Key focus: Know the expressions to solve triangular matrix using forward and backward substituting techniques and the FLOPS required for solving it.
Forward Substitution:
Consider a set of equations in a matrix form $latex 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 expressions
From the DSP implementation point of view, computation of $latex x_1 $ requires one FLoating Point Operation per Second (FLOPS) – only one division. Computing $latex x_2 $ will require 3 FLOPS – 1 multiplication, 1 division and 1 subtraction, $latex x_3 $ will require 5 FLOPS – 2 multiplications, 1 division and two subtractions. Thus the computation of $latex x_{mm} $ will require $latex (2n-1) $ FLOPS.
Thus the overall FLOPS required for forward substitution is $latex 1+3+5+\cdots+(2m-1)$ $latex = m^2 $ FLOPS
Backward substitution:
Consider a set of equations in a matrix form $latex 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 $latex m^2 $ FLOPS.
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