How to specify an approximate numerical result
It provides a new theoretical perspective on error propagation in numerical approximations, relevant to fields like finance and physics, but remains conceptual without concrete results.
The paper argues that for certain numerical approximations, an Ito-like second-order differential calculus is necessary to describe and propagate errors, and shows that measurement errors from graduated instruments are strongly stochastic based on the Poincaré-Hopf arbitrary functions principle.
The Dirichlet forms methods, in order to represent errors and their propagation, are particularly powerful in infinite dimensional problems such as models involving stochastic analysis encountered in finance or physics, cf. [5]. Now, coming back to the finite dimensional case, these methods give a new light on the very classical concept of 'numerical approximation' and suggest changes in the habits. We show that for some kinds of approximations only an Ito-like second order differential calculus is relevant to describe and propagate numerical errors through a mathematical model. We call these situations strongly stochastic. The main point of this work is an argument based on the arbitrary functions principle of Poincaré-Hopf showing that the errors due to measurements with graduated instruments are strongly stochastic. Eventually we discuss the consequences of this phenomenon on the specification of an approximate numerical result.