The Spike Conjecture - Some Counter-Examples

Götz Uebe
Institut für Statistik und quantitative Ökonomik, Universität de Bundeswehr Hamburg

Abstract

Solution and ordering of systems of linear and nonlinear equations is intimately related, as is well known in economics. An equally well known conjecture in this context has been, that the number of feedbacks is crucial, i.e. (i) if the computational structure is diagonal, - no feedbacks - computation is fastest, and (ii) - so the conjecture supported by numerical evidence has been extended: if the number of feedbacks, number of ``spikes" (variables) refering to the functional matrix is considerably less than n, n the number of equations, then the number of iterations is less than for the case if there are number of spikes approaching n. This claim can be shown to be false. The number of spikes need not be an indicator of fastness of computation. A number of counterexamples is given for innocently looking, i.e. quite usual linear and nonlinear system of equations.