Computing machine-efficient polynomial approximations

Abstract

Polynomial approximations are almost always used when implementing functions on a computing system. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite-and sometimes small-number of bits. This is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree-i coefficient has at most m i fractional bits; in other words, it is a rational number with denominator 2 m i . We provide a general and efficient method for finding the best polynomial approximation under this constraint. Moreover, our method also applies if some other constraints (such as requiring some coefficients to be equal to some predefined constants or minimizing relative error instead of absolute error) are required.

A Discrete Model for Correlation Between Quantization Noises

“…5 Horner), an L2-norm of an l-length vector (l = 4), a secondorder Voltera filter, and a function approximation. The function 1/(1 + x) is approximated in the interval [0, 1] with four thirdorder polynomials as described in [12].…”

Hardware operators for function evaluation using sparse-coefficient polynomials

“…to Remez have a major drawback: in most cases, their coefficients are not exactly representable using a finite number of bits. In [6], the authors propose an efficient method for computing a polynomial which minimizes the distance ||p − f || ∞, [a,b] among the polynomials p ∈ R d [X] that fulfill some given constraints on the format of the coefficients. The result polynomials q are of the form:…”

“…The coefficients are such that q i ∈ Z. The method presented in [6] provides result polynomials q such that ||q − f || ∞, [a,b] is minimal among the polynomials that fulfill the constraints. Those polynomials can be represented as the integer points of a polytope (cf.…”

“…Those polynomials can be represented as the integer points of a polytope (cf. [6]) and efficient scanning of all the polytope points is performed using linear programming tools.…”

Generating function approximations at compile time

“…To do that, we use two approaches, based on the reduction of our initial problem to -enumerating integer points in a polytope [2]; -using the LLL algorithm (lattice reduction) [? ].…”