2,166 research outputs found
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
Diagrams in Mathematics: history and philosophy: Introduction
The present version is slightly different from the final published one.International audienceThis is the introduction to a Special issue of Synthese, "Diagrams in Mathematics: History and Philosophy", edited by the authors together with Gabriel Sandu
Pierre Duhemâs philosophy and history of science
LEITE (FĂĄbio Rodrigo) â STOFFEL (Jean-François), Introduction (pp. 3-6). BARRA (Eduardo Salles de O.) â SANTOS (Ricardo Batista dos), Duhemâs analysis of Newtonian method and the logical priority of physics over metaphysics (pp. 7-19). BORDONI (Stefano), The French roots of Duhemâs early historiography and epistemology (pp. 20-35). CHIAPPIN (JosĂ© R. N.) â LARANJEIRAS (CĂĄssio Costa), Duhemâs critical analysis of mechaÂniÂcism and his defense of a formal conception of theoretical phyÂsics (pp. 36-53). GUEGUEN (Marie) â PSILLOS (Stathis), Anti-Âscepticism and epistemic humility in Pierre Duhemâs philosophy of science (pp. 54-72). LISTON (Michael), Duhem : images of science, historical continuity, and the first crisis in physics (pp. 73-84). MAIOCCHI (Roberto), Duhem in pre-war Italian philosÂophy : the reasons of an absence (pp. 85-92). HERNĂNDEZ MĂRQUEZ (VĂctor Manuel), Was Pierre Duhem an «esprit de finesse» ? (pp. 93-107). NEEDHAM (Paul), Was Duhem justified in not distinguishing between physical and chemical atomism ? (pp. 108-111). OLGUIN (Roberto Estrada), «Bon sens» and «noĂ»s» (pp. 112-126). OLIVEIRA (Amelia J.), Duhemâs legacy for the change in the historiography of science : An analysis based on Kuhnâs writings (pp. 127-139). PRĂNCIPE (JoĂŁo), PoincarĂ© and Duhem : Resonances in their first epistemological reflecÂtions (pp. 140-156). MONDRAGON (DamiĂĄn Islas), Book review of «Pierre Duhem : entre fĂsica y metafĂsica» (pp. 157-159). STOFFEL (Jean-François), Book review of P. Duhem : «La thĂ©orie physique : son objet, sa structure» / edit. by S. Roux (pp. 160-162). STOFFEL (Jean-François), Book review of St. Bordoni : «When historiography met epistemology» (pp. 163-165)
05391 Abstracts Collection -- Algebraic and Numerical Algorithms and Computer-assisted Proofs
From 25.09.05 to 30.09.05, the Dagstuhl Seminar 05391 ``Algebraic and Numerical Algorithms and Computer-assisted Proofs\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper.
Links to extended abstracts or full papers are provided, if available
Difference Methods and Deferred Corrections for Ordinary Boundary Value Problems
Compact as possible difference schemes for systems of nth order equations are developed. Generalizations of the Mehrstellenverfahren and simple theoretically sound implementations of deferred corrections are given. It is shown that higher order systems are more efficiently solved as given rather than as reduced to larger lower order systems. Tables of coefficients to implement these methods are included and have been derived using symbolic computations
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