[General boards] [Winter 2019 courses] [Fall 2018 courses] [Summer 2018 courses] [Older or newer terms]

Can't find an upper bound for an error


2017 Spring Exam Question 4

From what I have got in question (d) and (e),
p(x) = x + 7/6x(x-1)
0 < ksi < 1
Error formula = (15/8 ksi(^-1/2))/(6) x(x-1)(x-1/4)

With the above information, I’ve concluded that we can’t find the bound in question (f) as ksi = 0 will lead to the bound being undefined. Can someone verify this is a valid conclusion and if not,please explain under what condition we won’t be able to find the upper bound. Thanks!


You cannot use the polynomial error interpolation formula in this case,
since the assumptions of that theorem are not satisfied.
You need f to belong to C^3 in [0,1] to apply the theorem, and f is x^(5/2),
so it is not C^3.
Of course, if you try to apply the formula, with ksi -> 0, you will get infinity.


If we can’t use that theorem, what should we use for the error formula in (e)?


I meant to say that you can’t use the error formula to calculate a bound for the error.
You can still try to write the formula, but when ksi->0 the f^{(3)} is not defined,
so the formula says the error could grow to infinity (unbounded).