# Solving a Triangular Matrix using Forward & Backward Substitution

## Forward Substitution:

Consider a set of equations in a matrix form , 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 requires one FLoating Point Operation per Second (FLOPS) – only one division. Computing will require 3 FLOPS – 1 multiplication, 1 division and 1 subtraction, will require 5 FLOPS – 2 multiplications, 1 division and two subtractions. Thus the computation of will require FLOPS.

Thus the overall FLOPS required for forward substitution is FLOPS

## Backward substitution:

Consider a set of equations in a matrix form , 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 FLOPS.