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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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