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We consider algebraic iterative reconstruction methods with applications inimage reconstruction. In particular, we are concerned with methods based on anunmatched projector/backprojector pair; i.e., the backprojector is not theexact adjoint or transpose of the forward projector. Such situations are commonin large-scale computed tomography, and we consider the common situation wherethe method does not converge due to the nonsymmetry of the iteration matrix. Wepropose a modified algorithm that incorporates a small shift parameter, and wegive the conditions that guarantee convergence of this method to a fixed pointof a slightly perturbed problem. We also give perturbation bounds for thisfixed point. Moreover, we discuss how to use Krylov subspace methods toefficiently estimate the leftmost eigenvalue of a certain matrix to select aproper shift parameter. The modified algorithm is illustrated with testproblems from computed tomography.

Fixing Nonconvergence of Algebraic Iterative Reconstruction with an Unmatched Backprojector