Numerical techniques for computing the inertia of products of matrices of rational numbers
Proceedings of the 2007 international workshop on Symbolic-numeric computation
A Jacobi-Davidson type method for the product eigenvalue problem
Journal of Computational and Applied Mathematics
The Rayleigh-Ritz method, refinement and Arnoldi process for periodic matrix pairs
Journal of Computational and Applied Mathematics
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Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix $A$ are wanted, but $A$ is not given explicitly. Instead it is presented as a product of several factors: $A = A_{k}A_{k-1}\cdots A_{1}$. Usually more accurate results are obtained by working with the factors rather than forming $A$ explicitly. For example, if we want eigenvalues/vectors of $B^{T}B$, it is better to work directly with $B$ and not compute the product. The intent of this paper is to demonstrate that the product eigenvalue problem is a powerful unifying concept. Diverse examples of eigenvalue problems are discussed and formulated as product eigenvalue problems. For all but a couple of these examples it is shown that the standard algorithms for solving them are instances of a generic $GR$ algorithm applied to a related cyclic matrix.