In this question we are asked to give the order of convergence for our iteration scheme from (c).
Is it enough to give the rate of convergence? To compute the asymptotic error constant I need to solve a difficult limit that seems too complicated for a 4 point question.
The short answer is that you do not need to find the asymptotic error constant (to get the 4 points).
The longer would be why is it so hard to find an approximation to the asymptotic error constant. But I prefer to leave this discussion for after the due date.
So it’s sufficient to provide just the rate of convergence, even though the question asks for the order of convergence (e.g. linear, sublinear, etc…)?
Yes, you need to provide the rate of convergence and a justification why it is that rate. You do not need to provide the constant to get full points.
By the way, rate and order are used interchangeably.
However, their definition through the limit comes with the asymptotic error constant (same formula combines the beta/order/rate and the kappa/asympt_constant), and I suppose this is why you asked the first question.
And I want to emphasize that, although the definition formula for rate involves a rather “incomprehensible” limit, for fixed-point iteration, there are easier ways to find the rate and this is what you can do in this question.