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A Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants. (English) Zbl 1349.65122

The authors study the solution of separable boundary value problems by means of a two-parameter eigenvalue problem, which is solved by a generalized eigenvalue problem with \(2\times 2\) operator determinants of the form \[ (B_1\otimes A_2-A_1\otimes B_2)z=\mu (B_1\otimes C_2-C_1\otimes B_2)z. \] Methods to compute a small subset of the eigenvalues, Arnoldi or Krylov-Schur iterations and subspace iterations based on low-rank approximations are explored. Three algorithms are presented and several numerical examples illustrate the work, furthermore a Matlab implementation are also provided.

MSC:

65F10 Iterative numerical methods for linear systems
15A18 Eigenvalues, singular values, and eigenvectors
15A24 Matrix equations and identities
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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