Bounding Variable Values and Round-Off Effects Using Handelman Representations

Abstract

Abstract-The precision used in an algorithm affects the error and performance of individual computations, the memory usage and the potential parallelism for a fixed hardware budget. This paper describes a new method to determine the minimum precision required to meet a given error specification for an algorithm that consists of the basic algebraic operations. Using this approach, it is possible to significantly reduce the computational word-length in comparison to existing methods, and this can lead to superior hardware designs. We demonstrate the proposed procedure on an iteration of the conjugate gradient algorithm, achieving proofs of bounds that can translate to global wordlength savings ranging from a few bits to proving the existence of ranges that must otherwise be assumed to be unbounded when using competing approaches. We also achieve comparable bounds to recent literature in a small fraction of the execution time, with greater scalability.

A Scalable Precision Analysis Framework

“…This analysis makes it clear that for existing work, only interval arithmetic has an execution time that scales well with the number of floating point operations. However, as shown in [12], [41], [42], interval arithmetic is unable to find tight bounds due to dependencies between variables. The use of interval splitting in conjunction with any of these approaches allows some trade-off between quality of bounds and execution time, but this scales particularly poorly in the number of variables.…”

“…The use of interval splitting in conjunction with any of these approaches allows some trade-off between quality of bounds and execution time, but this scales particularly poorly in the number of variables. Our aim is to create an approach where the execution time grows in proportion with the code size, of O(n o ), provides a user a very flexible level of control over the execution time per floating point operation and can still obtain bounds approaching the tightness of the approach described in [12]. We discuss the main method to obtain a control over the execution time in Section V, and our overall approach which uses this heuristic to calculate bounds on the range of variables in an algorithm in Section VI.…”

“…This was addressed by the technique using Handelman representations [12], which also had added benefits by retaining correlation between a numerator and denominator polynomial in the case of division. As such, in this section, we describe how we combine this algorithm with the technique to bound rational functions using Handelman representations to create a scalable framework to find tight bounds for the range or relative error for variables in an algorithm.…”

“…We note that because each numerator polynomial for a vector will in general be different, each of these are simplified individually, and in order to capture round-off uncertainty, we first apply the model error described in Section III to the polynomial n ǫ . Finally, once we have created a rational function to bound the range of the desired output variable, we calculate bounds using Handelman representations [12] to try to improve the bounds by taking into account any dependencies in the rational function.…”

“…Recently, a novel approach has been suggested which is capable of reducing the effect of widening of bounds due to dependencies in a multivariate polynomial by bounding the polynomial using Handelman representations [12]. This work demonstrated that using this approach it is possible to obtain superior bounds to those achievable by interval arithmetic, and also showed significant advantages in the case of division, by using a rational expression instead of a polynomial.…”