Let $A$ be an $mn$ matrix. Determine the exact numbers of floating point additions, subtractions, multiplications, and divisions involved in computing the factorization $A=QR$ by Algorithm $10\mathrm{.}1($Trefethen$).$

Algorithm $10\mathrm{.}1.$ Householder QR Factorization

$|)|v_k|{|}_2|$

$A_{k\text{:}m,k\text{:}n}=A_{k\text{:}m,k\text{:}n}-2v_k(v_k^{**}{A}_{k\text{:}m,k\text{:}n})$