Solve Equations 2x+5y=16,3x+y=11 using LU decomposition using Gauss Elimination method
Solution:
Total Equations are 2
2x+5y=16 -> (1)
3x+y=11 -> (2)
Now converting given equations into matrix form
[[2,5],[3,1]] [[x],[y]]=[[16],[11]]
Now, A = [[2,5],[3,1]], X = [[x],[y]] and B = [[16],[11]]
LU decomposition : If we have a square matrix A, then an upper triangular matrix U can be obtained without pivoting under Gaussian Elimination method, and there exists lower triangular matrix L such that A=LU.
Using Gaussian Elimination method
R_2 larr R_2-(3/2)xx R_1 [:.L_(2,1)=color{blue}{3/2}]
L is just made up of the multipliers we used in Gaussian elimination with 1s on the diagonal.
:. LU decomposition for A is
Now, Ax=B, and A=LU => LUx=B
let Ux=y, then Ly=B =>
Now use forward substitution method
From (1)
y_1=16
From (2)
3/2y_1+y_2=11
=>(3(16))/(2)+y_2=11
=>24+y_2=11
=>y_2=11-24
=>y_2=-13
Now, Ux=y
Now use back substitution method
From (2)
-13/2y=-13
=>y=-13xx-2/13=2
From (1)
2x+5y=16
=>2x+5(2)=16
=>2x+10=16
=>2x=16-10
=>2x=6
=>x=(6)/(2)=3
Solution by LU decomposition method is
x=3 and y=2
This material is intended as a summary. Use your textbook for detail explanation.
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