實(shí)驗(yàn)3 插值與數(shù)值積分

1、數(shù)學(xué)實(shí)驗(yàn)作業(yè)郭明釗 2012011880 化21一、機(jī)翼剖面積問(wèn)題1、問(wèn)題分析：求剖面積，可以根據(jù)上下兩邊界線分別與坐標(biāo)軸圍成的面積之差來(lái)求。先插 值，然后再用梯形公式求積分。2、matlab編寫(xiě)如下x1=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;x2=0 3 5 7 9 11 12 13 14 15;y2=0 1.2 1.7 2.0 2.1 2.0 1.8 1.2 1.0 1.6;xa=0:0.1:15;%按0.1的間隔在x方向產(chǎn)生插值點(diǎn)ya=spline(x1,y1,xa);%在 y方向計(jì)算三次樣條插

2、值xb=0:0.1:15;yb=spline(x2,y2,xb);xa ya yb%輸出插值后的結(jié)果subplot(1,2,1),plot(x1,y1,x2,y2);subplot(1,2,2),plot(xa,ya,xb,yb);s1=trapz(xa,ya) %上線與坐標(biāo)軸圍成的面積s2=trapz(xb,yb)%下線與坐標(biāo)軸圍成的面積s=s1-s2%機(jī)翼的剖面積3、運(yùn)行結(jié)果如下插值前后圖像對(duì)比插值后數(shù)組結(jié)果輸出（加工所需數(shù)據(jù)）ans =0000.10000.10890.04990.20000.21340.09900.30000.31370.14740.40000.40970.19510.

3、50000.50180.24210.60000.58980.28840.70000.67400.33400.80000.75450.37880.90000.83140.42301.00000.90470.46651.10000.97470.50941.20001.04130.55151.30001.10470.59301.40001.16510.63381.50001.22250.67391.60001.27700.71341.70001.32870.75231.80001.37780.79041.90001.42440.82802.00001.46850.86492.10001.51040.

4、90122.20001.54990.93682.30001.58740.97192.40001.62291.00632.50001.65651.04012.60001.68841.07322.70001.71851.10582.80001.74711.13782.90001.77421.16923.00001.80001.20003.10001.82451.23023.20001.84801.25993.30001.87041.28893.40001.89181.31743.50001.91251.34543.60001.93251.37273.70001.95191.39953.80001.

5、97081.42583.90001.98941.45154.00002.00761.47674.10002.02581.50144.20002.04391.52554.30002.06201.54914.40002.08031.57224.50002.09891.59474.60002.11791.61684.70002.13741.63834.80002.15751.65944.90002.17841.67995.00002.20001.70005.10002.22251.71965.20002.24591.73875.30002.27001.75735.40002.29481.77545.

6、50002.32011.79305.60002.34591.81025.70002.37201.82695.80002.39841.84305.90002.42491.85886.00002.45151.87406.10002.47811.88876.20002.50451.90306.30002.53071.91686.40002.55661.93016.50002.58211.94306.60002.60711.95536.70002.63151.96726.80002.65521.97866.90002.67801.98957.00002.70002.00007.10002.72102.

7、01007.20002.74112.01957.30002.76022.02857.40002.77862.03707.50002.79612.04507.60002.81302.05257.70002.82912.05957.80002.84462.06607.90002.85952.07198.00002.87392.07738.10002.88782.08228.20002.90132.08658.30002.91442.09028.40002.92722.09338.50002.93972.09598.60002.95202.09798.70002.96412.09948.80002.

8、97612.10028.90002.98812.10049.00003.00002.10009.10003.01192.09909.20003.02382.09749.30003.03552.09529.40003.04692.09259.50003.05782.08939.60003.06832.08579.70003.07822.08159.80003.08732.07709.90003.09562.072110.00003.10292.066810.10003.10922.061110.20003.11432.055210.30003.11812.049010.40003.12062.0

9、42510.50003.12152.035810.60003.12092.028910.70003.11852.021910.80003.11432.014710.90003.10822.007411.00003.10002.000011.10003.08971.992411.20003.07721.984111.30003.06261.974211.40003.04591.962111.50003.02691.946911.60003.00591.928011.70002.98261.904611.80002.95731.875911.90002.92971.841312.00002.900

10、01.800012.10002.86821.751612.20002.83421.697012.30002.79841.637712.40002.76061.574912.50002.72111.509912.60002.67981.444212.70002.63701.379012.80002.59271.315712.90002.54701.255613.00002.50001.200013.10002.45181.150113.20002.40261.106313.30002.35271.068713.40002.30211.037713.50002.25131.013413.60002

11、.20040.996013.70002.14960.985713.80002.09910.982813.90002.04910.987514.00002.00001.000014.10001.95191.020514.20001.90491.049214.30001.85941.086314.40001.81561.132014.50001.77371.186614.60001.73391.250314.70001.69631.323314.80001.66141.405714.90001.62921.497915.00001.60001.6000上邊線與坐標(biāo)軸圍成的面積S1 =33.9237

12、下邊線與坐標(biāo)軸圍成的面積s2 =22.5793最終的結(jié)果s =34444、結(jié)果分析：通過(guò)圖像可以直觀看出，經(jīng)過(guò)三次樣條插值后，圖像變得更加圓滑，直觀可 以看出擬合得很好，計(jì)算結(jié)果是11.3444二、地圖面積問(wèn)題1、問(wèn)題分析：這也是求面積問(wèn)題，可以分別用分段線性插值法和三次樣條插值法分別來(lái)做， 在計(jì)算面積的時(shí)候用梯形公式來(lái)積分。2、matlab編寫(xiě)公式x1=7.0 10.5 13.0 17.5 34.0 40.5 44.5 48.0 56.0 61.0 68.5 76.5 80.5 91.0 96.0 101.0 104.0 106.5 .5 118.0 123.5 136.5 142.0 14

13、6.0 150.0 157.0 158.0;y1=44 45 47 50 50 38 30 30 34 36 34 41 45 46 43 37 33 28 32 65 55 54 52 50 66 66 68;x2=7.0 10.5 13.0 17.5 34.0 40.5 44.5 48.0 56.0 61.0 68.5 76.5 80.5 91.0 96.0 101.0 104.0 106.5 .111.5 118.0 123.5 136.5 142.0 146.0 150.0 157.0 158.0;y2=44 59 70 72 93 100 110 110 110 117 118 11

14、6 118 118 121 124 121 121 121 122 116 83 81 82 86 85 68;xa=7.0:0.1:158;%按0.1的間隔在x方向產(chǎn)生插值點(diǎn)ya1=interp1(x1,y1,xa);%在 y方向計(jì)算分段線性插值ya2=spline(x1,y1,xa);%在 y方向計(jì)算三次樣條插值xb=7.0:0.1:158;%按0.1的間隔在x方向產(chǎn)生插值點(diǎn)yb1=interp1(x2,y2,xb);%在 y方向計(jì)算分段線性插值yb2=spline(x2,y2,xb);%在y方向計(jì)算三次樣條插值subplot(1,3,1),plot(x1,y1,x2,y2);%畫(huà)出圖像對(duì)

15、比不同插值法的效果gtext(原輪廓線,FontSize,12)subplot(1,3,2),plot(xa,ya1,xb,yb1);gtext(分段線性插值,FontSize,12)subplot(1,3,3),plot(xa,ya2,xb,yb2);gtext(三次樣條插值,FontSize,12)s1=(-trapz(xa,ya1)+trapz(xb,yb1)*(40/18)A2% 上下面積之差vpa(s1,8)%控制有效數(shù)字八位s2=(-trapz(xa,ya2)+trapz(xb,yb2)*(40/18)A2vpa(s2,8)%控制有效數(shù)字八位3、結(jié)果：不同的插值方法圖像對(duì)比分段線性

16、插值三次樣條插值原輪廓線si =4.2414e+004ans =42413.58 (分段線性插值，梯形公式，單位是平方千米)s2 =4.2468e+004ans =42468.246 (三次樣條插值，梯形公式，單位是平方千米)4、結(jié)果分析：在這個(gè)題目當(dāng)中，兩種插值方法得到的結(jié)果是差不多的，但是都和真實(shí)值 41288km2有著比較大的差距，這應(yīng)該是給出的樣點(diǎn)不夠多或者分布不夠好。三、車(chē)流量問(wèn)題1、問(wèn)題分析：題目中給出的數(shù)據(jù)是一分鐘內(nèi)的車(chē)流量，那么這可以看作是瞬時(shí)的車(chē)流量， 而一天當(dāng)中總的車(chē)流量是瞬時(shí)的車(chē)流量對(duì)時(shí)間的積分，所以思路就是先分別用分段線性插值 和三次樣條插值法進(jìn)行插值，然后計(jì)算出插值點(diǎn)

17、的插值，然后再用梯形公式進(jìn)行積分。2、用matlab進(jìn)行編寫(xiě)計(jì)算：x0=0:60:24*60;x0(:,2 4 14 16)=;y0=2 2 0 2 5 8 25 12 5 10 12 7 9 28 22 10 9 11 8 9 3;x=0:1:24*60;%間隔一分鐘插值y1=interp1(x0,y0,x);%在 y方向計(jì)算分段線性插值y2=spline(x0,y0,x);%在y方向計(jì)算三次樣條插值subplot(1,3,1),plot(x0,y0);grid on;xlabel(time/h)ylabel( 一天過(guò)橋的車(chē)的數(shù)量)gtext(原輪廓線,FontSize,12);subplot(1,3,2),plot(x,y1);grid on;xlabel(time/h)ylabel( 一天過(guò)橋的車(chē)的數(shù)量)gtext(分段線性插值,FontSize,12);subplot(1,