Topic summary
Newton iteration
Extracted from the Wikipedia article Newton's method.
Convergence dependent on initialization
From this, it can be seen that there are three possible phenomena for a Newton iteration. If initialized strictly between ±1, the Newton iteration will converge (super-)quadratically to 0; if initialized exactly at 1 or −1, the Newton iteration will oscillate endlessly between ±1; if initialized anywhere else, the Newton iteration will diverge. This same trichotomy occurs for f(x) = arctan x.