清华大学数学实验报告1.docx

数学实验报告1插值与数值积分电13苗键强2011010645一、实验目的 1. 掌握用 MATLAB 计算拉格朗日、分段线性、三次样条三种插值的方法，改变结点的数目，对三种插值结果进行初步分析。 2. 掌握用 MATLAB 及梯形公式、辛普森公式计算数值积分。 3. 通过实例学习用插值和数值积分解决实际问题。二、实验内容题目1 考虑函数 fx=11+x2 ,x-5,5 Q1：取不同的节点数目分别用Lagrange 插值、分段线性、三次插值以及三次样条插值近似 f(x)，分析插值效果。Q2：选取非均匀插值节点，xk=5cos2k-12n,i=1,2,n尝试进行插值，并分析插值效果。Q1：【分析与解】对于x-5,5，可以通过n个节点先将其均分为n-1段，根据fx以及相应的x得到相应的y，在确定插值的节点（xk，yk），从而在matlab中通过Lagrange 插值、分段线性、三次插值以及三次样条插值进行相应的插值。其公式分别为：Lagrange 插值：Ln=i=0nyili，其中li=(x-x0)(x-x1)(x-xi-1)(x-xi+1)(x-xn)(xi-x0)(xi-x1)(xi-xi-1)(xi-xi+1)(xi-xn)分段线性：Ln=i=0nyili，其中： li=x-xj-1xj-xj-1 ,（xj-1xxj，j=0舍去）；li=x-xj+1xj-xj+1,（xjx-Nt tN时），可以得到当N-5时N=-N12e-t22dt-N12e-Nt2dt=-122Ne-N22v v=z(n)-yy(n); end end v通过运行程序得到如下数值积分结果：节点值数值积分值-50-4.53.11E-06-43.14E-05-3.50.0002323-30.0013496-2.50.0062094-20.0227497-1.50.0668069-10.158655-0.50.308537400.50.50.691462210.84134441.50.933192420.97724952.50.993789930.99864953.50.999767240.99996824.50.999996551.0000001在这些节点的基础上，通过三次样条插值，对于每一个输入的x，matlab可以通过求得相应的y，即为x的近似值。而在误差分析比较中，通过选取一定数量的自变量，通过比较数值积分求得的x值与normcdf中求得的x值进行比较，得出其中最大的误差项为1.0685e-04，可以认为误差上限为10-4。在一些特定的自变量取值情况下的数值积分值与实际值的相对误差如下：自变量数值积分值实际值误差-50.00E+002.87E-072.87E-07-4.92.06E-064.79E-07-1.58E-06-4.82.82E-067.93E-07-2.03E-06-4.72.87E-061.30E-06-1.57E-06-4.62.77E-062.11E-06-6.62E-07-4.53.11E-063.40E-062.87E-07-4.44.45E-065.41E-069.59E-07-4.37.38E-068.54E-061.16E-06-4.21.25E-051.33E-058.88E-07-4.12.03E-052.07E-053.89E-07-43.14E-053.17E-052.87E-07-3.94.70E-054.81E-051.11E-06-3.87.07E-057.23E-051.67E-06-3.71.07E-040.00010781.14E-06-3.60.000159150.000159109-3.98E-08-3.50.000232350.0002326292.83E-07-3.40.000332810.0003369294.12E-06-3.30.000476480.0004834246.95E-06-3.20.000681650.0006871385.49E-06-3.10.000966610.0009676039.97E-07-30.001349650.0013498982.52E-07-2.90.001855720.0018658131.01E-05-2.80.002536450.002555131.87E-05-2.70.003450110.0034669741.69E-05-2.60.004654990.0046611886.20E-06-2.50.006209360.0062096653.09E-07-2.40.008183390.0081975361.41E-05-2.30.010694830.010724112.93E-05-2.20.013873270.0139034483.02E-05-2.10.017848350.0178644211.61E-05-20.022749680.0227501324.49E-07-1.90.028715850.028716567.13E-07-1.80.035921260.0359303199.05E-06-1.70.044549320.0445654631.61E-05-1.60.054783390.0547992921.59E-05-1.50.066806870.0668072013.32E-07-1.40.08079090.080756659-3.42E-05-1.30.096857730.096800485-5.72E-05-1.20.115117360.11506967-4.77E-05-1.10.135679790.135666061-1.37E-05-10.158655050.1586552542.07E-07-0.90.184112830.184060125-5.27E-05-0.80.211961710.211855399-1.06E-04-0.70.242069920.241963652-1.06E-04-0.60.274305740.274253118-5.26E-05-0.50.308537420.3085375391.20E-07-0.40.344593040.344578258-1.48E-05-0.30.382139940.382088578-5.14E-05-0.20.42080530.420740291-6.50E-05-0.10.460216280.460172163-4.41E-0500.500000050.5-4.60E-080.10.539783780.5398278374.41E-050.20.579194660.5792597096.51E-050.30.617859880.6179114225.15E-050.40.655406650.6554217421.51E-050.50.691462150.6914624613.11E-070.60.725693760.7257468825.31E-050.70.757929540.7580363481.07E-040.80.788037750.7881446011.07E-040.90.815886620.8159398755.33E-0510.841344410.8413447463.34E-071.10.864319650.8643339391.43E-051.20.884882070.884930334.83E-051.30.903141660.9031995155.79E-051.40.919208450.9192433413.49E-051.50.933192440.9331927993.55E-071.60.945215880.945200708-1.52E-051.70.955449920.955434537-1.54E-051.80.964077940.964069681-8.26E-061.90.971283340.971283449.66E-0820.97724950.9772498683.67E-072.10.982150840.982135579-1.53E-052.20.986125940.986096552-2.94E-052.30.98930440.98927589-2.85E-052.40.991815850.991802464-1.34E-052.50.993789880.9937903354.54E-072.60.995344230.995338812-5.42E-062.70.996549080.996533026-1.61E-052.80.997462710.99744487-1.78E-052.90.998143430.998134187-9.25E-0630.998649520.9986501025.79E-073.10.999032610.999032397-2.15E-073.20.999317640.999312862-4.78E-063.30.99952290.999516576-6.32E-063.40.999666650.999663071-3.58E-063.50.999767170.9997673711.98E-073.60.999840410.9998408914.86E-073.70.999892910.9998922-7.06E-073.80.999928890.999927652-1.24E-063.90.999952570.999951904-6.64E-0740.999968160.9999683291.69E-074.10.999979270.9999793426.90E-084.20.999987090.999986654-4.38E-074.30.999992190.99999146-7.30E-074.40.999995150.999994587-5.58E-074.50.999996530.9999966026.86E-084.60.999996930.9999978889.57E-074.70.999996910.9999986991.79E-064.80.999997060.9999992072.15E-064.90.999997940.9999995211.59E-0651.000000130.999999713-4.16E-07【结论】通过采用数值积分的方法，可以求得x在一点的近似值。比较近似值与实际值的误差可以发现，在积分曲线的二阶导数比较小的时候，数值积分值与原函数值