1、實驗3插值與數值積分化工系 分0班2010011805 張亞清【實驗目的】1、掌握用MATLAB十算拉格朗日、分段線性、三次樣條三種插值的方法，改變節點的數目,對三種插值結果進行初步分析。2、掌握用MATLA吸梯形公式、辛普森公式計算數值積分。3、通過實例學習用插值和數值積分解決實際問題。【實驗內容】1、題目1 （3）選擇一些函數，在 n個節點上（n不要太大，如511）用拉格朗日、分段線性、三次樣條三種插值方法，計算 m個插值點的函數值（m要適中，如50100）。通過數值和圖形輸出，將三種插值結果與精確值進行比較。適當增加n,再作比較，由此作初步分析。10（3） y= cos x , -2 &

2、lt;x<2.【問題分析】由于對稱性，只分析函數在 x正半軸的情況即可。取 9個節點，即以-2為首項，公 差為0.8的等差數列。所求插值點設為81個，即以-2為首項，公差為0.05的等差數列。【問題解答】 程序如下：x0=-2:0.8:2;y0=cos(xG).A1G;x=-2:0.05:2;y=cos(x).A10;y1=lagr(x0,y0,x);y2=interp1(x0,y0,x);y3=spline(x0,y0,x);for k=1:21xx(k)=x(39+2*k);yy(k)=y(39+2*k);yy1(k)=y(39+2*k);yy2(k)=y2(39+2*k);yy3(

3、k)=y3(39+2*k);endA=xx;yy;yy2;yy3'z=0*x;plot(x,z, 'k:' ,x,y, 'k-',x,y1, 'm' ,x,y2, 'r' ,x,y3, 'b')title（'圖1.三種插值方法的對比'）;gtext（'原函數'）gtext（拉格朗日插值）；gtext（'分段線性插值'）;gtext（'三次樣條插值）;（1） 當節點數為6,插值點數為81時，輸出的計算結果如表1,其中y=cos10x , y1 ,y2,

4、y3依次是拉格朗日、分段線性、三次樣條插值，與精確值y相比，顯然他們在節點處相等。X在0,0.6的范圍內，三種方法的偏差都很大，在0.7 , 1,2的范圍內，三種方法得到的插值結果很接近，但和原函數還是有一定偏差，在 1,2,2的范圍內，分段線性插值的結果和原函數吻合的最好，三次樣條插值和 拉格朗日插值都出現了龍格現象。由于原函數是偶函數，所以在負數范圍內結 論一樣。表1xyy1y2y301.00000.51490.43940.51630.10000.95110.51010.43940.51150.20000.81760.49560.43940.49710.30000.63320.47190.

5、43940.47310.40000.43940.43940.43940.43940.50000.27090.39900.38450.39650.60000.14670.35150.32960.34610.70000.06850.29810.27470.29010.80000.02690.24030.21970.23070.90000.00860.17970.16480.17001.00000.00210.11800.10990.10991.10000.00040.05740.05500.05261.20000.00000.00000.00000.00001.30000.0000-0.05160

6、.0001-0.04561.40000.0000-0.09480.0001-0.08231.50000.0000-0.12670.0001-0.10811.60000.0000-0.14410.0001-0.12081.70000.0000-0.14380.0001-0.11831.80000.0000-0.12220.0001-0.09881.90000.0000-0.07550.0001-0.05992.00000.00020.00020.00020.0002圖1.三種插值方法的對比a fiJ51.5從圖1中看出，-0.6,0.6范圍內，分段線性插值法得到的曲線水平，與原函數差異很大,這是

7、由于分段插值法兩節點之間的數治愈兩節點的值有關，缺失了中間的信息造成的。(2)當節點數增加為9,插值點數仍為81時，輸出的計算結果如下：表2xyy1y2y301.00001.00001.00001.00000.10000.95110.96080.85420.94800.20000.81760.84880.70840.81480.30000.63320.67970.56260.63470.40000.43940.47740.41680.44200.50000.27090.27090.27090.27090.60000.14670.09030.21720.14840.70000.0685-0.03

8、810.16340.07110.80000.0269-0.09640.10960.02810.90000.0086-0.07940.05590.00871.00000.00210.00210.00210.00211.10000.00040.12050.0017-0.00081.20000.00000.23210.0013-0.00201.30000.00000.28300.0008-0.00211.40000.00000.21870.0004-0.00131.50000.00000.00000.00000.00001.60000.0000-0.37350.00000.00141.70000.0

9、000-0.83180.00010.00241.80000.0000-1.19300.00010.00281.90000.0000-1.10920.00010.00222.00000.00020.00020.00020.0002圖1.三種插值方法的對比-2-1.5 J -0.50D,511.52由圖可見，三次樣條插值的圖形幾乎與原函數完全重合，分段線性插值的結果比上一次更接近原函數，拉格朗日插值在-0.5,0.5范圍內與原函數結果接近，在 1,5,2范圍內，出現了龍格現象，插入點數值偏離了原函數的值，與情況(1)相比，偏離更大。(3)當適當增加節點數，使得節點數目達到41個，插值點個數應相應增

10、加為 200個時，由于節點增多，所以這種情況三種方法得到的結果都與原函數很接近，三種方法得到的曲線幾乎都與原函數重合，如下圖：由此可見，當增加節點個數時，插值函數均更加逼近原函數。2、題目10下表給出的x, y數據位于機翼剖面的輪廓線上，y1和y2分別對應輪廓的上下線。 假設需要得到x坐標每改變0.1時的y坐標。試完成加工所需數據，畫出曲線，求機翼剖面 的面積。x035791112131415y101.82.22.73.03.12.92.52.01.6y201.21.72.02.12.01.81.21.01.6【問題分析】給定的數據點是有限的，要想確定更多的數據，就要運用插值方法。而加工斷面的

11、面積,則應通過數值積分分別求出上下輪廓線對x軸圍城的面積，然后做差求得。【問題解答】運用插值方法，程序如下：漁目中給的節點數據x=0,3:2:11,12:15;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;u=0:0.1:15;L1=lagr(x,y1,u);L2=lagr(x,y2,u);I1=interp1(x,y1,u);I2=interp1(x,y2,u);S1=spline(x,y1,u);S2=spline(x,y2,u);subplot(1,3,1),plot(u

12、,L1,'m',u,L2,'m')title(' 圖1.拉格朗日插值);subplot(1,3,2),plot(u,I1,'r',u,I2,'r')title('圖2.分段線性插值');subplot(1,3,3),plot(u,S1,'b',u,S2,'b')title('圖3.三次樣條插值');%弟形求積公式z1=trapz(u,S1);z2=trapz(u,S2);z=z1-z2普森求積公式k=length(S1);S11=S1(2:2:k-1);s1=

13、sum(S11);S12=S1(3:2:k-1);s2=sum(S12);zz1=(S1(1)+S1(k)+4*s1+2*s2)*0.1/3;l=length(S2);S21=S2(2:2:l-1);ss1=sum(S21);S22=S2(3:2:l-1);ss2=sum(S22); zz2=(S2(1)+S2(l)+4*ss1+2*ss2)*0.1/3; zz=zz1-zz211.3444 ,用辛普森公式得到加工剖繪圖得到如下三幅圖，可以看出拉格朗日插值法得出的圖形與實際不符，不予考慮。 次樣條插值法得到的圖形比分段線性插值法得到的曲線更光滑。A=u;I1;I2;S1;S2'xY1（

14、分段）Y2 （分段）Y1 (樣條)Y2 （樣條）0.00000.00000.00000.00000.00000.10000.06000.04000.10890.04990.20000.12000.08000.21340.09900.30000.18000.12000.31370.14740.40000.24000.16000.40970.19510.50000.30000.20000.50180.24210.60000.36000.24000.58980.28840.70000.42000.28000.67400.33400.80000.48000.32000.75450.37880.9000

15、0.54000.36000.83140.4230用matlab計算，梯形求積公式得到加工剖面面積為 面面積為11.3460 ,兩種方法得到的結果近似。插值后得到的加工數據如下表：1.00000.60000.40000.90470.46651.10000.66000.44000.97470.50941.20000.72000.48001.04130.55151.30000.78000.52001.10470.59301.40000.84000.56001.16510.63381.50000.90000.60001.22250.67391.60000.96000.64001.27700.71341

16、.70001.02000.68001.32870.75231.80001.08000.72001.37780.79041.90001.14000.76001.42440.82802.00001.20000.80001.46850.86492.10001.26000.84001.51040.90122.20001.32000.88001.54990.93682.30001.38000.92001.58740.97192.40001.44000.96001.62291.00632.50001.50001.00001.65651.04012.60001.56001.04001.68841.07322

17、.70001.62001.08001.71851.10582.80001.68001.12001.74711.13782.90001.74001.16001.77421.16923.00001.80001.20001.80001.20003.10001.82001.22501.82451.23023.20001.84001.25001.84801.25993.30001.86001.27501.87041.28893.40001.88001.30001.89181.31743.50001.90001.32501.91251.34543.60001.92001.35001.93251.37273

18、.70001.94001.37501.95191.39953.80001.96001.40001.97081.42583.90001.98001.42501.98941.45154.00002.00001.45002.00761.47674.10002.02001.47502.02581.50144.20002.04001.50002.04391.52554.30002.06001.52502.06201.54914.40002.08001.55002.08031.57224.50002.10001.57502.09891.59474.60002.12001.60002.11791.61684

19、.70002.14001.62502.13741.63834.80002.16001.65002.15751.65944.90002.18001.67502.17841.67995.00002.20001.70002.20001.70005.10002.22501.71502.22251.71965.20002.25001.73002.24591.73875.30002.27501.74502.27001.75735.40002.30001.76002.29481.77545.50002.32501.77502.32011.79305.60002.35001.79002.34591.81025

20、.70002.37501.80502.37201.82695.80002.40001.82002.39841.84305.90002.42501.83502.42491.85886.00002.45001.85002.45151.87406.10002.47501.86502.47811.88876.20002.50001.88002.50451.90306.30002.52501.89502.53071.91686.40002.55001.91002.55661.93016.50002.57501.92502.58211.94306.60002.60001.94002.60711.95536

21、.70002.62501.95502.63151.96726.80002.65001.97002.65521.97866.90002.67501.98502.67801.98957.00002.70002.00002.70002.00007.10002.71502.00502.72102.01007.20002.73002.01002.74112.01957.30002.74502.01502.76022.02857.40002.76002.02002.77862.03707.50002.77502.02502.79612.04507.60002.79002.03002.81302.05257

22、.70002.80502.03502.82912.05957.80002.82002.04002.84462.06607.90002.83502.04502.85952.07198.00002.85002.05002.87392.07738.10002.86502.05502.88782.08228.20002.88002.06002.90132.08658.30002.89502.06502.91442.09028.40002.91002.07002.92722.09338.50002.92502.07502.93972.09598.60002.94002.08002.95202.09798

23、.70002.95502.08502.96412.09948.80002.97002.09002.97612.10028.90002.98502.09502.98812.10049.00003.00002.10003.00002.10009.10003.00502.09503.01192.09909.20003.01002.09003.02382.09749.30003.01502.08503.03552.09529.40003.02002.08003.04692.09259.50003.02502.07503.05782.08939.60003.03002.07003.06832.08579

24、.70003.03502.06503.07822.08159.80003.04002.06003.08732.07709.90003.04502.05503.09562.072110.00003.05002.05003.10292.066810.10003.05502.04503.10922.061110.20003.06002.04003.11432.055210.30003.06502.03503.11812.049010.40003.07002.03003.12062.042510.50003.07502.02503.12152.035810.60003.08002.02003.1209

25、2.028910.70003.08502.01503.11852.021910.80003.09002.01003.11432.014710.90003.09502.00503.10822.007411.00003.10002.00003.10002.000011.10003.08001.98003.08971.992411.20003.06001.96003.07721.984111.30003.04001.94003.06261.974211.40003.02001.92003.04591.962111.50003.00001.90003.02691.946911.60002.98001.

26、88003.00591.928011.70002.96001.86002.98261.904611.80002.94001.84002.95731.875911.90002.92001.82002.92971.841312.00002.90001.80002.90001.800012.10002.86001.74002.86821.751612.20002.82001.68002.83421.697012.30002.78001.62002.79841.637712.40002.74001.56002.76061.574912.50002.70001.50002.72111.509912.60

27、002.66001.44002.67981.444212.70002.62001.38002.63701.379012.80002.58001.32002.59271.315712.90002.54001.26002.54701.255613.00002.50001.20002.50001.200013.10002.45001.18002.45181.150113.20002.40001.16002.40261.106313.30002.35001.14002.35271.068713.40002.30001.12002.30211.037713.50002.25001.10002.25131

28、.013413.60002.20001.08002.20040.996013.70002.15001.06002.14960.985713.80002.10001.04002.09910.982813.90002.05001.02002.04910.987514.00002.00001.00002.00001.000014.10001.96001.06001.95191.020514.20001.92001.12001.90491.049214.30001.88001.18001.85941.086314.40001.84001.24001.81561.132014.50001.80001.3

29、0001.77371.186614.60001.76001.36001.73391.250314.70001.72001.42001.69631.323314.80001.68001.48001.66141.405714.90001.64001.54001.62921.497915.00001.60001.60001.60001.60003、題目12在橋梁的一端每隔一段時間記錄1min有幾輛車過橋，得到下表的過橋車輛數量:時間車輛數/輛時間車輛數/輛時間車輛數/輛0:0029:001218:00222:00210:30519:00104:00011:301020:0095:00212:3012

30、21:00116:00514:00722:0087:00816:00923:0098:002517:002824:003試估計一天通過橋梁的車流量。【問題分析】講一天的1440分鐘逐一表示為x0,x 1,x 1440.與每個xi對應的yi表示從第i分鐘到第 i+1分鐘的車輛數。通過三種不同的插值方法得到各個插值點(令插值點為所有的 xi),再辛普森公式求積即可。【問題解答】1、三次樣條插值法程序如下：砧中數據t=0 2 4:9 10.5 11.5 12.5 14 16:24;car=2 2 0 2 5 8 25 12 5 10 12 7 9 28 22 10 9 11 8 9 3;x=0:1/60:24;獷次樣條插值法q=spline(t,car,x);plot(x,q,'r')gtext('時間-流量圖)xlabel（' 時間/h'）,ylabel（'流量/輛'）求一天的流量，輸入： S=60*trapz(x,q)得到結果：S=1.2668e+0042、分段線性插值法程序如下：砧中數據t=0 2 4:9 10.5 11.5 12.5 14