Numerische Mathematik

, Volume 40, Issue 1, pp 137–141

Exact solution of linear equations usingP-adic expansions

  • John D. Dixon

DOI: 10.1007/BF01459082

Cite this article as:
Dixon, J.D. Numer. Math. (1982) 40: 137. doi:10.1007/BF01459082


A method is described for computing the exact rational solution to a regular systemAx=b of linear equations with integer coefficients. The method involves: (i) computing the inverse (modp) ofA for some primep; (ii) using successive refinements to compute an integer vector\(\bar x\) such that\(A\bar x \equiv b\) (modpm) for a suitably large integerm; and (iii) deducing the rational solutionx from thep-adic approximation\(\bar x\). For matricesA andb with entries of bounded size and dimensionsn×n andn×1, this method can be implemented in timeO(n3(logn)2) which is better than methods previously used.

Subject classifications

(MR 1980) AMS(MOS) 65F05, 15A06, 10M10, 10A30 CR: 5.14 

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • John D. Dixon
    • 1
  1. 1.Department of Mathematics and StatisticsCarleton UniversityOttawaCanada

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