On some topological properties of numerical algorithms

Abstract

Abstract.A result quantity in a numerical algorithm is considered as a function of the input data, roundoff and truncation errors. In order to investigate this functional relationship using the methods of mathematical analysis a structural model of the numerical algorithm called R-automaton is introduced. It is shown that the functional dependence defined by an R-automaton is a continuous rational function in a neighborhood of any data point except in a point set, the Lebesgue measure of which is zero. An effective general-purpose algorithm is presented to compute the derivative of any result quantity with respect to the individual roundoff and truncation errors. Some ways of generalizing the R-automaton model without losing the results achieved are finally suggested.Introduction.

Error linearization as an effective tool for experimental analysis of the numerical stability of algorithms

“…Mutually equivalent approaches to solve this problem are presented by S. Linnainmaa [7] and M. Tienari [18], and later by J. Larson and A. Sameh [3]. The following example describes briefly the idea of this error linearization method, Consider the expression u 1 +u2/u r Its value u4 can be computed using the algorithm U 3 ~ U2/b/l~ 12 4 ~ U 1 +U 3.…”

An analysis of the effect of rounding errors on the flow of control in numerical processes

“…As a main result of this chapter we have (cf. [14]): In order to prove assertion (i) suppose that ~ has no degenerate decision functions. Then the set Z constructed for ~ contains every a such that s(a,O) has a nontrivial decision value equal to zero.…”

“…If the order of sensitivity of a is finite, the execution s(a,0) has the desired property that some decision variable of it vanishes (cf. [14] In this section we consider the measure of sensitive points. We use the notations S;, S~ and S A given in Lemma 4.1 for the sets of sensitive points of ~, and the notation L for the n-dimensional Lebesgue measure.…”

“…In this paper, which is related to the work by Tienari [14] and also inspired by Knuth [6], we derive in a rather general setting some topological and measure theoretical results associated with rounding errors and the flow of control. Our approach is based on a model of numerical computation, i.e., on a formalism to define numerical processes.…”

“…On the other hand, the uncertainty of tests is discussed as a characteristic feature of numerical computation by some authors [5], [11], [16]. A more extensive analysis of the flow of control in computing processes involving rounding errors is presented in [14].…”