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Several strategies for solving a nonlinear eigenvalue problem are evaluated. This problem stems from the boundary integral equation solution of propagation in photonic crystal fibers. The origin and specificities of the eigenvalue problem are recalled before considering the solution of this eigenvalue problem. The first strategy, which is the starting point to illustrate the difficulties, is to solve the problem using Muller's method. We then look at more recent techniques based on contour integrals or a rational interpolant that can be used to compute several eigenmodes simultaneously and considerably reduce the volume of computations. [ABSTRACT FROM AUTHOR]