S', S(x)d«zaf(-1)+ S() is exact for all polynomials of degree < 1 then,
Answers
Answer:
Comparison between 2-point Gaussian and trapezoidal quadrature. The blue line is the polynomial {\textstyle y(x)=7x^{3}-8x^{2}-3x+3}{\textstyle y(x)=7x^{3}-8x^{2}-3x+3}, whose integral in [−1, 1] is 2⁄3. The trapezoidal rule returns the integral of the orange dashed line, equal to {\textstyle y(-1)+y(1)=-10}{\textstyle y(-1)+y(1)=-10}. The 2-point Gaussian quadrature rule returns the integral of the black dashed curve, equal to {\textstyle y\left(-{\sqrt {\frac {1}{3}}}\right)+y\left({\sqrt {\frac {1}{3}}}\right)={\frac {2}{3}}}{\textstyle y\left(-{\sqrt {\frac {1}{3}}}\right)+y\left({\sqrt {\frac {1}{3}}}\right)={\frac {2}{3}}}. Such a result is exact, since the green region has the same area as the sum of the red regions.
{\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i}),}{\displaystyle \int _{-1}^{1}f(x)\,dx\approx \sum _{i=1}^{n}w_{i}f(x_{i}),}
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
{\displaystyle f(x)=\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x),\quad \alpha ,\beta >-1,}{\displaystyle f(x)=\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x),\quad \alpha ,\beta >-1,}
where g(x) is well-approximated by a low-degree polynomial, then alternative nodes {\displaystyle x_{i}'}x_{i}' and weights {\displaystyle w_{i}'}w_{i}' will usually give more accurate quadrature rules. These are known as Gauss-Jacobi quadrature rules, i.e.,
{\displaystyle \int _{-1}^{1}f(x)\,dx=\int _{-1}^{1}\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x)\,dx\approx \sum _{i=1}^{n}w_{i}'g\left(x_{i}'\right).}{\displaystyle \int _{-1}^{1}f(x)\,dx=\int _{-1}^{1}\left(1-x\right)^{\alpha }\left(1+x\right)^{\beta }g(x)\,dx\approx \sum _{i=1}^{n}w_{i}'g\left(x_{i}'\right).}
Common weights include {\textstyle {\frac {1}{\sqrt {1-x^{2}}}}}{\textstyle {\frac {1}{\sqrt {1-x^{2}}}}} (Chebyshev–Gauss) and {\displaystyle {\sqrt {1-x^{2}}}}{\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.
PLZ MARK ME AS A BRAINLIST