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\(A\bar x \equiv b\) (modp m) 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(n 3(logn)2) which is better than methods previously used.
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References
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)
Khinchin, A.Ya.: Continued Fractions, 3rd ed. Chicago: Univ. Chicago Press 1961
Knuth, D.: The Art of Computer Programming, Volume 2. Reading, MA: Addison-Wesley, 1969
Krishnamurthy, E.V., Rao, T.M., Subramanian, K.:P-adic arithmetic procedures for exact matrix computations. Proc. Indian Acad. Sci.82A, 165–175 (1975)
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This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (Grant No. A 7171)
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Dixon, J.D. Exact solution of linear equations usingP-adic expansions. Numer. Math. 40, 137–141 (1982). https://doi.org/10.1007/BF01459082
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DOI: https://doi.org/10.1007/BF01459082