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

This theorem requires a proof. In particular: Graphical approach based on approximating the area under the curve as a series of parabolas. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also known as

Simpson's Rule 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