Gaussian quadrature
Approximation of the definite integral of a function / From Wikipedia, the free encyclopedia
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In numerical analysis, an n-point Gaussian quadrature rule, named after Carl Friedrich Gauss,[1] is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n.
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The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826.[2] The most common domain of integration for such a rule is taken as [−1, 1], so the rule is stated as
which is exact for polynomials of degree 2n − 1 or less. This exact rule is known as the Gauss–Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if f (x) is well-approximated by a polynomial of degree 2n − 1 or less on [−1, 1].
The Gauss–Legendre quadrature rule is not typically used for integrable functions with endpoint singularities. Instead, if the integrand can be written as
where g(x) is well-approximated by a low-degree polynomial, then alternative nodes xi' and weights wi' will usually give more accurate quadrature rules. These are known as Gauss–Jacobi quadrature rules, i.e.,
Common weights include (Chebyshev–Gauss) and . One may also want to integrate over semi-infinite (Gauss–Laguerre quadrature) and infinite intervals (Gauss–Hermite quadrature).
It can be shown (see Press et al., or Stoer and Bulirsch) that the quadrature nodes xi are the roots of a polynomial belonging to a class of orthogonal polynomials (the class orthogonal with respect to a weighted inner-product). This is a key observation for computing Gauss quadrature nodes and weights.
For the simplest integration problem stated above, i.e., f(x) is well-approximated by polynomials on , the associated orthogonal polynomials are Legendre polynomials, denoted by Pn(x). With the n-th polynomial normalized to give Pn(1) = 1, the i-th Gauss node, xi, is the i-th root of Pn and the weights are given by the formula[3]
Some low-order quadrature rules are tabulated below (over interval [−1, 1], see the section below for other intervals).
Number of points, n | Points, xi | Weights, wi | ||
---|---|---|---|---|
1 | 0 | 2 | ||
2 | ±0.57735... | 1 | ||
3 | 0 | 0.888889... | ||
±0.774597... | 0.555556... | |||
4 | ±0.339981... | 0.652145... | ||
±0.861136... | 0.347855... | |||
5 | 0 | 0.568889... | ||
±0.538469... | 0.478629... | |||
±0.90618... | 0.236927... |
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:
with
Applying the point Gaussian quadrature rule then results in the following approximation:
Use the two-point Gauss quadrature rule to approximate the distance in meters covered by a rocket from to as given by
Change the limits so that one can use the weights and abscissae given in Table 1. Also, find the absolute relative true error. The true value is given as 11061.34 m.
Solution
First, changing the limits of integration from to gives
Next, get the weighting factors and function argument values from Table 1 for the two-point rule,
Now we can use the Gauss quadrature formula
since
Given that the true value is 11061.34 m, the absolute relative true error, is