### Abstract

A continuous-time distributed algorithm is studied for solving linear equations of the form Ax = b with at least one solution. The equation is simultaneously solved by a network of m agents with the assumption that each agent knows only a subset of the rows of the partitioned matrix [A b ], the current estimates of the equation's solution generated by its current neighbors, and nothing more. Neighbor relationships among the agents are described by a piecewise-constant switching directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships. It is shown that for any matrix-vector pair (A, b) for which the equation has a solution and any sequence of repeatedly jointly strongly connected graphs, the algorithm causes all agents' estimates to asymptotically converge to the same solution to Ax = b. The limiting behavior of the algorithm in the case when Ax = b does not have a solution is also studied. It is shown that for any static strongly connected graph, the algorithm causes all agents' estimates to asymptotically converge to different values, and therefore enables the agents to detect the no-solution case distributively.

Original language | English (US) |
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Title of host publication | 2016 American Control Conference, ACC 2016 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 5551-5556 |

Number of pages | 6 |

ISBN (Electronic) | 9781467386821 |

DOIs | |

State | Published - Jul 28 2016 |

Event | 2016 American Control Conference, ACC 2016 - Boston, United States Duration: Jul 6 2016 → Jul 8 2016 |

### Publication series

Name | Proceedings of the American Control Conference |
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Volume | 2016-July |

ISSN (Print) | 0743-1619 |

### Other

Other | 2016 American Control Conference, ACC 2016 |
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Country | United States |

City | Boston |

Period | 7/6/16 → 7/8/16 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

*2016 American Control Conference, ACC 2016*(pp. 5551-5556). [7526540] (Proceedings of the American Control Conference; Vol. 2016-July). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ACC.2016.7526540

**A continuous-time distributed algorithm for solving linear equations.** / Liu, Ji; Chen, Xudong; Baar, Tamer; Nedic, Angelia.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2016 American Control Conference, ACC 2016.*, 7526540, Proceedings of the American Control Conference, vol. 2016-July, Institute of Electrical and Electronics Engineers Inc., pp. 5551-5556, 2016 American Control Conference, ACC 2016, Boston, United States, 7/6/16. https://doi.org/10.1109/ACC.2016.7526540

}

TY - GEN

T1 - A continuous-time distributed algorithm for solving linear equations

AU - Liu, Ji

AU - Chen, Xudong

AU - Baar, Tamer

AU - Nedic, Angelia

PY - 2016/7/28

Y1 - 2016/7/28

N2 - A continuous-time distributed algorithm is studied for solving linear equations of the form Ax = b with at least one solution. The equation is simultaneously solved by a network of m agents with the assumption that each agent knows only a subset of the rows of the partitioned matrix [A b ], the current estimates of the equation's solution generated by its current neighbors, and nothing more. Neighbor relationships among the agents are described by a piecewise-constant switching directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships. It is shown that for any matrix-vector pair (A, b) for which the equation has a solution and any sequence of repeatedly jointly strongly connected graphs, the algorithm causes all agents' estimates to asymptotically converge to the same solution to Ax = b. The limiting behavior of the algorithm in the case when Ax = b does not have a solution is also studied. It is shown that for any static strongly connected graph, the algorithm causes all agents' estimates to asymptotically converge to different values, and therefore enables the agents to detect the no-solution case distributively.

AB - A continuous-time distributed algorithm is studied for solving linear equations of the form Ax = b with at least one solution. The equation is simultaneously solved by a network of m agents with the assumption that each agent knows only a subset of the rows of the partitioned matrix [A b ], the current estimates of the equation's solution generated by its current neighbors, and nothing more. Neighbor relationships among the agents are described by a piecewise-constant switching directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships. It is shown that for any matrix-vector pair (A, b) for which the equation has a solution and any sequence of repeatedly jointly strongly connected graphs, the algorithm causes all agents' estimates to asymptotically converge to the same solution to Ax = b. The limiting behavior of the algorithm in the case when Ax = b does not have a solution is also studied. It is shown that for any static strongly connected graph, the algorithm causes all agents' estimates to asymptotically converge to different values, and therefore enables the agents to detect the no-solution case distributively.

UR - http://www.scopus.com/inward/record.url?scp=84992093392&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992093392&partnerID=8YFLogxK

U2 - 10.1109/ACC.2016.7526540

DO - 10.1109/ACC.2016.7526540

M3 - Conference contribution

AN - SCOPUS:84992093392

T3 - Proceedings of the American Control Conference

SP - 5551

EP - 5556

BT - 2016 American Control Conference, ACC 2016

PB - Institute of Electrical and Electronics Engineers Inc.

ER -