Gaussian quadrature

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 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n. Modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi 1826.. The most common domain of integration for such a rule is taken as , so the rule is stated as w i = a n a n − 1 ∫ a b ω ( x ) p n − 1 ( x ) 2 d x p n ′ ( x i ) p n − 1 ( x i ) {displaystyle w_{i}={frac {a_{n}}{a_{n-1}}}{frac {int _{a}^{b}omega (x)p_{n-1}(x)^{2}dx}{p'_{n}(x_{i})p_{n-1}(x_{i})}}}     (1) w i = 1 p n ′ ( x i ) ∫ a b ω ( x ) p n ( x ) x − x i d x {displaystyle w_{i}={frac {1}{p'_{n}(x_{i})}}int _{a}^{b}omega (x){frac {p_{n}(x)}{x-x_{i}}}dx}     (2) 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 or less by a suitable choice of the nodes xi and weights wi for i = 1, ..., n. Modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi 1826.. The most common domain of integration for such a rule is taken as , 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 . 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 x i ′ {displaystyle x_{i}'} and weights w i ′ {displaystyle w_{i}'} will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e., Common weights include 1 / 1 − x 2 {displaystyle 1/{sqrt {1-x^{2}}}} (Chebyshev–Gauss) and 1 − x 2 {displaystyle {sqrt {1-x^{2}}}} . 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 [ − 1 , 1 ] {displaystyle } , 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 (Abramowitz & Stegun 1972, p. 887) Some low-order quadrature rules are tabulated below (over interval , see the section below for other intervals). An integral over must be changed into an integral over before applying the Gaussian quadrature rule. This change of interval can be done in the following way: