# Nodes and Weights of Gauss-Jacobi Calculator Calculator

## Calculates the nodes and weights of the Gauss-Jacobi quadrature.

 $\normal Gauss-Jacobi\ quadrature\\ {\large\int_{\small -1}^{\hspace{25}\small 1}}(1-x)^{\alpha}(1+x)^{\beta}f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i)\\\ nodes\hspace{30} x_i:\ the\ i-th\ zeros\ of\ J_n^{\alpha,\beta}(x)\\\ weights\\\hspace{15} w_i=-{\normal\frac{(2n+\alpha+\beta+2)\Gamma(n+\alpha+1)\Gamma(n+\beta+1)2^{\alpha+\beta}}{(n+\alpha+\beta+1)\Gamma(n+\alpha+\beta+1)(n+1)!P_n^'(x_i)P_{n+1}(x_i)}}\\\hspace{15} {\large \frac{w_i}{w(x)}}={\large \frac{w_i}{(1-x)^{\alpha}(1+x)^{\beta}}}\\$
 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 α β 6digit10digit14digit18digit22digit26digit30digit34digit38digit42digit46digit50digit
 $\normal Gaussian\ quadrature\\\hspace{20} {\large\int_{\small a}^{\hspace{25}\small b}}w(x)f(x)dx\simeq{\large\sum_{\small i=1}^{n}}w_{i}f(x_i), \hspace{20} {\large\int_{\small a}^{\hspace{25}\small b}}g(x)dx\simeq{\large\sum_{\small i=1}^{n}}{\large\frac{w_{i}}{w(x)}}g(x_i)\\Gauss-Jacobi\ quadrature\\\hspace{30} interval(a,b):\hspace{20} (-1,\ 1)\\\hspace{30} w(x):\hspace{80} (1-x)^{\alpha}(1+x)^{\beta}\\\hspace{30} polynomialsl:\hspace{10} J_n^{\alpha,\beta} (x)\\$
Nodes and Weights of Gauss-Jacobi Calculator
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