Gaussian quadrature Guide, Meaning , Facts, Information and Description
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
| Table of contents |
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2 Change of interval for Gaussian quadrature 3 Other forms of Gaussian quadrature 4 References 5 External links |
For the integration problem stated above,
the associated polynomials are Legendre polynomials.
Some low-order rules for solving the integration problem are listed below.
Rules for the basic problem
| Number of points, n | Weights, wi | Points, xi |
|---|---|---|
| 1 | 2 | 0 |
| 2 | 1, 1 | -√(1/3), √(1/3) |
| 3 | 5/9, 8/9, 5/9 | -√(3/5), 0, √(3/5) |
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:
The integration problem can be expressed in a slightly more general way by introducing a weight function ω into the integrand,
and allowing an interval other than [-1, 1].
That is, the problem is to calculate
Change of interval for Gaussian quadrature
After applying the Gaussian quadrature rule, the following approximation is obtained:Other forms of Gaussian quadrature
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).
| Interval | ω(x) | Orthogonal polynomials | A&S |
| [-1, 1] | Legendre polynomials | Eq. 25.4.29 | |
| [-1, 1] | Chebyshev polynomials | Eq. 25.4.38 | |
| [0, ∞) | Laguerre polynomials | Eq. 25.4.45 | |
| (-∞, ∞) | Hermite polynomials | Eq. 25.4.46 |
The error of a Gaussian quadrature rule can be stated as follows (theorem 3.6.24 in Stoer and Bulirsch).
For an integrand which has 2n continuous derivatives,
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.
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 algorithms in QUADPACK (see below) are based on Gauss-Kronrod rules.
This is an Article on Gaussian quadrature. Page Contains Information, Facts Details or Explanation Guide About Gaussian quadrature Error estimates
for some ξ in (a, b), where pn is the orthogonal polynomial of order n.Gauss-Kronrod rules
References
External links