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Nodes and weights for Gaussian integration methods (selection of integration method)

TOP/Numerical integration/Gaussian Numerical Integration (Selection of Integration Methods)
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Calculates nodes and weights for various Gaussian integration methods.
(Gauss–Legendre integral, Chebyshev integral of the first kind, Chebyshev integral of the second kind, Laguerre integral, Hermite integral, Jacobi integral, Lobatto integral, Kronrod integral)
\(\normalsize Gauss-Legendre\ quadrature\\ {\large\int_{\small -1}^{\small 1}}f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\\ nodes\hspace{30px} x_i:\ the\ i-th\ zeros\ of\ P_n(x)\\\ weights\hspace{15px} w_i={\large\frac{2(1-x_i^2)}{[nP_{n-1}(x_i)]^2}}={\large \frac{2}{[P_n^1(x_i)]^2}}\\\)
kinds
order n
n=2,3,4,..,100
\(\normalsize Gaussian\ quadrature\\ \hspace{30px} {\large\int_{\small a}^{\small b}}w(x)f(x)dx\simeq{\large\displaystyle \sum_{\small i=1}^{n}}w_{i}f(x_i)\\ \hspace{20px} Quadrature \hspace{30px} interval \hspace{10px} w(x) \hspace{30px} polynomials\\ (1)\ Legendre \hspace{50px} [-1,\ 1] \hspace{25px} 1 \hspace{90px} P_n(x)\\ (2)\ Chebyshev\ 1st \hspace{5px} (-1,\ 1) \hspace{10px} {\large\frac{1}{\sqrt{1-x^2}}} \hspace{70px} T_n(x) \\ (3)\ Chebyshev\ 2nd \hspace{5px} [-1,\ 1] \hspace{10px} \sqrt{1-x^2} \hspace{65px} U_n(x) \\ (4)\ Laguerre \hspace{50px} [0,\infty) \hspace{20px} x^{\alpha} e^{-x} \hspace{65px} L_n^\alpha (x) \\ (5)\ Hermite \hspace{50px} (-\infty,\infty) \hspace{10px} e^{-x^2} \hspace{70px} H_n (x) \\ (6)\ Jacobi \hspace{75px} (-1,\ 1) \hspace{10px} (1-x)^{\alpha}(1+x)^{\beta} \hspace{5px} J_n^{\alpha,\beta} (x) \\ (7)\ Lobatto \hspace{60px} [-1,\ 1] \hspace{25px} 1 \hspace{80px} P_{n-1}^{'} (x)\\ (8)\ Kronrod \hspace{50px} [-1,\ 1] \hspace{25px} 1 \hspace{90px} P_n(x)\\ \)

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