Forward substitution: an example

To provide an example of forward substitution for solving a lower triangular system like , let’s apply the forward substitution method to a specific 4x4 system.

We start with the lower triangular matrix , the vector , and the unknown vector . In this case, becomes:

We will now apply the forward substitution steps to each row (i.e., each equation):

1. First Equation :

The first equation is simple because only appears:

2. Second Equation :

The second equation involves and :

And we can substitute from the first equation.

Third Equation :

The third equation involves , , and :

And we can substitute and from previous equations.

4. Fourth Equation :

The fourth equation involves , , , and :

And we can substitute , , and from the previous equations.

Summary of the Solution:

We now have the solution for the vector :

that can be generalised as: