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francois |
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\newpage |
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\begin{subappendices} |
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\section{Table d'intégration simple} |
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\begin{tabular}{|c|c|c|c||c|c|c|c|} |
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\hline |
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\multicolumn{4}{|c||}{\textbf{Quadrature de Gauss-Legendre}} & \multicolumn{4}{|c|}{\textbf{Newton-Cotes}}\\ |
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\hline |
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$r$ & $t_i$ & $w_i$ & Degré de précision & $r$ & $t_i$ & $w_i$ & Degré de précision \\ |
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\hline |
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$1$ & $0$ & $2$ & $1$ & \multicolumn{4}{|c|}{}\\ |
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\hline |
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$2$ & $-\sqrt{\frac{1}{3}}$ & $1$ & $3$ & $2$ & $-1$ & $1$ & $1$\\ |
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& $\sqrt{\frac{1}{3}}$ & $1$ & & & $1$& $1$ & \\ |
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\hline |
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$3$ & $0$ & $\frac{8}{9}$ & $5$ & $3$ & $0$ & $\frac{4}{3}$ & $2$ \\ |
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& $-\frac{\sqrt{15}}{5}$ & $\frac{5}{9}$ & & & $-1$ & $\frac{1}{3}$ &\\ |
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& $\frac{\sqrt{15}}{5}$ & $\frac{5}{9}$ & & & $1$ & $\frac{1}{3}$&\\ |
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\hline |
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$4$ & $-\frac{\sqrt{525+70\sqrt{30}}}{35}$ & $\frac{18-\sqrt{30}}{36}$ & $7$ & $4$ & $-\frac{1}{3}$ & $\frac{3}{4}$ & $3$\\ |
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& $-\frac{\sqrt{525-70\sqrt{30}}}{35}$ & $\frac{18+\sqrt{30}}{36}$ && & $\frac{1}{3}$ & $\frac{3}{4}$ & \\ |
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& $\frac{\sqrt{525-70\sqrt{30}}}{35}$ & $\frac{18+\sqrt{30}}{36}$ & & & $-1$ & $\frac{1}{4}$ & \\ |
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& $\frac{\sqrt{525+70\sqrt{30}}}{35}$ & $\frac{18-\sqrt{30}}{36}$ & & & $1$ & $\frac{1}{4}$ & \\ |
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\hline |
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$5$ & $0$ & $\frac{128}{225}$ & $9$&$5$ & $0$ & $\frac{12}{45}$ & $5$\\ |
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& $-\frac{\sqrt{245+14\sqrt{70}}}{35}$ & $\frac{322-13\sqrt{70}}{900}$ & & & $-\frac{1}{2}$ & $\frac{32}{45}$ &\\ |
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& $-\frac{\sqrt{245-14\sqrt{70}}}{35}$ & $\frac{322+13\sqrt{70}}{900}$ & & & $\frac{1}{2}$ & $\frac{32}{45}$ &\\ |
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& $\frac{\sqrt{245-14\sqrt{70}}}{35}$ & $\frac{322+13\sqrt{70}}{900}$ & & & $-1$ & $\frac{7}{45}$ &\\ |
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& $\frac{\sqrt{245+14\sqrt{70}}}{35}$ & $\frac{322-13\sqrt{70}}{900}$ && & $1$ & $\frac{7}{45}$ &\\ |
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\hline |
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\end{tabular} |
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\begin{equation*} |
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\int_{-1}^1f(x)dx=\sum_{i=1}^rw_if(t_i) |
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\end{equation*} |
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\begin{eqnarray*} |
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\int_a^b f(x)dx&=&\sum_{i=0}^{n-1} \frac{x_{i+1}-x_i}{2} \sum_{j=1}^{r} w_jf\left(\frac{x_{i+1}-x_i}{2}t_j+\frac{x_{i+1}+x_i}{2}\right)\\ |
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x_i&=&a+\frac{i}{n}(b-a)\\ |
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\end{eqnarray*} |
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\end{subappendices} |
