6.3: Local Versus Global Error
- Page ID
- 96061
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\mathrm{O}\left(h^{4}\right)\).
Figure 6.1: Adaptive Simpson quadrature: Level \(1 .\)