PASCA , Ioana

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...

We 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...

This thesis deals with the formalization of mathematics in the proof assistant Coq with the purpose of verifying numerical methods. We focus in particular on formalizing concepts involved in solving systems of equations, both linear and non-linear. We analyzed Newton's method which is a numerical method widely used for approximating solutions...

Kantorovitch'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...

We 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...

In 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...

We 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),...

One of the most common and practical ways of representing a real function on machines is by using a polynomial approximation. It is then important to properly handle the error introduced by such an approximation. The purpose of this work is to offer guaranteed error bounds for a specific kind of rigorous polynomial approximation called Taylor...