the value of $q$ in order to obtain the magnetized equilibria. We also fix one of $\hat{\mu}_0$ and $\hat{\Omega}_0$. If we fix $\hat{\mu}_0$, we will obtain the value of $\hat{\Omega}_0$ after the relaxation and iteration. If we fix $\hat{\Omega}_0$, we will obtain $\hat{\mu}_0$. Then, we will obtain one magnetized equilibrium state.

Numerical accuracy check
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![The virial quantity VC, plotted against the number of grid points in the $r$-direction.[]{data-label="fig:VC"}](img/fig1.eps)

In order to check the accuracy of converged solutions, we compute a relative value of the virial relation as follows: $$\begin{aligned}
   \mathrm{VC} \equiv \frac{|2T + W + 3\Pi +{\cal H}|}{|W|}.\end{aligned}$$ Since this quantity VC must vanish for exact equilibrium configurations, we can check the global accuracies of the numerically obtained models as a whole (see e.g. @Hachisu_1986). Since the numerical results depend on mesh size, we have computed the same model by changing the number of grid points in the $r$-coordinate but fixing the number of grid points in the $\theta$-direction as $n_\theta = 513$. Fig. \[fig:VC\] shows VC as a function of the number of grid points in the $r$-coordinate for polytropic models. Since we use schemes of second-order accuracy, VC decreases as the square inverse of the number of 