Algorithm 594: Software for Relative Error Analysis

Abstract

DESCRIPTIONThis package performs an automatic round-off error analysis of numerical algorithms, which is useful in determining the numerical properties (stability) of such algorithms. The approach taken with this software differs from that of Miller [4,5] by dealing with relative as opposed to absolute error analysis. This is the natural approach taken for dealing with floating-point arithmetic. In addition, this software gives an accurate analysis of an algorithm for some fixed data, rather than searches for data for which an approximate analysis indicates that the algorithm performs poorly. It uses one of two methods, either an exact method [1] or a heuristic algorithm [1-3]. T h e following paragraphs summarize the previous work in [2], and it is assumed that the reader is familiar with the concepts and techniques presented there.To perform the error analysis, the algorithm being analyzed is represented as a directed graph with each numeric quantity being a node in this graph. Directed arcs lead from operands to their results. Each node's value is subject to error, called local relative error. This error affects those subsequent nodes that are computed using the contaminated node as an operand. All of the local relative errors together are used to produce a total relative error for each node. F r o m the

On the Dimensional Estimate of Rounding-Errors of a typical Computing Process

“…[9] highlights those portions of a priori roundoff error analysis which can possibly be automated. More specific attempts are those which aim at developing software for error analysis in a numerical computational setting [10], [11], [12] and [13]. As regards algebraic computing, the concept of error analysis based on computer algebra system (CAS) for differential error-propagation model is introduced in [14].…”

Optimization by Direct Search in Matrix Computations

“…We by Miller [37], [38] and by recent work of Rowan [47]. Miller In [38] Miller and Spooner extend the work in [37] [37], [38], and [39] A different approach to algorithm analysis is taken in [35], [36]. Here errors are measured in a relative rather than an absolute sense, and the stability is analyzed at fixed data instead of attempting to maximize instability over all data; however, the analysis is still linearized.…”