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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
##### Software:
NLEVP; eigs; ARPACK; Matlab; Chebfun
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