Simple estimator for biggest singular value

Question

I was reading a paper where they used $\sqrt{\|A\|_\infty \|A\|_1 }$ as an estimate of the biggest singular value $\sigma_{max}(A)$ of a non-singular matrix $A \in \mathbb{R}^{n \times n}$.

However, this claim is not justified, and I cannot understand the reasoning behind it.

Recall that $\|A\|_{\infty}$ is defined as $\text{max}_{1 \leq i \leq n} \sum_j^{n} |a_{ij}|$ while $\|A\|_{1}$ is defined as $\text{max}_{1 \leq j\leq n} \sum_i^{n} |a_{ij}|$.

Answer 1

In algorithm 2, $\sigma_\max$ is an upperbound for the singular value.

To obtain an upper bound for the singular values:

$$\rho(A^TA)\le \|A^TA\|_1=\|A^T\|_1\|A\|_1 = \|A\|_\infty \|A\|_1$$