Question 10.4: Use MATLAB to compute the LU factorization and find the solu...
Use MATLAB to compute the LU factorization and find the solution for the same linear system analyzed in Examples 10.1 and 10.2:
\begin{bmatrix} 3 & -0.1 & -0.2 \\ 0.1 & 7 & -0.3 \\ 0.3 & -0.2 & 10 \end{bmatrix} \begin{Bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{Bmatrix} = \begin{Bmatrix} 7.85 \\ -19.3 \\ 71.4 \end{Bmatrix}The coefficient matrix and the right-hand-side vector can be entered in standard fashion as
>> A = [3 −.1 −.2;.1 7 −.3;.3 −.2 10];
>> b = [7.85; −19.3; 71.4];
Next, the LU factorization can be computed with
>> [L,U] = lu(A)
L =
1.0000 0 0
0.0333 1.0000 0
0.1000 −0.0271 1.0000
U =
3.0000 −0.1000 −0.2000
0 7.0033 −0.2933
0 0 10.0120
This is the same result that we obtained by hand in Example 10.1. We can test that it is correct by computing the original matrix as
>> L*U
ans =
3.0000 −0.1000 −0.2000
0.1000 7.0000 −0.3000
0.3000 −0.2000 10.0000
To generate the solution, we first compute
>> d = L\b
d =
7.8500
−19.5617
70.0843
And then use this result to compute the solution
>> x = U\d
x =
3.0000
−2.5000
7.0000
These results conform to those obtained by hand in Example 10.2.