  We consider the class of non-Hermitian operators represented by infinite
tridiagonal matrices, selfadjoint in an indefinite inner product space with one
negative square. We approximate them with their finite truncations. Both
infinite and truncated matrices have eigenvalues of nonpositive type: either a
single one on the real axis or a couple of complex conjugate ones. As a tool to
evaluate the reliability of the use of truncations in numerical simulations, we
give bounds for the rate of convergence of their eigenvalues of nonpositive
type. Numerical examples illustrate our results.
