6.3: Local Versus Global Error

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Consider the elementary formula (6.4) for the trapezoidal rule, written in the form

\[\int_{0}^{h} f(x) d x=\frac{h}{2}(f(0)+f(h))-\frac{h^{3}}{12} f^{\prime \prime}(\xi) \nonumber \]

where $\xi$ is some value satisfying $0 \leq \xi \leq h$, and we have used Taylor’s theorem with the mean-value form of the remainder. We can also represent the remainder as

\[-\frac{h^{3}}{12} f^{\prime \prime}(\xi)=\mathrm{O}\left(h^{3}\right) \nonumber \]

where $\mathrm{O}\left(h^{3}\right)$ signifies that when $h$ is small, halving of the grid spacing $h$ decreases the error in the elementary trapezoidal rule by a factor of eight. We call the terms represented by $\mathrm{O}\left(h^{3}\right)$ the Local Error.

More important is the Global Error which is obtained from the composite formula (6.7) for the trapezoidal rule. Putting in the remainder terms, we have

\[\int_{a}^{b} f(x) d x=\frac{h}{2}\left(f_{0}+2 f_{1}+\cdots+2 f_{n-1}+f_{n}\right)-\frac{h^{3}}{12} \sum_{i=0}^{n-1} f^{\prime \prime}\left(\xi_{i}\right) \nonumber \]

where $\xi_{i}$ are values satisfying $x_{i} \leq \xi_{i} \leq x_{i+1}$. The remainder can be rewritten as

\[-\frac{h^{3}}{12} \sum_{i=0}^{n-1} f^{\prime \prime}\left(\xi_{i}\right)=-\frac{n h^{3}}{12}\left\langle f^{\prime \prime}\left(\xi_{i}\right)\right\rangle \nonumber \]

where $\left\langle f^{\prime \prime}\left(\xi_{i}\right)\right\rangle$ is the average value of all the $f^{\prime \prime}\left(\xi_{i}\right)^{\prime}$ s. Now,

\[n=\frac{b-a}{h} \nonumber \]

so that the error term becomes

\[\begin{aligned} -\frac{n h^{3}}{12}\left\langle f^{\prime \prime}\left(\xi_{i}\right)\right\rangle &=-\frac{(b-a) h^{2}}{12}\left\langle f^{\prime \prime}\left(\xi_{i}\right)\right\rangle \\ &=\mathrm{O}\left(h^{2}\right) \end{aligned} \nonumber \]

Therefore, the global error is $\mathrm{O}\left(h^{2}\right)$. That is, a halving of the grid spacing only decreases the global error by a factor of four.

Similarly, Simpson’s rule has a local error of $\mathrm{O}\left(h^{5}\right)$ and a global error of $\mathrm{O}\left(h^{4}\right)$.

$Screen Shot 2022-05-31 at 12.20.51 AM.png$

Figure 6.1: Adaptive Simpson quadrature: Level $1 .$

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