Evaluating Rational Functions: Infinite Precision is Finite Cost and Tractable on Average

Abstract

If one is interested in the computational complexity of problems whose natural domain of discourse is the reals, then one is led to ask: what is the “cost” of obtaining solutions to within a prescribed absolute accuracy$\varepsilon = {1 / {2^s }}$ (or precision$s = - \log _2 \varepsilon $)? The loss of precision intrinsic to solving a problem, independent of method of solution, gives a lower bound on the cost. It also indicates how realistic it is to assume that basic (arithmetic) operations are exact and/or take one step for complexity analyses. For the relative case, the analogous notion is the loss of significance. Let ${{P(X)} / {Q(X)}}$ be a rational function of degree d, dimension n and real coefficients of absolute value bounded by $\rho $. The loss of precision in evaluating ${P / Q}$ will depend on the input x, and, in general, can be arbitrarily large. We show that, w.r.t. normalized Lebesgue measure on $B_r $, the ball of radius r about the origin in $R^n $, the average loss is small: loglinear in d, n, $\rho $, r; and $K_Q $, a simple constant. To get this, we use techniques of integral geometry and geometric measure theory to estimate the volume of the set of points causing the denominator values to be small. Suppose $\varepsilon > 0$ and $d \geqq 1$. Then: Theorem. Normalized volume$\{ x \in B_r | {|Q(x)| < \varepsilon } \} < \varepsilon ^{{1 / d}} K_Q^{{1 / d}} {{nd(d + 1)} / {2r}}$. An immediate application is a loglinear upper bound on the average loss of significance for solving systems of linear equations.