1. Solve Equations 2x+5y=21,x+2y=8 using LU decomposition using Gauss Elimination method
Solution:
Total Equations are 2
2x+5y=21 -> (1)
x+2y=8 -> (2)
Now converting given equations into matrix form
[[2,5],[1,2]] [[x],[y]]=[[21],[8]]
Now, A = [[2,5],[1,2]], X = [[x],[y]] and B = [[21],[8]]
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-(1/2)xx R_1 [:.L_(2,1)=color{blue}{1/2}]
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=21
From (2)
1/2y_1+y_2=8
=>((21))/(2)+y_2=8
=>21/2+y_2=8
=>y_2=8-21/2
=>y_2=-5/2
Now, Ux=y
Now use back substitution method
From (2)
-1/2y=-5/2
=>y=-5/2xx-2=5
From (1)
2x+5y=21
=>2x+5(5)=21
=>2x+25=21
=>2x=21-25
=>2x=-4
=>x=(-4)/(2)=-2
Solution by LU decomposition method is
x=-2 and y=5
This material is intended as a summary. Use your textbook for detail explanation.
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