I want to make sure I understand when the secant method will not converge as compared to the Newton's method.
When I look at $\arctan(x)$ and try to determine the initial guesses for which it will converge, and those for which it won't, I've come up with the following:
For Newton's method, when $|x_0| < 2$, the method will converge. and diverges otherwise.
For Secant method, when both $|x_0| < 2$, and $|x_1| < 2$ (since it requires 2 initial guesses) the the method converges, and diverges otherwise.
Can someone help me determine if this is correct? thanks very much.