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Gaussian Integration Methods (Select Integration Method)

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

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