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A linear eigenvalue algorithm for the nonlinear eigenvalue problem. (English) Zbl 1256.65043
A nonlinear matrix eigenvalue problem (NMEP) \(T(\lambda)x=0\) is transformed without loss of generality into a standard form \(\lambda B(\lambda)x=x\) (\(T\) and \(B\) analytic in \(\Omega\subset\mathbb{C}\)). This is then transformed into a linear operator eigenvalue problem (LOEP) of the form \(\lambda\mathcal{B}\varphi=\varphi\) (\(\varphi\in C_\infty(\mathbb{R},\mathbb{C}^n)\)). The eigenvalues of \(\mathcal{B}\) in LOEP are the reciprocals of the eigenvalues of \(B\) in the NMEP and the eigenfunction \(\varphi\) in LOEP is related to the NMEP eigenvector \(x\) by \(\varphi(\theta)=xe^{\lambda \theta}\). The classical Arnoldi method is translated in the operator terminology, hence keeping all its nice properties. Because of the definition of \(\mathcal{B}\), starting the Krylov sequence with a constant results in a sequence of (vector) polynomials. Defining an appropriate inner product allows for an efficient implementation. The generated polynomials can be expressed as a linear combination of either monomials or Chebyshev polynomials. Both of these are discussed and presented in algorithmic form and applied to numerical examples like a delay eigenvalue problem with quadratic term or an eigenvalue problem involving square roots.

MSC:
65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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