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Computing a partial Schur factorization of nonlinear eigenvalue problems using the infinite Arnoldi method. (English) Zbl 1319.65040
The authors consider the nonlinear eigenvalue problem (NEP) of finding $$\lambda \in \Omega\subseteq\mathbb{C}$$ and $$v\in\mathbb{C}^n\backslash\{0\}$$ such that $$M(\lambda)v=0$$, where $$M:\Omega\leftarrow\mathbb{C}^{n\times n}$$ is analytic in $$\Omega$$, which is an open disc centered at the origin. They introduce a new technique to compute a partial Schur factorization of the NEP based on the infinite Arnoldi method (cf. [E. Jarlebring et al., Numer. Math. 122, No. 1, 169–195 (2012; Zbl 1256.65043)]). The modification applies the fact that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. Using the characterization of the structure of the invariant pairs of the operator, they show how to modify the infinite Arnoldi method by respecting this structure. Finally, they present two numerical examples to justify the usability of the main algorithm.

##### MSC:
 65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
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