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.


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