The fundamental theorem of Gaussian quadrature states that an -point Gaussian quadrature rule integrates polynomials through polynomial degree exactly, and its abscissas are the roots of the orthogonal polynomial of polynomial degree for the same measure. Assume a finite positive measure on the real line with infinite support and finite moments of all orders.

Let be the monic polynomial of polynomial degree orthogonal to all lower-degree polynomials, with roots , ..., . The Lagrange interpolating polynomials are

with , where is the Kronecker delta. Define the weights by . Then

for every polynomial of polynomial degree at most .

To prove this, polynomial division gives

where and have polynomial degree at most . The first term has zero integral by orthogonality and vanishes at every . The Lagrange interpolating polynomial is integrated exactly by the definition of the weights. Conversely, exactness through polynomial degree forces the monic polynomial with the quadrature nodes as its roots to be orthogonal to every polynomial of polynomial degree at most , so it is .

Exactness applied to gives

No rule using real nodes can also integrate every polynomial of polynomial degree exactly. The square of its nodal polynomial vanishes at every node but has positive integral. For Gaussian nodes, the Christoffel function also gives .

For example, with on , the two-point Legendre-Gauss quadrature rule has nodes and both weights equal to 1. It is exact through polynomial degree 3, but gives for , whose integral is .

See also

Christoffel Function, Gaussian Quadrature, Lagrange Interpolating Polynomial, Orthogonal Polynomials

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References

National Institute of Standards and Technology. "Quadrature." §3.5 in NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/3.5.v.Simon, B. "The Christoffel-Darboux Kernel." In Perspectives in Partial Differential Equations, Harmonic Analysis and Applications (Ed. D. Mitrea and M. Mitrea). Providence, RI: Amer. Math. Soc., pp. 295-335, 2008. https://doi.org/10.1090/pspum/079/2500498. https://arxiv.org/abs/0806.1528.

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Fundamental Theorem of Gaussian Quadrature

Cite this as:

Weisstein, Eric W. "Fundamental Theorem of Gaussian Quadrature." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FundamentalTheoremofGaussianQuadrature.html

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