 Solution

ID: past-exam-of-the-numerical-analysis-course-of-the-university-of-oxford/2015/1/c/i/solution

a = x_{0} < x_{1} < \dots < x_{N} = b

, apply the trapezoidal rule separately to each panel and add:

T_{N} (f) = j = 1 \sum N \frac{x _{j} - x _{j - 1}}{2} (f (x_{j - 1}) + f (x_{j})) .

For uniform spacing

h = (b - a) / N

T_{N} (f) = h [\frac{1}{2} f (a) + j = 1 \sum N - 1 f (a + jh) + \frac{1}{2} f (b)] .

Each interior sample occurs in two panels; their half-quadrature weights combine to

h

. The endpoint quadrature weights are

h /2

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