Gaussian quadrature

In <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a>, an n-point Gaussian quadrature rule, named after <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a>, is a <a href="/facts/Quadrature_rule/MwJnvcDV">quadrature rule</a> constructed to yield an exact result for <a href="/facts/Polynomial/Lzak8VVx">polynomials</a> 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 <a href="/facts/Orthogonal_polynomial/9cyRbDYU">orthogonal polynomials</a> was developed by <a href="/facts/Carl_Gustav_Jacobi/4Syeauo5">Carl Gustav Jacobi</a> in 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 <a href="/facts/Gauss%25E2%2580%2593Legendre_quadrature/fRzdycEk">Gauss–Legendre quadrature</a> 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–<a href="/facts/Adrien-Marie_Legendre/2a7CAolM">Legendre</a> quadrature rule is not typically used for integrable functions with endpoint <a href="/facts/Singularity_(math)/Y5sMQY4I">singularities</a>. 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 xi' and weights wi' will usually give more accurate quadrature rules. These are known as <a href="/facts/Gauss%25E2%2580%2593Jacobi_quadrature/tcJshwa6">Gauss–Jacobi quadrature</a> 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}}}}}
 
 (<a href="/facts/Chebyshev%25E2%2580%2593Gauss_quadrature/dfWEokpd">Chebyshev–Gauss</a>) and 
 
 
 
 
 
 1
 −
 
 x
 
 2
 
 
 
 
 
 
 {\textstyle {\sqrt {1-x^{2}}}}
 
. One may also want to integrate over semi-infinite (<a href="/facts/Gauss%25E2%2580%2593Laguerre_quadrature/sX0r0Jge">Gauss–Laguerre quadrature</a>) and infinite intervals (<a href="/facts/Gauss%25E2%2580%2593Hermite_quadrature/lx8DJ1Iw">Gauss–Hermite quadrature</a>).
It can be shown (see Press et al., or Stoer and Bulirsch) that the quadrature nodes xi are the <a href="/facts/Root_of_a_function/mlK3dhoT">roots</a> of a polynomial belonging to a class of <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a> (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.

Gaussian quadrature open-in-new

Gaussian quadrature