“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.
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