Approximations: from symbolic to numerical computation, and applications

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


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

Lecture notes of the course

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




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


Lecture notes


  • 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



[Logo CNRS] [Logo ENS de Lyon] [Logo Inria] [Logo LIP]
[Logo UCB Lyon 1] [Logo Université de Lyon] [Logo Labex MILYON] [Logo Fédération Informatique de Lyon]

AriC project – Arithmetic and Computing