Master d’informatique fondamentale of École Normale Supérieure de Lyon, Fall-Winter 2015.

This course covers part of approximation theory from the point of view of effective computation. Computation of polynomial or rational approximants, efficient computation of Taylor series expansions or series expansions based on families of orthogonal polynomial, use of these expansions to produce approximations to a prescribed accuracy (Remez’ algorithm, Taylor and Chebyshev models). The power of these techniques is illustrated with three applications from different scientific domains: irrationality proofs in number theory, efficient evaluation of numerical functions, computational issues related to Near-Earth Objects.

Lecture notes of the course

Organisation

Evaluation

References

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Informations

• Next meeting: ?? ??:00 ??m, room ?? (4th floor of the GN1 building)
• The schedule will be the following: lectures on the ??
  

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Lecture notes of the course

You can have a look at a draft version of the lecture notes.

 

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Organisation

• Hours : ??
• Person in charge: Nicolas Brisebarre, Office n.175, LUG.
• Teachers: Nicolas Brisebarre and Bruno Salvy.
• The lectures will be delivered in French or English on request.

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Evaluation

• Evaluation: there will be a take-home exam (consisting of a practical session on a computer) to turn in and an exam.

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References

Lecture notes

• Mathématiques Expérimentales (in French), lecture notes by A. Bostan and B. Salvy. ENS Paris first year of Master.
• Algorithmes efficaces en calcul formel (in French), lecture notes by A. Bostan, F. Chyzak, G.Lecerf and B. Salvy. Parisian Master of Research in Computer Science.

Books

• E. B. Burger and R. Tubbs. Making Transcendence Transparent: An intuitive approach to classical transcendental number theory. Springer.
• J.-P. Demailly. Analyse numérique et équations différentielles. EDP Sciences.
• J. von zur Gathen and J. Gerhard. Modern Computer Algebra, Cambridge University Press.
• J. C. Mason and D. C. Handscomb. Chebyshev Polynomials. Chapman & Hall/CRC.
• M. Schatzman. Analyse numérique, une approche mathématique. Dunod. And in English: Numerical Analysis: A Mathematical Introduction. Oxford University Press.
• L. N. Trefethen. Approximation Theory and Approximation Practice.
• W. Tucker. Validated Numerics, a short introduction to rigorous computations. Princeton University Press.

More advanced level.

• C. M. Bender and S. A. Orszag. Advanced mathematical methods for scientists and engineers, McGraw-Hill Book Co..
• D. Bini and V. Y. Pan. Polynomial and Matrix Computations, Volume 1: Fundamental Algorithms, Birkhäuser.
• J. P. Boyd. Chebyshev and Fourier Spectral Methods. Dover.
• E. W. Cheney. Introduction to Approximation Theory. AMS Chelsea Pub.
• A. Gil, J. Segura and N. M. Temme, Numerical methods for special functions, Society for Industrial and Applied Mathematics (SIAM).
• E. W. Kaucher et W. L. Miranker. Self-validating numerics for function space problems. Academic Press, 1984.
• R. Moore and M. J. Cloud. Computational functional analysis. Horwood Pub.
• M. Powell. Approximation theory and methods. Cambridge University Press.
• L. B. Rall. Computational Solution of Nonlinear Operator Equations. John Wiley & Sons Ltd.

Master theses

PhD theses

• A. Benoit, Algorithmique semi-numérique rapide des séries de Tchebychev. Thèse de doctorat, École Polytechnique, juillet 2012
• S. Chevillard, Évaluation efficace de fonctions numériques – Outils et exemples. Thèse de doctorat, École Normale Supérieure de Lyon, septembre 2011
• P. Di Lizia, Robust Space Trajectory and Space System Design using Differential Algebra, Politecnico di Milano, 2008
• M. Joldeş, Rigorous Polynomial Approximations and Applications. Thèse de doctorat, École Normale Supérieure de Lyon, septembre 2011
• M. Mezzarobba, Autour de l’évaluation numérique des fonctions D-finies, Thèse de doctorat, École Polytechnique, octobre 2011

Softwares

• Chebfun, numerical computing with functions.
• Maple
• Mathemagix
• Pari/GP
• Sage, a free open-source mathematics software. A nice presentation (in French) is available
  here.
• Scilab

Others

• History of Approximation Theory
• Surveys in Approximation Theory