“Gaussian quadrature”的概念、定义、翻译、参考文献-科学参考

Gaussian quadrature

Mathematics

An approach in numerical analysis to the approximation of integrals. The aim is to approximate integrals of the form

$I (f) = \int_{a}^{b} f (x) w (x) d x$

where f,w are continuous functions and the weight function w(x) is positive. Then

$〈 f, g 〉 = \int_{a}^{b} f (x) g (x) w (x) d x$

defines an inner product on the space of continuous functions, and an orthonormal sequence p_n(x) of polynomials may be constructed such that p_n(x) has degree n. Gauss then showed that p_n + 1(x) has n + 1 distinct real roots x₀,…,x_n in (a,b) and that there are unique solutions W₀,…,W_n to the equations

$\int_{a}^{b} x^{k} w (x) d x = \sum_{i = 0}^{n} W_{i} x_{i}^{k} for 0 \leq k \leq 2 n + 1.$

It is then the case that

$Q_{n} (f) = \sum_{i = 0}^{n} W_{i} f (x_{i})$

exactly equals I(f) for polynomials of degree 2n + 1 or less, and more generally for continuous functions Q_n(f) tends to I(f) as n→∞.

Computer

See numerical integration.

单词	Gaussian quadrature
释义	Gaussian quadrature Mathematics An approach in numerical analysis to the approximation of integrals. The aim is to approximate integrals of the form $I (f) = \int_{a}^{b} f (x) w (x) d x$ where f,w are continuous functions and the weight function w(x) is positive. Then $〈 f, g 〉 = \int_{a}^{b} f (x) g (x) w (x) d x$ defines an inner product on the space of continuous functions, and an orthonormal sequence p_n(x) of polynomials may be constructed such that p_n(x) has degree n. Gauss then showed that p_n + 1(x) has n + 1 distinct real roots x₀,…,x_n in (a,b) and that there are unique solutions W₀,…,W_n to the equations $\int_{a}^{b} x^{k} w (x) d x = \sum_{i = 0}^{n} W_{i} x_{i}^{k} for 0 \leq k \leq 2 n + 1.$ It is then the case that $Q_{n} (f) = \sum_{i = 0}^{n} W_{i} f (x_{i})$ exactly equals I(f) for polynomials of degree 2n + 1 or less, and more generally for continuous functions Q_n(f) tends to I(f) as n→∞. Computer See numerical integration.
随便看	Betelgeuse BET equation Bethe ansatz Bethe, Hans Albrecht Bethe, Hans Albrecht (1906–2005) Bethe–Weizsäcker cycle Beth's definability theorem BET isotherm Betts process between-groups design(between-subjects design) between-groups estimator Bevan, Aneurin (1897–1960) Bevatron bevel bevelled cliff Beveridge curve Beveridge Report Beveridge, William Henry, 1st Baron (1879–1963) Bevin, Ernest (1881–1951) beyond the banner B F Goodrich Company BFSK BFT BGA BGP