A system of the linear equation has a unique solution if the number of variables is equal to the number of unique equations.
You must be wondering why we need numerical analysis to solve simple equations that can be solved analytically.
Now imagine if you have about 100 equations. This means that there are 100 unique variables present in that system and if you were to solve for these then it would take a lot of time. Thus, this calls for an algorithm to solve the system of equations.
Now your system may have no solution, unique solution or infinitely many solutions. There are multiple algorithms available that take into account the limitations in the previous one.
One important thing to note is when you’re dealing with a fixed number of significant digits and the number is very small then it may lead to blowing up(i.e. large error).
Take the example of predicting the basketball winners for the year 2020.
Assume that you’ve provided with the database of all the matches played by a team in the year 2019 with parameters such as opponent team, stadium, top player, player fitness, pay, traveling time before the match, etc.
There can be N number of parameters and Z number of combinations of these lead to their victory. You notice that you have an extensive network of the system of equations. It would take a very long time for you to identify the important parameters that affect a match outcome from the redundant parameters.
Generating a general script on Matlab for this particular problem, you can plot the effect of various sets of parameters and then develop a finite number of systems of the linear equation. A similar approach is followed in the RNN machine learning algorithm.
You can train the model from the previous data to predict future outcomes. There are multiple places where a good understanding of the system of the equation comes handy.
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