Analysis on the Levi-Civita field and computational applications

Abstract

In this paper, we present an overview of some of our research on the Levi-Civita fields R and C. R (resp. C) is the smallest non-Archimedean field extension of the real (resp. complex) numbers that is Cauchy-complete and real closed (resp. algebraically closed); in fact, R is small enough to allow for the calculus on the field to be implemented on a computer and used in applications such as the fast and accurate computation of the derivatives of real functions as “differential quotients” up to very high orders. We summarize the convergence and analytical properties of power series, showing that they have the same smoothness behavior as real and complex power series; we present a Lebesgue-like measure and integration theory on the Levi-Civita field R; we discuss solutions to one-dimensional and multi-dimensional optimization problems based on continuity and differentiability concepts that are stronger than the topological ones; and we give a brief summary of the results of our ongoing work on developing a non-Archimedean operator theory on a Banach space over C.