  Magnetic resonance electrical impedance tomography (MREIT) aims to recover
the electrical conductivity distribution of an object using partial information
of magnetic flux densities inside the tissue which can be measured using an MRI
scanner, with the advantage that a higher spatial resolution of conductivity
image can be provided than existing EIT techniques involving surface
measurements. Traditional MREIT reconstruction algorithms use two data sets
obtained with two linearly independent injected currents. However, injection of
two currents is often not possible in applications such as transcranial
electrical stimulation. Recently, we proposed an iterative conductivity
reconstruction algorithm called the single current harmonic $B_z$ algorithm
that demonstrated satisfactory performance in numerical and phantom tests. In
this paper, we provide a rigorous mathematical analysis of the convergence of
the iterative sequence for realizing this algorithm. We prove that, applying
some mild conditions on the exact conductivity, the iterative sequence
converges to the true solution within an explicit error bound. Such theoretical
results substantiate the reasonability and efficiency of the proposed
algorithm. We also provide more numerical evidence to validate these
theoretical results.
