Decentralized gradient algorithm for solution of a linear equation

Brian D.O. Anderson*, Shaoshuai Mou, A. Stephen Morse, Uwe Helmke

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    74 Citations (Scopus)

    Abstract

    The paper develops a technique for solving a linear equation Ax = b with a square and nonsingular matrix A, using a decentralized gradient algorithm. In the language of control theory, there are n agents, each storing at time t an n-vector, call it xi(t), and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph G with vertex set and edge set V and E respectively. We provide differential equation update laws for the xi with the property that each xi converges to the solution of the linear equation exponentially fast. The equation for xi includes additive terms weighting those xj for which vertices in G corresponding to the i-th and j-th agents are adjacent. The results are extended to the case where A is not square but has full row rank, and bounds are given on the convergence rate.

    Original languageEnglish
    Pages (from-to)319-328
    Number of pages10
    JournalNumerical Algebra, Control and Optimization
    Volume6
    Issue number3
    DOIs
    Publication statusPublished - Sept 2016

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