Formal Proofs for Theoretical Properties of Newton's Method

We discuss a formal development for the certification of Newton's method. We address several issues encountered in the formal study of numerical algorithms: developing the necessary libraries for our proofs, adapting paper proofs to suit the features of a proof assistant, and designing new proofs based on the existing ones to deal with optimizations of the method. We start from Kantorovitch's theorem that states the convergence of Newton's method in the case of a system of equations. To formalize this proof inside the proof assistant Coq we first need to code the necessary concepts from multivariate analysis. We also prove that rounding at each step in Newton's method still yields a convergent process with an accurate correlation between the precision of the input and that of the result. An algorithm including rounding is a more accurate model for computations with Newton's method in practice

Formally Verified Conditions for Regularity of Interval Matrices

The final publication is available at www.springerlink.comInternational audienceWe propose a formal study of interval analysis that concentrates on theoretical aspects rather than on computational ones. In particular we are interested in conditions for regularity of interval matrices. An interval matrix is called regular if all scalar matrices included in the interval matrix have non-null determinant and it is called singular otherwise. Regularity plays a central role in solving systems of linear interval equations. Several tests for regularity are available and widely used, but sometimes rely on rather involved results, hence the interest in formally verifying such conditions of regularity. In this paper we set the basis for this work: we define intervals, interval matrices and operations on them in the proof assistant Coq, and verify criteria for regularity and singularity of interval matrices

A Formal Verification for Kantorovitch's Theorem

National audienceKantorovitch's theorem gives sufficient conditions for the convergence of Newton's method. We present a full formalization of this theorem in the case of a real function. The work is accomplished inside the Coq proof assistant and it is based on the Reals library provided by the theorem prover. For the general case of the theorem we first describe a way to represent concepts from multivariate analysis, as Coq does not offer such a library. We then discuss the proof of Kantorovitch's theorem based on this representation

Formal verification of exact computations using Newton's method

International audienceWe are interested in the certification of Newton's method. We use a formalization of the convergence and stability of the method done with the axiomatic real numbers of Coq's Standard Library in order to validate the computation with Newton's method done with a library of exact real arithmetic based on co-inductive streams. The contribution of this work is twofold. Firstly, based on Newton's method, we design and prove correct an algorithm on streams for computing the root of a real function in a lazy manner. Secondly, we prove that rounding at each step in Newton's method still yields a convergent process with an accurate correlation between the precision of the input and that of the result. An algorithm including rounding turns out to be much more efficient

Certified, Efficient and Sharp Univariate Taylor Models in COQ

International audienceWe present a formalisation, within the COQ proof assistant, of univariate Taylor models. This formalisation being executable, we get a generic library whose correctness has been formally proved and with which one can effectively compute rigorous and sharp approximations of univariate functions composed of usual functions such as 1/x, sqrt(x), exp(x), sin(x) among others. In this paper, we present the key parts of the formalisation and we evaluate the quality of our certified library on a set of examples

Canonical Big Operators

The original publication is available at http://www.springerlink.com/content/16v67m7248714568/International audienceIn this paper, we present an approach to describe uniformly iterated “big” operations and to provide lemmas that encapsulate all the commonly used reasoning steps on these constructs. We show that these iterated operations can be handled generically using the syntactic notation and canonical structure facilities provided by the Coq system. We then show how these canonical big operations played a crucial enabling role in the study of various parts of linear algebra and multi-dimensional real analysis, as illustrated by the formal proofs of the properties of determinants, of the Cayley-Hamilton theorem and of Kantorovitch's theorem

The Influence of Solar Radiation on the Antioxidant Systems in Blood of Dairy Cows and the Processing of the Data Using Wavelets Transforms

The purpose of this paper was to observe if in cattle, exposed to the solar radiation, could be noticed a certain reaction of the organism related to the oxidative stress. The study was made in the period May - August 2014, on a group of 16 Romanian Simmental dairy cows, kept on pasture during the day. The processing of the determined data were made using wavelet transforms (In-Place Fast Haar Wavelets Transform). The results of the experiment shown that when Temperature Humidity Indexes (THI) are higher than 72 (the superior limit for thermal comfort in cattle), the oxidative stress appeared in dairy cows. This oxidative stress was mainly manifested by the increasing of superoxide dismutase with 95% in August compared to May, followed by the increasing of the catalase with 79% and of glutathione peroxidase with 13%. The increasing of the antioxidant enzymes level was directly co-related with THI. We considered that the determination of the antioxidant enzymes level was an appropriate model for studying the influence of hot environment on the oxidative status of dairy cows. The wavelets transforms can be easier applied to practical data compared to the classical statistical methods