A continuation method for a weakly elliptic two-parameter eigenvalue problem (Q2713124)

The method of solving eigenvalue problems for real non-symmetric matrices as proposed by \textit{T. Y. Li, Z. Zeng} and \textit{L. Cong} [SIAM J. Numer. Anal. 29, No. 1, 229-248 (1992; Zbl 0749.65028)] is generalized to the two-parametric matrix pencils. The key idea remains the same (using an auxiliary parametric ``move'' to a facilitated problem where all eigenvalues are algebraically simple) but the generalization is not easy (e.g., there exist no ``easy'' curves of solutions anymore). A careful discussion is required and offered both on the theoretical level (paying attention to the cases where the matrices are large and/or where the algorithms have to be parallelized) and on the level of implementation (where, e.g., the numerical process can switch from one solution curve to another). In tests, a subtle balance is shown to exist between the number of re-calculated continuation curves and the number of operations in the initial stage.