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

Abstract

Abstract.After introducing briefly the principles of the error linearization method, which is able to determine the coefficients of the first order error approximation, a collection of examples is presented to demonstrate its efficiency as a test bench tbr analyzing numerical algorithms. These examples illustrate the propagation of initial errors, the effect of cancellation, the easy location of the most unstable parts of an algorithm, calculation of condition numbers, approximating the statistical behavior of accumulated errors and the convergence of iterative methods.

A mathematical view of automatic differentiation

“…This relation has been used by many authors for the estimation of round-off errors (Iri, Tsuchiya and Hoshi 1988). In particular, it led Seppo Linnainmaa to the first publication of the reverse mode in English (Linnainmaa 1983). The factor l i=1 |v i v i | has been extensively used by Stummel (1981) and others as a condition estimate for the function evaluation procedure.…”