BIT Numerical Mathematics

, Volume 50, Issue 2, pp 395–403

Randomized Kaczmarz solver for noisy linear systems


DOI: 10.1007/s10543-010-0265-5

Cite this article as:
Needell, D. Bit Numer Math (2010) 50: 395. doi:10.1007/s10543-010-0265-5


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 Axb+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.


Randomized algorithms Kaczmarz method Algebraic reconstruction technique 

Mathematics Subject Classification (2000)

65F10 65F20 65F22 

Copyright information

© Springer Science + Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of StatisticsStanford UniversityStanfordUSA

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