@PhdThesis{ChevillardPhDThesis,
  author = 	 {Chevillard, Sylvain},
  title = 	 {{Évaluation efficace de fonctions numériques -- Outils et exemples}},
  school = 	 {École normale supérieure de Lyon -- Université de Lyon},
  year = 	 {2009},
  address = 	 {46, allée d'Italie, 69~364 Lyon Cedex~07},
  abstract =     {With computers, it is possible to evaluate some numerical 
                  functions such as $f=\exp$, $\sin$, $\arccos$, etc. The purpose
                  of this thesis is to study how these functions can be 
                  implemented. Depending on the target architecture (software or 
                  hardware, small or high accuracy required), different problems 
                  must be addressed, but the final goal is always to obtain 
                  eventually an implementation as efficient as possible. We first
                  study with an example the problems that arise in the case when
                  the precision is arbitrary.

                  When, on the contrary, the precision is known in advance, the
                  function $f$ is often replaced by an approximation polynomial
                  $p$. Such a polynomial can then be evaluated very efficiently. 
                  In practice, the coefficients of $p$ must be representable on 
                  a given finite number of bits. We propose several algorithms 
                  (some of them are heuristic and others are rigorous) for 
                  finding very good approximation polynomials satisfying this
                  constraint. Our results apply also in the case when the 
                  approximant is a rational fraction. Once $p$ has been found,
                  one must prove that the error $|p-f|$ is not greater than a 
                  given bound. The particular form of the function $p-f$ 
                  (subtraction between two very close functions) makes this 
                  property hard to prove rigorously. We propose an algorithm 
                  for overcoming this difficulty.

                  All these algorithms have been integrated into a software tool
                  called Sollya, developed during the thesis. In the beginning,
                  it was created for making the implementation of functions 
                  easier. Now, it may be interesting for anyone who needs to 
                  perform numerical computations in a safe environment.
                 },
  keywords =     {polynomial approximation, floating-point arithmetic, numerical
                  functions, rounding errors, supremum norm, euclidean lattices,
                  software library, arbitrary precision.}
}