The quadratic eigenvalue problem. (English) Zbl 0985.65028

The authors review current knowledge of the matrix quadratic eigenvalue problem, \[ (\lambda ^2 M+\lambda C+K)x=0, \qquad y^* (\lambda ^2 M+\lambda C+K)=0, \tag{1} \] including its main applications and its numerical solution, and give an excellent guide to the literature. They give a good introduction to the theory, with emphasis on those parts important for the numerical solution of (1), including results on pseudospectra. Properties of the matrices \(M,\;C\) and \(K\) arising in various applications, and the relevance of these properties for the choice of numerical method, are also considered. The rest of the paper deals with specific methods for the numerical solution of (1), with detailed discussion of the relative advantages of alternative methods for problems with particular structure, such as sparsity or various types of symmetry. Available software is catalogued and extensions to related problems (such as problems with more general nonlinear dependence on the eigenparameter) are mentioned.


65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65F30 Other matrix algorithms (MSC2010)
15A22 Matrix pencils
65G50 Roundoff error
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