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Consider solving the following system of simultaneous equations using LU decomposition.
x1 + x2 - 2x3 = 4
x1 + 3x2 - x3 = 7
2x1 + x2 - 5x3 = 7
where L and U are denoted as
L = (L11 0 0
L21 L22 0
L31 L32 L33), U = (U11 U12 U13
0 U22 U23
0 0 U33)
Which one of the following is the correct combination of values for L32, U33, and x1?
- L32 = 2, U33 = -1/2, x1 = -1
- L32 = 2, U33 = 2, x1 = -1
- L32 = -1/2, U33 = 2, x1 = 0
- L32 = -1/2, U33 = -1/2, x1 = 0
Correct answer: L32 = -1/2, U33 = -1/2, x1 = 0
Solution
Doolittle LU gives U22=2, U23=1, L32=(1-2)/2=-1/2, and U33=-5+4+1/2=-1/2. Forward solving Ly=b gives y=(4,3,1/2); back substitution gives x3=-1, x2=2, x1=0. So L32=-1/2, U33=-1/2, x1=0, index 3, not the stored option.
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