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We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate|λk-λk,h|≤Ck⁢h2{\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}}in the case of lowest order approximation for both Galerkin and Nyström methods, where  h  is the mesh size,λk{\lambda_{k}}andλk,h{\lambda_{k,h}}are the exact and approximate  k th largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that|λk,h(G)-λk,h(N)|≤C⁢h2{|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}}, whereλk,h(G){\lambda^{(G)}_{k,h}}andλk,h(N){\lambda^{(N)}_{k,h}}denote the  k th largest eigenvalues computed by Galerkin and Nyström methods, respectively, and  C  is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.

Eigenvalue Problems for Exponential-Type Kernels