计算科学 - 高斯积分(√x )(x) - 吾爱随笔录

我想为权重函数构建一个高斯积分 $w(x) = x^{1/2}$ 为了

\int_{0}^{1} x^{1 / 2} f (x) d x = a_{1} f (x_{1}) + a_{2} f (x_{2})

$\int_{0}^{1}x^{1/2}f(x)dx = a_{1}f(x_{1})+a_{2}f(x_{2})$

求解

\begin{aligned} a_{1} + a_{2} = & \int_{0}^{1} x^{1 / 2} d x = \frac{2}{3} \\ a_{1} x_{1} + a_{2} x_{2} = & \int_{0}^{1} x^{1 / 2} x d x = \frac{2}{5} \end{aligned}

$\begin{align*} a_{1}+a_{2} =& \int_{0}^{1}x^{1/2} dx = \frac{2}{3} \\ a_{1}x_{1}+a_{2}x_{2}=&\int_{0}^{1}x^{1/2}x dx =\frac{2}{5}\\ \end{align*}$

解决这个系统将产生 $a_{1},a_{2}$ ? 然后什么？