Rational reconstruction (mathematics)

In mathematics, rational reconstruction is a method that allows one to recover a

Using a method from Paul S. Wang, it is possible to recover ${\displaystyle r/s}$ from ${\displaystyle n}$ and ${\displaystyle m}$ using the Euclidean algorithm, as follows.^[1][2]

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

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

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