Algorithm 4: Power method with the proposed Monte Carlo AMM

Require: $\mathit{A}$:  $m\times m$ symmetric positive-semidefinite matrix, $\mathit{x}\ne \mathbf{0}$: m-dimensional randomized unit vector, $c\in \mathbb{N}$

Ensure:  $\lambda$: the largest absolute eigenvalue of $\mathit{A}$

1:  $\mathit{z}\leftarrow \mathit{A}\mathbf{\alpha }$, $\mathit{v}\leftarrow \mathit{z}-{\left|\mathbf{\alpha }\right|}^2{\mathit{e}}_m$

2:  Set $p_1,\dots ,p_m$, such that ${\sum }_{l=1}^m{p}_l=1$

3:  for while not converged do

4:    $\mathit{y}\leftarrow \left({\mathit{e}}_m^{\top }\mathit{x}\right)\mathit{v}+\left({\mathit{z}}^{\top }\mathit{x}\right){\mathit{e}}_m$

5:    for $t=1,\dots ,c$ do

6:        Pick $i_t$ with $\mathbb{P}[i_t=l]=p_l$

7:        $\mathit{y}\leftarrow \mathit{y}+{\displaystyle \frac{{\mathit{A}}^{\prime \left(i_t\right)}{\mathit{A}}_{\left(i_t\right)}^{\prime \top }\mathit{x}}{c{p}_{i_t}}}$

8:    end for

9:    $\lambda \leftarrow {\mathit{x}}^{\top }\mathit{y}$

10:    $\mathit{x}\leftarrow \mathit{y}/\parallel \mathit{y}\parallel$

11:  end for

12:  Return $\sqrt{\lambda }$