we can find initial guesses for which Newton's method fails to converge to any root. Analyzing these failures reveals a familiar shape. |
Return to the Mandelbrot set and Julia sets.