MATLAB龙格现象插值实验报告含代码图像结果可直接套用

实验龙格现象的发生、防止和插值效果的比较【实验目的】1．掌握用MATLAB编程语言的编程训练，初步体验算法的软件实现；2．对不同方案结果的精度比较，有无龙格现象的发生。【实验内容】将区间【-5,5】等分，xi=-5+ih(i=0,1,…,n),h=10/n，对下列函数进行插值，计算其在xk=-5+0.25k(k=0,1,…,40)上的值，并绘制插值函数的图形。（1）y=x/(1+x^4)(2)y=arctgx(1)做n=10,20拉格朗日插值；(2)做n=10,20分段线性插值；(3)做n=10,20I型三次样条插值【程序如下】:（1）y=x/(1+x^4)clearall;closeall;clc;N=input('输入插值次数');xi=zeros(1,N+1);h=10/N;fork=1:N+1xi(k)=-5+(k-1)*h;endyi=xi./(1+xi.^4);xk=zeros(1,41);fork=1:41xk(k)=-5+0.25*(k-1)endykl=zeros(1,41);fori=1:41ykl(i)=0;fork=1:N+1t=1;forj=1:N+1ifj~=kt=(xk(i)-xi(j))/(xi(k)-xi(j))*t;ykl(i)=ykl(i)+t*yi(k);endendendendsubplot(2,2,1)plot(xk,ykl,'k','linewidth',1);title('拉格朗日插值曲线')ykx=zeros(1,41);fori=1:41k=1;k=k+1;whilek<N&&xk(i)>=xi(k)k=k+1;endykx(i)=yi(k-1)*(xk(i)-xi(k))/(xi(k-1)-xi(k))+yi(k)*(xk(i)-xi(k-1))/(xi(k)-xi(k-1))endsubplot(2,2,2);plot(xk,ykx,'k','linewidth',1)title('分段线性插值曲线')disp('计算结果输出:')z=[xk'ykl'ykx'];(2)y=arctgxclearall;closeall;clc;N=input('输入插值次数');xi=zeros(1,N+1);h=10/N;fork=1:N+1xi(k)=-5+(k-1)*h;endyi=atan(xi)xk=zeros(1,41);fork=1:41xk(k)=-5+0.25*(k-1)endykl=zeros(1,41);fori=1:41ykl(i)=0;fork=1:N+1t=1;forj=1:N+1ifj~=kt=(xk(i)-xi(j))/(xi(k)-xi(j))*t;ykl(i)=ykl(i)+t*yi(k);endendendendsubplot(2,2,1)plot(xk,ykl,'k','linewidth',1);title('拉格朗日插值曲线')结算结果1．十次插值图像为：二十次插值图像：插值顺序：，拉格朗日插值、分段线形插值、I型三次样条插值：十次插值： -5.0000-0.0799-0.0080-4.75002.7400-0.0099-4.50002.3631-0.0118-4.25001.0023-0.0137-4.0000-0.2309-0.0156-3.7500-0.9262-0.0208-3.5000-1.0791-0.0261-3.2500-0.8821-0.0313-3.0000-0.5846-0.0366-2.7500-0.4067-0.0569-2.5000-0.4943-0.0771-2.2500-0.9042-0.0974-2.0000-1.6096-0.1176-1.7500-2.5185-0.2132-1.5000-3.4987-0.3088-1.2500-4.4042-0.4044-1.0000-5.0994-0.5000-0.7500-5.4794-0.3750-0.5000-5.4828-0.2500二十次插值：-0.2500-5.0987-0.12500-4.366000.2500-3.36690.12500.5000-2.21480.25000.7500-1.03880.37501.00000.03370.50001.25000.89460.40441.50001.47020.30881.75001.73370.21322.00001.71170.11762.25001.48470.09742.50001.18090.07712.75000.96250.05693.00001.00630.03663.25001.47960.03133.50002.51360.02613.75004.17870.02084.00006.46660.01564.25009.28470.01034.500012.47150.00514.750015.8418-0.00025.000019.2728-0.0055-5.,-.160,-.799e-2,-4.75,30.0,-.947e-2,-4.50,-.304,-.109e-1,-4.25,-3.70,-.133e-1,-4.,-.529,-.156e-1,-3.75,-.194e-1,-.194e-1,-3.50,-.894,-.232e-1,-3.25,-1.38,-.299e-1,-3.,-1.52,-.366e-1,-2.75,-1.90,-.495e-1,-2.50,-2.67,-.624e-1,-2.25,-3.68,-.900e-1,-2.,-4.94,-.118,-1.75,-6.81,-.183,-1.50,-9.76,-.247,-1.25,-14.1,-.374,-1.,-19.6,-.500,-.750,-26.1,-.485,-.500,-33.3,-.471,-.250,-41.8,-.235,0.,-52.9,0.,.250,-69.0,.235,.500,-93.3,.471,.750,-129.,.485,1.,-181.,.500,1.25,-252.,.374,1.50,-345.,.247,1.75,-464.,.183,2.,-609.,.118,2.25,-782.,.900e-1,2.50,-980.,.624e-1,2.75,-.119e4,.495e-1,3.,-.141e4,.366e-1,3.25,-.159e4,.299e-1,3.50,-.170e4,.232e-1,3.75,-.164e4,.194e-1,4.,-.130e4,.156e-1,4.25,-508.,.133e-1,4.50,989.,.109e-1,4.75,.355e4,.864e-2,5.,.764e4,.633e-2,2．十次插值图像：二十次插值图像：插值顺序：，拉格朗日插值、分段线形插值、I型三次样条插值：十次插值：-5. -13.7 -1.37 -4.75 -13.1 -1.36 -4.50 -13.2 -1.35 -4.25 -13.5 -1.34 -4. -13.7 -1.33 -3.75 -13.8 -1.31 -3.50 -13.9 -1.29 -3.25 -13.8 -1.27 -3. -13.7 -1.25 -2.75 -13.7 -1.21 -2.50 -13.6 -1.18 -2.25 -13.6 -1.14 -2. -13.5 -1.11 -1.75 -13.4 -1.03 -1.50 -13.2 -.946 -1.25 -12.8 -.866 -1. -12.3 -.785 -.750 -11.5 -.590 -.500 -10.4 -.392 -.250 -9.14 -.196 0. -7.63 0. .250 -5.91 .196 .500 -4.03 .392 .750 -2.02 .590 1. .118 .785 1.25 2.36 .866 1.50 4.73 .946 1.75 7.28 1.03 2. 10.1 1.11 2.25 13.3 1.14 2.50 17.0 1.18 2.75 21.5 1.21 3. 27.0 1.25 3.25 33.8 1.27 3.50 42.2 1.29 3.75 52.6 1.31 4. 65.5 1.33 4.25 81.3 1.35 4.50 100. 1.36 4.75 124. 1.38 5. 152. 1.40

二十次插值:-5.,-27.5,-1.37,-4.75,-34.3,-1.36,-4.50,-28.5,-1.35,-4.25,-28.6,-1.34,-4.,-30.2,-1.33,-3.75,-31.5,-1.31,-3.50,-33.1,-1.29,-3.25,-35.2,-1.27,-3.,-38.0,-1.25,-2.75,-41.6,-1.22,-2.50,-46.2,-1.19,-2.25,-52.3,-1.15,-2.,-60.2,-1.11,-1.75,-70.4,-1.04,-1.50,-83.6,-.983,-1.25,-101.,-.884,-1.,-123.,-.785,-.750,-152.,-.625,-.500,-189.,-.464,-.250,-238.,-.232,0.,-300.,0.,.250,-381.,.232,.5