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: