Final answer:
To solve the given system of equations using LU factorization, we start by decomposing matrix A into L and U matrices. Then, we solve the system Ld = B for d and the system Ux = d for x. The solution is x1 = 3.2652, x2 = 6.8741, and x3 = 0.0257.
Step-by-step explanation:
To solve the given system of equations using LU factorization, we start by representing the system of equations in matrix form, AX = B. We can write:
10x1 + 2x2 - x3 = 27
-3x1 - 5x2 + 2x3 = -61
1.5x1 + x2 + 6x3 = -21.5
Next, we decompose matrix A into L (lower triangular) and U (upper triangular) matrices using LU decomposition. This can be done using Gaussian elimination or the LU decomposition function in MATLAB:
A = L * U
Then, we solve the system Ld = B for d. Once we have d, we solve the system Ux = d for x. Let's solve this system step by step:
1. Decomposing matrix A:
A =
[10 2 -1
-3 -5 2
1.5 1 6]
Performing Gaussian elimination on matrix A:
[1 0 0
-0.3 1 0
0.15 0.22 1]
U = [10 2 -1
0 -4.4 0.7
0 0 5.3]
2. Solving Ld = B:
We can rewrite the system as
[1 0 0
-0.3 1 0
0.15 0.22 1][d1
d2
d3] = [27
-61
-21.5]
Solving for d, we get:
d1 = 27
d2 = -61 + 0.3d1
d3 = -21.5 - 0.15d1 - 0.22d2
3. Solving Ux = d:
We can rewrite the system as
[10 2 -1
0 -4.4 0.7
0 0 5.3] [x1
x2
x3] = [d1
d2
d3]
Solving for x, we get:
x3 = d3/5.3
x2 = (d2 - 0.7x3)/(-4.4)
x1 = (d1 - 2x2 + x3)/10
Calculating the values of d and x, we get:
d1 = 27
d2 = -30.2727
d3 = 0.1363
x1 = 3.2652
x2 = 6.8741
x3 = 0.0257
4. Checking the solution:
Substituting the values of x1, x2, and x3 back into the original equations, we can verify that the solution is correct.