## 2 Theoretical Background
### Waveguides of Arbitrary Cross Section Filled by M Metallic Conductors
It is proved that: for every CN \(k^{\prime}\in\sigma_{H}^{M}\) there exists a sequence \(\left\{k_{N}^{(0)}\right\}_{N\geq N_{0}>1}^{\infty}\) of approximate CNs of the OVF \(H(k)\) or \(H^{M}(k)\) such that \(k_{N}^{(0)}\in\sigma_{A}^{(N)}\) and \(\left|k_{N}^{(0)}-k^{\prime}\right|\to 0\) as \(N\rightarrow\infty\), where \(\sigma_{A}^{(N)}\) denotes the spectrum of finite-dimensional operators with truncated matrix \(A_{N}=A_{N}(k)\) (a collection of approximate CNs of the OVF \(H(k)\) or \(H^{M}(k)\) corresponding to the truncation index \(N\)). On the other hand, if \(k_{N}^{(0)}\in\sigma_{A}^{(N)}\), \(N\geq N_{0}>1\), and there exists an infinite subsequence of complex numbers \(\left\{k_{N}^{(s,0)}\right\}_{N\geq N_{0}>1}^{\infty}\subset\left\{k_{N}^{(0)} \right\}_{N=0}^{\infty}\) such that \(\left|k_{N}^{(s,0)}-k^{\prime}\right|\to 0,\ N\to\infty\) then \(k^{\prime}\) is a CN of the infinite-matrix OVF \(H(k)\) or \(H^{M}(k)\). According to the definition of \"approximate\" spectrum \(\sigma_{A}^{(N)}\), the numbers \(k_{N}^{(0)}\in\sigma_{A}^{(N)}\) are zeros of the determinants \(\det A_{N}(k)\). The condition number \(\mathit{cond}\ A_{N}(k)\) of the matrix \(A_{N}(k)\) is defined to be the product \(\parallel A_{N}(k)\parallel\parallel A_{N}^{-1}(k)\parallel\) of the \(l_{2}\)-matrix norms of \(A_{N}(k)\) and its inverse; the maximal values of \(\mathit{cond}\ A_{N}(k)\) are used to localize and calculate the zeros of the determinants \(\det A_{N}(k)\). The theory elaborated in the preceding paragraph ensures that these zeros do indeed converge (as \(N\to\infty\)) to the cut-off wavenumbers sought for the waveguide. The second point concerns the structure of the matrix \(A_{N}\), which is formed by the truncation of the system to a finite size. In contrast to the Method of Moments, where the matrix elements are computed by numerical integration, the solution presented here has no need to use such a time-consuming operation since the matrix elements, derived from the Fourier coefficients \(h_{n,p}^{(s,s)}\) and \(q_{n,p}^{(m,s)}\)\(\left(-\frac{1}{2}N\leq n,p\leq\frac{1}{2}N\right)\) where \(N\) is the truncation number, are computed with the use of the Fast Fourier Transform. The third point in finding the cut-off wavenumbers (i.e., the CNs) is the role played by the spectral dependence of the condition number \(\mathit{cond}\ A_{N}(k)\) of the matrix \(A_{N}\) of the truncated algebraic system on the spectral parameter \(k\). If \(A_{\infty}(k)\) denotes the matrix associated with infinite system (36)-(38) and the infinite-matrix (summation) OVF \(H^{M}(k)\), the spectral set \(\sigma_{H}^{M}\subset\ \mathcal{R}\) of the sought CNs can be determined as roots of the dispersion equation \(\det A_{\infty}(k)=0\). (This determinant is well defined because \(A_{\infty}\) is a compact perturbation of the identity matrix and has the property that \(\det A_{N}(k)\to\det A_{\infty}(k)\) as \(N\to\infty\).) The algorithm based on the MAR, with the above justification using discrete convergence, provides fast convergence (to the exact spectral values of \(k\)) of the solutions (i.e., the approximations to spectral values, or the exact cut-off wavenumbers) of the corresponding characteristic equation \(\det A_{N}(k)=0\) of the truncated system as the truncation number \(N\) increases. We illustrate these arguments by one simple and instructive example, demonstrating how the spectrum of the cut-off wavenumbers for a hollow elliptical waveguide is transformed as its eccentricity \(e\) increases. We consider the circular waveguide (\(e=0\)) and three elliptical waveguides of increasing eccentricity: small (\(e=0.1\)), intermediate (\(e=0.5\)) and high (\(e=0.9\)); if \(a\) and \(b\) denote the major and minor semi-axes of the ellipse, the corresponding aspect ratios \(b/a\) are 0.995, 0.886 and 0.436, respectively. We restrict ourselves to consideration of the first three \(TM\)-modes: \(TM_{c01}\), \(TM_{c11}\) and \(TM_{s11}\).