Section: New Results
Efficient Polynomial Approximation
We have developed methods for generating the best, or at least very good, polynomial approximations (with respect to the infinite norm or the L2 norm) to functions, among polynomials whose coefficients follow size constraints (e.g., the degree-i coefficient has at most mi fractional bits, or it has a given, fixed, value). Regarding to the infinite norm on a given compact interval, one of these methods, due to N. Brisebarre, J.-M. Muller and A. Tisserand (CNRS, LIRMM, U. Montpellier) reduces to scanning the integer points in a polytope  , another one, due to N. Brisebarre and S. Chevillard, uses the LLL lattice reduction algorithm  . About the L2 norm, N. Brisebarre and G. Hanrot (LORIA, INRIA Lorraine) started a complete study of the problem  and proposed theoretical and algorithmic results that make it possible to get optimal approximants.
Potential applications to hardwired operators have been studied  by N. Brisebarre, J.-M. Muller, A. Tisserand and S. Torres. We are developing a library for generating such polynomial approximations (see Section 5.5 ).