A Kaczmarz Algorithm for Solving Tree Based Distributed Systems of Equations

Chinmay Hegde, Fritz Keinert, Eric S. Weber

    Research output: Chapter in Book/Report/Conference proceedingChapter


    The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a modified Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e., the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We prove that the modified algorithm converges under no additional assumptions on the equations. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also demonstrate that in the case of an inconsistent system of equations, the modified relaxed Kaczmarz algorithm converges to a weighted least-squares solution as the relaxation parameter approaches 0.

    Original languageEnglish (US)
    Title of host publicationApplied and Numerical Harmonic Analysis
    Number of pages27
    StatePublished - 2021

    Publication series

    NameApplied and Numerical Harmonic Analysis
    ISSN (Print)2296-5009
    ISSN (Electronic)2296-5017

    ASJC Scopus subject areas

    • Applied Mathematics


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