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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.

More on Estimation Theory:

[1]An Introduction to Estimation Theory
[2]Bias of an Estimator
[3]Minimum Variance Unbiased Estimators (MVUE)
[4]Maximum Likelihood Estimation
[5]Maximum Likelihood Decoding
[6]Probability and Random Process
[7]Likelihood Function and Maximum Likelihood Estimation (MLE)
[8]Score, Fisher Information and Estimator Sensitivity
[9]Introduction to Cramer Rao Lower Bound (CRLB)
[10]Cramer Rao Lower Bound for Scalar Parameter Estimation
[11]Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE)
[12]Efficient Estimators and CRLB
[13]Cramer Rao Lower Bound for Phase Estimation
[14]Normalized CRLB - an alternate form of CRLB and its relation to estimator sensitivity
[15]Cramer Rao Lower Bound (CRLB) for Vector Parameter Estimation
[16]The Mean Square Error – Why do we use it for estimation problems
[17]How to estimate unknown parameters using Ordinary Least Squares (OLS)
[18]Essential Preliminary Matrix Algebra for Signal Processing
[19]Why Cholesky Decomposition ? A sample case:
[20]Tests for Positive Definiteness of a Matrix
[21]Solving a Triangular Matrix using Forward & Backward Substitution
[22]Cholesky Factorization and Matlab code
[23]LTI system models for random signals – AR, MA and ARMA models
[24]Comparing AR and ARMA model - minimization of squared error
[25]Yule Walker Estimation
[26]AutoCorrelation (Correlogram) and persistence – Time series analysis
[27]Linear Models - Least Squares Estimator (LSE)
[28]Best Linear Unbiased Estimator (BLUE)

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