Meaning of the cost of constructing a matrix

Question

In a text I saw written, $``$The dominant costs in this method are the construction of the Jacobian matrix $\textbf{f}_\textbf{x}(\textbf{x}_n)$ (typically $\mathcal{O}(m^2)$) and solution of the linear system (typically $\mathcal{O}(m^3)$). What do the $``$orders of cost" mean? The text is not so clear about them. The method is the Newton-Raphson method. The linear system talked about here is $\textbf{f}_\textbf{x}(\textbf{x}_n)\Delta \textbf{x}_\textbf{n}=\textbf{f}(\textbf{x}_n)$ where $\Delta\textbf{x}_\textbf{n}=\textbf{x}_n-\textbf{x}_{n+1}$

Answer 1

Order of cost means the number of elementary operations. Typically the number of additions and multiplications.

For example $\mathcal O(m³)$ means that there is a constant $M >0$ such that the number of required operations for the algorithm is around $M m³$.