Rational reconstruction (mathematics)

In mathematics, rational reconstruction is a method that allows one to recover a rational number from its value modulo an integer. If a problem with a rational solution ${\frac {r}{s}}$ is considered modulo a number m, one will obtain the number $n=r\times s^{{-1}}{\pmod {m}}$ . If |r| < N and 0 < s < D then r and s can be uniquely determined from n if m > 2ND using the Euclidean algorithm, as follows.^[1]

One puts $v=(m,0)$ and $w=(n,1)$ . One then repeats the following steps until the first component of w becomes $\leq N$ . Put $q=\left\lfloor {{\frac {v_{{1}}}{w_{{1}}}}}\right\rfloor$ , put z = v − qw. The new v and w are then obtained by putting v = w and w = z.

Then with w such that $w_{{1}}\leq N$ , one makes the second component positive by putting w = −w if $w_{{2}}<0$ . If $w_{2}<D$ and $\gcd(w_{1},w_{2})=1$ , then the fraction ${\frac {r}{s}}$ exists and $r=w_{{1}}$ and $s=w_{{2}}$ , else no such fraction exists.

References

↑ P. S. Wang, a p-adic algorithm for univariate partial fractions, Proceedings of SYMSAC ´81, ACM Press, 212 (1981); P. S. Wang, M. J. T. Guy, and J. H. Davenport, p-adic reconstruction of rational numbers, SIGSAM Bulletin 16 (1982).

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[1] P. S. Wang, a p-adic algorithm for univariate partial fractions, Proceedings of SYMSAC ´81, ACM Press, 212 (1981); P. S. Wang, M. J. T. Guy, and J. H. Davenport, p-adic reconstruction of rational numbers, SIGSAM Bulletin 16 (1982).