数学实验作业二.docx
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数学实验作业二.docx
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数学实验作业二
数学实验作业二
题目:
P72.1.d);6);8)
日期:
2003-3-9
【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作线性最小二乘的方法。
3、通过实例学习如何用插值方法与拟合方法解决实际问题,注意二者的联系和区别。
【实验内容】:
一:
对函数
在n个节点上(n不要太大,如5至11)用拉格朗日、分段线性、三次样条三种插值方法,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进行比较。
适当增加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:
%数学实验作业二.1-d
clear;
n=5;
%在n个节点上进行插值
m=75;
%产生m个插值点,计算函数在插值点处的精确值,将来进行对比
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
-1.20000.23690.17140.29800.2011
-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
-0.02670.99930.99950.98310.9993
0.02670.99930.99950.98310.9993
0.08000.99360.99510.94940.9937
0.13330.98240.98650.91570.9828
0.18670.96580.97360.88200.9670
0.24000.94400.95660.84830.9467
0.29330.91760.93550.81460.9221
0.34670.88680.91040.78090.8937
0.40000.85210.88150.74720.8617
0.45330.81420.84900.71340.8266
0.50670.77360.81310.67970.7888
0.56000.73080.77400.64600.7484
0.61330.68650.73200.61230.7060
0.66670.64120.68720.57860.6618
0.72000.59550.64010.54490.6162
0.77330.54990.59100.51120.5695
0.82670.50490.54010.47740.5222
0.88000.46100.48800.44370.4746
0.93330.41850.43490.41000.4269
0.98670.37780.38130.37630.3796
1.04000.33910.32770.35390.3330
1.09330.30260.27450.33530.2875
1.14670.26850.22220.31660.2434
1.20000.23690.17140.29800.2011
1.25330.20790.12260.27930.1609
1.30670.18130.07640.26070.1232
1.36000.15730.03340.24200.0883
1.41330.1357-0.00580.22340.0566
1.46670.1164-0.04060.20470.0284
1.52000.0992-0.07020.18610.0041
1.57330.0841-0.09400.1675-0.0159
1.62670.0709-0.11120.1488-0.0314
1.68000.0595-0.12110.1302-0.0419
1.73330.0496-0.12290.1115-0.0472
1.78670.0411-0.11580.0929-0.0468
1.84000.0339-0.09900.0742-0.0404
1.89330.0277-0.07170.0556-0.0276
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
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