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利普希兹条件漏了L
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xy3xy3 committed Jul 9, 2024
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**利普希兹条件**: $ |f(x,y_1)-f(x,y_2)|\leqslant|y_1-y_2|, L>0 $
**利普希兹条件**: $|f(x,y_1)-f(x,y_2)|\leqslant{L|y_1-y_2|}, L>0$

**前向欧拉法**: $y_{n+1}=y_n+hf(x_n,y_n) $ ,局部截断误差 $T_{n+1}=\displaystyle \frac{h^2}{2}y''(x_n)+O(h^3)$

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| 3 | 4 | $\displaystyle y_{n+3} = y_{n+2} + \frac{h}{24} (9f_{n+3} + 19f_{n+2} - 5f_{n+1} + f_n)$ | $\displaystyle -\frac{19}{720}$ |
| 4 | 5 | $\displaystyle y_{n+4} = y_{n+3} + \frac{h}{720} (251f_{n+4} + 646f_{n+3} - 264f_{n+2} + 106f_{n+1} - 19f_n)$ | $\displaystyle -\frac{3}{160}$ |

**米尔尼方法** $y_{n+4}=\displaystyle y_{n}+\frac{4h}{3}(2f_{n+3}-f_{n+2}+2f_{n+1}) $
**米尔尼方法** $y_{n+4}=\displaystyle y_{n}+\frac{4h}{3}(2f_{n+3}-f_{n+2}+2f_{n+1}) $

局部截断误差 $ T_{n+4}=\displaystyle \frac{14}{45}h^5y^{(5)}(x_n)+O(h^6) $

**辛普森方法** $y_{n+2}=\displaystyle y_n+\frac h 3 (f_n+4f_{n+1}+f_{n+2}).$
**辛普森方法** $y_{n+2}=\displaystyle y_n+\frac h 3 (f_n+4f_{n+1}+f_{n+2}).$

局部截断误差 $ T_{n+2}=\displaystyle -\frac{h^5}{90}y^{(5)}(x_n)+O(h^6). $

**汉明方法** $y_{n+3}=\displaystyle \frac 1 8 (9y_{n+2}-y_n)+\frac{3h}8(f_{n+3}+2f_{n+2}-f_{n+1})$
**汉明方法** $y_{n+3}=\displaystyle \frac 1 8 (9y_{n+2}-y_n)+\frac{3h}8(f_{n+3}+2f_{n+2}-f_{n+1})$

局部截断误差 $ T_{n+3}=\displaystyle -\frac{h^5}{40}y^{(5)}(x_n)+O(h^6). $

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