Integer Relation Detection and Lattice Reduction

By an integer relation algorithm, we mean a practical computational scheme that can recover the vector of integers ai, if it exists, or can produce bounds within which no integer relation exists. As we shall see, integer relation algorithms have a variety of interesting applications, including the recognition of a numeric constant in terms of the mathematical formula that it satisfies. The problem of finding integer relations is not new. It was first studied by Euclid, whose Euclidean algorithm solves this problem in the case n = 2. The generalization of this problem for n > 2 was attempted by Euler, Jacobi, Poincaré, Minkowski, Perron, Brun, Bernstein, among others. The first integer relation algorithm with the required properties mentioned above was discovered in 1977 by Ferguson and Forcade [18]. There is a close connection between integer lattice reduction and integer relation detection. Indeed, one common solution to the integer relation problem is to apply the Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm. However, there are some difficulties with this approach, notably the somewhat arbitrary selection of a required multiplier — if it is too small, or too large, the LLL solution will not be the desired integer relation. These difficulties were addressed in the “HJLS” algorithm [19], which is based on the LLL algorithm. Unfortunately, the HJLS algorithm suffers from numerical instability, and it fails as a result in many cases of practical interest.