Consider solving the following system of simultaneous equations using LU…
2022
Consider solving the following system of simultaneous equations using LU decomposition.
Which one of the following is the correct combination of values for\(L_{32} \), \(U_{33}\) , and \(x_1\)?
- A.
\(L_{32} = 2, U_{33} = - \frac {1}{2} , x_1 = -1\) - B.
\(L_{32} = 2, U_{33} = 2 , x_1 = -1\) - C.
\(L_{32} = - \frac {1}{2}, U_{33} = 2 , x_1 = 0\) - D.
\(L_{32} = - \frac {1}{2}, U_{33} = - \frac {1}{2} , x_1 = 0\)
Attempted by 48 students.
Show answer & explanation
Correct answer: D
Solution outline: perform LU decomposition (Doolittle with unit lower diagonal) on A = [ [1,1,-2], [1,3,-1], [2,1,-5] ] and solve LUx = b with b = [4,7,7].
Step 1: First row of U is the first row of A: u_{11}=1, u_{12}=1, u_{13}=-2. Multipliers: L_{21}=1, L_{31}=2.
Step 2: Eliminate below the first pivot: row2' = row2 - 1*row1 = [0,2,1], row3' = row3 - 2*row1 = [0,-1,-1]. Thus u_{22}=2 and the second-row of U is [0,2,1].
Step 3: Multiplier to eliminate the second column from row3 is L_{32} = (row3' second entry)/u_{22} = (-1)/2 = -1/2.
Step 4: Compute u_{33} = (row3' third entry) - L_{32}*u_{23} = (-1) - (-1/2)*(1) = -1/2.
So the relevant entries are L_{32} = -1/2 and U_{33} = -1/2.
Step 5: Forward substitution for Ly = b (L has 1s on the diagonal): y_1 = 4, y_2 = 7 - 1*4 = 3, y_3 = 7 - 2*4 - (-1/2)*3 = 1/2.
Step 6: Back substitution for Ux = y: from u_{33} x_3 = y_3 => (-1/2) x_3 = 1/2 => x_3 = -1. Then 2 x_2 + 1*x_3 = 3 => x_2 = 2. Finally x_1 + x_2 - 2 x_3 = 4 => x_1 + 2 - 2(-1) = 4 => x_1 = 0.
Conclusion: L_{32} = -1/2, U_{33} = -1/2, x_1 = 0.