Clarification on the computational complexity of $f = x + \frac{c - \left| a^* x \right|}{\left| a^* x \right|} a \left( a^* x \right)$?

Question

I am sorry for asking probably simple question regarding the computational complexity measure of the following function (I don't have a good background on these complexity measures). \begin{align} f = x + \frac{c - \left| a^* x \right|}{\left| a^* x \right|} a \left( a^* x \right)\end{align} where $()^*$ is complex conjugate transpose, $x, a \in \mathbb{C}^n$ vectors and $c \in \mathbb{R}$.

Is the computational complexity of function $f$ is $\mathcal{O}(n)$? If I am wrong, please correct me. Thank you so much in advance

Answer 1

Yes, it is $\mathcal{O}(n)$.

We first compute $a^*x$, do a scalar vector multiplication and matrix addition , each of which is $\mathcal{O}(n)$.