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Gaussian quadrature is a numerical integration rule that chooses abscissas and weights to integrate polynomials of the highest possible polynomial degree exactly. The fundamental theorem of Gaussian quadrature states that the optimal abscissas of the -point Gaussian quadrature formulas are precisely the roots of the orthogonal polynomials of polynomial degree for the same interval and weighting function. For a nonnegative weighting function with finite moments of all orders and positive integral against the square of every nonzero polynomial, Gaussian quadrature fits all polynomials up to polynomial degree exactly. Slightly less optimal fits are obtained when nodes are constrained to include endpoints, as in Radau quadrature or Lobatto quadrature.

The weighting function specifies the weighted integral approximated by the rule,

For a chosen rule, the interval and are fixed while varies. The weighting function is distinct from the discrete weights multiplying .

To determine the weights corresponding to the Gaussian abscissas , compute a Lagrange interpolating polynomial for by letting

(where Chandrasekhar 1967 uses instead of ), so

Then fitting a Lagrange interpolating polynomial through the points gives

for arbitrary points . We are therefore looking for a set of points and weights such that for a weighting function ,

with weight

The weights are sometimes also called the Christoffel numbers (Chandrasekhar 1967). For the member of polynomial degree in a family of orthogonal polynomials, set above. Then

where is its leading coefficient. The weights are given by

(Hildebrand 1956, p. 322), where

Using the relationship

(Hildebrand 1956, p. 323) gives

(Note that Press et al. 1992 omit the factor .) In Gaussian quadrature, the weights are all positive. The error is given by

where (Hildebrand 1956, pp. 320-321).

Other curious identities are

and

(Hildebrand 1956, p. 323).

In the notation of Szegö (1975), let be an ordered set of points in , and let , ..., be a set of real numbers. If is an arbitrary function on the closed interval , write the Gaussian quadrature as

Here are the abscissas and are the Cotes numbers.

The nodes and weights can also be obtained from the leading block of the Jacobi matrix for the normalized orthogonal polynomials. Its eigenvalues are the nodes . If is a unit eigenvector for and , then

This is the method of Golub and Welsch (1969, pp. 221-223). Equivalently, the Christoffel function gives .

For example, the three-point Legendre-Gauss quadrature rule has

Its nodes are , 0, and , with corresponding weights , , and . The rule is exact through polynomial degree 5.

The rule's higher-degree exactness is illustrated above by the quartic polynomial and its quadratic Lagrange interpolating polynomial ,

Their values agree at the three nodes. The blue and orange shaded regions lie where is positive and negative, respectively. Their signed integrals cancel. Consequently,

The numbers beneath the nodes are the corresponding weights.

See also

Chebyshev-Gauss Quadrature, Chebyshev Quadrature, Chebyshev-Radau Quadrature, Christoffel-Darboux Kernel, Christoffel Function, Discrete Variable Representation, Fundamental Theorem of Gaussian Quadrature, Hermite-Gauss Quadrature, Jacobi-Gauss Quadrature, Jacobi Matrix, Lagrange-Mesh Method, Laguerre-Gauss Quadrature, Legendre-Gauss Quadrature, Lobatto Quadrature, Orthogonal Polynomials, Radau Quadrature

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 887-888, 1972.Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 103, 1990.Arfken, G. "Appendix 2: Gaussian Quadrature." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 968-974, 1985.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 461, 1987.Chandrasekhar, S. An Introduction to the Study of Stellar Structure. New York: Dover, 1967.Gauss, C. F. "Methodus nova integralium valores per approximationem inveniendi." Commentationes Societatis regiae scientarium Gottingensis recentiores 3, 39-76, 1814. Reprinted in Werke, Vol. 3. New York: George Olms, p. 163, 1981.Golub, G. H. and Welsch, J. H. "Calculation of Gauss Quadrature Rules." Math. Comput. 23, 221-230, 1969. https://doi.org/10.1090/S0025-5718-69-99647-1.Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 319-323, 1956.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gaussian Quadratures and Orthogonal Polynomials." §4.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 140-155, 1992.Stroud, A. H. and Secrest, D. Gaussian Quadrature Formulas. Englewood Cliffs, NJ: Prentice-Hall, 1966.Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 37-48 and 340-349, 1975.Whittaker, E. T. and Robinson, G. "Gauss's Formula of Numerical Integration." §80 in The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 152-163, 1967.

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Gaussian Quadrature

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

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