Algorithm 5: Variance reduction 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:  $\mathit{z}\leftarrow \mathit{A}\mathbf{\alpha }$, $\mathit{v}\leftarrow \mathit{z}-{\left|\mathbf{\alpha }\right|}^2{\mathit{e}}_m$, $\overline{\mathit{x}}\leftarrow \mathbf{0}$, $\overline{\mathit{y}}\leftarrow \mathbf{0}$

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

3:  for while not converged do

4:    if criteria are met then

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

6:        for $t=1,\dots ,c$ do

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

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

9:           end for

10:    else

11:        $\mathit{y}\leftarrow {\mathit{A}}^2\mathit{x}$, $\overline{\mathit{x}}\leftarrow \mathit{x}$, $\overline{\mathit{y}}\leftarrow \mathit{y}$

12:    end if

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

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

15:  end for

16:  Return $\sqrt{\lambda }$