哈工大-计算方法实验-1拉格朗日

./实验题目1 Lagrange插值摘要给定平面上n+1个不同的数据点:则满足条件的n次拉格朗日插值多项式是存在唯一的。若,且充分光滑,则当时,有误差估计式前言利用拉格朗日插值多项式求的近似值程序设计流程拉格朗日插值框图拉格朗日插值框图开始结束输入<xi,yi>,ni=0,1,2,…,nL=0i=0xl=1xl=*xlj=0,1,…,i-1,i+1,…,nL=L+xl*i=n?输出y否是i=i+1问题1〔1N=5时,程序运行如下：TestLag<inline<'1./<1+x.^2>'>,-5,5,5,0.75:4.75>;将区间[-5,5]分为了5段计算插值的点xi=0.75001.75002.75003.75004.7500计算出的插值yi=0.90540.52580.0096-0.3568-0.1595插值点处函数值yFact=0.64000.24620.11680.06640.0424计算误差err=-0.2654-0.27960.10720.42320.2020N=10时,程序运行如下：TestLag<inline<'1./<1+x.^2>'>,-5,5,10,0.75:4.75>;将区间[-5,5]分为了10段计算插值的点xi=0.75001.75002.75003.75004.7500计算出的插值yi=0.69070.23300.11220.1084-0.2360插值点处函数值yFact=0.64000.24620.11680.06640.0424计算误差err=-0.05070.01320.0045-0.04200.2785N=20时,程序运行如下：TestLag<inline<'1./<1+x.^2>'>,-5,5,20,0.75:4.75>;将区间[-5,5]分为了20段计算插值的点xi=0.75001.75002.75003.75004.7500计算出的插值yi=0.64130.24910.12820.19036.4150插值点处函数值yFact=0.64000.24620.11680.06640.0424计算误差err=-0.0013-0.0029-0.0114-0.1239-6.3726问题1〔2N=5时,程序运行如下：TestLag<inline<'exp<x>'>,-1,1,5,[-0.95-0.050.050.95]>;将区间[-1,1]分为了5段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.38630.95131.05122.5863插值点处函数值yFact=0.38670.95121.05132.5857计算误差err=1.0e-003*0.4471-0.10510.1069-0.6129N=10时,程序运行如下：TestLag<inline<'exp<x>'>,-1,1,10,[-0.95-0.050.050.95]>;将区间[-1,1]分为了10段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.38670.95121.05132.5857插值点处函数值yFact=0.38670.95121.05132.5857计算误差err=1.0e-008*-0.3126-0.0055-0.0055-0.3714N=20时,程序运行如下：TestLag<inline<'exp<x>'>,-1,1,20,[-0.95-0.050.050.95]>;将区间[-1,1]分为了20段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.38670.95121.05132.5857插值点处函数值yFact=0.38670.95121.05132.5857计算误差err=1.0e-012*0.73390-0.0002-0.5671问题2〔1N=5时,程序运行如下：TestLag<inline<'1./<1+x.^2>'>,-1,1,5,[-0.95-0.050.050.95]>;将区间[-1,1]分为了5段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.51360.99780.99780.5136插值点处函数值yFact=0.52560.99750.99750.5256计算误差err=0.0121-0.0002-0.00020.0121N=10时,程序运行如下：TestLag<inline<'1./<1+x.^2>'>,-1,1,10,[-0.95-0.050.050.95]>;将区间[-1,1]分为了10段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.52430.99750.99750.5243插值点处函数值yFact=0.52560.99750.99750.5256计算误差err=0.00140.00000.00000.0014N=20时,程序运行如下：TestLag<inline<'1./<1+x.^2>'>,-1,1,20,[-0.95-0.050.050.95]>;将区间[-1,1]分为了20段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.52560.99750.99750.5256插值点处函数值yFact=0.52560.99750.99750.5256计算误差err=1.0e-005*-0.70230.00000.0000-0.7023实验2〔2N=5时,程序运行如下：TestLag<inline<'exp<x>'>,-5,5,5,[-4.75-0.250.254.75]>;将区间[-5,5]分为了5段计算插值的点xi=-4.7500-0.25000.25004.7500计算出的插值yi=-1.93211.42750.5882123.7146插值点处函数值yFact=0.00870.77881.2840115.5843计算误差err=1.9408-0.64870.6958-8.1303N=10时,程序运行如下：TestLag<inline<'exp<x>'>,-5,5,10,[-4.75-0.250.254.75]>;将区间[-5,5]分为了10段计算插值的点xi=-4.7500-0.25000.25004.7500计算出的插值yi=0.04250.77961.2848115.6630插值点处函数值yFact=0.00870.77881.2840115.5843计算误差err=-0.0339-0.0008-0.0008-0.0788N=20时,程序运行如下：TestLag<inline<'exp<x>'>,-5,5,20,[-4.75-0.250.254.75]>;将区间[-5,5]分为了20段计算插值的点xi=-4.7500-0.25000.25004.7500计算出的插值yi=0.00870.77881.2840115.5843插值点处函数值yFact=0.00870.77881.2840115.5843计算误差err=1.0e-007*-0.09140.00000.0000-0.1434问题3〔1N=5时,程序运行如下：TestLag2<inline<'1./<1+x.^2>'>,-1,1,5,[-0.95-0.050.050.95]>;将区间[-1,1]分为了5段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.52540.99780.99780.5254插值点处函数值yFact=0.52560.99750.99750.5256计算误差err=1.0e-003*0.2071-0.3011-0.30110.2071N=10时,程序运行如下：TestLag2<inline<'1./<1+x.^2>'>,-1,1,10,[-0.95-0.050.050.95]>;将区间[-1,1]分为了10段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.52550.99720.99720.5255插值点处函数值yFact=0.52560.99750.99750.5256计算误差err=1.0e-003*0.15620.26030.26030.1562N=20时,程序运行如下：TestLag2<inline<'1./<1+x.^2>'>,-1,1,20,[-0.95-0.050.050.95]>;将区间[-1,1]分为了20段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.52560.99750.99750.5256插值点处函数值yFact=0.52560.99750.99750.5256计算误差err=1.0e-007*0.23180.23810.23810.2318问题3〔2N=5时,程序运行如下：TestLag2<inline<'exp<x>'>,-1,1,5,[-0.95-0.050.050.95]>;将区间[-1,1]分为了5段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.38670.95141.05112.5857插值点处函数值yFact=0.38670.95121.05132.5857计算误差err=1.0e-003*0.0079-0.13170.1339-0.0108N=10时,程序运行如下：TestLag2<inline<'exp<x>'>,-1,1,10,[-0.95-0.050.050.95]>;将区间[-1,1]分为了10段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.38670.95121.05132.5857插值点处函数值yFact=0.38670.95121.05132.5857计算误差err=1.0e-009*-0.5045-0.4791-0.4835-0.5994N=20时,程序运行如下：TestLag2<inline<'exp<x>'>,-1,1,20,[-0.95-0.050.050.95]>;将区间[-1,1]分为了20段计算插值的点xi=-0.9500-0.05000.05000.9500计算出的插值yi=0.38670.95121.05132.5857插值点处函数值yFact=0.38670.95121.05132.5857计算误差err=1.0e-015*0.16650.3331-0.4441-0.8882问题4程序运行如下：TestLag3<[149],[550115185]>计算插值的点xi=550115185计算出的插值yi=2.2667-20.2333-171.9000-492.7333插值点处函数值yFact=2.23617.071110.723813.6015计算误差err=-0.030627.3044182.6238506.3348程序运行如下：TestLag3<[364964],[550115185]>计算插值的点xi=550115185计算出的插值yi=3.11587.071810.167010.0388插值点处函数值yFact=2.23617.071110.723813.6015计算误差err=-0.8797-0.00070.55683.5626程序运行如下：TestLag3<[100121144],[550115185]>计算插值的点xi=550115185计算出的插值yi=4.43917.285010.722813.5357插值点处函数值yFact=2.23617.071110.723813.6015计算误差err=-2.2030-0.21390.00100.0658程序运行如下：TestLag3<[169196225],[550115185]>计算插值的点xi=550115185计算出的插值yi=5.49727.800110.800513.6006插值点处函数值yFact=2.23617.071110.723813.6015计算误差err=-3.2611-0.7291-0.07670.0009实验所用函数functionyh=LagInterp<x,y,xh>%LagInterp计算拉格朗日插值%%Synopsis:yh=LagInterp<x,y,xh>%%Input:x=一维向量,将要做插值x的值%y=一维向量,将要做插值y的值%xh=数值或一维向量,计算插值的位置,支持计算一列xh的值%%Output:yh=数值或一维向量,通过计算插值的位置算出的插值ifmin<size<x>>>1|min<size<y>>>1 %判断x,y是否为一维向量error<'x,ymustbevectors!'>;elseiflength<x>~=length<y> %判断x,y是否有同样多的元素error<'xandymustagree!'>;endyh=zeros<size<xh>>;L=zeros<length<x>-1>;forj=1:length<xh>fori=1:length<x>xCal=x;xCal<i>=[];%prod<xh<j>-xCal>/prod<x<i>-xCal>为拉格朗日基函数L<i>=prod<xh<j>-xCal>/prod<x<i>-xCal>;yh<j>=yh<j>+L<i>*y<i>;%yh=sum<L<i>*y<i>>endendfunctionTestLag<fx,a,b,n,xi>%TestLag实验题目11,2%%Synopsis:TestLag<fun,a,b,n,xi>%%Input:fx=用来验证插值计算准确率的函数%a,b=节点选取上下限%n=多项式次数,固定区间[-a,b]分段数%xi=要计算插值的点x=linspace<a,b,n>;y=feval<fx,x>;yi=LagInterp<x,y,xi>;yFact=feval<fx,xi>;err=yFact-yi;fprintf<'将区间[%d,%d]分为了%d段\n',a,b,n>;fprintf<'计算插值的点xi=\n'>;disp<xi>;fprintf<'计算出的插值yi=\n'>;disp<yi>;fprintf<'插值点处函数值yFact=\n'>;disp<yFact>;fprintf<'计算误差err=\n'>;disp<err>;functionTestLag2<fx,a,b,n,xi>%TestLag2实验题目13%%Synopsis:TestLag2<fun,a,b,n,xi>%%Input:fx=用来验证插值计算准确率的函数%a,b=节点选取上下限%n=多项式次数,固定区间[-a,b]分段数%xi=要计算插值的点x=zeros<1,n>;fork=1:nx<k>=cos<<2*k-1>*pi/<2*n>>;%构造非等距节点endy=feval<fx,x>;yi=LagInterp<x,y,xi>;yFact=feval<fx,xi>;err=yFact-yi;fprintf<'将区间[%d,%d]分为了%d段\n',a,b,n>;fprintf<'计算插值的点xi=\n'>;disp<xi>;