Why is efficency of Gaussian Elimination $O(n^3)$? See Big O Notations.


Wikipedia states that it is $O(n^3)$ because it requires:

  • $\frac{n(n+1)}{2}$ divisions
  • $\frac{2n^3 + 3n^2 − 5n}{6}$ multiplications
  • $\frac{2n^3 + 3n^2 − 5n}{6}$ substractions

i.e. an average of $\frac{2n^3}{3}$ operations. And the proof is in Farebrother, R.W. (1988), Linear Least Squares Computations, which I was not able to find for free on the internet.

So here is my interpretation of the result:

  • $n$ equations with $n$ variables so it requires $n$ pivots -> $O(n)$
  • for each pivot we substract a multiple of row from each other row - >$O(n^2)$
  • Total runtime is $O(n^2 n) =O(n^3)$

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