Nodes and Weights of Gauss-Legendre

Calculates the nodes and weights of the Gauss-Legendre integration.

 $\normal{\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\\ nodes\hspace{30} x_i:\hspace{10} P_n(x_i)=0\\\ weights\hspace{15} w_i={\large\frac{2(1-x^2)}{n[P_{n-1}(x)]^2}}={\large \frac{-2}{[P_n^1(x_i)]^2}\\$
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
 6dgt10dgt14dgt18dgt22dgt26dgt30dgt34dgt38dgt42dgt46dgt50dgt
 $\normal\ i=1,2,...,n\\(1)\ x_{i0}=\left(1-{\large\frac{n-1}{8n^3}}\right)cos(\frac{4i-1}{4n+2}\pi)\\(2)\ solve\ x_i\hspace{20}P_n(x_i)=0\\ \hspace{15}Halley's\ method\hspace{10}x\leftarrow x-{\large\frac{2yy^'}{2[y^']^2-yy^{''}}}\\ \hspace{20}y^'=P^'_n(x)={\large\frac{n(P_{n-1}(x)-xP_n(x))}{1-x^2}}\\ \hspace{20}y^{''}=P^{''}_n(x)={\large\frac{2xP^'_n(x)-n(n+1)P_n(x)}{1-x^2}}\\[10](3)\ w_i={\large\frac{2(1-x^2)}{n[P_{n-1}(x)]^2}}={\large \frac{-2}{[P_n^1(x_i)]^2}\\$
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