determine the order of convergence of steffensen's method

Hey guys. This numerical analysis section is really throwing me. I don't quite know how to handle this order of convergence stuff. Here's just one example of a problem that's giving me trouble:

Quote:

Let

, where

and

is smooth.

Suppose a sequence

satisfies

and converges to

, where

is a simple zero of

. We define "order of convergence" to be the largest real number

such that

exists and is nonzero. Determine the order of convergence.

Any help would be much appreciated !

Re: determine the order of convergence of steffensen's method

Re: determine the order of convergence of steffensen's method

Thanks, but I get , not .

Oh, and I forgot to say is smooth.

I guess I could maybe try computing and using Taylor's series, but to compute would take perhaps an hour or more---assuming I didn't make any elementary mistakes. Perhaps there is a more efficient method of solution ... ?

Re: determine the order of convergence of steffensen's method

Yes, I misread a sign, but the argument still works: The limit is then for and infinity for . Have you asked a professor for clarification? Maybe there's something we're missing.