數(shù)學(xué)實(shí)驗(yàn)――插值與數(shù)值積分

1、實(shí)驗(yàn)3插值和數(shù)值積分1黃浩第一，實(shí)驗(yàn)?zāi)康?)用matlab掌握四種插值方法：拉格朗日插值、分段線性插值、分段三次插值和三次樣條插值，并通過(guò)改變節(jié)點(diǎn)數(shù)來(lái)分析插值結(jié)果2)掌握matlab數(shù)值積分的計(jì)算，梯形公式和辛普森公式3)通過(guò)實(shí)例學(xué)習(xí)，利用插值和數(shù)值積分解決實(shí)際問(wèn)題二、實(shí)驗(yàn)內(nèi)容1.(1)問(wèn)題：?jiǎn)栴}描述：考慮到FX=11 x2 x-5，5，采用不同的節(jié)點(diǎn)數(shù)，采用拉格朗日插值、分段線性插值、三次插值和三次樣條插值來(lái)逼近f(x)，并分析插值效果方案實(shí)施：由于matlab中沒(méi)有預(yù)設(shè)的拉格朗日插值函數(shù)，所以我首先編寫了拉格朗日(程序見4.1)和分段線性插值函數(shù)(程序見4.2)，并對(duì)y=x2函數(shù)進(jìn)行了插值

2、測(cè)試(程序見4.3)，測(cè)試編寫的函數(shù)是否準(zhǔn)確，如下圖所示：在上圖中，紅線和綠線是自行設(shè)計(jì)的拉格朗日和分段線性函數(shù)的插值圖，藍(lán)色交叉點(diǎn)是matlab自帶的分段線性函數(shù)的插值圖。藍(lán)色和綠色完全一致，紅色和綠色基本一致，證明了自行設(shè)計(jì)的功能是正確的。然后進(jìn)行本課題所需的插值運(yùn)算：(x0，y0)作為節(jié)點(diǎn)，節(jié)點(diǎn)均勻分布在區(qū)間內(nèi)，節(jié)點(diǎn)數(shù)分別調(diào)整為6、11和21；x為插值點(diǎn)，繪圖時(shí)使用101個(gè)插值點(diǎn)(總是大于節(jié)點(diǎn)數(shù))，制表時(shí)使用的插值點(diǎn)數(shù)量根據(jù)節(jié)點(diǎn)數(shù)進(jìn)行調(diào)整；y為真值，y1、y2、y3和y4分別為拉格朗日、分段線性、分段三次和三次樣條插值方法得到的插值結(jié)果。1)當(dāng)節(jié)點(diǎn)數(shù)為6時(shí)：(即節(jié)點(diǎn)為-5，-3，3，5)

3、xyy1y2y3y401.00000.56730.50000.50000.56850.50000.80000.55010.50000.50000.55131.00000.50000.50000.50000.50000.50001.50000.30770.42130.40000.44250.41672.00000.20000.32120.30000.31330.31312.50000.13790.20970.20000.17750.20293.00000.10000.10000.10000.10000.10003.5萬(wàn)0.07550.00780.08460.07540.01834.00000.0

4、588-0.04810.06920.0559-0.02854.5萬(wàn)0.0471-0.04600.05380.0431-0.02635.00000.03850.03850.03850.03850.0385圖像有：從上圖可以看出，當(dāng)節(jié)點(diǎn)數(shù)為6時(shí)，插值效果不理想，節(jié)點(diǎn)間插值誤差較大，在x=0附近誤差最大2)當(dāng)節(jié)點(diǎn)數(shù)為11時(shí)：(即節(jié)點(diǎn)為-5，-4，4，5)xyy1y2y3y401.00001.00001.00001.00001.00000.50000.80000.84340.75000.79690.82051.00000.50000.50000.50000.50000.50001.50000.3077

5、0.23530.35000.32190.29732.00000.20000.20000.20000.20000.20002.50000.13790.25380.15000.13850.14013.00000.10000.10000.10000.10000.10003.5萬(wàn)0.0755-0.22620.07940.07550.07454.00000.05880.05880.05880.05880.05884.5萬(wàn)0.04711.57870.04860.04650.04845.00000.03850.03850.03850.03850.0385圖像有：由上圖可以看出，當(dāng)節(jié)點(diǎn)數(shù)為11時(shí)，分段三次和三

6、次樣條插值較好，其次是分段線性插值，而拉格朗日插值在-2，2區(qū)間有較高的精度，但當(dāng)|x|3時(shí)會(huì)發(fā)生龍格振蕩。3)當(dāng)節(jié)點(diǎn)數(shù)為21時(shí)(程序見4.4):(即節(jié)點(diǎn)為-5，-4.5，4.5，5)xyy1y2y3y401.00001.00001.00001.00001.00000.10000.99010.99040.96000.98690.98910.20000.96150.96260.92000.95260.95940.30000.91740.91890.88000.90500.91520.40000.86210.86320.84000.85150.86060.50000.80000.80000.800

7、00.80000.80000.60000.73530.73370.74000.74560.73720.70000.67110.66820.68000.68230.67420.80000.60980.60650.62000.61630.61270.90000.55250.55030.56000.55350.55411.00000.50000.50000.50000.50000.50001.10000.45250.45500.46150.45440.45171.20000.40980.41430.42310.41180.40891.3萬(wàn)0.37170.37650.38460.37280.37111

8、40000.33780.34110.34620.33790.33751.50000.30770.30770.30770.30770.30771.6萬(wàn)0.28090.27700.28620.28130.28101.70000.25710.25010.26460.25750.25721.8萬(wàn)0.23580.22810.24310.23600.23591.90000.21690.21150.22150.21690.21702.00000.20000.20000.20000.20000.20002.10000.18480.19180.18760.18500.18482.20000.17120.1843

9、0.17520.17130.1712230000.15900.17420.16280.15900.15892.40000.14790.15900.15030.14790.14792.50000.13790.13790.13790.13790.13792.60000.12890.11300.13030.12890.12892.7萬(wàn)0.12060.08960.12280.12070.1206280000.11310.07500.11520.11310.11312.90000.10630.07700.10760.10630.10633.00000.10000.10000.10000.10000.10

10、003.10000.09430.14080.09510.09430.09423.20000.08900.18580.09020.08900.0890330000.08410.21010.08530.08410.08413.40000.07960.18230.08040.07960.07963.5萬(wàn)0.07550.07550.07550.07550.07553.60000.0716-0.11430.07210.07160.0716370000.0681-0.34590.06880.06810.0681380000.0648-0.51360.06550.06480.06483.90000.0617

11、-0.44590.06220.06170.06174.00000.05880.05880.05880.05880.05884.10000.05611.13530.05650.05620.05614.20000.05362.67440.05410.05370.0536430000.05134.06910.05180.05130.05134.40000.04913.94510.04940.04910.04914.5萬(wàn)0.04710.04710.04710.04710.0471460000.0451-10.33450.04530.04510.0451470000.0433-28.66260.0436

12、0.04330.0433480000.0416-50.86440.04190.04150.04164.90000.0400-58.23810.04020.03990.04005.00000.03850.03850.03850.03850.0385圖像有：從上圖可以看出，當(dāng)節(jié)點(diǎn)數(shù)為21時(shí)，分段三次和三次樣條插值的精度很高，分段線性度稍差。然而，拉格朗日插值在x=0附近的區(qū)間中具有良好的精度，但是當(dāng)|x|3.5時(shí)會(huì)發(fā)生更嚴(yán)重的龍格振蕩(參見上表中藍(lán)色粗體字，當(dāng)x=4.9時(shí)，插值實(shí)際上達(dá)到-58.2381)為了定量比較各種算法的誤差，當(dāng)節(jié)點(diǎn)數(shù)為21時(shí)，上表的數(shù)據(jù)在1.1，2.0區(qū)間內(nèi)進(jìn)行處理，沒(méi)有龍

13、格振蕩，結(jié)果如下：(其中y1、y2、y3、y4分別是通過(guò)拉格朗日、分段線性、分段三次和三次樣條插值方法獲得的相對(duì)誤差的絕對(duì)值)xy1y2y3y41.10000.00550.01430.01540.00591.20000.01100.02120.02670.00701.3萬(wàn)0.01290.02150.03070.0046140000.00980.01500.02400.00121.50000.00000.00000.00000.00001.6萬(wàn)0.01390.03320.01710.00111.70000.02720.05800.02680.00121.8萬(wàn)0.03270.06580.02920

14、.00041.90000.02490.04730.02080.00052.00000.00000.00000.00000.0000總額0.13790.27620.19070.0218由上表可以看出，在無(wú)龍格振蕩的條件下，采用不同的插值方法，誤差由大到小是：分段線性分段三次拉格朗日三次樣條插值。得出結(jié)論：從以上比較中，我們可以得出以下結(jié)論：一般來(lái)說(shuō)，當(dāng)節(jié)點(diǎn)數(shù)增加時(shí)，插值精度一般會(huì)提高b)當(dāng)龍格振蕩不發(fā)生時(shí)，分段三次、三次樣條插值和拉格朗日插值精度相近，插值效果較好，而分段線性插值不如上述三種，具體誤差為分段線性分段三次拉格朗日樣條插值c)當(dāng)節(jié)點(diǎn)數(shù)增加時(shí)，拉格朗日插值方法的龍格振蕩會(huì)更加劇烈，在區(qū)間的末端會(huì)出現(xiàn)更大的誤差，但區(qū)間內(nèi)的插值效果更好基于以上三點(diǎn)，我們還可以對(duì)四種插值方法進(jìn)行綜合評(píng)價(jià)：a)拉格朗日插值方法：當(dāng)節(jié)點(diǎn)數(shù)較少時(shí)，插值精度較差，當(dāng)節(jié)點(diǎn)數(shù)增加時(shí)，可能會(huì)發(fā)生龍格振蕩，導(dǎo)致誤差較大，這是其作為高階插值多項(xiàng)式的固有缺陷。因此，當(dāng)我們不能評(píng)價(jià)節(jié)點(diǎn)數(shù)是否適合所選區(qū)間，不能判斷插值的突變是否為龍格振蕩時(shí)，其精度就不能保證，可信度就低。b)分段線性插值：其精度隨節(jié)點(diǎn)數(shù)的增加而單調(diào)增加，但在相同情況下，其精度低于分段三次和三次樣條插值。因此，在計(jì)算機(jī)計(jì)算中，后兩者更有利于保證精