Thread: Cubic Convergence

Question :
Consider an iteration function of the form F(x) = x + f(x)g(x), where f(r) = 0 and f'(r) != 0. Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r.

Any help with this problem would be greatly appreciated! Thanks!

Originally Posted by theamazingjenny

Question :
Consider an iteration function of the form F(x) = x + f(x)g(x), where f(r) = 0 and f'(r) != 0. Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r.

Any help with this problem would be greatly appreciated! Thanks!

I'm a bit confused here. Function iteration produces a sequence of functions:

f_{n+1}(x) = f(f_{n}(x))

for all x in some domain D.

But as far as I can see you are interested in the iteration:

x_{n+1} = x_{n} + f(x_{n})g(x_{n})

RonL