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RatReconstructByContFrac, RatReconstructByLattice

rational reconstruction from modular image
RatReconstructByContFrac(X: INT, M: INT): RECORD
RatReconstructByContFrac(X: INT, M: INT, threshold: INT): RECORD
RatReconstructByLattice(X: INT, M: INT): RECORD
RatReconstructByLattice(X: INT, M: INT, threshold: INT): RECORD 
These functions attempt to reconstruct rational numbers from a modular
image
X mod M
.
The algorithms are
faulttolerant: they will succeed provided that
X
is correct modulo a sufficiently large factor of
M
.
The result is a record: the boolean field
failed
is
true
if no
convincing result was found; otherwise it is
false
,
and a second field, called
ReconstructedRat
, contains the
value reconstructed.
An optional third argument,
threshold
, determines what
convincing
means: a higher value gives a more reliable answer, but may need a larger
modulus before the answer is found.
There are two different underlying heuristic algorithms: a faster one
based on continued fractions, and a slower one based on 2dimensional
lattice reduction. See the JSC paper by John Abbott:
"Faulttolerant modular reconstruction of rational numbers",
http://www.sciencedirect.com/science/article/pii/S0747717116300773
;
a nearfinal version is at
http://arxiv.org/abs/1303.2965
.
NOTE: so that the heuristic can work, the modulus must be a bit larger
than strictly necessary; indeed, reconstruction will always fail if
M
is too small.
/**/ X := 3333333333;
/**/ M := 10^10;
/**/ RatReconstructByContFrac(X,M);
record[ReconstructedRat := 1/3, failed := false]
/**/ X := 3141592654;
/**/ M := 10^10;
/**/ RatReconstructByContFrac(X,M);
record[failed := true]
