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. This proof is based on Kantorovitch’s theorem but it represents an original result. An algorithm including rounding is a more accurate model for computations with Newton’s method in practice. Key-words: proof assistants, formalization of mathematics, multivariate analysis, Kantorovitch’s theorem, Newton’s method with rounding Preuves formelles pour les propriétés théoriques de la méthode de Newton Résumé : Ce rapport présente un développement formel pour la certification de la méthode de Newton. On s’intéresse à plusieurs problèmes rencontrés dans l’étude formelle des algorithmes numériques : développer les bibliothèques nécessaires pour nos preuves, adapter des preuves papier aux caractéristiques d’un assistant à la preuve, concevoir des nouvelles preuves basées sur les preuves existantes pour certifier des optimisations de la méthode. Notre point de départ est le théorème de Kantorovitch qui établit la convergence de la méthode de Newton dans le cas d’un système d’équations. Pour formaliser ce théorème dans l’assistant à la preuve Coq on a besoin d’abord de coder les concepts nécessaires d’analyse multivariée. On démontre aussi qu’arrondir à chaque itération de la méthode de Newton donne lieu à un processus qui est encore convergent, avec une corrélation bien determinée entre la précision des données d’entrée et celle du résultat. Un algorithme avec des arrondis est un modèle plus fidèle pour les calculs pratiques par la méthode de Newton. Mots-clés : assistants à la preuve, formalisation des mathématiques, analyse multivariée, théorème de Kantorovitch, méthode de Newton avec arrondis Formal Proofs for Theoretical Properties of Newton’s Method 3 1 Formal systems and numerical methods Often, in verifying mathematical theorems in proof assistants we start with a paper proof of some (famous) theorem and try to obtain a formal model of the theorem inside the system. The concepts are coded in a manner that keeps the balance between mathematical accuracy and handiness of use. This encoding process is not always trivial as the mathematical concepts, expressed in general in set theory, need to be translated into type theory or higher order logic. The limitations and benefits of the formal framework need to be taken into account. Once the concepts inside the system, we try to reproduce the reasoning steps to get the desired proof. Automatization is often possible for some (small) parts of the problem, depending on the field. A good example of a field where mechanization is wide spread is algebra while calculus is less prone to automatization. As a consequence formal developments in real or numerical analysis are more tedious. This is a setback for proof assistants in comparison to computer algebra system which support a wide variety of numerical methods. However, it is sometimes the case that these systems produce erroneous output [16, 6]. So, when a high level of correctness is required, choosing a proof assistant for the task could be a good solution. The aim of this paper is to describe several aspects of a formal development around a numerical algorithm. We discuss the problems encountered, possible solutions and potential applications, in order to allow a better understanding of what such a development entails. Among others, we address the following issues: ◦ providing the proof assistant with the necessary concepts to support all reasoning steps that we are interested in; ◦ formalizing a piece of mathematics stating the desired properties for our algorithms; ◦ designing new proof based on the existing ones to offer theoretical basis for optimizations of our algorithms. We do a case study on Newton’s method and detail all the points above. Widely used as an approximation method to determine the root of a given function or, equivalently, the solution of a system of equations, Newton’s method has good performance with respect to the speed of convergence and the stability of the process. These performances are backed by theoretical results in numerical analysis establishing sufficient conditions for the convergence of the method. Among such results we have Kantorovitch’s theorem, which we chose as a basis for our formal development. The statement of the theorem according to [8] is as follows: Theorem 1 (Kantorovitch). Consider a system of non-linear algebraic or transcendent equations f(x) = 0, where the vector function f : R → R has continuous first and second partial derivatives in a certain domain ω, i.e. f(x) ∈ C(ω). Let x0 be a point with its closed ε-neighborhood Uε(x0) = {‖x−x0‖ ≤ ε} included in ω. If the following conditions hold: 1. the Jacobian matrix W (x) = [ ∂fi(x) ∂xj ] has an inverse for x = x0, Γ0 = W(x0) with ‖Γ0‖ ≤ A0;