Gaussian quadrature

In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An n-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n. The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi 1826. The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as





∫

−
1


1


f
(
x
)

d
x
≈

∑

i
=
1


n



w

i


f
(

x

i


)
,


{\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i}),}
which is exact for polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f(x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1].
The Gauss-Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as




f
(
x
)
=


(

1
−
x

)


α




(

1
+
x

)


β


g
(
x
)
,

α
,
β
>
−
1
,


{\displaystyle f(x)=\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x),\quad \alpha ,\beta >-1,}
where g(x) is well-approximated by a low-degree polynomial, then alternative nodes




x

i

′



{\displaystyle x_{i}'}
and weights




w

i

′



{\displaystyle w_{i}'}
will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e.,





∫

−
1


1


f
(
x
)

d
x
=

∫

−
1


1




(

1
−
x

)


α




(

1
+
x

)


β


g
(
x
)

d
x
≈

∑

i
=
1


n



w

i

′

g

(

x

i

′

)

.


{\displaystyle \int _{-1}^{1}f(x)\,dx=\int _{-1}^{1}\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x)\,dx\approx \sum _{i=1}^{n}w_{i}'g\left(x_{i}'\right).}
Common weights include





1

1
−

x

2







{\textstyle {\frac {1}{\sqrt {1-x^{2}}}}}
(Chebyshev–Gauss) and





1
−

x

2






{\displaystyle {\sqrt {1-x^{2}}}}
. One may also want to integrate over semi-infinite (Gauss-Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature).
It can be shown (see Press, et al., or Stoer and Bulirsch) that the quadrature nodes xi are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.

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