积分近似 - 辛普森的规则据/h1>
测验据/h4>
相关......据/h4>
- 结石据/span>>据/span>
辛普森的规则据/strong>是一种近似函数积分的方法。通常(但并不总是)比使用近似更准确据一种href="//www.parkandroid.com/wiki/riemann-sums/" class="wiki_link" title="riemann Sum." target="_blank">riemann Sum.据/一种>或者据一种href="//www.parkandroid.com/wiki/integral-approximation-trapezium-rule/" class="wiki_link" title="梯形规则" target="_blank">梯形规则据/一种>,并确切地用于线性和二次函数。近似于四次积分的错误 -据一种href="//www.parkandroid.com/wiki/differentiable-function/" class="wiki_link" title="可怜的" target="_blank">可怜的据/一种>SIMPSON的规则的功能与间隔某些点的函数的第四个导数成比例。据/p>
(比较据一种href="//www.parkandroid.com/wiki/integral-approximation-trapezium-rule/" class="wiki_link" title="梯形规则" target="_blank">梯形规则据/一种>,这近似据span class="katex">
F据/span>(据/span>X据/span>)据/span>通过线性函数通过间隔的端点。)据/p>
定义据/h2>
认为据span class="katex"> F据/span>(据/span>X据/span>)据/span>在间隔内定义据span class="katex"> [据/span>一种据/span>那据/span>B.据/span>]据/span>。据/span>然后SIMPSON整个间隔的规则近似于明确积分据span class="katex"> F据/span>(据/span>X据/span>)据/span>在公式的间隔据span class="katex-display"> ∫据/span>一种据/span>B.据/span>F据/span>(据/span>X据/span>)据/span>D.据/span>X据/span>≈据/span>6.据/span>B.据/span>-据/span>一种据/span>(据/span>F据/span>(据/span>一种据/span>)据/span>+据/span>4.据/span>F据/span>(据/span>2据/span>一种据/span>+据/span>B.据/span>)据/span>+据/span>F据/span>(据/span>B.据/span>)据/span>)据/span>。据/span>这个想法是,如果据span class="katex"> F据/span>(据/span>X据/span>)据/span>=据/span>1据/span>那据/span>X据/span>那据/span>或者据span class="katex"> X据/span>2据/span>那据/span>该公式是一个确切的平等。因此,辛普森的规则给出了任何二次函数的正确积分。一般来说,辛普森的规则近似据span class="katex"> F据/span>(据/span>X据/span>)据/span>通过抛物线通过图表上的点据span class="katex"> F据/span>(据/span>X据/span>)据/span>和据span class="katex"> X据/span>- 建结据span class="katex"> 一种据/span>那据/span>2据/span>一种据/span>+据/span>B.据/span>那据/span>B.据/span>。据/span>
辛普森的规则通常通过将间隔打破进入据span class="katex"> N据/span>平等大小的子场,在哪里据span class="katex"> N据/span>使用上述估计是一种偶数,并近似于每对相邻子内部的积分。据/p>
也就是说,让据span class="katex"> X据/span>0.据/span>=据/span>一种据/span>那据/span>X据/span>1据/span>=据/span>一种据/span>+据/span>N据/span>B.据/span>-据/span>一种据/span>那据/span>X据/span>2据/span>=据/span>一种据/span>+据/span>2据/span>N据/span>B.据/span>-据/span>一种据/span>那据/span>......据/span>那据/span>X据/span>N据/span>-据/span>1据/span>=据/span>一种据/span>+据/span>(据/span>N据/span>-据/span>1据/span>)据/span>N据/span>B.据/span>-据/span>一种据/span>那据/span>X据/span>N据/span>=据/span>B.据/span>。据/span>然后据span class="katex-display"> ∫据/span>一种据/span>X据/span>2据/span>F据/span>(据/span>X据/span>)据/span>D.据/span>X据/span>∫据/span>X据/span>2据/span>X据/span>4.据/span>F据/span>(据/span>X据/span>)据/span>D.据/span>X据/span>≈据/span>3.据/span>N据/span>B.据/span>-据/span>一种据/span>(据/span>F据/span>(据/span>一种据/span>)据/span>+据/span>4.据/span>F据/span>(据/span>X据/span>1据/span>)据/span>+据/span>F据/span>(据/span>X据/span>2据/span>)据/span>)据/span>≈据/span>3.据/span>N据/span>B.据/span>-据/span>一种据/span>(据/span>F据/span>(据/span>X据/span>2据/span>)据/span>+据/span>4.据/span>F据/span>(据/span>X据/span>3.据/span>)据/span>+据/span>F据/span>(据/span>X据/span>4.据/span>)据/span>)据/span>等等。添加这些给予据span class="katex-display"> ∫据/span>一种据/span>B.据/span>F据/span>(据/span>X据/span>)据/span>D.据/span>X据/span>≈据/span>3.据/span>N据/span>B.据/span>-据/span>一种据/span>(据/span>F据/span>(据/span>一种据/span>)据/span>+据/span>4.据/span>F据/span>(据/span>X据/span>1据/span>)据/span>+据/span>2据/span>F据/span>(据/span>X据/span>2据/span>)据/span>+据/span>4.据/span>F据/span>(据/span>X据/span>3.据/span>)据/span>+据/span>2据/span>F据/span>(据/span>X据/span>4.据/span>)据/span>+据/span>⋯据/span>+据/span>4.据/span>F据/span>(据/span>X据/span>N据/span>-据/span>1据/span>)据/span>+据/span>F据/span>(据/span>B.据/span>)据/span>)据/span>。据/span>
让据span class="katex"> F据/span>(据/span>X据/span>)据/span>=据/span>X据/span>4.据/span>那据/span> 一种据/span>=据/span>0.据/span>那据/span> B.据/span>=据/span>4.据/span>。据/span>
将间隔划分为据span class="katex"> 4.据/span>平等的子宫内壁给提供据span class="katex-display"> ∫据/span>0.据/span>4.据/span>X据/span>4.据/span>D.据/span>X据/span>=据/span>3.据/span>1据/span>(据/span>0.据/span>4.据/span>+据/span>4.据/span>(据/span>1据/span>)据/span>4.据/span>+据/span>2据/span>(据/span>2据/span>)据/span>4.据/span>+据/span>4.据/span>(据/span>3.据/span>)据/span>4.据/span>+据/span>4.据/span>4.据/span>)据/span>=据/span>2据/span>0.据/span>5.据/span>3.据/span>1据/span>。据/span>与积分的实际值进行比较,即据span class="katex"> 4.据/span>5.据/span>/据/span>5.据/span>=据/span>2据/span>0.据/span>4.据/span>。据/span>8.据/span>。据/span>
估计误差据/h2>
似乎合理的是,SIMPSON的规则估计的错误对间隔的误差应与函数的第三衍生物成比例,类似于错误据一种href="//www.parkandroid.com/wiki/integral-approximation-trapezium-rule/" class="wiki_link" title="梯形规则" target="_blank">梯形规则据/一种>与第二衍生物成比例。但事实上,辛普森的规则是额外的力量据span class="katex"> X据/span>“免费”:据/p>
表明辛普森的规则给出了确切的整体据span class="katex"> X据/span>3.据/span>在任何间隔。据/p>
辛普森的规则近似据span class="katex"> [据/span>一种据/span>那据/span>B.据/span>]据/span>是据span class="katex-display"> 6.据/span>B.据/span>-据/span>一种据/span>(据/span>一种据/span>3.据/span>+据/span>4.据/span>(据/span>2据/span>一种据/span>+据/span>B.据/span>)据/span>3.据/span>+据/span>B.据/span>3.据/span>)据/span>=据/span>6.据/span>B.据/span>-据/span>一种据/span>(据/span>2据/span>3.据/span>一种据/span>3.据/span>+据/span>2据/span>3.据/span>一种据/span>2据/span>B.据/span>+据/span>2据/span>3.据/span>一种据/span>B.据/span>2据/span>+据/span>2据/span>3.据/span>B.据/span>3.据/span>)据/span>=据/span>4.据/span>B.据/span>-据/span>一种据/span>(据/span>一种据/span>3.据/span>+据/span>一种据/span>2据/span>B.据/span>+据/span>一种据/span>B.据/span>2据/span>+据/span>B.据/span>3.据/span>)据/span>=据/span>4.据/span>B.据/span>4.据/span>-据/span>一种据/span>4.据/span>那据/span>哪个等于据span class="katex"> ∫据/span>一种据/span>B.据/span>X据/span>3.据/span>D.据/span>X据/span>。据/span>
因此,辛普森的规则错误与第四个衍生物成比例:据/p>
让据span class="katex"> E.据/span>是近似值的错误据span class="katex"> ∫据/span>一种据/span>B.据/span>F据/span>(据/span>X据/span>)据/span>D.据/span>X据/span>通过辛普森的规则据span class="katex"> N据/span>平等大小的子宫。认为据span class="katex"> F据/span>四次可分辨方向据span class="katex"> (据/span>一种据/span>那据/span>B.据/span>)据/span>。据/span>然后,如果有一些常数据span class="katex"> K.据/span>这样据span class="katex"> F据/span>'据/span>'据/span>'据/span>'据/span>(据/span>C据/span>)据/span>据据/span>K.据/span>对所有人据span class="katex"> C据/span>∈据/span>(据/span>一种据/span>那据/span>B.据/span>)据/span>那据/span> |据/span>E.据/span>|据/span>据据/span>1据/span>8.据/span>0.据/span>N据/span>4.据/span>K.据/span>(据/span>B.据/span>-据/span>一种据/span>)据/span>5.据/span>。据/span>
需要多少相等大小的子宫内地以保证近似值的错误据span class="katex"> ∫据/span>0.据/span>2据/span>X据/span>4.据/span>D.据/span>X据/span>由辛普森的规则是据span class="katex"> ≤.据/span>9.据/span>6.据/span>0.据/span>1据/span>还是据/p>
我们想要据span class="katex-display"> 1据/span>8.据/span>0.据/span>N据/span>4.据/span>K.据/span>⋅据/span>2据/span>5.据/span>≤.据/span>9.据/span>6.据/span>0.据/span>1据/span>那据/span>在哪里据span class="katex"> K.据/span>是第四衍生物的上限据span class="katex"> X据/span>4.据/span>。据/span>但这是一个常量,据span class="katex"> 2据/span>4.据/span>那据/span>所以据span class="katex"> K.据/span>=据/span>2据/span>4.据/span>。据/span>插入和简化给出据span class="katex-display"> N据/span>4.据/span>≥据/span>1据/span>8.据/span>0.据/span>9.据/span>6.据/span>0.据/span>⋅据/span>2据/span>4.据/span>⋅据/span>3.据/span>2据/span>=据/span>4.据/span>0.据/span>9.据/span>6.据/span>那据/span>所以据span class="katex"> N据/span>≥据/span>8.据/span>。据/span>答案是据span class="katex"> 8.据/span>。据/span>
⎩据/span>⎪据/span>⎪据/span>⎨据/span>⎪据/span>⎪据/span>⎧据/span>F据/span>(据/span>1据/span>7.据/span>2据/span>9.据/span>)据/span>=据/span>4.据/span>F据/span>(据/span>1据/span>7.据/span>3.据/span>0.据/span>)据/span>=据/span>5.据/span>F据/span>(据/span>1据/span>7.据/span>3.据/span>1据/span>)据/span>=据/span>6.据/span>
让据span class="katex"> F据/span>(据/span>X据/span>)据/span>是满足以下方程式的立方多项式。找据span class="katex"> ∫据/span>1据/span>7.据/span>2据/span>9.据/span>1据/span>7.据/span>3.据/span>1据/span>F据/span>(据/span>X据/span>)据/span>D.据/span>X据/span>。据/p>