Simpson's Rule

Theorem

Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.

$\ds \int_a^b \map f x \rd x \approx \dfrac {b - a} 6 \paren {\map f a + 4 \map f {\dfrac {a + b} 2} + \map f b}$

Hence the area under the curve is approximated by the area under the quadratic polynomial passing through $\tuple {x, \map f x}$ for the $3$ values $x = a$, $x = \dfrac {a + b} 2$ and $x = b$.

Repeated Simpson's Rule

Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.

Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$:

$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$

where $n$ is even.

Then the definite integral of $f$ with respect to $x$ from $a$ to $b$ can be approximated as:

$\ds \int_a^b \map f x \rd x$

$\approx$

$\ds \dfrac h 3 \paren {y_0 + \sum_{r \mathop = 1}^{n / 2 - 1} 4 y_{2 r - 1} + \sum_{r \mathop = 1}^{n / 2 - 1} 2 y_{2 r} + y_n}$

$\ds $

$=$

$\ds \dfrac h 3 \paren {y_0 + 4 y_1 + 2 y_2 + 4 y_3 + 2 y_4 + \cdots + 2 y_{n - 2} + 4 y_{n - 1} + y_n}$

where:

$\forall i \in \set {0, 1, 2, \ldots, n}: y_i = \map f {x_i}$

$h = \dfrac {b - a} n$

This is known as the repeated Simpson's rule.

Proof

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In particular: Graphical approach based on approximating the area under the curve as a series of parabolas.
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Also known as

is also known as the Parabolic Formula.

It can also be seen as Simpson's Formula, but this may be confused with Simpson's Formulas, which is a set of completely different results.

Source of Name

This entry was named for Thomas Simpson.

Sources

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Simpson's rule
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Simpson's rule