 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability/4/c/i/solution

Take

n

to be a positive even integer, as required by composite Simpson's rule. There are

n /2

two-strip panels, each of width

2/ n

. Applying the supplied panel bound with the global fourth-derivative minimum

0

and maximum

1

, and summing, gives

0 \leq E_{n} \leq \frac{n}{2} \frac{( 2/ n ) ^{5}}{2880} = \frac{1}{180 n ^{4}} .

Here

E_{n}

is the nonnegative absolute error. In this example Simpson's rule overestimates the integral, so

E_{n} = S_{n} - I

. Each panel has that same sign because its fourth derivative is nonnegative; cancellation is not being assumed.^[ref]

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