My definition of band matrix: $A_{ij} = 0$ for $|i-j|>m$.
I have to count the number of additions, multiplications and divisions for the LU decomposition without pivoting of a matrix $A$ with bandwidth $m$. And I have to count this for the solution $L \cdot Ux = b$.
I have already a solution, but I don't understand it.
Solution:
(1) in the $k$th step of the elimnationpart we need $m-1,...,1$ operations for the rows $k+1,...,k+m$.
(2) So this step need $\frac{1}{2}m(m-1)$ operations.
(3) The complete eliminationspart needs $\frac{1}{2}nm(m-1)$ operations.
(4) For $L$ and $U$ we need $m$ operation for every variable.
(5) So the complete resolutionpart needs $2nm$ operations.
I hope you can help me with (1) (2) and (4). I really tried to understand this solution, but I failed. So I need a detailed explanation.