矩阵分解与线性方程组求解

本文格式为Word版，下载可任意编辑——矩阵分解与线性方程组求解一、用列主元素高斯削去法求解下述线性方程组：

?x1?13x2?2x3?34x4?13?2x?6x?7x?10x??22?1234??10x?x?5x?9x?141234????3x1?5x2?15x4??36程序：

functionx=gaussa(a)

m=size(a);n=m(1);x=zeros(n,1);fork=1:n-1

[c,i]=max(abs(a(k:n,k)));q=i+k-1;ifq~=k

d=a(q,:);a(q,:)=a(k,:);a(k,:)=dend

fori=k+1:n

a(i,:)=a(i,:)-a(k,:)*a(i,k)/a(k,k)endend

forj=n:-1:1

x(j)=(a(j,n+1)-a(j,j+1:n)*x(j+1:n))/a(j,j)end

执行过程：

>>a=[113-2-3413;26-7-10-22;-10-15914;-3-5015-36]a=

-10-1591426-7-10-22113-2-3413-3-5015-36>>gaussa(a)a=

-10.0000-1.00005.00009.000014.000005.8000-6.0000-8.2000-19.20001.000013.0000-2.0000-34.000013.0000-3.0000-5.0000015.0000-36.0000a=

-10.0000-1.00005.00009.000014.000005.8000-6.0000-8.2000-19.2000012.9000-1.5000-33.100014.4000-3.0000-5.0000015.0000-36.0000a=

-10.0000-1.00005.00009.000014.000005.8000-6.0000-8.2000-19.2000012.9000-1.5000-33.100014.40000-4.7000-1.500012.3000-40.2000

a=

-10.0000-1.00005.00009.000014.0000012.9000-1.5000-33.100014.400005.8000-6.0000-8.2000-19.20000-4.7000-1.500012.3000-40.2000a=

-10.0000-1.00005.00009.000014.0000012.9000-1.5000-33.100014.400000.0000-5.32566.6822-25.67440a=

-10.0000000a=

-10.0000000x=

00010.7786x=

0018.345210.7786x=

030.906218.345210.7786x=

14.382730.906218.345210.7786ans=

14.382730.906218.3452

-4.7000-1.000012.90000.00000-1.000012.90000.0000-0.0000-1.500012.3000-40.20005.00009.000014.0000-1.5000-33.100014.4000-5.32566.6822-25.6744-2.04650.2403-34.95355.00009.000014.0000-1.5000-33.100014.4000-5.32566.6822-25.67440-2.3275-25.087310.7786

二、用矩阵A的杜丽特尔三角分解A=LU，求解方程组：

?157010??x1??????618159??x2??010287??x?=

3?????50635??x????4??8????6??4????2???程序：

functionx=lua(a,b)

m=size(a);n=m(1);x=zeros(n,1);y=zeros(n,1);fork=1:n

a(k,k:n)=a(k,k:n)-a(k,1:k-1)*a(1:k-1,k:n);

a(k+1:n,k)=(a(k+1:n,k)-a(k+1:n,1:k-1)*a(1:k-1,k))/a(k,k);end

U=triu(a,0)

L=eye(n)+tril(a,-1)fori=1:n

y(i)=b(i)-L(i,1:i-1)*y(1:i-1);end

fori=n:-1:1

x(i)=(y(i)-U(i,i+1:n)*x(i+1:n))/U(i,i);end执行过程：

>>a=[157010;618159;010287;50635]a=

15701061815901028750635>>b=[8;6;4;2]b=8642>>lua(a,b)U=

15.00007.0000010.0000015.200015.00005.00000018.13163.710500030.7351L=

1.00000000.40001.000000

00.65791.000000.3333-0.15350.45791.0000ans=

0.52640.07180.1272-0.0399

三、用乔列斯基分解计算下述线性方程组：

00??x1??5??4?10??????0??x2??8???14?10?0?14?10??x?=?16????3???0?14?1??x4??24??0?????000?14????x5??36?程序：

functionx=choleskey(a,b)

m=size(a);n=m(1);x=zeros(n,1);y=zeros(n,1);G=zeros(m);forj=1:n

G(j,j)=sqrt(a(j,j)-G(j,1:j-1)*G(j,1:j-1)')

G(j+1:n,j)=(a(j+1:n,j)-G(j+1:n,1:j-1)*G(j,1:j-1)')/G(j,j)end

fori=1:n

y(i)=(b(i)-G(i,1:i-1)*y(1:i-1))/G(i,i);end

fori=n:-1:1

x(i)=(y(i)-G(i+1:n,i)'*x(i+1:n))/G(i,i);end运行过程：

>>a=[4-1000;-14-100;0-14-10;00-14-1;000-14]a=

4-1000-14-1000-14-1000-14-1000-14>>b=[5;8;16;24;36]b=58162436

>>choleskey(a,b)G=

2000000000000000000000000G=

2.00000000-0.5000000G=

2.0000-0.5000000G=

2.0000-0.5000000G=

2.0000-0.5000000G=

2.0000-0.5000000G=

2.0000-0.5000000G=

000001.936500001.9365-0.51640001.9365-0.51640001.9365-0.51640001.9365-0.51640000000000000000001.932200001.9322-0.51750001.9322-0.517500000000000000000000000000000000000000000000000000000000001.93192.00000000-0.50001.93650000-0.51641.93220000-0.51751.93190000-0.51760G=

2.00000000-0.50001.93650000-0.51641.93220000G=

2.0000-0.5000000ans=

2.39104.56417.865410.897411.7244

0001.9365-0.516400-0.51750001.9322-0.517501.93190-0.51761.93190000001.93190-0.51761.9319

2.00000000-0.50001.9365