列主元高斯消去法、LU三角分解法、龙贝格(Romberg)算法、最小二乘法的Matlab程序及运行结果

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1、1列主元高斯消去法M文件functionx=gauss(a,b)n=length(a);x=zeros(n,1);a=ab;fork=1:n-1max=k;fori=k+1:nifa(i,k)a(max,k)max=i;endendtemp=a(k,k:n+1);a(k,k:n+1)=a(max,k:n+1);a(max,k:n+1)=temp;fori=k+1:na(i,k)=-a(i,k)/a(k,k);a(i,k+1:n+1)=a(i,k+1:n+1)+a(i,k)*a(k,k+1:n+1);endendx(n,1)=a(n,n+1)/a(n,n);fori=n-1:-1:1sum=0;

2、forj=i+1:nsum=sum+x(j,1)*a(i,j);endx(i,1)=(a(i,n+1)-sum)/a(i,i);endMatlab运行结果CommandWindowa=034;1-11;212a=0341-11212b=l；2；3b-123gaussb)ans=1.1667-0.3333.50002.LU三角分解法M文件functiony=LU(A,B);n=length(A);A=AB;fork=1:n-1;fori=k:n;if(abs(A(i,k)=max(abs(A(k:n,k)P(k)=i;temp=A(k,:);A(k,:)=A(i,:);A(i,:)=temp;e

3、ndendforj=k+1:n;A(j,k)=A(j,k)/A(k,k);A(j,k+1:n+1)=A(j,k+1:n+1)-A(j,k)*A(k,k+1:n+1);endendP(n)=n;L(1,1)=1;L(2:n,1)=A(2:n,1);L(1,2:n)=0;U(1,1)=A(1,1);U(2:n,1)=0;U(1,2:n)=A(1,2:n);fori=2:n;L(i,1:i-1)=A(i,1:i-1);L(i,i)=1;L(i,i+1:n)=0;U(i,1:i-1)=0;U(i,i:n)=A(i,i:n);endx(n)=A(n,n+1)/U(n,n);fork=n-1:-1:1x(

4、k)=A(k,n+1);forp=n:-1:k+1;x(k)=x(k)-U(k,p)*x(p);endx(k)=x(k)/U(k,k);endxLUPEndMatlab运行结果a=EO34;1-11;212a.341-11212b=Ll;2;3b=123LU(a,b)x-1.1667-0.33330.5000L=1.00000n1.0000.5000-0.50001.00003.龙贝格(Romberg)算法M文件functiont=romberg(f,a,b,e)t=zeros(15,4);t(1,1)=(b-a)/2*(f(a)+f(b);fork=2:4sum=0;fori=l:2人(k-

5、2)sum=sum+f(a+(2*i-l)*(b-a)/2人(k-1);endt(k,1)=0.5*t(k-1,1)+(b-a)/2人(k-1)*sum;fori=2:kt(k,i)=(4A(i-1)*t(k,i-1)-t(k-1,i-1)/(4A(i-1)-1);endendfork=5:15sum=0;fori=1:2A(k-2)sum=sum+f(a+(2*i-1)*(b-a)/2A(k-1);endt(k,1)=0.5*t(k-1,1)+(b-a)/2A(k-1)*sum;fori=2:4t(k,i)=(4A(i-1)*t(k,i-1)-t(k-1,i-1)/(4A(i-1)-1);e

6、ndifk6ifabs(t(k,4)-t(k-1,4)=15disp(溢出);endMatlab运行结果CommaHdWindowf=inlinesin(s)/KJj3kj)=Inlinefunct1on:f(k)二sin(K)/zrombergff,10-100,1,5*10n(-?)普案0.94608ans=0.92070000.93930.946100.54450.?4610.94100.94570.9461a.94610.94610.94600.9461a.94610.94610.94610.9461a.94610.94610.94610.9461a.94610.9461nn0nn00

7、0c000000000nn0nn000004最小二乘法M文件functiona,max,det=zuixiaoerchengfa(x,y,r)n=length(x);c=ones(n,r+1);fori=2:r+1forj=1:nc(j,i)=x(j)人(i-1);endendA=c*c;b=c*y;a=inv(A)*b;det=0;max=0;fori=1:nsum=a(1);forj=2:r+1sum=sum+a(j)*x(i)人(j-1);endcc=abs(y(i)-sum);ifccmaxmax=cc;enddet=det+ccA2;enddet=sqrt(det);Matlab运行结果CommandWindowA=2468A二2468B=L2112S40B=2112840zuisiaoerchengfa(AjBj2)ans=-S.75004.5750.1875zuisiaoerchengfa(AjBj1)arts=-12.50006.5500