New elements of analysis on the levi-civita field (computational differentiation)

New elements of analysis on the Levi-Civita field R , are presented. First we prove general results about skeleton groups and field automorphisms that will enhance the understanding of the structure of the field. We show that while the identity map is the only field automorphism on R , there can be nontrivial automorphisms on non-Archimedean field extensions of R like R . We also show that every automorphism on R is order preserving and that if P is such an automorphism and r a real number then P(r) is approximately equal to r, moreover, if q is a rational number, then P(q) = q. After reviewing the algebraic, order, and topological structures of the field R [3, 5, 7], we review two types of convergence and prove new results about the convergence of the sums and products of sequences and infinite series. A weak convergence criterion [5] for power series is then enhanced and proved, and we show that power series can be reexpanded around any point of their domain of convergence. Knowledge of weak convergence of power series allows the extension to the new field and the study of all transcendental functions. This also will allow the extension of all the real functions that can be represented on a computer and is thus of great importance for the implementation of the R calculus on computers [38, 39, 40, 42]. We review two different definitions of continuity and differentiability [5, 10]. We show that these smoothness criteria are preserved under addition, multiplication and composition of functions. We show with several examples that topological continuity and differentiability are not sufficient to assure that a function be bounded or satisfy any of the common theorems of real calculus on a closed interval of R . We derive a result which allows for an easy check of the differentiability of functions. Then, based on the stronger concept of differentiability, we present a detailed study of a large class of functions for which we generalize the intermediate value theorem in [5] and prove an inverse function theorem. Based on our knowledge of convergence of power series, we study a large class of functions which are given locally by power series with R coefficients and which generalize the normal functions discussed in [5]. We show that the so-called expandable functions [41, 43] form an algebra and have all the nice properties of real power series. In particular, they satisfy the intermediate value theorem, the maximum theorem and the mean value theorem. Moreover, they are infinitely often differentiable and integrable; and the derivative functions of all orders are themselves expandable functions. The existence of infinitely small numbers in the non-Archimedean field R allows the use of the old numerical algorithm for computing derivatives of real functions, but now with an error that in a rigorous way can be shown to become infinitely small (and hence irrelevant). Using calculus on R , we formulate a necessary and sufficient condition for the derivatives of real functions representable on a computer to exist at any given real point, and we show how to accurately compute the derivatives up to very high orders if they exist, even when the coding exhibits branch points or nondifferentiable pieces [38, 39, 40, 42].