Expert: Paul Klarreich - 2/5/2007

Question
Both Gaussian Elimination and Cramer's Rule are used for solving linear systems. If you need to solve large systems with more than three variables, which method is more efficient and why?

Answer
Questioner:   Chad Chase
Category:  Advanced Math
Private:  no
 
Subject:  College math
Question:  Both Gaussian Elimination and Cramer's Rule are used for solving linear systems. If you need to solve large systems with more than three variables, which method is more efficient and why?
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Hi, Chad,

You have to tell me exactly what you mean by 'more efficient'.  In the 'algorithmic' sense, I believe that both methods are considered to be O(n^3) in 'complexity'.

That's a fancy way of saying that, given that the system has 'n' equations in 'n' variables, the time needed is proportional to  n^3.  [The constant of proportionality would depend on lots of things, of course.]

There are, however, some sophisticated methods for computing determinants that have complexity that is lower than O(n^3), so if one of those is used, Cramer's rule becomes more efficient.

HOWEVER, this really only matters if you are talking about methods of solving linear systems with a few thousand variables.  If all you want is to find a simple way to do a small system and, perhaps, to program it, then Gaussian elimination is fairly simple to work out.