The first practical algorithm for determining if there exist integers 
for given real numbers 
such that
 |
or else establish bounds within which no such integer relation can exist (Ferguson and Forcade 1979). The algorithm therefore became
the first viable generalization of the Euclidean
algorithm to 
variables.
A nonrecursive variant of the original algorithm was subsequently devised by Ferguson (1987). The Ferguson-Forcade algorithm has been shown to be polynomial time in the logarithm in the size of a smallest relation, but has not been shown to be polynomial in dimension (Ferguson et al. 1999).
, 
, and Euler's Constant." Math. Comput. 50,
275-281, 1988.Bergman, G. "Notes on Ferguson and Forcade's Generalized
Euclidean Algorithm." Unpublished notes. Berkeley, CA: University of California
at Berkeley, Nov. 1980.Ferguson, H. R. P. "A Short Proof
of the Existence of Vector Euclidean Algorithms." Proc. Amer. Math. Soc. 97,
8-10, 1986.Ferguson, H. R. P. "A Non-Inductive GL(
)
Algorithm that Constructs Linear Relations for 
-Linearly Dependent Real Numbers." J. Algorithms 8,
131-145, 1987.Ferguson, H. R. P.; Bailey, D. H.; and
Arno, S. "Analysis of PSLQ, An Integer Relation Finding Algorithm." Math.
Comput. 68, 351-369, 1999.Ferguson, H. R. P. and
Forcade, R. W. "Generalization of the Euclidean Algorithm for Real Numbers
to All Dimensions Higher than Two." Bull. Amer. Math. Soc. 1,
912-914, 1979.Ferguson, H. R. P. and Forcade, R. W. "Multidimensional
Euclidean Algorithms." J. reine angew. Math. 334, 171-181, 1982.Gibbs,
W. W. "A Digital Slice of Pi. The New Way to do Pure Math: Experimentally."
Sci. Amer. 288, 23-24, May 2003.