Definition:Gaussian Integration Rule

Definition

A Gaussian integration rule is a numerical integration rule of the form:

$\ds \int_a^b \map w x \map f x \rd x \approx \sum_{i \mathop = 1}^n w_i \map f {x_i}$

where $\map w x$ is a non-negative weight function on the interval $\closedint a b$ such that both:

the $n$ nodes $x_i$

the weights $w_i$

are chosen to make the approximation exact when $f$ is a polynomial of degree less than or equal to $2 n - 1$.

The purpose of the weight function is to build into the rule any special behaviour of the integrand.

Common choices for the weight function are:

$\map w x = 1$ with $\closedint a b = \closedint {-1} 1$

$\map w x = e^x$ with $\closedint a b = \closedint 0 \to$

Examples

Three-Point Gauss-Chebyshev Rule

An example of a Gaussian integration rule is the three-point Gauss-Chebyshev rule:

The three-point Gauss-Chebyshev rule is a Gaussian integration rule of the form:

$\ds \int_{-1}^1 \dfrac {\map f x} {\sqrt {1 - x^2} } \rd x \approx \dfrac 1 3 \pi \paren {\map f {-\dfrac {\sqrt 3} 2} + \map f 0 + \map f {\dfrac {\sqrt 3} 2} }$

Also see

• Results about Gaussian integration rules can be found here.

Source of Name

This entry was named for Carl Friedrich Gauss.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Gaussian integration rule
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Gaussian integration rule