# Throwaway Digits

Tomorrow, my PhD student He Li will present our paper Digit Elision for Arbitrary-accuracy Iterative Computation (joint work with James Davis and John Wickerson) at the IEEE Symposium on Computer Arithmetic in Amherst, MA.

Readers of this blog may remember that we previously came up with a neat way of computing arbitrarily precise values of arbitrarily deep iterations of an iterative real-number computation, while only using constant-area compute hardware. This latest paper extends our previous work in the following way.

In our previous work, we computed every digit of every iteration of the computation. While for any computable real function this will give a correct result, it tends to be wasteful in practice. There are two reasons it’s wasteful. Firstly, often the reason we’re computing an iteration is because that iteration converges. Convergence can be seen as agreement in most-significant digits – after a while they don’t change. So why do we recompute them? We see this again and again in standard numerical computing – each iteration might add just a couple of new correct digits, but we still end up wasting time and energy computing all of the digits in each iteration, even the stable ones. Secondly, not all iterations may contribute equally to the overall error resulting from early termination. This paper addresses these two issues.

The first, and more general, issue is the wastefulness of computing stabilised digits. But just because they look stable, are they really stable? Maybe we’ve stabilised to 0.9, 0.99, 0.999, 0.999, and then one more iteration might kick us over to 1.0001. So can we really afford not to recompute most-significant digits? Ercegovac‘s Online Arithmetic comes to our rescue again! If we compute in an appropriate redundant number representation, then we can prove that stability of digits means we don’t need to consider them any more. This is our first contribution – to recognise this and utilise it within an appropriately modified computational architecture.

The second, and more specific, issue is that some digits are effectively ‘don’t care’. In this paper, we only analyse the specific case of stationary iterative methods (Jacobi, SOR, etc.) for this kind of digit. We show that, in these cases, for a fixed digit budget (e.g. “compute at most D digits across all iterations”), you should allocate these digits by computing a constant more digits each iteration. This constant can be estimated from the infinity norm of a certain matrix involved in the computation. Again, we modify our hardware architecture to take advantage of this pattern.

The end result is that we end up tracing out a corridor of digits, shown in the figure below, where the vertical axis is iteration and the horizontal axis is precision / digit number. Some digits have provably stabilised and no longer need computation (marked “), some are irrelevant don’t cares (marked X). This corridor radically improves the storage requirements of the original ARCHITECT scheme.