Nonlinearrank-onemodificationofthesymmetriceigenvalueproblemarisesfromeigen-vibrationsofmechanicalstructureswithelasticallyattachedloadsandcalculationofthepropagationmodesinopticalfiber.Inthispaper,wefirststudytheexistenceanduniquenessofeigenvalues,andtheninvestigatethreenumericalalgorithms,namelyPicarditeration,nonlinearRayleighquotientiterationandsuccessivelinearapproximationmethod(SLAM).TheglobalconvergenceoftheSLAMisprovenundersomemildassumptions.NumericalexamplesillustratethattheSLAMisthemostrobustmethod.