## 2 Theoretical Background
### Waveguides of Arbitrary Cross Section Filled by M Metallic Conductors
\begin{array}{l}\left(\mathfrak{x}_{l,p}^{(s,s)}\right)^{(+,+)}=\frac{1}{2} \Big{[}\left(h_{l,p}^{(s,s)}+h_{l,-p}^{(s,s)}\right)+\left(h_{-l,p}^{(s,s)}+h_{- l,-p}^{(s,s)}\right)\Big{]};\\ \left(\mathfrak{x}_{l,p}^{(m,s)}\right)^{(+,+)}=\frac{1}{2}\Big{[}\left(q_{l,p }^{(m,s)}+q_{l,-p}^{(m,s)}\right)+\left(q_{-l,p}^{(m,s)}+q_{-l,-p}^{(m,s)} \right)\Big{]};\\ \left(\mathfrak{x}_{l,p}^{(s,s)}\right)^{(+,-)}=\frac{1}{2}\Big{[}\left(h_{l,p }^{(s,p)}-h_{l,-p}^{(s,s)}\right)+\left(h_{-l,-p}^{(s,s)}-h_{-l,-p}^{(m,s)} \right)\Big{]};\\ \left(\mathfrak{x}_{l,p}^{(m,s)}\right)^{(+,-)}=\frac{1}{2}\Big{[}\left(q_{l,p }^{(m,s)}-q_{l,-p}^{(m,s)}\right)+\left(q_{-l,p}^{(m,s)}-q_{-l,-p}^{(m,s)} \right)\Big{]};\\ \left(\mathfrak{x}_{l,p}^{(s,s)}\right)^{(-,+)}=\frac{1}{2}\Big{[}\left(h_{l, p}^{(s,s)}+h_{l,-p}^{(s,s)}\right)-\left(h_{-l,p}^{(s,s)}+h_{-l,-p}^{(s,s)} \right)\Big{]};\\ \left(\mathfrak{x}_{l,p}^{(m,s)}\right)^{(-,+)}=\frac{1}{2}\Big{[}\left(q_{l, p}^{(m,s)}+q_{l,-p}^{(m,s)}\right)-\left(q_{-l,p}^{(m,s)}+q_{-l,-p}^{(m,s)} \right)\Big{]};\\ \left(\mathfrak{x}_{l,p}^{(s,s)}\right)^{(-,-)}=\frac{1}{2}\Big{[}\left(h_{l, p}^{(s,s)}-h_{l,-p}^{(s,s)}\right)-\left(h_{-l,p}^{(s,s)}-h_{-l,-p}^{(s,s)} \right)\Big{]};\\ \left(\mathfrak{x}_{l,p}^{(m,s)}\right)^{(-,-)}=\frac{1}{2}\Big{[}\left(q_{l, p}^{(m,s)}-q_{l,-p}^{(m,s)}\right)-\left(q_{-l,p}^{(m,s)}-q_{-l,-p}^{(m,s)} \right)\Big{]}.\end{array}  \tag{39}\]  
The unknown coefficients are  
\[ \begin{array}{l}\left(\mathfrak{x}_{l}^{(s)}\right)=\frac{1}{2\sqrt{l}} \begin{pmatrix}\mathfrak{x}_{l}^{(s)}+\mathfrak{x}_{-}^{(s)}\\ \mathfrak{x}_{l}^{(s)}-\mathfrak{x}_{-}^{(s)}\end{pmatrix};\end{array}  \tag{40}\]  
This is a Fredholm matrix system of the second kind. More precisely, in a similar way to that for a single boundary contour, one can show that the left-hand sides of ISLAE (36)-(38) define an infinite-matrix (summation) OVF \(H^{M}(k):\;I_{2}^{2M+2}\to I_{2}^{2M+2}\), \(k\in\mathcal{R}\), where \(I_{2}^{2M+2}\) is a Cartesian product of \(2M+2\) copies of spaces \(l_{2}\) and Equations (36)-(38) form a Fredholm infinite-matrix (summation) equation system of the second kind equivalent to the initial homogeneous boundary IE (30). This system, written as \(H^{M}(k)=0\), constitutes, in turn, the problem on CNs for the OVF \(H^{M}(k)\). Since \(H^{M}(k)\) is a Fredholm OVF considered in Hilbert space \(I_{2}^{2M+2}\) and its resolvent set is nonempty, its \"spectral\" set \(\sigma_{H}^{M}\subset\;\mathcal{R}\) of the sought CNS is discrete, i.e., consists of isolated points on the complex plane with the only accumulation point at infinity. Based on the equivalence, one can verify that every \(k\in\sigma_{H}^{M}\) is a CN of the initial integral OVF defined by the initial IE (30) and is a sought eigenvalue of the initial Dirichlet BVP. Some Aspects of the Numerical Implementation of the MAR and Cross-Validation with Results Obtained by Other Methods  
Inhomogeneous well-conditioned systems of the type (18)-(20) and (36)-(38), which arise as a result of employing the MAR, have been discussed in many publications (see, for example, [1; 23; 26]). Homogeneous systems possessing similar properties to those obtained in the present paper were used in [28], where the complex eigenvalues of a sound-soft elliptic cavity with a variably placed longitudinal slit were calculated. Without repeating all the arguments contained in these and other publications, we wish to discuss a few points. The first point concerns numerical methods aimed at the calculation of approximate values of CNs of infinite-matrix OVF \(H(k)\) and \(H^{M}(k)\) and a justification of this method using the theory of discrete convergence. These two issues are described in detail in [1] and summarized in Theorem 1.17 of Section 1.11.