Distributed Algorithm to Solve a System of Linear Equations With Unique or Multiple Solutions From Arbitrary Initializations

A discrete-time distributed algorithm to solve a system of linear equations <inline-formula><tex-math notation="LaTeX">$Ax=b$</tex-math></inline-formula> is proposed with <inline-formula><tex-math notation="LaTeX">$M$</tex-math></inline-formula>-Fejer mappings. The algorithm can find a solution of <inline-formula><tex-math notation="LaTeX">$Ax=b$</tex-math></inline-formula> from arbitrary initializations at a geometric rate when <inline-formula><tex-math notation="LaTeX">$Ax=b$</tex-math></inline-formula> has either unique or multiple solutions. When <inline-formula><tex-math notation="LaTeX">$Ax=b$</tex-math></inline-formula> has a unique solution, the geometric convergence rate of the algorithm is proved by analyzing the mixed norm of homogeneous <inline-formula><tex-math notation="LaTeX">$M$</tex-math></inline-formula>-Fejer mappings. Then, when <inline-formula><tex-math notation="LaTeX">$Ax=b$</tex-math></inline-formula> has multiple solutions, the geometric convergence rate is proved through orthogonal decompositions of the agents’ estimates onto the row space and null space of <inline-formula><tex-math notation="LaTeX">$A$</tex-math></inline-formula>, and the relationship between the initializations and the final convergence point is also specified. Quantitative upper bounds of the convergence rates for two special cases are given. Finally, some simulation examples are adopted to illustrate the effectiveness of the proposed algorithm.

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