Edited by: F. Magoulès

Keywords: algebraic eigenvalue problems, generalised eigenvalue problem, quadratic eigenvalue problem, Lanczos method, Arnoldi method, spectral transformation.

This chapter reviews the theory on the algebraic eigenvalue problem, and in particular, the theory on the linear definite generalised eigenvalue problem (stiffness-mass), and the quadratic eigenvalue problem (stiffness-damping-mass). Numerical methods are presented for solving large scale problems, where the focus is on Krylov methods: Lanczos and Arnoldi, and the spectral transformation. Important notions such as inertia and sparse LDL^T factorisation are also touched on. Examples are included to illustrate the theory, as well as a bibliographical note with references to techniques other than those discussed in this chapter.

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