### Abstract

In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form Ax=b assuming that the equation has at least one solution. The equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix Ab, the current estimates of the equation's solution generated by each of its neighbors, and nothing more. Neighbor relationships are represented by a time-dependent directed graph N(t) whose vertices correspond to agents and whose arcs characterize neighbor relationships. Sufficient conditions on N(t) were derived under which the algorithm can cause all agents’ estimates to converge exponentially fast to the same solution to Ax=b. These conditions were also shown to be necessary for exponential convergence, provided the data about Ab available to the agents is “non-redundant”. The aim of this paper is to relax this “non-redundant” assumption. This is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand; the conditions are based on a new notion of graph connectivity. An improved bound on the convergence rate is also derived.

Original language | English (US) |
---|---|

Pages (from-to) | 37-46 |

Number of pages | 10 |

Journal | Automatica |

Volume | 83 |

DOIs | |

State | Published - Sep 1 2017 |

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### ASJC Scopus subject areas

- Control and Systems Engineering
- Electrical and Electronic Engineering

### Cite this

*Automatica*,

*83*, 37-46. https://doi.org/10.1016/j.automatica.2017.05.004

**Exponential convergence of a distributed algorithm for solving linear algebraic equations.** / Liu, Ji; Stephen Morse, A.; Nedich, Angelia; Başar, Tamer.

Research output: Contribution to journal › Article

*Automatica*, vol. 83, pp. 37-46. https://doi.org/10.1016/j.automatica.2017.05.004

}

TY - JOUR

T1 - Exponential convergence of a distributed algorithm for solving linear algebraic equations

AU - Liu, Ji

AU - Stephen Morse, A.

AU - Nedich, Angelia

AU - Başar, Tamer

PY - 2017/9/1

Y1 - 2017/9/1

N2 - In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form Ax=b assuming that the equation has at least one solution. The equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix Ab, the current estimates of the equation's solution generated by each of its neighbors, and nothing more. Neighbor relationships are represented by a time-dependent directed graph N(t) whose vertices correspond to agents and whose arcs characterize neighbor relationships. Sufficient conditions on N(t) were derived under which the algorithm can cause all agents’ estimates to converge exponentially fast to the same solution to Ax=b. These conditions were also shown to be necessary for exponential convergence, provided the data about Ab available to the agents is “non-redundant”. The aim of this paper is to relax this “non-redundant” assumption. This is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand; the conditions are based on a new notion of graph connectivity. An improved bound on the convergence rate is also derived.

AB - In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form Ax=b assuming that the equation has at least one solution. The equation is presumed to be solved by m agents assuming that each agent knows a subset of the rows of the matrix Ab, the current estimates of the equation's solution generated by each of its neighbors, and nothing more. Neighbor relationships are represented by a time-dependent directed graph N(t) whose vertices correspond to agents and whose arcs characterize neighbor relationships. Sufficient conditions on N(t) were derived under which the algorithm can cause all agents’ estimates to converge exponentially fast to the same solution to Ax=b. These conditions were also shown to be necessary for exponential convergence, provided the data about Ab available to the agents is “non-redundant”. The aim of this paper is to relax this “non-redundant” assumption. This is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand; the conditions are based on a new notion of graph connectivity. An improved bound on the convergence rate is also derived.

UR - http://www.scopus.com/inward/record.url?scp=85019967959&partnerID=8YFLogxK

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U2 - 10.1016/j.automatica.2017.05.004

DO - 10.1016/j.automatica.2017.05.004

M3 - Article

VL - 83

SP - 37

EP - 46

JO - Automatica

JF - Automatica

SN - 0005-1098

ER -