Improved error bounds for floating-point products and Horner’s scheme

Abstract

Let u denote the relative rounding error of some floating-point format.Recently it has been shown that for a number of standard Wilkinson-type bounds the typical factors γ k := ku/(1−ku) can be improved into ku, and that the bounds are valid without restriction on k. Problems include summation, dot products and thus matrix multiplication, residual bounds for LU-and Cholesky-decomposition, and triangular system solving by substitution.In this note we show a similar result for the product k i=0 x i of real and/or floatingpoint numbers x i , for computation in any order, and for any base β 2. The derived error bounds are valid under a mandatory restriction of k. Moreover, we prove a similar bound for Horner's polynomial evaluation scheme.

Tight Interval Inclusions with Compensated Algorithms

“…Recently, it has been shown that classic Wilkinsontype error bounds for summation, dot product and polynomial evaluation [17], [18], [19] can be slightly improved by replacing the factor γ n (u) by nu with no condition on n (for summation, dot product and Horner scheme). It is likely that the error bounds given in this paper could also be slightly improved by replacing all the γ n (u) by nu.…”

Error Bounds for Computer Arithmetics

“…Note that the index i starts from 0. In particular, if β " 2 and all the x i are in F, then pK, ωq " pk, 2q and (10) becomešˇr´k As another consequence, the classical factor γ 2n for Horner's scheme was improved as well in [35]. Let x, a 0 , a 1 , .…”

Exploiting Structure in Floating-Point Arithmetic

“…The new values of α shown in Table 1 are free of any O(u 2 ) term and thus simpler and sharper than the classical ones. In the last three cases, the price to be paid for those refined constants is some mild restriction on n; we refer to [38] for a precise condition and an example showing that it is indeed necessary.…”

“…mul. nu + O(u 2 ) nu [24] Euclidean norm ( n 2 + 1)u + O(u 2 ) ( n 2 + 1)u [25] T x = b, A = LU nu + O(u 2 ) nu [39] A = R T R (n + 1)u + O(u 2 ) (n + 1)u [39] x [38] poly. eval.…”

“…eval. (Horner) 2nu + O(u 2 ) 2nu ( ) [38] the one by Brunie [4, §3.3.2]), this operation will in general be implemented in software using basic floating-point arithmetic. When doing so, however, some care is needed and a classical scheme like RN(RN(ab) + RN(cd)) or, if an FMA is available, RN(ab + RN(cd)) can produce a highly inaccurate answer.…”