6.2: Composite Rules
- Page ID
- 96060
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We now use our elementary formulas obtained for (6.2) to perform the integral given by \((6.1)\)
6.2.1. Trapezoidal rule
We suppose that the function \(f(x)\) is known at the \(n+1\) points labeled as \(x_{0}, x_{1}, \ldots, x_{n}\), with the endpoints given by \(x_{0}=a\) and \(x_{n}=b\). Define
\[f_{i}=f\left(x_{i}\right), \quad h_{i}=x_{i+1}-x_{i} \nonumber \]
Then the integral of (6.1) may be decomposed as
\[\begin{aligned} \int_{a}^{b} f(x) d x &=\sum_{i=0}^{n-1} \int_{x_{i}}^{x_{i+1}} f(x) d x \\ &=\sum_{i=0}^{n-1} \int_{0}^{h_{i}} f\left(x_{i}+s\right) d s \end{aligned} \nonumber \]
where the last equality arises from the change-of-variables \(s=x-x_{i}\). Applying the trapezoidal rule to the integral, we have
\[\int_{a}^{b} f(x) d x=\sum_{i=0}^{n-1} \frac{h_{i}}{2}\left(f_{i}+f_{i+1}\right) \nonumber \]
If the points are not evenly spaced, say because the data are experimental values, then the \(h_{i}\) may differ for each value of \(i\) and (6.6) is to be used directly.
However, if the points are evenly spaced, say because \(f(x)\) can be computed, we have \(h_{i}=h\), independent of \(i\). We can then define
\[x_{i}=a+i h, \quad i=0,1, \ldots, n \nonumber \]
and since the end point \(b\) satisfies \(b=a+n h\), we have
\[h=\frac{b-a}{n} \nonumber \]
The composite trapezoidal rule for evenly space points then becomes
\[\begin{aligned} \int_{a}^{b} f(x) d x &=\frac{h}{2} \sum_{i=0}^{n-1}\left(f_{i}+f_{i+1}\right) \\ &=\frac{h}{2}\left(f_{0}+2 f_{1}+\cdots+2 f_{n-1}+f_{n}\right) \end{aligned} \nonumber \]
The first and last terms have a multiple of one; all other terms have a multiple of two; and the entire sum is multiplied by \(h / 2\).
6.2.2. Simpson’s rule
We here consider the composite Simpson’s rule for evenly space points. We apply Simpson’s rule over intervals of \(2 h\), starting from \(a\) and ending at \(b\) :
\[\begin{aligned} \int_{a}^{b} f(x) d x=\frac{h}{3}\left(f_{0}+4 f_{1}+f_{2}\right)+\frac{h}{3}\left(f_{2}+4 f_{3}+f_{4}\right)+& \ldots \\ &+\frac{h}{3}\left(f_{n-2}+4 f_{n-1}+f_{n}\right) \end{aligned} \nonumber \]
Note that \(n\) must be even for this scheme to work. Combining terms, we have
\[\int_{a}^{b} f(x) d x=\frac{h}{3}\left(f_{0}+4 f_{1}+2 f_{2}+4 f_{3}+2 f_{4}+\cdots+4 f_{n-1}+f_{n}\right) \nonumber \]
The first and last terms have a multiple of one; the even indexed terms have a multiple of \(2 ;\) the odd indexed terms have a multiple of \(4 ;\) and the entire sum is multiplied by \(h / 3\).