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Backward error, condition numbers, and pseudospectra for the multiparameter eigenvalue problem. (English) Zbl 1048.65034

The authors define and evaluate the normwise backward error and condition numbers for the multiparameter eigenvalue problem (MEP) \((V_{i0}-\sum _{j=1}^k \lambda _j V_{ij})x_i =0 \neq x_i , \quad i=1,\dots ,k,\) where the \(V_{ij}\) are \(n_i \times n_i\) complex matrices. They define and characterize the pseudospectrum for the MEP and show that the distance from a right definite MEP to the closest non right definite MEP is related to the smallest unbounded pseudospectrum.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
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