Simpson's Rule/Repeated

Theorem

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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Also known as

The repeated Simpson's rule is also known as just Simpson's rule.

It 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 are a set of completely different results.

Source of Name

This entry was named for Thomas Simpson.

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Approximate Formulas for Definite Integrals: $15.17$
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
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 18$: Definite Integrals: Approximate Formulas for Definite Integrals: $18.17$