The iterative process converges for a large class of smooth initial guesses[^2]. Convergence is attained when the $L_\infty$ norm between two successive iterations is less than a prescribed tolerance $\varepsilon$ (usually of the order of machine precision). Additionally, the residual error $E_r$ in approximating the nonlinear equation is checked by substituting in the converged numerical solution. In the present work, the convergence of the algorithm is checked numerically in the extended floating point arithmetics using $30$ significant digits [@MATLAB2012].

For example, consider the domain $\Omega := [-\ell_x/2, \ell_x/2] \times [-\ell_y/2, \ell_y/2] \subset {\mathds{R}}^2$ to be a square with the side length equal to $210$ ($\Omega = [-105, 105]^2$). For $\omega = 0.012$, the Petviashvili method on a $1024 \times 1024$ Fourier grid yields the strictly periodic regular pattern shown in Figure \[fig:f0\](a). The convergence is attained in $N\sim 80$ iterations as clearly seen in Figure \[fig:f0\](b) and the associated error $L_\infty \sim {\mathcal{O}}(10^{-33})$, and the residual $E_r \sim {\mathcal{O}}(10^{-32})$. Note that $L_\infty \sim~{\mathrm{e}}^{-0.095 N}$ decays exponentially in agreement with the theoretical geometric convergence rate [@Alvarez2014]. For the sake of efficiency, the numerical solutions presented below are computed on the same Fourier grid using standard double-precision arithmetics. It is verified that 