Finding the eigenvalues of a Sturm–Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behavior of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. There are many approaches to the numerical solution of the Sturm–Liouville problem: finite differences and variational methods, both leading to a matrix eigenvalue problem; shooting methods using an initial value solver; and coefficient approximation methods.
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This is highly problem-dependent. For structural mechanics or calculating the stability of a construction one has to know only those eigenvalues related to first fundamental eigenmodes. On the other hand, when solving for wave propagation problems in waveguides, one has to usually solve for all eigenvalues, which correspond to propagating and evanescent (radiating or leaky) modes and all contribute to the solution. But, usually, for the latter problems, the problem size is not as large (due to the dimensional reductions and symmetries, but the matrix is denser) as for the former one (where all details in 3D are frequently needed to be taken into account, but the matrix is sparse). This leads to different techniques of the solution of such eigenproblems, delivering only some or all eienvalues.
Very often the Fourier method of separation of variables leads to representations for solutions of partial differential equations in the form of slowly converging series. Thus, in order to obtain a reasonably accurate solution one needs a considerable number of eigendata computed with a good accuracy.
In addition to the previous answers, I shall mention a method that provides rapidly and safely all eigenvalues that are less than a positive given threshold. The methods actually provides the number of eigenvalues that between 0 and this limit. With this means one can be sure not missing any eivenvalue during the exploration. Combining coarse and fine steps can then accelarate the search (instead of using only fine steps).
The method is described here:
Mikhailov, M. D., Özişik, M. N., & Vulchanov, N. L. (1983). Diffusion in composite layers with automatic solution of the eigenvalue problem. International Journal of Heat and Mass Transfer, 26(8), 1131-1141.
An application to thermal inversion (defect characterization) is here:
Krapez, J. C., & Cielo, P. (1991). Thermographic Nondestructive Evaluation: Data Inversion Procedures Part I: 1-D Analysis. Journal of Research in Nondestructive Evaluation, 3(2), 81-100.
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