Solution of Nonlinear Generalised Eigenvalue Problems
The algorithm for solving nonlinear waveguiding problems can be justified in the following way. Firstly, a comparison is made with linear problems of known analytical solutions as our eigenvalue problem is a linear one at each nonlinear iteration step. Then a check is performed to ensure that the nonlinear iterative solution method converges properly and the error accumulated during the nonlinear iteration is within an acceptable range by evaluating the residuals of the nonlinear algebraic equations for each eigenpair obtained. The choice of an efficient solution method for a given problem depends on many aspects, such as size, symmetry, positive definiteness, sparsity pattern, available RAM, and the number of required eigenvalues and eigenvectors. An algorithm is useful only when the error resulting from it is within an acceptable range. The successive overrelaxation method is one of the oldest and most successful linear iterative methods for sparse system computations.