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The fast multipole method (FMM) was developed by Rokhlin to solve acoustic scattering problems very efficiently. We have modified and adapted it to the second-kind-integral-equation formulation of electromagnetic scattering problems in two dimensions. The present implementation treats the exterior Dirichlet (TM) problem for two-dimensional closed conducting objects of arbitrary geometry. The FMM reduces the operation count for solving the second-kind integral equation (SKIE) from O(n 3) for Gaussian elimination to O(n 4/3) per conjugated-gradient iteration, where n is the number of sample points on the boundary of the scatterer. We also present a simple technique for accelerating convergence of the iterative method: "complexifying" k, the wavenumber. This has the effect of bounding the condition number of the discrete system; consequently, the operation count of the entire FMM (all iterations) becomes O(n 4/3). We present computational results for moderate values of ka, where a is the characteristic size of the scatterer. Comments

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The Prony method is extended to handle the nonsymmetric algebraic eigenvalue problem and improved to search automatically for the number of dominant eigenvalues. A simple iterative algorithm is given to compute the associated eigenvectors. Resolution studies using the QR method are made in order to determine the accuracy of the matrix approximation. Numerical results are given for both simple well defined resonators and more complex advanced designs containing multiple propagation geometries and misaligned mirrors.