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If I do a trapezoidal rule estimate and get 0.6386 and the true value of the integral is 0.636294, then the error is 0.002306.

If I was asked to find the order of error, does it just mean the error or has it a different meaning? Also, if the estimate and integral were correct to S.F, how many S.F should I put the error to. Is 3 reasonable?

Thanks

user137090
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1 Answers1

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For numerical quadrature, in general, the integral is approximated as a weighted sum of function values:

$$\int_{a}^{b}f(x)\,dx \approx \sum_{k=1}^{n}w_kf(x_k).$$

If the absolute global error satifies the following as $n \rightarrow \infty$

$$\left|\int_{a}^{b}f(x)\,dx - \sum_{k=1}^{n}w_kf(x_k)\right|=O(n^{-p})$$

then the error is of order $p$. Depending on the smoothness of $f$, the trapezoidal rule is second-order accurate.

RRL
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