The quadratic eigenvalue problem. (English) Zbl 0985.65028

Tisseur, Françoise; Meerbergen, Karl

SIAM Rev. 43, No. 2, 235-286 (2001).

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.

Reviewer: Alan L.Andrew (Bundoora)

MSC:

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

Keywords:

quadratic eigenvalue problem; eigenvalue; eigenvector; \(\lambda\)-matrix; matrix polynomial; second-order differential equations; damped vibrations; overdamped system; gyroscopic system; linearization; conditioning; pseudospectrum; Krylov method; Arnoldi method; Lanczos method; Jacobi-Davidson method; millenium footbridge