  We consider algebraic iterative reconstruction methods with applications in
image reconstruction. In particular, we are concerned with methods based on an
unmatched projector/backprojector pair; i.e., the backprojector is not the
exact adjoint or transpose of the forward projector. Such situations are common
in large-scale computed tomography, and we consider the common situation where
the method does not converge due to the nonsymmetry of the iteration matrix. We
propose a modified algorithm that incorporates a small shift parameter, and we
give the conditions that guarantee convergence of this method to a fixed point
of a slightly perturbed problem. We also give perturbation bounds for this
fixed point. Moreover, we discuss how to use Krylov subspace methods to
efficiently estimate the leftmost eigenvalue of a certain matrix to select a
proper shift parameter. The modified algorithm is illustrated with test
problems from computed tomography.
