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LU分解高斯消元列主元高斯消元matlab代码

更新时间：2024-04-13 09:01:01 阅读量： 综合文库 文档下载

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数学实验 作业

一、矩阵LU分解：

function [L,U,p]=lutx(A) [n,n]=size(A); p=(1:n)'; for k=1:n-1

[r,m]=max(abs(A(k:n,k))); m=m+k-1; if (A(m,k)~=0) if (m~=k)

A([k m],:)=A([m k],:); p([k m])=p([m k]); end i=k+1:n;

A(i,k)=A(i,k)/A(k,k); j=k+1:n;

A(i,j)=A(i,j)-A(i,k)*A(k,j); end end

L=tril(A,-1)+eye(n,n) U=triu(A) p end

高斯消元法求解方程： n=3;

a=[1 2 3 ;4 5 6 ;7 8 9 ]; b=[17 18 19]; l=eye(n); y=1;

for i=1:(n-1) for j=1:(n-i) if a(j+(i-1)*n+y)~=0 l(j+(i-1)*n+y)=a(j+(i-1)*n+y)/a(j+(i-1)*n+y-j) for k=1:(n-i+1) a(j+(i-1)*n+y+(k-1)*n)=a(j+(i-1)*n+y+(k-1)*n)-a(j+(i-1)*n+y+(k-1)*n-j)*l(j+(i-1)*n+y) end b(j+y-1)=b(j+y-1)-b(y)*l(j+(i-1)*n+y); end end y=y+1;

end

sum=0; for j=1:n sum=sum+x(j)+a(k,j); end

sum=0; for j=1:n x(j)=0; end

for k=n:-1:1 for j=1:n

sum=sum+x(j)*a(k,j) end

x(k)=(b(k)-sum)/a(k,k) sum=0; end

列主元高斯消元法代码：

n=3;

a=[1 2 3 ;4 5 6 ;7 8 9 ];

b=[17 18 19];

l=eye(n); p=eye(n); ma=0

for i=1:(n-1) for j=i:n if a(j,i)>ma; ma=a(j,i) end end

for k=i:n if a(k,i)==ma m=k; end end

for j=1:n a1=a(m,j); a(m,j)=a(i,j);

a(i,j)=a1 p1=p(m,j); p(m(1),j)=p(i,j); p(i,j)=p1; end b1=b(m); b(m)=b(i); b(i)=b1; ma=0; end y=1;

for i=1:(n-1) for j=1:(n-i)

if a(j+(i-1)*n+y)~=0

l(j+(i-1)*n+y)=a(j+(i-1)*n+y)/a(j+(i-1)*n+y-j) for k=1:(n-i+1)

a(j+(i-1)*n+y+(k-1)*n)=a(j+(i-1)*n+y+(k-1)*n)-a(j+(i-1)*n+y+(k-1)*n-j)*l(j+(i-1)*n+y) end

b(j+y-1)=b(j+y-1)-b(y)*l(j+(i-1)*n+y); end end y=y+1; end sum=0; for j=1:n x(j)=0; end

for k=n:-1:1 for j=1:n

sum=sum+x(j)*a(k,j) end

x(k)=(b(k)-sum)/a(k,k) sum=0; end

全主元高斯消元法代码：

n=3;

a=[1 2 3 ;4 5 6 ;7 8 9 ];

b=[17 18 19]; l=eye(n); p=eye(n); q=eye(n); max=0; for i=1:(n-1) for j=i:n for k=i:n

if max

if max==abs(a(j,k)) m=[j,k]; end end end for j=1:n a1=a(m(1),j); a(m(1),j)=a(i,j); a(i,j)=a1; p1=p(m(1),j); p(m(1),j)=p(i,j); p(i,j)=p1; end

b1=b(m(1)); b(m(1))=b(i); b(i)=b1; for j=1:n a1=a(j,m(2)); a(j,m(2))=a(j,i); a(j,i)=a1; q1=q(j,m(2)); q(j,m(2))=q(j,i); q(j,i)=q1; end max=0; end y=1;

for i=1:(n-1) for j=1:(n-i)

if a(j+(i-1)*n+y)~=0

l(j+(i-1)*n+y)=a(j+(i-1)*n+y)/a(j+(i-1)*n+y-j) for k=1:(n-i+1)

a(j+(i-1)*n+y+(k-1)*n)=a(j+(i-1)*n+y+(k-1)*n)-a(j+(i-1)*n+y+(k-1)*n-j)*l(j+(i-1)*n+y); end

b(j+y-1)=b(j+y-1)-b(y)*l(j+(i-1)*n+y); end end y=y+1; end

sum=0; for j=1:n x(j)=0; end

for k=n:-1:1 for j=1:n

sum=sum+x(j)*a(k,j) end

x(k)=(b(k)-sum)/a(k,k) sum=0; end

解：编写矩阵：

0 0 a a 0 0

A=

0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

0 0

0 0

-1 0 0

0 0 0 0 1 0 1 0 0 0

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 -1 0

-1 -a 0

-a 0 0 0 a a 0 0 0

0 0 1 0 0 0 0

1 0 0 0 0 0 0

-1 0

-1 0

-a -1 0 -a 0 0 0 a

1 0 0

-a 0

0 0

B= F=

0 0

0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0

a 0

0 0

1 0

a a 0

0 1 0)’

(0 10 0 15 0 20 0

(f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13)’

AF=B F=A\\B

程序及运算结果： >> a=sym(1/sqrt(2)) a =

2^(1/2)/2

>> A=[0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 a 0 0 -1 -a 0 0 0 0 0 0 0 0 a 0 -1 0 -a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 a 1 0 0 -a -1 0 0 0 0 0 0 0 a 0 1 0 -a 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 a 0 0 -a 0 0 0 0 0 0 0 0 0 a 0 1 a 0 0 0 0 0 0 0 0 0 0 0 0 a -1 ] A =

[ 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0] [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] [ 2^(1/2)/2, 0, 0, -1, -2^(1/2)/2, 0, 0, 0, 0, 0, 0, 0, 0] [ 2^(1/2)/2, 0, -1, 0, -2^(1/2)/2, 0, 0, 0, 0, 0, 0, 0, 0] [ 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0] [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0] [ 0, 0, 0, 0, 2^(1/2)/2, 1, 0, 0, -2^(1/2)/2, -1, 0, 0, 0] [ 0, 0, 0, 0, 2^(1/2)/2, 0, 1, 0, -2^(1/2)/2, 0, 0, 0, 0] [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1] [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0] [ 0, 0, 0, 0, 0, 0, 0, 1, 2^(1/2)/2, 0, 0, -2^(1/2)/2, 0] [ 0, 0, 0, 0, 0, 0, 0, 0, 2^(1/2)/2, 0, 1, 2^(1/2)/2, 0] [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2^(1/2)/2, 1]

>> B=[0;10;0;0;0;0;0;15;0;20;0;0;0] B =

0 10 0 0 0 0 0 15 0 20 0 0 0

>> F=A\\B F =

10*2^(1/2) -10 10 10 0 -10 0 10 -15*2^(1/2) 5 20 -5*2^(1/2) 5

LU分解： >> a=2^(1/2)/2 a =

0.7071

>> A=[0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 a 0 0 -1 -a 0 0 0 0 0 0 0 0 a 0 -1 0 -a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 a 1 0 0 -a -1 0 0 0 0 0 0 0 a 0 1 0 -a 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 a 0 0 -a 0 0 0 0 0 0 0 0 0 a 0 1 a 0 0 0 0 0 0 0 0 0 0 0 0 a 1 ] A =

Columns 1 through 8

0 1.0000 0 0 0 -1.0000 0 0 1.0000 0 0 0

0.7071 0 0 -1.0000 -0.7071 0 0.7071 0 -1.0000 0 -0.7071 0 0 0 0 1.0000 0 0 0 0 0 0 0 0

0 0 0 0 0.7071 0

0 0 0 0 0.7071 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 1.0000

0 0 0 0 0 0

0 0 0 0 0 0

Columns 9 through 13

0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0

0 0 0 -1.0000

1.0000 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.7071 -1.0000 0 0 0 -0.7071 0 0 0 0 0 1.0000 0 0 -1.0000 0 0 1.0000 0 0 0.7071 0 0 -0.7071 0 0.7071 0 1.0000 0.7071 0 0 0 0 0.7071 1.0000

>> [L,U,P]=lu(A) L =

Columns 1 through 8

1.0000 0 0 0 0 0

0 1.0000 0 0 0 0

0 0 1.0000 0 0 0

1.0000 0 -1.0000 1.0000 0 0 0 0 0 0 1.0000 0

0 0 0 0 1.0000 0

0 0 0 0 0 0

0 0 0 1.0000 0 1.0000

0 0 0 0 0 -1.0000

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0

0 0 1.0000 0 0 0 0 0 0

0

Columns 9 through 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 1.0000 0 1.0000 1.0000 0 0 0 0 0.5000 1.0000 U =

Columns 1 through 8

0.7071 0 0 -1.0000 -0.7071 0 0 1.0000 0 0 0 -1.0000 0 0 1.0000 0 0 0

0 0 0 1.0000 0 0

0 0 0 0 0.7071 0

0 0 0 0 0 -1.0000 0 0 0 0 0 0

0 0 0 0 0 -1.0000

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1.0000 1.0000 0 0 0 0 0 0 0 0

0 0 0 0

1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0

Columns 9 through 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.7071 -1.0000 0 1.0000 0 0 0 0 0.7071 0 0 1.0000 0 0 0 0 0 0 P =

0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

>> F=U\\(L\\B) F =

10.6066 37.5000 0 0

0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.7071 0 0 -1.0000 0 0 1.4142 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 10.6066 27.5000 0 -15.0000 10.6066 27.5000 0 -10.6066 7.5000

解：

程序及运算结果： Lutx.m文件：

function [L,U,p,sig] = lutx(A) %LU Triangular factorization

% [L,U,p,sig] = lutx(A) computes a unit lower triangular % matrix L, an upper triangular matrix U, a permutation % vector p, and a scalar sig, so that L*U = A(p,:) and % sig = +1 or -1 if p is an even or odd permutation. [n,n] = size(A); p = (1:n)'; w=0

for k = 1:n-1

% Find largest element below diagonal in k-th column [r,m] = max(abs(A(k:n,k))); m = m+k-1;

% Skip elimination if column is zero if (A(m,k) ~= 0)

% Swap pivot row if (m ~= k)

A([k m],:) = A([m k],:); p([k m]) = p([m k]); w=w+1; end

% Compute multipliers i = k+1:n;

A(i,k) = A(i,k)/A(k,k);

% Update the remainder of the matrix j = k+1:n;

A(i,j) = A(i,j) - A(i,k)*A(k,j); end end

% Separate result L = tril(A,-1) + eye(n,n) U = triu(A) p

sig=(-1)^w

mydet.m文件：

function det=mydet(A) [L,U,p,sig] = lutx(A) det=sig*prod(diag(U))

运行结果： >> a=2^(1/2)/2 a =

0.7071

>> A=[0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 a 0 0 -1 -a 0 0 0 0 0 0 0 0 a 0 -1 0 -a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 a 1 0 0 -a -1 0 0 0 0 0 0 0 a 0 1 0 -a 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 a 0 0 -a 0 0 0 0 0 0 0 0 0 a 0 1 a 0 0 0 0 0 0 0 0 0 0 0 0 a 1 ] A =

Columns 1 through 8

0 1.0000 0 0 0 -1.0000 0 0 1.0000 0 0 0

0.7071 0 0 -1.0000 -0.7071 0 0.7071 0 -1.0000 0 -0.7071 0 0 0 0 1.0000 0 0 0 0 0 0 0 0

0 0 0 0 0.7071 0

0 0 0 0 0.7071 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 1.0000

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0

0 0 0 -1.0000

1.0000 0 1.0000 0 0 0 0 0

Columns 9 through 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.7071 -1.0000 0 0 0 -0.7071 0 0 0 0 0 1.0000 0 0 -1.0000 0 0 1.0000 0 0 0.7071 0 0 -0.7071 0 0.7071 0 1.0000 0.7071 0 0 0 0 0.7071 1.0000

>> mydet(A) w =

0 L =

Columns 1 through 8

1.0000 0 0 0 0 0

0 1.0000 0 0 0 0

0 0 1.0000 0 0 0

1.0000 0 -1.0000 1.0000 0 0 0 0 0 0 1.0000 0

0 0 0 0 1.0000 0

0 0 0 0 0 0

0 0 0 1.0000 0 1.0000

0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0

0 0 1.0000 0 0

-1.0000

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Columns 9 through 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 1.0000 0 1.0000 1.0000 0 0 0 0 0.5000 1.0000 U =

Columns 1 through 8

0.7071 0 0 -1.0000 -0.7071 0 0 1.0000 0 0 0 -1.0000 0 0 1.0000 0 0 0

0 0 0 1.0000 0 0

0 0 0 0 0.7071 0

0 0 0 0 0 -1.0000 0 0 0 0 0 0

0 0 0 0 0 -1.0000

0 0 0 0 1.0000 1.0000 0 0 0 0

0 0 0 0

1.0000 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0

Columns 9 through 13

0 0 0 0 0 0 0 0 -0.7071 -1.0000 0 1.0000 0 0 0 0 0.7071 0 0 1.0000 0 0 0 0 0 0 p =

3 1 2 4 7 8 6 5 11 9 10 12 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.7071 0 0 1.0000 0 0 1.4142 0 0 0 0 0 0 0 0 0 0 0 -1.0000 0 0 1.0000 sig =

-1 L =

Columns 1 through 8

1.0000 0 0 0 0 0

0 1.0000 0 0 0 0

0 0 1.0000 0 0 0

1.0000 0 -1.0000 1.0000 0 0 0 0 0 0 1.0000 0

0 0 0 0 1.0000 0

0 0 0 0 0 0

0 0 0 1.0000 0 1.0000

0 0 0 0 0 -1.0000

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

Columns 9 through 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0

0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 1.0000 0 1.0000 1.0000 0 0 0 0 0.5000 1.0000 U =

Columns 1 through 8

0.7071 0 0 1.0000 0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 -1.0000

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

Columns 9 through 13

0 0 0 0 0 0 0 0 -0.7071 -1.0000 0 -1.0000 -0.7071 0 0 0 0 -1.0000 1.0000 0 0 0 1.0000 0 0 0 0.7071 0 0 0 -1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1.0000 1.0000 0 0 0 0 0 0 0 0 0

0 0 0 0

1.0000 0 0 0 0 0 0

0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0.7071 0 0 -0.7071 0 0 1.0000 0 0 -1.0000 0 0 1.0000 0 0 0 0 0 1.4142 0 0 0 0 0 1.0000 p =

3 1 2 4 7 8 6 5 11 9 10 12 13 sig =

-1

ans =

-0.5000

解：

程序及运算结果：

function [L,U,p] = lutx1(A) [n,n] = size(A); p = (1:n)'; for k = 1:n-1

[r,m] = max(abs(A(k:n,k))); m = m+k-1;

if (A(m,k) ~= 0) if (m ~= k)

A([k m],:) = A([m k],:); p([k m]) = p([m k]); end

for i=k+1:n

A(i,k)=A(i,k)/A(k,k); end

for j=k+1:n

A(i,j)=A(i,j)-A(i,k)*A(k,j); end end end

L=tril(A,-1)+eye(n,n) U=triu(A)

运行结果： >> lutx1(A) L =

1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 U =

Columns 1 through 8

0.7071 0 0 1.0000 0 0 0

0 0 0 0 0

0 0 0

0 0 0

0 0 1.0000

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

Columns 9 through 13

0 0 0 0 0 0 0 0 0 -1.0000 0 0 1.0000 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0.7071 0 0 -1.0000 0 0 0 0 0 -0.7071 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0

0 -1.0000

0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 -0.7071 -1.0000 0 0 0 0 0 0 0 0 0.7071 0 0 -0.7071 0 -0.7071 0 0 0 0 0 1.0000 0 0 -1.0000 0 0 1.0000 0 0 0 0 0 0.7071 0 0 0 0 0 1.0000

ans =

1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 -1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 -1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1

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