## Forward Substitution:

Consider a set of equations in a matrix form \(Ax=b \), where A is a lower triangular matrix with non-zero diagonal elements. The equation is re-written in full matrix form as

It can be solved using the following algorithm

From the DSP implementation point of view, computation of \(x_1 \) requires one FLoating Point Operation per Second (FLOPS) – only one division. Computing \(x_2 \) will require 3 FLOPS – 1 multiplication, 1 division and 1 subtraction, \( x_3 \) will require 5 FLOPS – 2 multiplications, 1 division and two subtractions. Thus the computation of \(x_{mm} \) will require \((2n-1) \) FLOPS.

Thus the overall FLOPS required for forward substitution is \(1+3+5+\cdots+(2m-1)\) \(= m^2 \) FLOPS

## Backward substitution:

Consider a set of equations in a matrix form \(Ax=b \), where A is a upper triangular matrix with non-zero diagonal elements. The equation is re-written in full matrix form as

Solved using the following algorithm

This one also requires \(m^2 \) FLOPS.