I have this question: what is the cost of computing LU decomposition for a symmetric matrix.
I tried to compute it, however, I calculated it as $2n^2$ as follows: I considered the LDL decomposition, and I concluded for the $i$-th row, I need $4(i-1)$ multiplications. So at the end, I need $\sum_{i=1}^n 4(i-1)\approx2n^2$. But I saw the cost of LDL decomposition grows as $1/3n^3$.