Solve Triangular Matrix – Forward & Backward Substitution

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 , 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 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.

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[9]	Introduction to Cramer Rao Lower Bound (CRLB)
[10]	Cramer Rao Lower Bound for Scalar Parameter Estimation
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[18]	Essential Preliminary Matrix Algebra for Signal Processing
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[21]	Solving a Triangular Matrix using Forward & Backward Substitution
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