(f, p) Representations of $\mathcal{O}$-ideals

We assume in this chapter and the next that $\mathcal{O}$ is an order of $\mathbb{K}$ with positive discriminant Δ. As we have seen in the previous chapter, we can greatly improve the speed of determining R_Δ when we make use of the infrastructure technique of Shanks. Unfortunately, however, this requires that we compute distances, and as such quantities are logarithms of quadratic irrationals, they must be transcendental numbers.¹ This means, of course, that we cannot compute them to full accuracy but must instead be content with approximations to a fixed number of figures. When Δ is small, this is not likely to cause many difficulties, but when Δ becomes large, we have no real handle on how much round-off or truncation error might accumulate. Numerical analysts pay a great deal of attention to this problem, but, frequently, computational number theorists ignore it, hoping or believing that their techniques are sufficiently robust that serious deviations of their results from the truth will not occur. It must be admitted that this is usually what happens, but if a computational algorithm is to produce a numerical answer that is to be formally accepted as correct, it must contain within it the same aspects of rigour that one would expect within any mathematical proof. This means that we must provide provable bounds on the possible errors in our results.