## 2 Theoretical Background
### Waveguides with Circular Outer Conductors and Eccentric Circular Inner Conductors  
In this section, we consider the structure shown in Figure 4 of a circular waveguide of radius \(a\) enclosing an inner circular conductor of radius \(b\), electrically placed with its centre offset by a distance \(d\). We compare our results obtained by the MAR with those obtained in [3] by the classical method of separation of variables and the use of addition theorems. From the variety of the results contained in [3], we choose for cross validation the set of parameters \(b/a\ =d/a=0.25\). The spectral dependence \(cond\ A_{N}(ka)\) for this set of parameters is shown in Figure 5. It allows us to find the cut-off wavenumbers with nearly the same accuracy for both symmetric and antisymmetric modes; these are presented in [3], with four significant decimal digits of accuracy. We present magnified views of the spectral dependence in narrow intervals near the cut-off wavenumbers, belonging to a pair of symmetric modes and a pair of antisymmetric modes (Figure 6). \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(\gamma_{c01}\) & \(\gamma_{c01}\) & \(\gamma_{c11}\) & \(\gamma_{c11}\) & \(\gamma_{s11}\) & \(\gamma_{s11}\) \\ \hline
**Method** & **MAR** & [28] & **MAR** & [28] & **MAR** & [28] \\ \hline \(e=0.1\) & 2.41088987 & 2.41088987 & 3.83653893 & 3.83653893 & 3.84619062 & 3.84619061 \\ \hline \(e=0.5\) & 2.59677924 & 2.59677924 & 3.98640797 & 3.98640797 & 4.28221524 & 4.28221523 \\ \hline \(e=0.9\) & 4.21513438 & 4.21513437 & 5.41326079 & 5.41326080 & 7.76364414 & 7.76364415 \\ \hline \end{tabular}
\end{table} 
Table 2: Normalized cut-off wavenumbers \(k_{c}b\ \left(\gamma_{c(s)ml}\right)\) of an elliptic waveguide. Taking the approximate resonance values _ka_ as initial values, the root finding code produces the cut-off wavenumbers with a proven accuracy of 8 significant decimal digits (see Table 3). One observes near-perfect coincidence of the results with those of [3]; there is a tiny deviation of less than 0.0001 for both symmetric and antisymmetric modes. For the computations displayed in Figure 5, the interval \(3\leq ka\leq 8\) was subdivided into 500 intervals with step \(\Delta ka=0.001\). The total computational time was 73.036 s; for each individual value of _ka_, the time was 73.036/501 = 0.146 s. Upon doubling the truncation number \(N\) to 256, the corresponding times were 296.846 s (total) and 0.592 s (individual). The convergence of the sequence of approximations \(\gamma_{\rm col}^{(N)}\) to the wavenumber \(\gamma_{\rm col}\) of the lowest even mode \(TM_{\rm c01}\) is displayed in Table 4. The computational time associated with each truncation number is also displayed. Figure 4: Geometry of a circular waveguide with an eccentrically placed inner circular conductor. Figure 5: Spectral dependence of the condition number \(cond\ A_{N}(ka),\ N=128\ (b/a\ =\ d/a=0.25)\). Similar to that observed in Section 3.1, application of [PERSON]'s delta-squared process [31] shows that \(\gamma_{c01}^{(N)}\approx\gamma_{c01}+cN^{-p}\) for some constant \(c\) and an exponent \(p\) very close to 3; the 8 decimal place extrapolated value of 3.47226068, achieved with the values \(\gamma_{c01}^{(N)}\) computed for \(N=64,\ 128\) and \(256\), agrees, to 8 decimal places, with that achieved in the computation of \(\gamma_{c01}^{(4096)}\) with the much larger truncation number of 4096. This offers yet a good approximation of the \(\gamma_{c01}^{(N)}\).