ON RELATIVE ERRORS OF FLOATING-POINT OPERATIONS: OPTIMAL BOUNDS AND APPLICATIONS

摘要

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 fl and barring underflow and overflow, such models bound the relative errors E-1(t) = vertical bar t -fl(t)vertical bar/vertical bar t vertical bar and E-2(t) - vertical bar t -fl(t)vertical bar/vertical bar fl(t)vertical bar by the unit roundoff u. This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We show that E-1(t) and E-2(t) are optimally bounded by u/(1 + u) and u, respectively, when t is real or, under mild assumptions on the base and the precision, when t = x +/- y or t = xy with x, y two floating-point numbers. We prove that while this remains true for division in base beta 2, smaller, attainable bounds can be derived for both division in base 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.