On relative errors of floating-point operations: Optimal bounds and applications

Abstract

Rounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function f l \mathrm {fl} and barring underflow and overflow, such models bound the relative errors E 1 ( t ) = | t − f l ( t ) | / | t | E_1(t) = |t-\mathrm {fl}(t)|/|t| and E 2 ( t ) = | t − f l ( t ) | / | f l ( t ) | E_2(t) = |t-\mathrm {fl}(t)|/|\mathrm {fl}(t)| by the unit roundoff u u . This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t t and in the case where t t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 ( t ) E_1(t) and E 2 ( t ) E_2(t) are optimally bounded by u / ( 1 + u ) u/(1+u) and u u , respectively, when t t is real or, under mild assumptions on the base and the precision, when t = x ± y t = x \pm y or t = x y t = xy with x , y x,y two floating-point numbers. We prove that while this remains true for division in base β > 2 \beta > 2 , smaller, attainable bounds can be derived for both division in base β = 2 \beta =2 and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves.