  Purpose: We analyze a recently proposed polyenergetic version of the
simultaneous algebraic reconstruction technique (SART). This algorithm, denoted
pSART, replaces the monoenergetic forward projection operation used by SART
with a post-log, polyenergetic forward projection, while leaving the rest of
the algorithm unchanged. While the proposed algorithm provides good results
empirically, convergence of the algorithm was not established mathematically in
the original paper.
  Methods: We analyze pSART as a nonlinear fixed point iteration by explicitly
computing the Jacobian of the iteration. A necessary condition for convergence
is that the spectral radius of the Jacobian, evaluated at the fixed point, is
less than one. A short proof of convergence for SART is also provided as a
basis for comparison.
  Results: We show that the pSART algorithm is not guaranteed to converge, in
general. The Jacobian of the iteration depends on several factors, including
the system matrix and how one models the energy dependence of the linear
attenuation coefficient. We provide a simple numerical example that shows that
the spectral radius of the Jacobian matrix is not guaranteed to be less than
one. A second set of numerical experiments using realistic CT system matrices,
however, indicates that conditions for convergence are likely to be satisfied
in practice.
  Conclusion: Although pSART is not mathematically guaranteed to converge, our
numerical experiments indicate that it will tend to converge at roughly the
same rate as SART for system matrices of the type encountered in CT imaging.
Thus we conclude that the algorithm is still a useful method for reconstruction
of polyenergetic CT data.
