1. Code
%% LU decomposition method
function LUDM = LU_Decomposition_method(A,b)
global n global B; global to, 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 ( 'equation has no solution');
elseif (R_A == R_B) && (R_A == n)
DISP ( 'This equation has a unique solution');
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 decomposition of L =');
L
disp ( 'LU decomposition U =');
The
else
DISP ( 'equation has infinite number of solutions');
end
DISP ( 'solution vector is:');
LUDM = X;
LU %% matrix decomposition
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) = 50;
LUD (:,:, 2) = U;
end
%% lower triangular elementary transformation
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
%% lower triangular iterative method
function LTIS = Lower_trig_iterative_solution(M)
[m,n] = size(M);
B = M (:, 1: n-1); b = M (:, n);
y = zeros(1,m);
y (1) = three (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
%% upper triangular iterative method
function UTIS = Upper_trig_iterative_solution(M)
[m,n] = size(M);
B = M (:, 1: n-1); b = 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. Examples
clear all clc M = rand (9) b = reshape(rand(3),9,1) S = LU_Decomposition_method(M,b) M\b
result
M =
Column 1-7
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
Column 8-9
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
This equation has a unique solution
LU decomposition L =
L =
Column 1-7
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
Column 8-9
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1.0000 0
5.1107 1.0000
LU decomposition U =
U =
Column 1-7
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
Column 8-9
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
Solution vector is:
S =
-0.9496
2.2130
0.5483
1.9595
-3.8859
-0.4632
0.4453
0.3978
2.6573
years =
-0.9496
2.2130
0.5483
1.9595
-3.8859
-0.4632
0.4453
0.3978
2.6573
>>