Example 2

Use LU decomposition method to solve the following simultaneous linear equations.

Solution

Recall that

and if

then first solving

and then

gives the solution vector .

Now in the previous example, we showed

First solve

to give

z₁ = 106.8

2.56z₁ +z₂ = 177.2

5.76 z₁ + 3.5 z₂ + z₃ = 279.2

Forward substitution starting from the first equation gives

z₁ =106.8

z₂ = 177.2 – 2.56 z₁

= 177.2 – 2.56 (106.8)

= -96.21

z₃ = 279.2 – 5.76 z₁ – 3.5z₂

= 279.2 – 5.76(106.8) – 3.5(-96.21)

= 0.735

Hence

This matrix is same as the right hand side obtained at the end of the forward elimination steps of Naïve Gauss elimination method. Is this a coincidence?

Now solve

From the third equation

Substituting the value of a₃ in the second equation,

Substituting the value ofin the first equation,

The solution vector is