Numerische Mathematik

, Volume 40, Issue 1, pp 137–141 | Cite as

Exact solution of linear equations usingP-adic expansions

  • John D. Dixon
Article

Summary

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 

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References

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    Cabay, S., Lam, T.P.L.: Congruence techniques for the exact solution of integer systems of linear equations. ACM Trans. Math. Software3, 386–397 (1977)Google Scholar
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    Khinchin, A.Ya.: Continued Fractions, 3rd ed. Chicago: Univ. Chicago Press 1961Google Scholar
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    Knuth, D.: The Art of Computer Programming, Volume 2. Reading, MA: Addison-Wesley, 1969Google Scholar
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    Krishnamurthy, E.V., Rao, T.M., Subramanian, K.:P-adic arithmetic procedures for exact matrix computations. Proc. Indian Acad. Sci.82A, 165–175 (1975)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

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

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