Aug 19, 2017

How many points will be used in total and what is the expected order of the resulting formula?

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1. Derive a difference formula for the fourth derivative of f at x0 using Taylor’s expansions at x0 ±h and x0 ±2h. How many points will be used in total and what is the expected order of the resulting formula?

2. Let f (x) be a given function that can be evaluated at points x0 ± jh, j = 0,1,2,…, for any fixed value of h, 0 ≪ 1.

(a) Find a second order formula (i.e., truncation error O(h2)) approximating the third derivative f”(x0). Give the formula, as well as an expression for the truncation error, i.e., not just its order.

(b) Use your formula to find approximations to f”(0) for the function f (x) = ex employing values h = 10−1,10−2,…,10−9, with the default MATLAB arithmetic. Verify that for the larger values of h your formula is indeed second order accurate. Which value of h gives the closest approximation to e0 = 1?

(c) For the formula that you derived in (a), how does the roundoff error behave as a function of h, as h → 0?

(d) How would you go about obtaining a fourth order formula for f”(x0) in general? (You don’t have to actually derive it: just describe in one or two sentences.) How many points would this formula require?


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