# Nodes and Weights of Gauss-Laguerre Calculator

## Calculates the nodes and weights of the Gauss-Laguerre integration.

 $\normal(1)\ {\large\int_{\small 0}^{\hspace{25}\small \infty}}x^\alpha e^{-x}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\[10](2)\ g(x)=x^\alpha e^{-x}f(x)\\\hspace{20}{\large\int_{\small 0}^{\hspace{25}\small \infty}}g(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}x_i^{-\alpha} e^{x_i}g(x_i)\\[10]\hspace{5}nodes\hspace{35}x_i:\hspace{10} L^{\alpha}_n(x_i)=0\\\hspace{5}weights\hspace{20} w_i={\large\frac{\Gamma(n+\alpha+1)x_i}{{\large n!}[(n+1)L^\alpha_{n+1}(x_i)]^2}}\\$
α
order n
 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 n=2,3,4,..,100
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal\ i=1,2,...,n\\(1)\ x_{10}={\large\frac{(1+\alpha)(3+0.92\alpha)}{1+2.4n+1.8\alpha}}\\\hspace{20}x_{20}=x_{1}+{\large\frac{15+6.25\alpha}{1+2.5n+0.9\alpha}}\\\hspace{20}x_{i0}=x_{i-1}+\left({\large\frac{1+2.55(i-2)}{1.9(i-2)}+\frac{1.26(i-2)\alpha}{1+3.5(i-2)}\right)\\\hspace{210}\times{\large\frac{x_{i-1}-x_{i-2}}{1+0.3\alpha}}\\\hspace{40} .......\\[10](2)\ solve\ x_i\hspace{20}L^\alpha_n(x_i)=0\\ \hspace{20}Halley's\ method\hspace{20}x\leftarrow x-{\large\frac{2yy^'}{2[y^']^2-yy^{''}}}\\ \hspace{20}y=L^\alpha_n(x)\\ \hspace{20}y^'=L^{\alpha'}_n(x)={\large\frac{nL^\alpha_n(x)-(n+a)L^\alpha_{n-1}(x)}{x}}\\ \hspace{20}y^{''}=L^{\alpha''}_n(x)={\large\frac{(x-1-\alpha)y^'-ny}{x}}\\[10](3)\ w_i={\large\frac{\Gamma(n+\alpha)x_i}{n!(n+\alpha)[L^\alpha_{n-1}(x_i)]^2}}\\\hspace{45}={\large\frac{\Gamma(n+\alpha+1)x_i}{{\large n!}[(n+1)L^\alpha_{n+1}(x_i)]^2}}\\$

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