My apologies if this question makes no sense. I am trying to find an algorithm that can solve a linear system of equations. Unlike most problems like this, this algorithm does not need to find a solution set that solves the entire set of equations. It only needs to solve the MOST number of these equations. For example, if a given linear system has $n$ equations, then the solution set returned by the algorithm should "fit" the most possible number of equations in the system.
Example: if there are $N$ linear equations in a system $S$ of linear equations, then the algorithm should return a solution set that solves $m$ linear equations in $S$, where $m\le N$.
Based on what I have researched, none of the algorithms I know of will do this. Hopefully I am just missing something and I can get pointed into the right direction. Thanks.
Also, I forgot to add in that the given system of equations will have 10,000-1,000,000 variables, with x being Sparse. Is there a mod 2 algorithm or something similar to it that works on very large matricies?