We'll use a large banded matrix to observe the speedup possible in LU factorization.
n = 8000;
A = diag(1:n) + diag(n-1:-1:1,1) + diag(ones(n-1,1),-1);
If we factor the matrix as is, MATLAB has no idea that it could be exploiting the fact that it is tridiagonal.
tic, [L,U] = lu(A); toc
But if we convert the matrix to a sparse one, the time required gets a lot smaller.
tic, [L,U] = lu(sparse(A)); toc