Simulating relations Sample Clauses

Simulating relations. Before we start with the simulation, we run a short sieving test. In order to get a rep- resentative selection of the actual relations, we ensure that the points we are sieving in this test form a representative sample of the entire sieving area. The parameters for the sieving are set in such a way that we have at most two large primes both on the rational side and on the algebraic side. In the case of lattice sieving we have one additional special prime on one of the sides. In this section we describe the process of simulating relations both for line sieving and for lattice sieving. Note that we only simulate the large primes; for the primes in the factorbase we use a correction as will be explained in Section 4.3. ∈ { } The first step after the sieving test consists of splitting the relations according to the number of large primes occurring in the relation. The set of relations with i large primes on the rational side and j large primes on the algebraic side is denoted by riaj for i, j 0, 1, 2 . This leads to nine different sets and the mutual ratios of their cardinalities determine the ratios by which we will simulate the relations. In the case of lattice sieving we split the relations in the same way, treating the special prime separately. Our first experiments with simulating the large primes for the set r1a0 (and re- moving singletons) concentrated on the large primes at hand. We tried linear inter- polation between two consecutive large primes, Lagrange polynomials, and splines, but all these local approaches did not give a satisfying result: the number of relations after singleton removal was too far from the original data. We then tried a more global approach, looking at all the large primes and seeing if we could find a distribu- tion for them. We found in this case that an exponential distribution simulates best the distribution of these large primes over the interval [F, L] (cf. [7], Ch. 6) and the result after singleton removal was satisfying. The inverse of this distribution function is given by g(x) = F − a log . 1 − x . 1 − e F −L a ΣΣ , 0 ≤ x ≤ 1, (4.1) where a is the average of the large primes in the set r1a0. Note that g(0) = F and g(1) = L. In order to generate primes according to the actual distribution of the large primes, we generate a random number between 0 and 1, substitute this number in g(x), round the number g(x) to the nearest prime, and repeat this for each prime that we want to generate. To avoid expensive prime tests, we ...
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