A Jacobi–Davidson type method for the product eigenvalue problem

We propose a Jacobi–Davidson type technique to compute selected eigenpairs of the product eigenvalue problem Am · · ·A1x = λx, where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi–Davidson correction equation, and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers.