1. Solve Equations 6x+15y+55z=76,15x+55y+225z=295,55x+225y+979z=1259 using Cholesky decomposition method

Solution:
Total Equations are `3`

`6x+15y+55z=76 -> (1)`

`15x+55y+225z=295 -> (2)`

`55x+225y+979z=1259 -> (3)`

Now converting given equations into matrix form
`[[6,15,55],[15,55,225],[55,225,979]] [[x],[y],[z]]=[[76],[295],[1259]]`

Now, A = `[[6,15,55],[15,55,225],[55,225,979]]`, X = `[[x],[y],[z]]` and B = `[[76],[295],[1259]]`

Here matrix is symmetric positive definate, so Cholesky decomposition is possible.

A matrix is positive definite if it's symmetric and all its pivots are positive.

`A`

=

`6` `15` `55` `15` `55` `225` `55` `225` `979`

Test method 1: Existence of all Positive Pivots.
First apply Gaussian Elimination method to find Pivots

`A`

=

`6` `15` `55` `15` `55` `225` `55` `225` `979`

`R_2 larr R_2-5/2xx R_1`

=	`6` `15` `55`  `0` `0=15-5/2xx6` `R_2 larr R_2-5/2xx R_1`  `35/2` `35/2=55-5/2xx15` `R_2 larr R_2-5/2xx R_1`  `175/2` `175/2=225-5/2xx55` `R_2 larr R_2-5/2xx R_1` `55` `225` `979`

`R_3 larr R_3-55/6xx R_1`

=	`6` `15` `55` `0` `35/2` `175/2`  `0` `0=55-55/6xx6` `R_3 larr R_3-55/6xx R_1`  `175/2` `175/2=225-55/6xx15` `R_3 larr R_3-55/6xx R_1`  `2849/6` `2849/6=979-55/6xx55` `R_3 larr R_3-55/6xx R_1`

`R_3 larr R_3-5xx R_2`

=	`6` `15` `55` `0` `35/2` `175/2`  `0` `0=0-5xx0` `R_3 larr R_3-5xx R_2`  `0` `0=175/2-5xx35/2` `R_3 larr R_3-5xx R_2`  `112/3` `112/3=2849/6-5xx175/2` `R_3 larr R_3-5xx R_2`

Pivots are the first non-zero element in each row of this eliminated matrix.

`:.` Pivots are `6,35/2,112/3`

Here all pivots are positive, so matrix is positive definate.

---

Test method 2: Determinants of all upper-left sub-matrices are positive.

`A`

=

`6` `15` `55` `15` `55` `225` `55` `225` `979`

`6`

`=6`

`6`   `15`   `15`   `55`

`=105`

`6`   `15`   `55`   `15`   `55`   `225`   `55`   `225`   `979`

`=3920`

Dets are `6,105,3920`

Here all determinants are positive, so matrix is positive definate.

---

Formula
`l_(ki)=(a_(ki) - sum_{j=1}^{i-1} l_(ij) * l_(kj))/(l_(ii))`

`l_(kk)=sqrt(a_(kk)-sum_{j=1}^{k-1} l_(kj)^2)`

Here `A`

=

6 15 55 15 55 225 55 225 979

`l_(11)=sqrt(a_(11))=sqrt(6)=2.4495`

`l_(21)=(a_(21))/l_(11)=(15)/(2.4495)=6.1237`

`l_(22)=sqrt(a_(22)-l_(21)^2)=sqrt(55-(6.1237)^2)=sqrt(55-37.5)=4.1833`

`l_(31)=(a_(31))/l_(11)=(55)/(2.4495)=22.4537`

`l_(32)=(a_(32)-l_(31) xx l_(21))/l_(22)=(225-(22.4537)xx(6.1237))/(4.1833)=(225-137.5)/(4.1833)=20.9165`

`l_(33)=sqrt(a_(33)-l_(31)^2-l_(32)^2)=sqrt(979-(22.4537)^2-(20.9165)^2)=sqrt(979-941.6667)=6.1101`

So `L`

=

`l_(11)` `0` `0` `l_(21)` `l_(22)` `0` `l_(31)` `l_(32)` `l_(33)`

=

2.4495 0 0 6.1237 4.1833 0 22.4537 20.9165 6.1101

`L xx L^T`

=

2.4495 0 0 6.1237 4.1833 0 22.4537 20.9165 6.1101

`xx`

2.4495 6.1237 22.4537 0 4.1833 20.9165 0 0 6.1101

=

6 15 55 15 55 225 55 225 979

and `A`

=

6 15 55 15 55 225 55 225 979

Now, `Ax=B`, and `A=LL^T => LL^Tx=B`

let `L^Tx=y`, then `Ly=B =>`

`2.4495` `0` `0` `6.1237` `4.1833` `0` `22.4537` `20.9165` `6.1101`

`xx`

`y_1` `y_2` `y_3`

=

`76` `295` `1259`

			``	`2.4495y_1`					`=`	`76`	``
			``	`6.1237y_1`	`+`	`4.1833y_2`			`=`	`295`	``
			``	`22.4537y_1`	`+`	`20.9165y_2`	`+`	`6.1101y_3`	`=`	`1259`	``

Now use forward substitution method
From (1)
`2.4495y_1=76`

`=>2.4495y_1=76`

`=>y_1=76/2.4495=31.0269`

From (2)
`6.1237y_1+4.1833y_2=295`

`=>6.1237(31.0269)+4.1833y_2=295`

`=>190+4.1833y_2=295`

`=>4.1833y_2=295-190=105`

`=>y_2=105/4.1833=25.0998`

From (3)
`22.4537y_1+20.9165y_2+6.1101y_3=1259`

`=>22.4537(31.0269)+20.9165(25.0998)+6.1101y_3=1259`

`=>1221.6667+6.1101y_3=1259`

`=>6.1101y_3=1259-1221.6667=37.3333`

`=>y_3=37.3333/6.1101=6.1101`

Now, `L^Tx=y`

`2.4495` `6.1237` `22.4537` `0` `4.1833` `20.9165` `0` `0` `6.1101`

`xx`

`x` `y` `z`

=

`31.0269` `25.0998` `6.1101`

			``	`2.4495x`	`+`	`6.1237y`	`+`	`22.4537z`	`=`	`31.0269`	``
					``	`4.1833y`	`+`	`20.9165z`	`=`	`25.0998`	``
							``	`6.1101z`	`=`	`6.1101`	``

Now use back substitution method
From (3)
`6.1101z=6.1101`

`=>z=6.1101/6.1101=1`

From (2)
`4.1833y+20.9165z=25.0998`

`=>4.1833y+20.9165(1)=25.0998`

`=>4.1833y+20.9165=25.0998`

`=>4.1833y=25.0998-20.9165=4.1833`

`=>y=4.1833/4.1833=1`

From (1)
`2.4495x+6.1237y+22.4537z=31.0269`

`=>2.4495x+6.1237(1)+22.4537(1)=31.0269`

`=>2.4495x+28.5774=31.0269`

`=>2.4495x=31.0269-28.5774=2.4495`

`=>x=2.4495/2.4495=1`

Solution by Cholesky Decomposition method is
`x=1,y=1,z=1`