Hardware for Rational Functions

Next Tuesday, my collaborator Silviu-Ioan Filip will present some of our recent work with Nicolas Brisebarre, Miloš Ercegovac, Matei Istoan and Jean-Michel Muller at the IEEE International Symposium on Computer Arithmetic.

In the 1970s, Miloš invented a rather nice method called the E-method for evaluating rational functions, i.e. ratios of two polynomials.  The basic idea of his method is as follows. We may solve a system of linear equations $Ay = b$ where $A$ is a matrix of a special structure formed from constants $q_i$ together with variable $x$:

$A = \begin{bmatrix} 1 & -x & 0 & 0 & \cdots & 0 & 0 \\ q_1 & 1 & -x & 0 & \cdots & 0 & 0 \\ q_2 & 0 & 1 & -x & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ q_{n-1} & 0 & 0 & 0 & \cdots & 1 & -x \\ q_n & 0 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}$

If we further choose the vector $b = \begin{bmatrix} p_0 & \cdots & p_n \end{bmatrix}^T$, then it turns out that the first element of the solution vector is the rational function $\frac{p_n x^n + \cdots + p_0}{q_n x^n + \cdots + q_0}$.

So we can use this to evaluate such rational functions. On the face of it, that doesn’t seem very interesting: why would we go to the bother of solving a system of linear equations to evaluate a rational function?

The answer lies in the combination of this idea with another one of Miloš’s key contributions, the idea of online arithmetic – computing results most-significant-digit first. In fact, if the matrix $A$ is sufficiently well conditioned then we may use a stationary iterative method to solve the system of equations in such a way that it produces one new correct digit of the solution for each iteration of the method, leading to very efficient evaluation.

Our paper at ARITH makes two novel contributions. Firstly, we show how to find such a matrix $A$ that is sufficiently well conditioned and for which the solution is close to a given function we’re trying to approximate, improving on the previous technique of Brisebarre et al. Secondly, we show how this method can be efficiently implemented in modern FPGA hardware, when aiming for high throughput.

The main domain of interest will be functions where rational approximation provides a much better fit than polynomials, as the computation required essentially provides rational computation for the price of polynomial computation. A buy-one-get-one-free offer, if you will.

I’m pleased to say that both the rational approximation generator and the hardware IP core generator will soon be open-sourced. Watch this space! Edit: I’m pleased to say this is now available at https://github.com/sfilip/emethod.