2. LU Factorization Method :
This method is based on the fact that every square matrix can be represented as a
product of lower and upper triangular matrix; i.e. [A]=[L]*[U]
[L]=Lower triangular matrix; [U]=Upper triangular matrix
Provided all the principle minors are non singular .
Let us consider we have : 𝑎11 𝑥 + 𝑎12 𝑦 + 𝑎13 𝑧 = 𝑏1
𝑎21 𝑥 + 𝑎22 𝑦 + 𝑎23 𝑧 = 𝑏2
𝑎31 𝑥 + 𝑎32 𝑦 + 𝑎33 𝑧 = 𝑏3
the system of equation can be represented as : [A]*[X]=[B]
now matrix A can be represented as [A]=[L]*[U]
3. [A]=[L]*[U]=
𝑙11 0 0
𝑙21 𝑙22 0
𝑙31 𝑙32 𝑙33
*
1 𝑢12 𝑢13
0 1 𝑢23
0 0 1
From here we have to find the values of the elements of lower & upper triangular
matrix.
Then : [A]*[X]=[B];
[L]*[U]*[X]=[B]; As : [A]=[L]*[U]
Now we have to consider that [U]*[X]=[Y] ----------(i)
So that [L]*[Y]=[B]---------------------(ii)
Now using the (ii) equ. We have to find the elements of matrix [Y];where [L],[B]
Is known to us.
Then using the matrix [Y] in equ. (i) we can find the unknown variable matrix
[X] & get the set of solution of the given equation.
4. For example we have three equation given bellow and find the sol. By LU method :
8𝑥 − 3𝑦 + 2𝑧 = 20
4𝑥 + 11𝑦 − 𝑧 = 33
6x + 3𝑦 + 12𝑧 = 36
The given equ. Can be represented as : [A]*[X]=[B];
A=
8 −3 2
4 11 −1
6 3 12
,B=
20
33
36
,X=
𝑥
𝑦
𝑧
A=L*U;
𝑙11 0 0
𝑙21 𝑙22 0
𝑙31 𝑙32 𝑙33
*
1 𝑢12 𝑢13
0 1 𝑢23
0 0 1
=
8 −3 2
4 11 −1
6 3 12
Leading to : 𝑙11 = 8 ; 𝑙11 𝑢12 = −3 𝑢12 = −(
3
8
)