سوال ۱۶ کنکور ارشد ریاضی ۱۴۰۲

PASSAGE 2 (excerpt):

"Gaussian elimination, QR factorization, and most other algorithms of dense linear algebra fit the following pattern: there are $O(m)$ steps, each requiring $O(m^{2})$ work, for a total work estimate of $O(m^{3})$. (Of course these figures, especially the second, may change on a parallel computer.) For iterative methods, the same figures still apply, but now they represent a typical worst-case behavior. When these methods succeed, they may do so by reducing one or both of these factors. ...

The ideal iterative method in linear algebra reduces the number of steps from $m$ to $O(1)$ and the work per step from $O(m^{2})$ to $O(m)$, reducing the total work from $O(m^{3})$ to $O(m)$. Such extraordinary speedups do occur in practical problems, but a more typical improvement is perhaps from $O(m^{3})$ to $O(m^{2})$."

Successful iterative methods are expected to have a total work estimate ----------.