# Repeated matrix squaring for the parallel solution of linear systems

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## Abstract

Given a *n×n* nonsingular linear system *Ax*=b, we prove that the solution x can be computed in parallel time ranging from *Ω*(log *n*) to *O*(log^{2}*n*), provided that the condition number, μ(*A*), of *A* is bounded by a polynomial in *n*. In particular, if *μ(A*) *=**O*(1), a time bound *O*(log *n*) is achieved. To obtain this result, we reduce the computation of x to repeated matrix squaring and prove that a number of steps independent of *n* is sufficient to approximate x up to a relative error 2^{−d}, *d=O*(1). This algorithm has both theoretical and practical interest, achieving the same bound of previously published parallel solvers, but being far more simple.

## Keywords

Linear System Condition Number Spectral Radius Parallel Solution Arithmetic Circuit
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