# KEX-o[gϖ@̕_Əd

## KEX-o[glϕ̕_inodesjƏd݁iweightsjvZ܂B

 $\normal{\large\int_{\small -1}^{\hspace{25}\small 1}}f(x)dx\simeq w_{\small 1}f(-1)+w_nf(1)+{\large\sum_{\small i=2}^{n-1}}w_{i}f(x_i)\\\hspace{20}nodes\hspace{30} x_i:\hspace{10} P_{n-1}^{'}(x_i)=0\\\hspace{20}weights\hspace{15} w_{\small 1}=w_n={\large\frac{2}{n(n-1)}}\\\hspace{98} w_i={\large\frac{2}{n(n-1)[P_{n-1}(x_i)]^2}}\\$
 nodes half(x>=0) all 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
 61014182226303438424650
 $\normal\ i=2,3,...,n-1\\(1)\ initial\ x_{i}=\left(1-{\large\frac{3*(n-2)}{8(n-1)^3}}\right)cos(\frac{4i-3}{4(n-1)+1}\pi)\\(2)\ solve\hspace{20}P^'_{n-1}(x_i)=0\\ \hspace{20}Halley's\ method\hspace{20}x\leftarrow x-{\large\frac{2yy^'}{2[y^']^2-yy^{''}}}\\\hspace{10}y=P^'_{n-1}(x_i) ,\hspace{10}y^'=P^{''}_{n-1}(x_i) \\\hspace{10}y^{''}=P^{'''}_{n-1}(x_i) \\[10]\hspace{10} P^'_n(x)={\large\frac{n(P_{n-1}(x)-xP_n(x))}{1-x^2}}\\\hspace{10} P^{''}_n(x)={\large\frac{2xP^'_n(x)-n(n+1)P_n(x)}{1-x^2}}\\\hspace{10} P^{'''}_n(x)={\large\frac{2xP^{''}_n(x)-(n(n+1)-2)P^'_n(x)}{1-x^2}}\\[10](3)\ w_i={\large\frac{2}{n(n-1)[P_{n-1}(x_i)]^2}}\\$

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