@MASTERSTHESIS{Chevillard2006,
  author = {Chevillard, Sylvain},
  title = {Polyn\^omes de meilleure approximation \`a coefficients flottants},
  school = {\'Ecole Normale Sup\'erieure de Lyon},
  year = {2006},
  address = {46 All\'ee d'Italie, 69364 Lyon Cedex 07, France},
  abstract = {When one needs to compute approximated values of a function $f$ at
	many points of an interval $[a,\,b]$, one generally replaces the
	function with another function, which approximates well $f$ and which
	is simpler to evaluate. Polynomials are often used because one needs
	only additions and multiplications to evaluate them. The error made
	by replacing $f$ with a polynomial $p$ is $\eps = p-f$. In the worst
	case, when one replaces $f$ by $p$, one observes an error of $\max
	\{|\eps(x)|,\,x \in [a,\,b]\}$. This is the infinite norm of $\eps$
	and is denoted by $\ninf{\eps}$. The general problem of searching
	a real-coefficient polynomial of degree not greater than~$n$ which
	minimizes $\ninf{f-p}$ has been deeply studied in the early years
	of the~XX${th}$ century. Since every real number cannot be exactly
	stored in the finite memory of a computer, one generally uses the
	so-called floating-point numbers as an approximation of them. When
	implementing a function in a library, one uses polynomials with floating-point
	coefficients. But the best polynomial with such coefficients cannot
	be directly deduced from the best real-coefficient polynomial. The
	problem of searching the best polynomial with floating-point coefficients
	is thus an unsolved problem. We propose a heuristic to find a good
	polynomial to approximate $f$ in infinite norm. We describe the method
	---~which uses lattices theory and their reduction~--- and we illustrate
	its efficiency with a detailed example.},
  keywords = {polynomial approximation, floating-point numbers, lattices, LLL algorithm,
	CVP},
  url = {http://www.ens-lyon.fr/LIP/Pub/Rapports/DEA/DEA2006/DEA2006-03.ps.gz}
}