数学实验作业二

千里之行，始于足下让知识带有温度。第第2页/共2页精品文档推荐数学实验作业二数学试验作业二

题目：P72.1.d)；6)；8)日期：2022-3-9

【试验目的】：

1、把握用MATLAB计算拉格朗日、分段线性、三次样条三种插值的办法，转变节点的数目，对三种插值结果举行初步分析。

2、把握用MATLAB作线性最小二乘的办法。

3、通过实例学习如何用插值办法与拟合办法解决实际问题，注重二者的联系和区分。

【试验内容】：

一：对函数2

xye-=,22x-≤≤在n个节点上（n不要太大，如5

至11）用拉格朗日、分段线性、三次样条三种插值办法，计算m个插值点的函数值（m要适中，如50至100）。通过数值和图形输出，将三种插值结果与精确值举行比较。适当增强n，再作比较，由此作初步分析。

【MATLAB源程序】

先附上拉各朗日插值程序如下：

functiony=lagr1(x0,y0,x)

n=length(x0);m=length(x);

fori=1:m

z=x(i);

s=0.0;

fork=1:n

p=1.0;

forj=1:n

ifj~=k

p=p*(z-x0(j))/(x0(k)-x0(j));

end

end

s=s+p*y0(k);

end

y(i)=s;

end

函数插值比较程序如下：

%数学试验作业二.1-d

clear;

n=5;

%在n个节点上举行插值

m=75;

%产生m个插值点，计算函数在插值点处的精确值，未来举行对照x=-2:4/(m-1):2;

y=exp(-x.^2);

z=0*x;

x0=-2:4/(n-1):2;

y0=exp(-x0.^2);

y1=lagr1(x0,y0,x);

%y1为拉格朗日插值

y2=interp1(x0,y0,x);

%y2为分段线性插值

y3=spline(x0,y0,x);

%y3为三次样条插值

[x'y'y1'y2'y3']

plot(x,z,'k',x,y,'r:',x,y1,'g-.',x,y2,'b',x,y3,'y--')

gtext('Lagr.'),gtext('Pieces.linear'),gtext('Spline'),

gtext('y=exp(-x.^2)')

holdoff;

%比较插值所得结果与函数在插值点处的精确值

s='xyy1y2y3'

[x'y'y1'y2'y3']

【MATLAB计算结果】

n=5时，得到结果如下：

数值比较如下：

xyy1y2y3

－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－-2.00000.01830.01830.01830.0183

-1.94670.0226-0.03280.0370-0.0082

-1.89330.0277-0.07170.0556-0.0276

-1.84000.0339-0.09900.0742-0.0404

-1.78670.0411-0.11580.0929-0.0468

-1.73330.0496-0.12290.1115-0.0472

-1.68000.0595-0.12110.1302-0.0419

-1.62670.0709-0.11120.1488-0.0314

-1.57330.0841-0.09400.1675-0.0159

-1.52000.0992-0.07020.18610.0041

-1.46670.1164-0.04060.20470.0284

-1.41330.1357-0.00580.22340.0566

-1.36000.15730.03340.24200.0883

-1.30670.18130.07640.26070.1232

-1.25330.20790.12260.27930.1609

-1.20000.23690.17140.29800.2022

-1.14670.26850.22220.31660.2434

-1.09330.30260.27450.33530.2875

-1.04000.33910.32770.35390.3330

-0.98670.37780.38130.37630.3796

-0.93330.41850.43490.41000.4269

-0.88000.46100.48800.44370.4746

-0.82670.50490.54010.47740.5222

-0.77330.54990.59100.51120.5695

-0.72000.59550.64010.54490.6162

-0.66670.64120.68720.57860.6618

-0.61330.68650.73200.61230.7060

-0.56000.73080.77400.64600.7484

-0.50670.77360.81310.67970.7888

-0.45330.81420.84900.71340.8266

-0.40000.85210.88150.74720.8617-0.34670.88680.91040.78090.8937-0.29330.91760.93550.81460.9221-0.24000.94400.95660.84830.9467-0.18670.96580.97360.88200.9670-0.13330.98240.98650.91570.9828-0.08000.99360.99510.94940.9937-0.02670.99930.99950.98310.99930.02670.99930.99950.98310.99930.08000.99360.99510.94940.99370.13330.98240.98650.91570.98280.18670.96580.97360.88200.96700.24000.94400.95660.84830.94670.29330.91760.93550.81460.92210.34670.88680.91040.78090.89370.40000.85210.88150.74720.86170.45330.81420.84900.71340.82660.50670.77360.81310.67970.78880.56000.73080.77400.64600.74840.61330.68650.73200.61230.70600.66670.64120.68720.57860.66180.72000.59550.64010.54490.61620.77330.54990.59100.51120.56950.82670.50490.54010.47740.52220.88000.46100.48800.44370.47460.93330.41850.43490.41000.4269

0.98670.37780.38130.37630.3796

1.04000.33910.32770.35390.33301.09330.30260.27450.33530.28751.14670.26850.22220.31660.24341.20000.23690.17140.29800.20221.25330.20790.12260.27930.16091.30670.18130.07640.26070.12321.36000.15730.03340.24200.08831.41330.1357-0.00580.22340.05661.46670.1164-0.04060.20470.02841.52000.0992-0.07020.18610.00411.57330.0841-0.09400.1675-0.01591.62670.0709-0.11120.1488-0.03141.68000.0595-0.12110.1302-0.04191.73330.0496-0.12290.1115-0.04721.78670.0411-0.11580.0929-0.04681.84000.0339-0.09900.0742-0.04041.89330.0277-0.07170.0556-0.0276

1.94670.0226-0.03280.0370-0.0082

2.00000.01830.01830.01830.0183

函数图像如下：

n=7时，得到结果如下：

数值比较如下：

xyy1y2y3

－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－－-2.00000.01830.01830.01830.0183

-1.94670.02260.06030.03040.0115

-1.89330.02770.08880.04240.0085

-1.84000.03390.10710.05450.0091

-1.78670.04110.11800.06650.0133

-1.73330.04960.12380.07860.0208

-1.68000.05950.12670.09070.0316

-1.62670.07090.12830.10270.0454

-1.57330.08410.13020.11480.0621-1.52000.09920.13360.12680.0815-1.46670.11640.13950.13890.1036-1.41330.13570.14850.15090.1280-1.36000.15730.16120.16300.1548-1.30670.18130.17790.18790.1837-1.25330.20790.19880.22570.2146-1.20000.23690.22400.26340.2473-1.14670.26850.25330.30120.2817-1.09330.30260.28660.33900.3176-1.04000.33910.32340.37680.3549-0.98670.37780.36340.41450.3934-0.93330.41850.40620.45230.4329-0.88000.46100.45110.49010.4734-0.82670.50490.49770.52790.5146-0.77330.54990.54530.56560.5564-0.72000.59550.59340.60340.5986-0.66670.64120.64120.64120.6412-0.61330.68650.68820.66990.6838-0.56000.73080.73370.69860.7260-0.50670.77360.77730.72730.7671-0.45330.81420.81820.75600.8067-0.40000.85210.85600.78470.8442-0.34670.88680.89020.81340.8790-0.29330.91760.92040.84210.9105-0.24000.94400.94610.87080.9382-0.18670.96580.96710.89950.9614-0.13330.98240.98310.92820.9797-0.08000.99360.99390.95690.9925-0.02670.99930.99930.98560.99910.02670.99930.99930.98560.99910.08000.99360.99390.95690.99250.13330.98240.98310.92820.97970.18670.96580.96710.89950.96140.24000.94400.94610.87080.93820.29330.91760.92040.84210.91050.34670.88680.89020.81340.87900.40000.85210.85600.78470.84420.45330.81420.81820.75600.80670.50670.77360.77730.72730.76710.56000.73080.73370.69860.72600.61330.68650.68820.66990.68380.66670.64120.64120.64120.64120.72000.59550.59340.60340.5986

0.77330.54990.54530.56560.5564

0.82670.50490.49770.52790.5146

0.88000.46100.45110.49010.4734

0.93330.41850.40620.45230.4329

0.98670.37780.36340.41450.3934

1.04000.33910.32340.37680.3549

1.09330.30260.28660.33900.3176

1.14670.26850.25330.30120.2817

1.20000.23690.22400.26340.2473

1.25330.20790.19880.22570.2146

1.30670.18130.17790.18790.1837

1.36000.15730.16120.16300.1548

1.41330.13570.14850.15090.1280

1.46670.11640.13950.13890.1036

1.52000.09920.13360.12680.0815

1.57330.08410.13020.11480.0621

1.62670.07090.12830.10270.0454

1.68000.05950.12670.09070.0316

1.73330.04960.12380.07860.0208

1.78670.04110.11800.06650.0133

1.84000.03390.10710.05450.0091

1.89330.02770.08880.04240.0085

1.94670.02260.06030.03040.0115

2.00000.01830.01830.01830.0183函数图像如下：

n=9时，得到结果如下：

数值比较如下：

xyy1y2y3

－－－－－－－－－－－－－－－－－－－－－－－－－－－－－-2.00000.01830.01830.01830.0183

-1.94670.02260.00430.02760.0209

-1.89330.02770.00120.03690.0249

-1.84000.03390.00570.04620.0304

-1.78670.04110.01550.05550.0375

-1.73330.04960.02870.06480.0462

-1.68000.05950.04410.07400.0566

-1.62670.07090.06110.08330.0689

-1.57330.08410.07910.09260.0829

-1.52000.09920.09800.10190.0989

-1.46670.11640.11800.12290.1168

-1.41330.13570.13910.15090.1368

-1.36000.15730.16160.17890.1589

-1.30670.18130.18580.20690.1832

-1.25330.20790.21190.23490.2096

-1.20000.23690.24020.26290.2384

-1.14670.26850.27090.29090.2695-1.09330.30260.30400.31890.3031-1.04000.33910.33960.34690.3392-0.98670.37780.37760.37880.3778-0.93330.41850.41780.42270.4188-0.88000.46100.45990.46650.4618-0.82670.50490.50370.51030.5063-0.77330.54990.54870.55420.5516-0.72000.59550.59450.59800.5973-0.66670.64120.64040.64180.6429-0.61330.68650.68600.68570.6879-0.56000.73080.73060.72950.7316-0.50670.77360.77360.77330.7737-0.45330.81420.81440.79940.8135-0.40000.85210.85240.82300.8507-0.34670.88680.88710.84660.8848-0.29330.91760.91780.87020.9153-0.24000.94400.94430.89380.9418-0.18670.96580.96590.91740.9639-0.13330.98240.98250.94100.9811-0.08000.99360.99370.96460.9930-0.02670.99930.99930.98820.99920.02670.99930.99930.98820.99920.08000.99360.99370.96460.99300.13330.98240.98250.94100.98110.18670.96580.96590.91740.96390.24000.94400.94430.89380.94180.29330.91760.91780.87020.91530.34670.88680.88710.84660.88480.40000.85210.85240.82300.85070.45330.81420.81440.79940.81350.50670.77360.77360.77330.77370.56000.73080.73060.72950.73160.61330.68650.68600.68570.68790.66670.64120.64040.64180.64290.72000.59550.59450.59800.59730.77330.54990.54870.55420.55160.82670.50490.50370.51030.50630.88000.46100.45990.46650.46180.93330.41850.41780.42270.4188

0.98670.37780.37760.37880.3778

1.04000.33910.33960.34690.33921.09330.30260.30400.31890.30311.14670.26850.27090.29090.2695

1.20000.23690.24020.26290.2384

1.25330.20790.21190.23490.2096

1.30670.18130.18580.20690.1832

1.36000.15730.16160.17890.1589

1.41330.13570.13910.15090.1368

1.46670.11640.11800.12290.1168

1.52000.09920.09800.10190.0989

1.57330.08410.07910.09260.0829

1.62670.07090.06110.08330.0689

1.68000.05950.04410.07400.0566

1.73330.04960.02870.06480.0462

1.78670.04110.01550.05550.0375

1.84000.03390.00570.04620.0304

1.89330.02770.00120.03690.0249

1.94670.02260.00430.02760.0209

2.00000.01830.01830.01830.0183函数图像如下：

n=11时，得到结果如下：

数值比较如下：

xyy1y2y3

－－－－－－－－－－－－－－－－－－－－－－－－－－－-2.00000.01830.01830.01830.0183

-1.94670.02260.02960.02620.0223

-1.89330.02770.03670.03400.0273

-1.84000.03390.04210.04190.0334

-1.78670.04110.04740.04980.0406

-1.73330.04960.05360.05760.0492

-1.68000.05950.06150.06550.0592

-1.62670.07090.07150.07340.0708

-1.57330.08410.08370.08790.0842

-1.52000.09920.09830.10920.0994

-1.46670.11640.11520.13050.1166

-1.41330.13570.13460.15180.1359

-1.36000.15730.15650.17310.1575

-1.30670.18130.18080.19440.1814

-1.25330.20790.20760.21560.2079

-1.20000.23690.23690.23690.2369

-1.14670.26850.26870.27560.2687

-1.09330.30260.30280.31440.3030

-1.04000.33910.33930.35310.3397

-0.98670.37780.37800.39180.3784

-0.93330.41850.41870.43050.4190

-0.88000.46100.46110.46920.4613

-0.82670.50490.50490.50790.5050

-0.77330.54990.54990.54890.5499

-0.72000.59550.59540.59230.5955

-0.66670.64120.64110.63560.6414

-0.61330.68650.68640.67890.6868

-0.56000.73080.73070.72220.7312

-0.50670.77360.77360.76550.7740

-0.45330.81420.81420.80880.8145

-0.40000.85210.85210.85210.8521

-0.34670.88680.88680.87190.8864

-0.29330.91760.91760.89160.9168

-0.24000.94400.94400.91130.9431

-0.18670.96580.96580.93100.9649

-0.13330.98240.98240.95070.9817

-0.08000.99360.99360.97040.9933

-0.02670.99930.99930.99010.9992

0.02670.99930.99930.99010.9992

0.08000.99360.99360.97040.9933

0.13330.98240.98240.95070.9817

0.18670.96580.96580.93100.9649

0.24000.94400.94400.91130.9431

0.29330.91760.91760.89160.9168

0.34670.88680.88680.87190.8864

0.40000.85210.85210.85210.8521

0.45330.81420.81420.80880.8145

0.50670.77360.77360.76550.7740

0.56000.73080.73070.72220.7312

0.61330.68650.68640.67890.6868

0.66670.64120.64110.63560.6414

0.72000.59550.59540.59230.5955

0.77330.54990.54990.54890.5499

0.82670.50490.50490.50790.5050

0.88000.46100.46110.46920.4613

0.93330.41850.41870.43050.4190

0.98670.37780.37800.39180.3784

1.04000.33910.33930.35310.3397

1.09330.30260.30280.31440.3030

1.14670.26850.26870.27560.2687

1.20000.23690.23690.23690.2369

1.25330.20790.20760.21560.2079

1.30670.18130.18080.19440.1814

1.36000.15730.15650.17310.1575

1.41330.13570.13460.15180.1359

1.46670.11640.11520.13050.1166

1.52000.09920.09830.10920.0994

1.57330.08410.08370.08790.0842

1.62670.07090.07150.07340.0708

1.68000.05950.06150.06550.0592

1.73330.04960.05360.05760.0492

1.78670.04110.04740.04980.0406

1.84000.03390.04210.04190.0334

1.89330.02770.03670.03400.0273

1.94670.02260.02960.02620.0223

2.00000.01830.01830.01830.0183函数图像如下：

【结果分析】

通过图像及数值的对照都可以看到：随着n的增大，在区间[-2,2]上，插值函数都越来越靠近于原函数，而且当n为9、11时，几条插值曲线几乎重合。下面比较一下三种办法的异同和优劣。

拉格朗日多项式，当n增大时，并不能保证在全部区间都收敛于原函数——拉格朗日多项式在区间[-2,2]以外，因为拉格朗日多项式的次数增大，在收敛区间外的点上，高阶导数不为零。光洁性变差，从而产生了极大的振荡。也就是说惟独已知插值点落在收敛区间以内时，才可采纳。所以影

响了这种办法的有用价值。

分段线性插值的曲线不如拉格朗日和三次样条的曲线光洁。但是当n趋于无穷时，它总能到处收敛于原函数。因此，分段线性插值普通应用在需要迅速计算而又无特别要求的状况下。

三次样条插值，当n趋于无穷时，它也总能到处收敛于原函数。而且它的曲线更光洁，可以应用于机械加工等领域。

六：用电压V＝10伏的电池给电容器充电，电容器上t时刻的电压为0()()t

vtVVVeτ-=--，其中V0是电容器的初始电压，τ是充电常数。试由下面一组t,V数据确定V0和τ。

模型建立：

由题目中公式，两边同时取对数，得：

0ln(())ln()t

VvtVVτ

-=--

令1()ln(())vtVvt=-，得：

10()ln()t

vtVVτ

=--

可见1()vt与t成线性关系.

模型假设：

本题中将数据取对数后,假如真正希翼使数据点与曲线距离平方和最小，那么应当使用非线性最小二乘法拟合。但是这里为了简便而作一个近似，从而使用线性最小二乘法拟合的。

模型求解：

用Matlab作线性最小二乘法拟合，编程语句如下：

%数学试验作业二.6

t=[0.51234579];v=[6.366.487.268.228.668.999.439.63];V=10;

v1=log(V-v);formatbank;a=polyfit(t,v1,1)tao=-1/a(1)

V0=V-exp(1.48)tt=linspace(0,10,20);

vv=V-(V-V0)*exp(-tt/tao);plot(t,v,'+')holdonplot(tt,vv)

运行程序可以得到：10.28a=-

21.48a=

即01

0.28ln()1.48VVτ?-=-???

-=?于是解得：

03.535.61

Vτ=??

=?

同时得到v(t)曲线如下。与数据点相比较，得：

结论：

V＝5.61伏；τ＝3.53秒。

八：弹簧在力F的作用下伸长x,一定范围内听从虎克定律：F与x成正比，即Fkx

=,k为弹性系数，。现在得到下面一组x、F数据，并在（x,F）坐标下作图（如下图）。可以看出，当F大到一定数值后，就不听从这个定律了。试由数据确定k，并给出不听从虎克定律时的近似公式。

弹簧在力F下的伸长x

模型分析：

从上图可以看出，开头的几个点基本呈线性；而以后的几个点显然偏离了开头的直线。故而考虑开头的几个点用直线拟合；后面的点用二次曲线拟合。特殊应当注重题目中的按照库克定律拟合的直线必需过0点，这是实际问题所打算的。因此在建模和matlab实现时要充分考虑这个问题。

模型假设：

1)

0xx时，F

与x呈正比，又由于拟合的直线必需过0点

F(x)＝kx;

2)

0xx≥时，F

与x呈二次关系，2

()Fxax

bxc=++

3)F(x)在0x点延续且光洁

模型建立：

采纳最小二乘法，目标函数为

9

2

1

(,,,)(())iikJkabcFxF==-∑9

2

2

2

1

1

()()m

iii

iiiimkxFax

bxcF==+=-+

++-∑∑

其中m代表前面的m个点用直线拟合

解题目标：求出合适的k，a，b，c，使J取最小值。

模型求解：

为求出J的最小值，应保证

00J

kJJJabc??=????

????===?????

1

9

2219

21

9

21

2()02(02(02(0m

iiiiiiiiimiiiiimiiiimkxFxaxbxcFxaxbxcFxaxbxcF==+=

+=+??-=????++-=?????++-=????++-=??∑∑∑∑）））

即：2

143229932112()011miiiiiiiiiiiiiimimiikxFxxxxxaxxxbxFcxx==

+=+?-=?????????????????=???????????????????????∑∑∑

若记

4

32

2993

2

112

,11iiiii

i

iii

imimi

i