Convergence
The iterates of the secant method converge to a root of, if the initial values and are sufficiently close to the root. The order of convergence is α, where
is the golden ratio. In particular, the convergence is superlinear, but not quite quadratic.
This result only holds under some technical conditions, namely that be twice continuously differentiable and the root in question be simple (i.e., with multiplicity 1).
If the initial values are not close enough to the root, then there is no guarantee that the secant method converges. There is no general definition of "close enough", but the criterion has to do with how "wiggly" the function is on the interval . For example, if is differentiable on that interval and there is a point where on the interval, then the algorithm may not converge.
Read more about this topic: Secant Method