This is in the spirit of a previous question of mine about the efficiency of the QR algorithm. The reason for asking is that I want to motivate some students, and I'm also curious.
I do understand why LU decomposition is more efficient than Gaussian elimination, but I was wondering if it is still considered efficient for today's standards, or there are different methods that are usually prefered for applications.
So, is LU decomposition still relevant in today's applications?
Both LU decomposition and Gaussian elimination require
$$~\frac{2n^3}{3} \text{ FLOPs}$$
In fact, Gaussian elimination is LU factorization. If we start with a linear system $Ax=b$, assuming no degenerate pivots occur, we perform Gaussian elimination on it to make it look like $$ Ux=L^{-1}b $$
The reason we might want to use LU decomposition over Gaussian elimination is that once we have the LU factorization of a matrix $A$, solving linear systems involving $A$ is cheap. So, if you had $n$ linear systems to solve of the form
$$ Ax=b_{k}\text{ for }k=1,2,\ldots,n $$
this would be useful. See for example solving linear equations on Wikipedia.