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Computing all pairs \((\lambda , \mu )\) such that \(\lambda \) is a double eigenvalue of \(A+\mu B\). (English) Zbl 1245.65043
The authors propose, analyze and test a method for the accurate numerical computation of the set of all scalar pairs \((\lambda ,\mu )\) such that \(\lambda\) is a multiple eigenvalue of \(A+\mu B\), where \(A\) and \(B\) are arbitrary given complex \(n\times n\) matrices. Emphasis is on the generic case in which \(\lambda\) is a double eigenvalue and the number of such pairs is finite but non-zero. First a global method is used to obtain approximations of all solutions by finding pairs \((\lambda ,\mu )\) for which both \(\lambda\) and \((1+\varepsilon )\lambda\) are eigenvalues, for some carefully chosen small \(\varepsilon\). Then a quadratically convergent iterative method (with different forms for the semisimple and non-semisimple cases) is used to refine the approximations.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A22 Matrix pencils
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