Probabilistic error analysis of floating point and crd arithmetics

This paper discusses the longstanding, unsolved problem of the probability distribution of roundoff errors in computer arithmetics. Probabilistic models of floating point and logarithmic arithmetic are constructed using assumptions with both theoretical and empirical justification. To justify these assumptions we resolve open questions in Hamming (1970) and Bustoz et al. (1979). We also develop a probabilistic roundoff error model for fixed point arithmetic as discussed in Bareiss and Barlow (1980). The models for floating point and logarithmic arithmetic are applied to the error from sums, extended products, polynomials in one variable, inner products, matrix computations, linear equation solving procedures both direct and iterative, and linear multistep methods for the solution of ordinary differential equations. A comparison is made of the error analysis properties of floating point and logarithmic computers. We conclude that the logarithmic computer has smaller error confidence intervals for roundoff errors than a floating point computer with the same computer word size and approximately the same number range in any numerical computation.