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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, by a suitable choice of the n points xi and n weights wi. The domain of integration for such a rule is conventionally taken as [−1, 1], so the rule is stated as Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ...
In calculus, the integral of a function is an extension of the concept of a sum. ...
Partial plot of a function f. ...
A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a weight than others. ...
Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ...
(30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ...
In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...
 It can be shown (see Press, et al., or Stoer and Bulirsch) that the evaluation points are just the roots of a polynomial belonging to a class of orthogonal polynomials. In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...
In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...
In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner...
Rules for the basic problem
For the integration problem stated above, the associated polynomials are Legendre polynomials. Some low-order rules for solving the integration problem are listed below. In mathematics, Legendre functions are solutions to Legendres differential equation: They are named after Adrien-Marie Legendre. ...
| Number of points, n | Points, xi | Weights, wi | | 1 | 0 | 2 | | 2 |  | 1 | | 3 | 0 | 8/9 |  | 5/9 | | 4 | ±0.339981044 | 0.652145155 | | ±0.861136312 | 0.347854845 | | 5 | 0 | 0.568889 | | ±0.538469 | 0.478629 | | ±0.906180 | 0.236927 |
Change of interval for Gaussian quadrature An integral over [a, b] must be changed into an integral over [−1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way:  After applying the Gaussian quadrature rule, the following approximation is obtained: 
Other forms of Gaussian quadrature The integration problem can be expressed in a slightly more general way by introducing a positive weight function ω into the integrand, and allowing an interval other than [−1, 1]. That is, the problem is to calculate A weight function is a mathematical device used when performing a sum, integral, or average in order to give some elements more of a weight than others. ...
 for some choices of a, b, and ω. For a = −1, b = 1, and ω(x) = 1, the problem is the same as that considered above. Other choices lead to other integration rules. Some of these are tabulated below. Equation numbers are given for Abramowitz and Stegun (A & S). Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...
In mathematics, Legendre functions are solutions to Legendres differential equation: They are named after Adrien-Marie Legendre. ...
In mathematics, Jacobi polynomials are a class of orthogonal polynomials. ...
In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ...
In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834 - 1886), are a polynomial sequence defined by These polynomials are orthogonal to each other with respect to the inner product given by Also, for each n, Ln(x) is a solution of Laguerres equation which is a second-order...
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; and in physics, as the eigenstates of the quantum harmonic oscillator. ...
Fundamental theorem Let q be a nontrivial polynomial of degree n such that  If we pick the nodes to be the zeros of q, then there exist weights wi which make the computed integral exact for all polynomials of degree up to 2n − 1. Furthermore, all these nodes will lie in the open interval (a, b) (Stoer & Bulirsch 2002, pp. 172–175). 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. ...
Error estimates The error of a Gaussian quadrature rule can be stated as follows (Stoer & Bulirsch 2002, Thm 3.6.24). For an integrand which has 2n continuous derivatives, 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. ...
 for some ξ in (a, b), where pn is the orthogonal polynomial of order n.
Stoer and Bulirsch remark that this error estimate is inconvenient in practice, since it may be difficult to estimate the 2n'th derivative, and furthermore the actual error may be much less than a bound established by the derivative. Another approach is to use two Gaussian quadrature rules of different orders, and to estimate the error as the difference between the two results. For this purpose, Gauss-Kronrod rules can be useful.
Gauss-Kronrod rules If the interval [a, b] is subdivided, the evaluation points of the new subintervals generally do not coincide with the previous evaluation points, and thus the integrand must be evaluated at every point. Gauss-Kronrod rules are Gaussian quadrature rules that are modified to make some of the evaluation points coincide after subdivision. The difference between the results before and after subdivision can be taken as an estimate of the error of approximation, so such an approach can increase the accuracy achieved for a given number of function evaluations. The rules are named after Alexander Kronrod who invented them in the 1960s. The algorithms in QUADPACK (see below) are based on Gauss-Kronrod rules. Aleksandr (Alexander) Semenovich Kronrod (Russian ÐлекÑандр Семёнович Кронрод) (October 22, 1921 – October 6, 1986) was a Russian mathematician and computer scientist, best known for the Gauss-Kronrod quadrature which he published in 1964. ...
References - A.S. Kronrod, Nodes and weights of quadrature formulas. Sixteen-place tables, Authorized translation from the Russian, Consultants Bureau, New York, 1965.
- Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover: New York, 1972)
- Robert Piessens, Elise de Doncker-Kapenga, C.W. Überhuber, D.K. Kahaner. QUADPACK, A subroutine package for automatic integration. Springer Verlag, 1983. (Reference guide for QUADPACK.)
- William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, 1988. (See Section 4.5.)
- Stoer, Josef & Roland Bulirsch (2002), Introduction to Numerical Analysis (3rd ed.), Springer, ISBN 0-387-95452-X.
Aleksandr (Alexander) Semenovich Kronrod (Russian ÐлекÑандр Семёнович Кронрод) (October 22, 1921 – October 6, 1986) was a Russian mathematician and computer scientist, best known for the Gauss-Kronrod quadrature which he published in 1964. ...
Abramowitz and Stegun is the informal moniker of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the U.S. National Bureau of Standards. ...
Numerical Recipes is the generic term for the following books on algorithms and numerical analysis, all by William Press, Saul Teukolsky, William Vetterling and Brian Flannery: Numerical Recipes in C++. The Art of Scientific Computing, ISBN 0-521-75033-4. ...
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