Simpson's Rule

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Oct 27, 2024
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- Numerical Integration
- 4: Transcendental Functions

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The Trapezoidal and Midpoint estimates provided better accuracy than the Left and Right endpoint estimates. It turns out that a certain combination of the Trapezoid and Midpoint estimates is even better.

Definitions: Trapezoidal and Midpoint Estimates

Let $f(x)$ be a function defined on $[a,b]$ . Then

$S(n) = \dfrac{1}{3} T(n) + \dfrac{2}{3} M(n) \nonumber$

where $T(n)$ and $M(n)$ are the Trapezoidal and Midpoint Estimates.

Geometrically, if $n$ is an even number then Simpson's Estimate gives the area under the parabolas defined by connecting three adjacent points.

Let $n$ be even then using the even subscripted $x$ values for the trapezoidal estimate and the midpoint estimate, gives

$\begin{align} S(n) &= \dfrac{1}{3}\left[\dfrac{b-a}{2n} \left( f(x_0) +2f(x_2) +...+ f(x_{2n-2}) +f(x_{2n}) \right)\right] + \dfrac{2}{3}\left[\dfrac{b-a}{n}\left( f(x_1)f(x_3)+f(x_5)+...+f(x_{2n-1}) \right) \right] \label{Simp1} \\ &= \dfrac{b-a}{3n}\left(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+4f(x_5)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n) \right) \label{Simp2} \end{align}$

Notice the "1 2 4 2 4 ... 2 4 2 4 1" pattern in the coefficients.

Example 1

Use Simpson's Estimate to approximate

$\int_{0}^{2} e^{x^2} dx \nonumber$

Using $n = 6$

Solution

We partition

$0 < 1/3 < 2/3 < 1 < 4/3 < 5/3 < 2 \nonumber$

and calculate

$e^{0^2}=1, e^{(\frac{1}{3})^2}=1.12, e^{(\frac{2}{3})^2}=1.56, e^{(1)^2}=2.72 \\ e^{(\frac{4}{3})^2}=5.92, e^{(\frac{5}{3})^2}=16.08, e^{(2)^2}=54.60 \nonumber$

and put these into the formula for Simpson's Estimate (Equation $\ref{Simp2}$ )

$\dfrac{2-0}{6} [ 1 + 2 \cdot 1.12 + 4 \cdot 1.56 + 2 \cdot 2.72 + 4 \cdot 5.92 + 2 \cdot 16.08 + 54.60 ] =41.79 \nonumber$

Exercise

Approximate

$\int_{2}^{6} \dfrac{1}{1+x^3} dx \nonumber$

Error in Simpson's Estimate

Without proof, we state:

Let $M = \text{max }|f''''(x)|$ and let $E_s$ be the error in using Simpson's estimate then

$|E_s| \leq \dfrac{M(b - a)^5}{180n^4} \nonumber$

Contributors and Attributions

Larry Green (Lake Tahoe Community College)

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Definitions: Trapezoidal and Midpoint Estimates

Example 1

Exercise

Error in Simpson's Estimate

Contributors and Attributions

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