  The Kaczmarz method is an iterative algorithm for solving systems of linear
equations Ax=b. Theoretical convergence rates for this algorithm were largely
unknown until recently when work was done on a randomized version of the
algorithm. It was proved that for overdetermined systems, the randomized
Kaczmarz method converges with expected exponential rate, independent of the
number of equations in the system. Here we analyze the case where the system
Ax=b is corrupted by noise, so we consider the system where Ax is approximately
b + r where r is an arbitrary error vector. We prove that in this noisy
version, the randomized method reaches an error threshold dependent on the
matrix A with the same rate as in the error-free case. We provide examples
showing our results are sharp in the general context.

Numerical Analysis;Probability