Iterative scheme of finding a spectrum of the product of two non-commutative operators

We consider spectral features and an iterative scheme of finding a spectrum of the product of two non-commutative partially symmetric operators in the Hilbert space $H$. In this case it is assumed that one of operators is compact, the second not necessarily being compact and even restricted in $H$. Numerical implementation of the iterative scheme for finding the operator spectrum of the problem of eigen-oscillations of the Rayleigh beam is presented.