Differentiability of Polynomials over Reals

In this article, we formalize in the Mizar system [3] the notion of the derivative of polynomials over the field of real numbers [4]. To define it, we use the derivative of functions between reals and reals [9].Institute of Informatics, University of, Białystok, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015. doi: 10.1007/s10817-015-9345-1.Kazimierz Kuratowski. Rachunek rózniczkowy i całkowy - funkcje jednej zmiennej. Biblioteka Matematyczna. PWN - Warszawa (in polish), 1964.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.Konrad Raczkowski. Integer and rational exponents. Formalized Mathematics, 2(1):125-130, 1991.Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797-801, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990

Formal Verification of Medina's Sequence of Polynomials for Approximating Arctangent

The verification of many algorithms for calculating transcendental functions is based on polynomial approximations to these functions, often Taylor series approximations. However, computing and verifying approximations to the arctangent function are very challenging problems, in large part because the Taylor series converges very slowly to arctangent-a 57th-degree polynomial is needed to get three decimal places for arctan(0.95). Medina proposed a series of polynomials that approximate arctangent with far faster convergence-a 7th-degree polynomial is all that is needed to get three decimal places for arctan(0.95). We present in this paper a proof in ACL2(r) of the correctness and convergence rate of this sequence of polynomials. The proof is particularly beautiful, in that it uses many results from real analysis. Some of these necessary results were proven in prior work, but some were proven as part of this effort.Comment: In Proceedings ACL2 2014, arXiv:1406.123

Tracking p-adic precision

We present a new method to propagate $p$-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with many examples and give a toy application to the stable computation of the SOMOS 4 sequence

Effectively Open Real Functions

A function f is continuous iff the PRE-image f^{-1}[V] of any open set V is open again. Dual to this topological property, f is called OPEN iff the IMAGE f[U] of any open set U is open again. Several classical Open Mapping Theorems in Analysis provide a variety of sufficient conditions for openness. By the Main Theorem of Recursive Analysis, computable real functions are necessarily continuous. In fact they admit a well-known characterization in terms of the mapping V+->f^{-1}[V] being EFFECTIVE: Given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f^{-1}[V]. Analogously, EFFECTIVE OPENNESS requires the mapping U+->f[U] on open real subsets to be effective. By effectivizing classical Open Mapping Theorems as well as from application of Tarski's Quantifier Elimination, the present work reveals several rich classes of functions to be effectively open.Comment: added section on semi-algebraic functions; to appear in Proc. http://cca-net.de/cca200