LU Decomposition Method
Department of Mathematics
Faculty of Science & Technology
ICFAI Foundation for Higher Education
Hyderabad
Objectives
LU Decomposition is another method to solve a
set of simultaneous linear equations
LU Decomposition Method
For most non-singular matrix [A] that one can always
write it as
[A] = [L][U]
where
[L] = lower triangular matrix
[U] = upper triangular matrix
Example
Solve the following set of
linear equations using LU
Decomposition
 25 5 1  x1  106 .8 
 64 8 1  x  = 177 .2 

  2 

144 12 1  x3  279 .2
Using the procedure for finding the [L] and [U] matrices
0 0 25
5
1 
 1
A = LU  = 2.56 1 0  0 − 4.8 − 1.56
5.76 3.5 1  0
0
0.7 
Set [L][Z] = [B]
Solve for [Z]
0 0  z1  106.8 
 1
2.56 1 0  z  = 177 .2 

 2  

5.76 3.5 1  z 3  279.2
z1 = 10
2.56 z1 + z 2 = 177.2
5.76 z1 + 3.5 z 2 + z 3 = 279.2
Complete the forward substitution to solve for [Z]
z1 = 106 .8
z 2 = 177 .2 − 2.56 z1
= 177 .2 − 2.56(106 .8)
= −96.2
z3 = 279 .2 − 5.76 z1 − 3.5 z2
= 279 .2 − 5.76(106 .8) − 3.5(− 96.21)
= 0.735
 z1   106 .8 
Z  =  z2  = − 96.21
 z3   0.735 
Set [U][X] = [Z]
Solve for [X] =
1   x1   106 .8 
25 5
 0 − 4.8 − 1.56  x  = − 96.21

  2 

 0
0
0.7   x3   0.735 
Thank you