# Nodes and Weights of Gauss-Hermite

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

 $\normal(1)\ {\large\int_{\small -\infty}^{\hspace{25}\small \infty}}e^{-x^2}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\[10](2)\ g(x)=e^{-x^2}f(x)\\\hspace{20}{\large\int_{\small -\infty}^{\hspace{25}\small \infty}}g(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}e^{x_i^2}g(x_i)\\[20]\hspace{5}nodes\hspace{35} x_i:\hspace{10} H_n(x_i)=0\\\hspace{5}weights\hspace{20} w_i={\large\frac{2^{n+1}{\large n!\sqrt{\pi}}}{[H_{n+1}(x_i)]^2}}\\$
nodes
 half(x>=0) all
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

i xi wi wiexi2

 $\normal\ i=1,2,...,n\\(1)\ x_{10}=\sqrt{2n+1}-\frac{1.85575}{(2n+1)^{1/6}}\\\hspace{25}x_{20}=x_1-{\large\frac{1.14n^{0.426}}{x_1}},\\\hspace{25}x_{30}=1.86x_{2}-0.86x_{1}\\\hspace{25}x_{40}=1.91x_{3}-0.91x_{2},\\\hspace{25}x_{i0}=1.86x_{i-1}-0.86x_{i-2}\\\hspace{30}..........\\[15](2)\ solve\ x_i\hspace{20}H_n(x_i)=0\\ \hspace{20}Halley's\ method\hspace{20}x\leftarrow x-{\large\frac{2yy^'}{2[y^']^2-yy^{''}}}\\ \hspace{20}y=H_n(x)\\ \hspace{20}y^'=H^'_n(x)=2nH_{n-1}(x)\\ \hspace{20}y^{''}=H^{''}_n(x)=2xH^'_n(x)-2nH_n(x)\\[10](3)\ w_i={\large\frac{2^{n+1}{\large n!\sqrt{\pi}}}{[H_{n+1}(x_i)]^2}}\\$

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