数值分析实验作业-gauss消去法的数值稳定性分析

数值分析实验作业-gauss消去法的数值稳定性分析1/9实验3.1Gauss消去法的数值稳定性试验实验目的:观察和理解Gauss消元过程中出现小主元(即)(kkka很小)时引起的方程组解的数值不稳定性。实验内容：求解方程组bAx，其中（1）11212592.1121-130.6-291.51314.59103.015-1A，2178.4617.591b；（2）2010151526990999999999.23107102A,15019000000000.582b.实验要求：（1）计算矩阵的条件数，判断系数矩阵是良态的还是病态的。（2）用Gauss列主元消去法求得L和U及解向量421,Rxx.（3）用不选主元的Gauss消去法求得L~和U~及解向量421~,~Rxx.（4）观察小主元并分析其对计算结果的影响.程序如下:计算矩阵条件数及Gauss列主元消去法:formatlongengA1=[0.3e-1559.1431;5.291-6.130-12;11.2952;1211];b1=[59.17;46.78;1;2];n=4;k2=cond(A1)%k2为矩阵的条件数；数值分析实验作业-gauss消去法的数值稳定性分析2/9fork=1:n-1a=max(abs(A1(k:n,k)));[p,k]=find(A1==a);B=A1(k,:);c=b1(k);A1(k,:)=A1(p,:);b1(k)=b1(p);A1(p,:)=B;b1(p)=c;ifA1(k,k)~=0A1(k+1:n,k)=A1(k+1:n,k)/A1(k,k);A1(k+1:n,k+1:n)=A1(k+1:n,k+1:n)-A1(k+1:n,k)*A1(k,k+1:n);elsebreakendendL1=tril(A1,0);fori=1:nL1(i,i)=1;endL=L1U=triu(A1,0)forj=1:n-1b1(j)=b1(j)/L(j,j);b1(j+1:n)=b1(j+1:n)-b1(j)*L(j+1:n,j);endb1(n)=b1(n)/L(n,n);forj=n:-1:2b1(j)=b1(j)/U(j,j);b1(1:j-1)=b1(1:j-1)-b1(j)*U(1:j-1,j);endb1(1)=b1(1)/U(1,1);x1=b1运行结果如下：K2=68.43；14929.00202.00893.0011755.04724.00011079.26000118L801.0000231.1835.2001314.5902592.11U1x=[18.9882;3.3378;-34.747;-33.9865]数值分析实验作业-gauss消去法的数值稳定性分析3/9不选主元的Gauss消去法程序：clearformatlongengA1=[0.3e-1559.1431;5.291-6.130-12;11.2952;1211];b1=[59.17;46.78;1;2];n=4;fork=1:n-1A1(k+1:n,k)=A1(k+1:n,k)/A1(k,k);A1(k+1:n,k+1:n)=A1(k+1:n,k+1:n)-A1(k+1:n,k)*A1(k,k+1:n);endL1=tril(A1,0);fori=1:nL1(i,i)=1;endL=L1U=triu(A1,0)forj=1:n-1b1(j)=b1(j)/L(j,j);b1(j+1:n)=b1(j+1:n)-b1(j)*L(j+1:n,j);endb1(n)=b1(n)/L(n,n);forj=n:-1:2b1(j)=b1(j)/U(j,j);b1(1:j-1)=b1(1:j-1)-b1(j)*U(1:j-1,j);endb1(1)=b1(1)/U(1,1);x1=b1程序运行结果如下：10189.010333.3011168.21033.3700110637.170001~151515L5.00008160010637.171091.5210043.101314.59103.0~15151815U]0;0;0005.1;6848.23[~1x同理可得2A对应的系数矩阵条件数及Gauss列主元消去法求解结果：K2=8.994;数值分析实验作业-gauss消去法的数值稳定性分析4/91333.04.0001104.0-3.0-0015.0000112-L36667.300030.26005.155.2010710U]0.1;;0.1;0.1;10444.0[152x不选主元的Gauss消去法结果：1400.0109999.0-001104998.2-5.00013.0-0001~1212L3667.3000107495.5109987.14003.26100.1010710~121212U]000145.1;99994.0;000.1;1045.1[~52x实验4.5三次样条插值函数的收敛性问题提出：多项式插值不一定收敛的，即插值的节点多，效果不一定就好。对三次样条插值函数又如何呢？理论上证明三次样条插值函数的收敛性是比较困难的，也超过了本课程的内容。通过本实验可以验证这一理论结果.数值分析实验作业-gauss消去法的数值稳定性分析5/9实验内容：请按一定的规则分别选择等距或者非等距的插值节点，并不断增加插值节点的个数。考虑实验4.4中的函数或者选择其他感兴趣的函数，可用Matlab的函数“spline”作此函数的三次样条插值函数。实验要求：（1）随着节点个数的增加，比较被逼近函数和三次样条差值函数的误差变化情况。分析所得结果并与拉格朗日插值多项式比较。（2）三次样条插值函数的思想最早产生于工业部门。作为工业迎合用的例子，考虑如下例子：某汽车制造商根据三次样条差值函数设计车门曲线，其中一段的数据如下：kx012345678910ky0.00.791.532.192.713.033.272.893.063.193.29'ky0.80.2(3)计算实验4.4的样条插值.程序如下：formatshortx1=-1:0.5:1;y1=1./(1+25.*x1.*x1);x2=-1:0.25:1;y2=1./(1+25.*x2.*x2);x3=-1:0.1:1;y3=1./(1+25.*x3.*x3);x4=[-1,-0.82,-0.6,-0.53,-0.34,-0.2,0,0.04,0.2,0.25,0.5,0.8,1];y4=1./(1+25.*x4.*x4);xx=-1:0.01:1;yy1=spline(x1,y1,xx);yy2=spline(x2,y2,xx);yy3=spline(x3,y3,xx);yy4=spline(x4,y4,xx);数值分析实验作业-gauss消去法的数值稳定性分析6/9holdonfplot('1./(1+25.*x.*x)',[-1,1],'m')plot(xx,yy1,'g')plot(xx,yy2,'b')plot(xx,yy3,'k')plot(xx,yy4,'r')holdoff%比较被逼近函数与三次样条插值函数图像，直观表现不同插值节点处误差的变化xx=-1:0.2:1;y=1./(1+25.*xx.*xx)%函数在相应节点处的真实值；yy1=spline(x1,y1,xx)y1la=lagrange(x1,y1,xx)yy2=spline(x2,y2,xx)y2la=lagrange(x2,y2,xx)yy3=spline(x3,y3,xx)y3la=lagrange(x3,y3,xx)yy4=spline(x4,y4,xx)y4la=lagrange(x4,y4,xx)其中lagrange函数对应的m文件为：functiony=lagrange(x0,y0,x);n=length(x0);m=length(x);fori=1:mz=x(i);s=0.0;fork=1:np=1.0;forj=1:nifj~=kp=p*(z-x0(j))/(x0(k)-x0(j));endends=p*y0(k)+s;endy(i)=s;end程序运行结果如下：数值分析实验作业-gauss消去法的数值稳定性分析7/9插值结果比较：y0.03850.05880.10.20.510.50.20.10.05880.0385yy10.0385-0.252-0.06230.36870.802410.80240.3687-0.0623-0.2520.0385y1la0.0385-0.3793-0.11010.40050.834210.83420.4005-0.1101-0.37930.0385yy20.03850.05510.10850.17650.534210.53420.17650.10850.05510.0385y2la0.0385-0.25870.30490.07260.563610.56360.07260.3049-0.25870.0385yy30.03850.05880.10.20.510.50.20.10.05880.0385y3la0.03850.05880.10.20.510.50.20.10.05880.0385yy40.03850.05870.10.20160.510.50.19890.09930.05880.0385y4la0.03850.43120.10.24610.510.50.2640.23870.05880.0385车门曲线求解程序:x=0:10;y=[0,0.79,1.53,2.19,2.71,3.03,3.27,2.89,3.06,3.19,3.29];dx0=0.8;dx10=0.2;S=csfit(x,y,dx0,dx10)其中csfit函数的m文件为:functionS=csfit(X,Y,dx0,dxn)%ClampedCubicSpline%Input-Xisthe1xnabscissavector%-Yisthe1xnordinatevector%-dx0=S'(x0)firstderivativeboundarycondition%-dxn=S'(xn)firstderivativeboundarycondition%Output-S:rowsofSarethecoefficients,indescendingorder,forthe%cubicinterpolantsN=length(X)-1;H=diff(X);D=diff(Y)./H;数值分析实验作业-gauss消去法的数值稳定性分析8/9A=H(2:N-1);B=2*(H(1:N-1)+H(2:N));C=H(2:N);U=6*diff(D);%ClampedsplineendpointconstraintsB(1)=B(1)-H(1)/2;U(1)=U(1)-3*(D(1)-dx0);B(N-1)=B(N-1)-H(N)/2;U(N-1)=U(N-1)-3*(dxn-D(N));fork=2:N-1temp=A(k-1)/B(k-1);B(k)=B(k)-temp*C(k-1);U(k)=U(k)-temp*U(k-1);endM(N)=U(N-1)/B(N-1);fork=N-2:-1:1M(k+1)=(U(k)-C(k)*M(k+2))/B(k);endM(1)=3*(D(1)-dx0)/H(1)-M(2)/2;M(N+1)=3*(dxn-D(N))/H(N)-M(N)/2;fork=0:N-1S(k+1,1)=(M(k+