leave the calculations with a formal expression for the DOR.

Numerical Methods and Verification of The Eqs. (\[abdz7\], \[LargeNdx\], \[LargeNdy\])
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![Comparison of asymptotic results with numerics, for exact resonances where $N=100$, $W=0.015$ and $\eta=0.81$. In this figure, filled circles, squares and diamonds are the numerical results respectively for $\langle\,|\delta_{1}\tilde{\mathcal{Z}}_{\alpha}|^{2}\,\rangle/\sigma^{2}$, $\langle\,(\Re\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ and $\langle\,(\Im\{\delta_{1}\tilde{\mathcal{Z}}_{\alpha}\})^{2}\,\rangle/\sigma^{2}$ while open circles are the rescaled theories (\[abdz7\], \[LargeNdx\]) and (\[LargeNdy\]). In the inset we show a comparison for 14 indices near the middle of the energy band on a different scale of the plot.[]{data-label="ex_numericsvsLarge"}](ex-numericsvsLarge.eps){width="75.00000%"}

Numerical simulations for parametric resonance are always cost efficient. The reason being that there one deals with standard eigenvalue problem for which many fast subroutine packages are available, for instance LAPACK. On the other hand to verify the results (\[abdz7\], \[LargeNdx\], \[LargeNdy\]) for exact resonances, where one needs to obtain numerical solutions of a characteristic polynomial equation of order $N$ in a complex plane, there is no as good algorithm. In this paper we show results for the exact resonances by calculating roots of the characteristic polynomial where we have used a cost efficient numerical subroutine [*ezero*]{}. The subroutine is available on the CPC program library. There is one major advantage of 