Derivation of Simpson's Three-Eights Rule

To derive the formula for Simpson's Three-Eights Rule using the closed Newton-Cotes theorem, we follow these steps:

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1. Setting Up the Problem

We aim to approximate the integral:

$\int_{x_0}^{x_3} f(x) \, dx$

using a polynomial interpolant for the function $f(x)$ over four equally spaced points: $x_0, x_1, x_2, x_3$. The step size is denoted as $h$, so:

$x_1 = x_0 + h, \quad x_2 = x_0 + 2h, \quad x_3 = x_0 + 3h.$

2. Interpolating Polynomial

We use a cubic Lagrange polynomial $P_3(x)$ to interpolate $f(x)$ at $x_0, x_1, x_2, x_3$:

$P_3(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3.$

The integral of $f(x)$ is approximated by integrating this cubic polynomial:

$\int_{x_0}^{x_3} f(x) \, dx \approx \int_{x_0}^{x_3} P_3(x) \, dx.$

3. Newton-Cotes Coefficients

The Newton-Cotes theorem provides weights for the integrand values $f(x_0), f(x_1), f(x_2), f(x_3)$. Using the theorem for $n = 3$, the formula is:

$\int_{x_0}^{x_3} f(x) \, dx = \frac{3h}{8} \left[f(x_0) + 3f(x_1) + 3f(x_2) + f(x_3)\right] - \frac{3h^5}{80} f^{(4)}(\xi),$

where $x_0 < \xi < x_3$ and $f^{(4)}(\xi)$ is the fourth derivative of $f(x)$, representing the error term.

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4. Deriving the Formula

To see how the formula is derived:

1. Integration of Basis Polynomials: The Lagrange basis polynomials for $x_0, x_1, x_2, x_3$ are constructed, and their coefficients are determined.

   $\int_{x_0}^{x_3} P_3(x) \, dx = \sum_{i=0}^3 w_i f(x_i),$

   where the weights $w_i$ are determined from the basis polynomials.

2. Weights: The weights are calculated by integrating each basis polynomial:

   • $w_0 = \frac{3h}{8}, \, w_1 = \frac{9h}{8}, \, w_2 = \frac{9h}{8}, \, w_3 = \frac{3h}{8}.$

   This leads to:

   $\int_{x_0}^{x_3} f(x) \, dx \approx \frac{3h}{8} \left[f(x_0) + 3f(x_1) + 3f(x_2) + f(x_3)\right].$

3. Error Term: The error term is derived by subtracting the integral of $P_3(x)$ from $f(x)$, resulting in the remainder term:

   $E = -\frac{3h^5}{80} f^{(4)}(\xi),$

   where $f^{(4)}(\xi)$ is the fourth derivative of $f(x)$ evaluated at some point $\xi \in [x_0, x_3]$.

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5. Final Formula

Combining the approximation and the error term, we arrive at the Simpson’s Three-Eights Rule:

$\int_{x_0}^{x_3} f(x) \, dx = \frac{3h}{8} \left[f(x_0) + 3f(x_1) + 3f(x_2) + f(x_3)\right] - \frac{3h^5}{80} f^{(4)}(\xi),$

where $x_0 < \xi < x_3$.

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Questions for Further Study

1. How do Newton-Cotes formulas change for higher-order interpolants?
2. Why is Simpson's Three-Eights Rule considered more accurate than the Trapezoidal Rule for cubic functions?
3. Can the error term be reduced without increasing the number of points in the formula?
4. How does the step size $h$ influence the accuracy of Simpson's Three-Eights Rule?
5. Can Simpson’s Three-Eights Rule be generalized to unevenly spaced points?

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Tip

When using numerical integration, always consider the smoothness of the function $f(x)$ and choose a method with an error term suitable for its behavior.