Gaussian quadrature

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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

\int_{-1}^1 f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)

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.

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[edit] 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.

Number of points, n Points, xi Weights, wi
1 0 2
2 \pm\sqrt{1/3} 1
3 0 8/9
\pm\sqrt{3/5} 5/9
4 ±0.339981044 0.652145155
±0.861136312 0.347854845
5 0 0.568889
±0.538469 0.478629
±0.906180 0.236927

[edit] 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:

\int_a^b f(t)\,dt = \frac{b-a}{2} \int_{-1}^1 f\left(\frac{b-a}{2}x + \frac{a+b}{2}\right)\,dx

After applying the Gaussian quadrature rule, the following approximation is obtained:

\frac{b-a}{2} \sum_{i=1}^n w_i f\left(\frac{b-a}{2}x_i + \frac{a+b}{2}\right)

[edit] 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

\int_a^b \omega(x)\,f(x)\,dx

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] 1\, Legendre polynomials 25.4.29
(−1, 1) (1-x)^\alpha (1+x)^\beta,\quad \alpha, \beta > -1\, Jacobi polynomials 25.4.33 (β = 0)
(−1, 1) \frac{1}{\sqrt{1 - x^2}} Chebyshev polynomials (first kind) 25.4.38
[−1, 1] \sqrt{1 - x^2} Chebyshev polynomials (second kind) 25.4.40
[0, ∞) e^{-x}\, Laguerre polynomials 25.4.45
(−∞, ∞) e^{-x^2} Hermite polynomials 25.4.46

[edit] Fundamental theorem

Let q be a nontrivial polynomial of degree n such that

\int_a^b \omega(x) \, x^k q(x) \, dx = 0, \quad \text{for all }k=0,1,\ldots,n-1.

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).

[edit] 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,

\int_a^b \omega(x)\,f(x)\,dx - \sum_{i=1}^n w_i\,f(x_i) = \frac{f^{(2n)}(\xi)}{(2n)!} \, \|p_n\|^2

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.

[edit] 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.

[edit] 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.

[edit] External links

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