线性方程组的分解法——LU分解法
Posted guliangt
tags:
篇首语：本文由小常识网(cha138.com)小编为大家整理，主要介绍了线性方程组的分解法——LU分解法相关的知识，希望对你有一定的参考价值。
1.代码
%%LU分解法
function LUDM = LU_Decomposition_method(A,b)
global n;global B;global U;global L;global M;
[n,n] = size(A);
B = [A,b];
R_A = rank(A);R_B = rank(B);
if R_A ~= R_B
disp(‘方程无解‘);
elseif (R_A == R_B) && (R_A == n)
disp(‘此方程有唯一解‘);
M = LU_decomposition(A);
L = M(:,:,1);U = M(:,:,2);
matrix1 = [L b];
Y = Lower_trig_iterative_solution(matrix1);
matrix2 = [U Y];
X = Upper_trig_iterative_solution(matrix2);
disp(‘LU分解中L=‘);
L
disp(‘LU分解中U=‘);
U
else
disp(‘方程有无穷多组解‘);
end
disp(‘解向量为:‘);
LUDM = X;
%%矩阵的LU分解
function LUD = LU_decomposition(A)
[n,n] = size(A);
M = Elementary_transformation_of_the_lower_triangle(A);
L = M(:,:,n);U=A;
for i = 1:1:n-1
U = M(:,:,i)*U;
end
LUD(:,:,1) = L;
LUD(:,:,2) = U;
end
%%下三角初等变换
function ETLT = Elementary_transformation_of_the_lower_triangle(A)
[n,n] = size(A);
L = zeros(n,1,n);
for i = 1:1:n
for j = 1:1:n
for k = 1:1:n
if j == k
L(j,k,i) = 1;
end
end
end
end
for i = 1:1:n-1
for j = 1:1:n
for k = 1:1:n
if j > k
if i == k
L(j,k,i) = -A(j,k)/A(k,k);
end
L(i+1:n,i,n) = -L(i+1:n,i,i);
end
end
end
A = L(:,:,i)*A;
end
ETLT = L;
end
%%下三角迭代法
function LTIS = Lower_trig_iterative_solution(M)
[m,n] = size(M);
B =M(:,1:n-1);ba = M(:,n);
y = zeros(1,m);
y(1) = ba(1);
for i = 2:1:m
sum = 0;
for j = 1:1:i-1
sum = sum+B(i,j)*y(j);
end
y(i) = ba(i)-sum;
end
LTIS = y‘;
end
%%上三角迭代法
function UTIS = Upper_trig_iterative_solution(M)
[m,n] = size(M);
B = M(:,1:n-1);ba = M(:,n);
x = zeros(1,m);
x(m) =ba(m)/B(m,m);
for i = m-1:-1:1
sum = 0;
for j = i+1:1:m
sum = sum+B(i,j)*x(j);
end
x(i) = (ba(i)-sum)/B(i,i);
end
UTIS = x‘;
end
end
2.例子
clear all clc M = rand(9) b = reshape(rand(3),9,1) S = LU_Decomposition_method(M,b) M
结果
M =
列 1 至 7
0.5944 0.4709 0.4076 0.4235 0.5181 0.0680 0.6022
0.0225 0.6959 0.8200 0.0908 0.9436 0.2548 0.3868
0.4253 0.6999 0.7184 0.2665 0.6377 0.2240 0.9160
0.3127 0.6385 0.9686 0.1537 0.9577 0.6678 0.0012
0.1615 0.0336 0.5313 0.2810 0.2407 0.8444 0.4624
0.1788 0.0688 0.3251 0.4401 0.6761 0.3445 0.4243
0.4229 0.3196 0.1056 0.5271 0.2891 0.7805 0.4609
0.0942 0.5309 0.6110 0.4574 0.6718 0.6753 0.7702
0.5985 0.6544 0.7788 0.8754 0.6951 0.0067 0.3225
列 8 至 9
0.7847 0.1917
0.4714 0.7384
0.0358 0.2428
0.1759 0.9174
0.7218 0.2691
0.4735 0.7655
0.1527 0.1887
0.3411 0.2875
0.6074 0.0911
b =
0.5762
0.6834
0.5466
0.4257
0.6444
0.6476
0.6790
0.6358
0.9452
此方程有唯一解
LU分解中L=
L =
列 1 至 7
1.0000 0 0 0 0 0 0
0.0379 1.0000 0 0 0 0 0
0.7155 0.5352 1.0000 0 0 0 0
0.5261 0.5762 -74.4491 1.0000 0 0 0
0.2717 -0.1391 -136.4397 1.7669 1.0000 0 0
0.3008 -0.1074 -74.0359 0.9200 0.6765 1.0000 0
0.7115 -0.0228 42.5434 -0.5996 0.3838 -141.0829 1.0000
0.1585 0.6728 -1.3001 -0.0414 0.8852 -70.1396 0.4925
1.0070 0.2658 -39.5864 0.4476 1.3552 49.3425 -0.3788
列 8 至 9
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1.0000 0
5.1107 1.0000
LU分解中U=
U =
列 1 至 7
0.5944 0.4709 0.4076 0.4235 0.5181 0.0680 0.6022
0 0.6781 0.8045 0.0748 0.9240 0.2522 0.3640
0 0 -0.0039 -0.0765 -0.2275 0.0404 0.2903
0 0 0 -5.8101 -16.7848 3.4944 21.0900
-0.0000 0 0 0 -1.1550 0.1988 2.6992
0.0000 0 0 0 0 -0.0074 0.5483
0.0000 -0.0000 0 0 0 0 76.6535
0.0000 0.0000 0 -0.0000 0 0 0
-0.0000 -0.0000 0 0.0000 0 0 0
列 8 至 9
0.7847 0.1917
0.4416 0.7312
-0.7621 -0.2857
-57.2283 -20.8735
-2.2924 -1.7782
-1.9343 0.0429
-274.3037 6.4447
-1.9999 -0.0598
0 0.7768
解向量为:
S =
-0.9496
2.2130
0.5483
1.9595
-3.8859
-0.4632
0.4453
0.3978
2.6573
ans =
-0.9496
2.2130
0.5483
1.9595
-3.8859
-0.4632
0.4453
0.3978
2.6573
>>
以上是关于线性方程组的分解法——LU分解法的主要内容，如果未能解决你的问题，请参考以下文章