建模实验 拉格朗日插值

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1、建模实验 插值方法【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，改变节点的数目， 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。【实验内容】 题 1 (3 )选择一些函数，在n个节点上(n不要太大，如511)用拉格朗日、分段线性、三次样条三 种插值的方法，计算m个插值点的函数值(m要适中，如50100)。通过数值和图形输出， 将三种插值结果与精确值进行比较，适当增加n,再作比较，由此作初步分析。下列函数供 选择参考：(3) y=cos10x,2WxW2解：当 n=8，

2、 m=50 时，在 matlab 中编程如下：x=2:0.5:2;y=(cos(x).10; a=2:4/50:2;yy=(cos(a).10; y1=lagr(x,y,a);y2=interp1(x,y,a);y3=interp1(x,y,a,spline); a;yy;y1;y2;y3数值输出如下：20.000155760.000155760.000155760.000155761.922.1987e0050.992890.000130840.00192671.841.771e0061.23940.000105920.00273911.765.538e0081.0788.0997e0050

3、.00278671.682.3645e0100.740095.6075e0050.00226321.64.5057e0160.373453.1153e0050.00136211.521.1388e0130.0632966.2305e0060.000277241.441.4242e0090.15090.000254420.000797781.361.6085e0070.261290.000593640.00166931.283.7541e0060.280380.000932870.00214361.23.9028e0050.232090.00127210.00202711.120.0002464

4、30.144950.00161130.0011261.040.00110510.0471670.00195050.00075320.960.00385020.036860.0236260.00391550.880.0110260.0885930.0666380.0112230.80.0269460.0964010.109650.0280940.720.0576840.0557170.152660.06006-0.640.110220.0314770.195670.11265-0.560.190610.157660.238690.19139-0.480.301450.31110.300110.3

5、0176-0.40.439420.47740.416760.44195-0.320.593940.641020.53340.59603-0.240.747660.786850.650050.74643-0.160.879370.901590.76670.8756-0.080.968470.974830.883350.96597011110.080.968470.974830.883350.965970.160.879370.901590.76670.87560.240.747660.786850.650050.746430.320.593940.641020.53340.596030.40.4

6、39420.47740.416760.441950.480.301450.31110.300110.301760.560.190610.157660.238690.191390.640.110220.0314770.195670.112650.720.057684-0.0557170.152660.060060.80.026946-0.0964010.109650.0280940.880.011026-0.0885930.0666380.0112230.960.0038502-0.036860.0236260.00391551.040.00110510.0471670.00195050.000

7、75321.120.000246430.144950.0016113-0.0011261.23.9028e-0050.232090.0012721-0.00202711.283.7541e-0060.280380.00093287-0.00214361.361.6085e-0070.261290.00059364-0.00166931.441.4242e-0090.15090.00025442-0.000797781.521.1388e-013-0.0632966.2305e-0060.000277241.64.5057e-016-0.373453.1153e-0050.00136211.68

8、2.3645e-010-0.740095.6075e-0050.00226321.765.538e-008-1.0788.0997e-0050.00278671.841.771e-006-1.23940.000105920.00273911.922.1987e-005-0.992890.000130840.001926720.000155760.000155760.000155760.00015576再观察图形：plot(a,yy,k*,a,y1,r,a,y2,b,a,y3,c)值得到的函数（蓝色）符合的较差，而拉格朗日插值函数（红色）在小范围内符合的较好 但大范围内波动较大。改变 n 值，令

9、 n=16 得到数值输出为：-20.000155760.000155760.000155760.00015576-1.922.1987e-005-1.41530.000105937.8336e-005-1.841.771e-006-0.637935.6096e-0052.9027e-005-1.765.538e-008-0.0407146.2616e-0062.1084e-006-1.682.3645e-0100.124082.331e-008-8.1509e-006-1.64.5057e-0160.0807081.2952e-008-7.4801e-006-1.521.1388e-0130.

10、011722.5928e-009-1.6089e-006-1.441.4242e-009-0.0187582.3322e-0064.1326e-006-1.361.6085e-007-0.01635.4417e-0067.8924e-006-1.283.7541e-006-0.00383218.5513e-0069.6095e-006-1.23.9028e-0050.00401230.000431812.6208e-005-1.120.000246430.00475330.00110710.000305-1.040.00110510.00259710.00178250.0012593-0.96

11、0.00385020.00280270.00881790.003417-0.880.0110260.00947080.0222130.0098536-0.80.0269460.0262780.0356090.026193-0.720.0576840.0579940.0712170.058164-0.640.110220.110840.143840.11129-0.560.190610.190930.216470.19083-0.480.301450.301360.30760.30193-0.40.439420.439170.454250.44258-0.320.593940.593780.60

12、0890.59685-0.240.747660.747680.740030.74717-0.160.879370.879450.826690.87606-0.080.968470.968510.913340.96612011110.080.968470.968510.913340.966120.160.879370.879450.826690.876060.240.747660.747680.740030.747170.320.593940.593780.600890.596850.40.439420.439170.454250.442580.480.301450.301360.30760.3

13、01930.560.190610.190930.216470.190830.640.110220.110840.143840.111290.720.0576840.0579940.0712170.0581640.80.0269460.0262780.0356090.0261930.880.0110260.00947080.0222130.00985360.960.00385020.00280270.00881790.0034171.040.00110510.00259710.00178250.00125931.120.000246430.00475330.00110710.0003051.23

14、.9028e-0050.00401230.000431812.6208e-0051.283.7541e-006-0.00383218.5513e-0069.6095e-0061.361.6085e-007-0.01635.4417e-0067.8924e-0061.441.4242e-009-0.0187582.3322e-0064.1326e-0061.521.1388e-0130.011722.5928e-009-1.6089e-0061.64.5057e-0160.0807081.2952e-008-7.4801e-0061.682.3645e-0100.124082.331e-008-

15、8.1509e-0061.765.538e-008-0.0407146.2616e-0062.1084e-0061.841.771e-006-0.637935.6096e-0052.9027e-0051.922.1987e-005-1.41530.000105937.8336e-00520.000155760.000155760.000155760.00015576图形为：题 10F表给出的x, y数据位于机翼剖面的轮廓线上，y1和y2分别对应轮廓的上下线。假设需要得到X坐标每改变0.1时的y坐标。试完成加工所需数据，画出曲线，求机翼剖面的面积。X035791112131415Y101.82.

16、22.73.03.12.92.52.01.6Y201.21.72.02.12.01.81.21.01.6解：先用三种插值法得出完成加工所需数据如下： x=0 3 5 7 9 11 12 13 14 15; y1=0 1.8 2.2 2.7 3.0 3.1 2.9 2.5 2.0 1.6; y2=0 1.2 1.7 2.0 2.1 2.0 1.8 1.2 1.0 1.6; a=0:0.1:15; l1=lagr(x,y1,a); l2=lagr(x,y2,a); l3=interp1(x,y1,a); l4=interp1(x,y2,a); l5=spline(x,y1,a); l6=splin

17、e(x,y2,a);a;l1;l2;l3;l4;l5;l6得到X值y1 （拉格朗日）y2 （拉格朗日）y1 （分段）y2 （分段）y1 （三次）y2 （三次）00000000.10.55158-4.94370.060.040.108910.0498610.21.0117-8.820.120.080.213430.0989970.31.391-11.7730.180.120.313670.147410.41.6992-13.9310.240.160.409740.195110.51.9452-15.4120.30.20.501760.24210.62.1371-16.3190.360.240.5

18、89820.288380.72.2821-16.7460.420.280.674040.333960.82.3868-16.7750.480.320.754520.378840.92.4573-16.4790.540.360.831380.4230312.4989-15.9240.60.40.904720.466541.12.5165-15.1650.660.440.974650.509361.22.5143-14.2520.720.481.04130.551511.32.4961-13.2290.780.521.10470.592981.42.4654-12.1310.840.561.165

19、10.633791.52.4251-10.990.90.61.22250.673941.62.3778-9.83260.960.641.2770.713421.72.3258-8.68121.020.681.32870.752261.82.2711-7.55391.080.721.37780.790451.92.2154-6.46541.140.761.42440.8279922.16-5.42721.20.81.46850.0.0.0.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.86492.12.1061-4.44841.260.841.5104901182.22

20、.0549-3.53531.320.881.5499936832.32.0069-2.69231.380.921.5874971852.41.9629-1.9221.440.961.622900632.51.9233-1.22531.511.656504012.61.8886-0.601911.561.041.688407322.71.8587-0.0504091.621.081.718510582.81.8340.431581.681.121.747113782.91.81450.847161.741.161.7742169231.81.21.81.21.823.11.79051.49421

21、.821.2251.824523023.21.78581.73431.841.251.84825993.31.78571.92491.861.2751.870428893.41.792.07061.881.31.891831743.51.79832.17651.91.3251.912534543.61.81042.2471.921.351.932537273.71.82592.2871.941.3751.951939953.81.84462.30071.961.41.970842583.91.86612.29261.981.4251.9894451541.892.266621.452.0076

22、47674.11.91612.22652.021.4752.025850144.21.94412.17572.041.52.04391.52554.31.97352.11742.061.5252.0621.54914.42.00422.05442.081.552.08031.57224.52.03591.98942.11.5752.09891.59474.62.06821.92452.121.62.11791.61684.72.1011.86162.141.6252.13741.63834.82.1341.80242.161.652.15751.65944.92.16711.74812.181

23、.6752.17841.679952.21.72.21.72.21.75.12.23261.65872.2251.7152.22251.71965.22.26471.62492.251.732.24591.73875.32.29621.59882.2751.7452.271.75735.42.3271.58062.31.762.29481.77545.52.3571.57032.3251.7752.32011.7935.62.38621.56772.351.792.34591.81025.72.41451.57232.3751.8052.3721.82695.82.44181.58382.41

24、.822.39841.8435.92.46821.60152.4251.8352.42491.858862.49361.62492.451.852.45151.8746.12.5181.65312.4751.8652.47811.88876.22.54151.68552.51.882.50451.9036.32.5641.72122.5251.8952.53071.91686.42.58571.75952.551.912.55661.93016.52.60661.79952.5751.9252.58211.9436.62.62661.84052.61.942.60711.95536.72.64

25、591.88172.6251.9552.63151.96726.82.66451.92252.651.972.65521.97866.92.68261.96212.6751.9852.6781.989572.722.722.727.12.71692.03572.7152.0052.7212.017.22.73352.06862.732.012.74112.01957.32.74962.09842.7452.0152.76022.02857.42.76542.12492.762.022.77862.0377.52.7812.14772.7752.0252.79612.0457.62.79632.

26、16662.792.032.8132.05257.72.81142.18172.8052.0352.82912.05957.82.82642.19292.822.042.84462.0667.92.84142.20032.8352.0452.85952.071982.85622.2042.852.052.87392.07738.12.87092.20412.8652.0552.88782.08228.22.88562.2012.882.062.90132.08658.32.90032.19492.8952.0652.91442.09028.42.91492.18622.912.072.9272

27、2.09338.52.92942.17522.9252.0752.93972.09598.62.94392.16232.942.082.9522.09798.72.95822.14792.9552.0852.96412.09948.82.97232.13252.972.092.97612.10028.92.98632.11642.9852.0952.98812.1004932.132.132.19.13.01342.08383.0052.0953.01192.0999.23.02642.0683.012.093.02382.09749.33.0392.05313.0152.0853.03552

28、.09529.43.0512.03933.022.083.04692.09259.53.06252.02693.0252.0753.05782.08939.63.07332.01613.032.073.06832.08579.73.08332.00693.0352.0653.07822.08159.83.09241.99963.042.063.08732.0779.93.10051.99413.0452.0553.09562.0721103.10761.99043.052.053.10292.066810.13.11361.98833.0552.0453.10922.061110.23.118

29、31.98783.062.043.11432.055210.33.12161.98863.0652.0353.11812.04910.43.12361.99033.072.033.12062.042510.53.12391.99283.0752.0253.12152.035810.63.12271.99553.082.023.12092.028910.73.11981.9983.0852.0153.11852.021910.83.115123.092.013.11432.014710.93.10852.00083.0952.0053.10822.0074113.123.123.1211.13.

30、08951.9973.081.983.08971.992411.23.0771.99143.061.963.07721.984111.33.06241.98263.041.943.06261.974211.43.04571.97023.021.923.04591.962111.53.02681.953731.93.02691.946911.63.00581.93282.981.883.00591.92811.72.98261.90722.961.862.98261.904611.82.95721.87652.941.842.95731.875911.92.92971.84082.921.822

31、.92971.8413122.91.82.91.82.91.812.12.86821.75422.861.742.86821.751612.22.83441.70352.821.682.83421.69712.32.79861.64852.781.622.79841.637712.42.76091.58952.741.562.76061.574912.52.72141.52722.71.52.72111.509912.62.68011.46252.661.442.67981.444212.72.63721.39622.621.382.6371.37912.82.59281.32962.581.

32、322.59271.315712.92.5471.26372.541.262.5471.2556132.51.22.51.22.51.213.12.45191.13992.451.182.45181.150113.22.40291.08482.41.162.40261.106313.32.35311.03642.351.142.35271.068713.42.30270.996172.31.122.30211.037713.52.2520.965642.251.12.25131.013413.62.20110.946222.21.082.20040.9959713.72.15020.93913

33、2.151.062.14960.9857213.82.09960.945362.11.042.09910.9828413.92.04940.965572.051.022.04910.987531142121214.11.95151.04831.961.061.95191.020514.21.90421.10961.921.121.90491.049214.31.85831.18191.881.181.85941.086314.41.81421.26271.841.241.81561.13214.51.7721.3481.81.31.77371.186614.61.7321.43251.761.

34、361.73391.250314.71.69451.50951.721.421.69631.323314.81.65981.57051.681.481.66141.405714.91.62821.6051.641.541.62921.4979151.61.61.61.61.61.6接下来作图：subplot(1,3,1),plot(a,l1,m,a,l2,m)tit le (图1拉格朗日插值)；subplot(1,3,2),plot(a,l3,r,a,l4,r)title(图2.分段线性插值)；subplot(1,3,3),plot(a,l5,r,a,l6,r)tit le (图3.三次样条插值)；梯形公式s1=trapz(a,y1) s1 =211.14