I have a problem where I am trying to solve many systems of equations, that have very few variables per equation, but a lot of equations. For example potentially 10 variables max in a single equation, but a million equations. A lot of them are trivial in finding a solution, but some of the larger equations are difficult to solve.

I have a naive algorithm in mind to simply solve the trivial equations first, and hope those solutions can then just translate into solving more and more etc, until the more complicated problems are completely solved. Is this a solid initial approach? Is there a different way of thinking about this type of problem that I do not know about? I have tried looking through this SE and similar but I tend to get confused once people start mentioning sparse matrices verse dense ones and things such as that. If someone could point me in the right direction that would be awesome.

  • 2
    $\begingroup$ Are the equations linear? If you treat the whole set of equations (over all the variables) as one big matrix of coefficients, is it square or rectangular? $\endgroup$
    – Bill Barth
    May 26, 2015 at 16:17
  • $\begingroup$ when you ask if it is linear, do you mean linear as opposed to quadratic etc? if so yes it is linear. If it were to be treated as a singular matrix it is definitely rectangular by orders of magnitude. $\endgroup$
    – mpwaka
    May 26, 2015 at 16:47
  • $\begingroup$ Linear programming refers to optimization problems involving both linear equations and linear inequalities. If your problem involves only equations, it would not make sense to use linear programming solvers. $\endgroup$ May 27, 2015 at 18:34
  • $\begingroup$ yea sorry i got my terminology backwards. I think I need to use something like this: codeproject.com/Tips/388179/… that way I can try to solve my system of equations, and if I find a non complete answer, then I can ignore it $\endgroup$
    – mpwaka
    May 28, 2015 at 19:45

1 Answer 1


If your equations are linear you have a sparse linear system of equations, and you'd read some of the standard material about these (e.g., the book by Duff, Erisman, Reid, or the book Matrix computations by Golub and van Loan).

If your equations are nonlinear you have a sparse nonlinear system of equations, and you'd use a damped Newton method to solve it iteratively.


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