  We study in this paper the function approximation error of multivariate
linear extrapolation. While the sharp error bound of linear interpolation
already exists in the literature, linear extrapolation is used far more often
in applications such as derivative-free optimization, and its error is not
well-studied. A method to numerically compute the sharp error bound is
introduced, and several analytical bounds are presented along with the
conditions under which they are sharp. The approximation error achievable by
quadratic functions and the error bound for the bivariate case are analyzed in
depth. Additionally, we provide the convergence theories regarding the simplex
derivative-free optimization method as a demonstration of the utility of the
derived bounds. All results are under the assumptions that the function being
interpolated has Lipschitz continuous gradient and is interpolated on an
affinely independent sample set.
