We can find a root for a function
by equatingto zero and rearranging the equation to give
for some function
On putting
we obtain an iteration sequence which will under some circumstance converge to the desired root.
The condition for convergence to a root
is that
and that the starting point for the sequence,
be sufficiently close to
If
the sequence iterates to infinity.
Given a function f(x), many rearrangements may be possible. If f(x) has several roots, a sequence may iterate to some fixed point and some other rearrangement is found the sequence may iterate to some other fixed point.