1. Solve Equations 2x+5y=21,x+2y=8 using Doolittle's 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]]`

Doolittle's method for LU decomposition
Let `A=LU`

`2` `5` `1` `2`

=

`1` `0` `l_(21)` `1`

`xx`

`u_(11)` `u_(12)` `0` `u_(22)`

`2` `5` `1` `2`

=

`u_(11)` `u_(12)` `l_(21)u_(11)` `l_(21)u_(12) + u_(22)`

This implies
`u_(11)=2`

`u_(12)=5`



`l_(21)u_(11)=1=>l_(21)xx2=1=>l_(21)=1/2`

`l_(21)u_(12) + u_(22)=2=>1/2xx5 + u_(22)=2=>u_(22)=-1/2`



`:.A=L xx U=LU`

`2` `5` `1` `2`

=

`1` `0` `1/2` `1`

`xx`

`2` `5` `0` `-1/2`

=

`2` `5` `1` `2`

---

Now, `Ax=B`, and `A=LU => LUx=B`

let `Ux=y`, then `Ly=B =>`

`1` `0` `1/2` `1`

`xx`

`y_1` `y_2`

=

`21` `8`

			``	`y_1`			`=`	`21`	``
			``	`1/2y_1`	`+`	`y_2`	`=`	`8`	``

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`

`2` `5` `0` `-1/2`

`xx`

`x` `y`

=

`21` `-5/2`

			``	`2x`	`+`	`5y`	`=`	`21`	``
					`-`	`1/2y`	`=`	`-5/2`	``

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 Doolittle's method is
`x=-2 and y=5`