Optimal numerical differentiation using three function evaluations

Abstract:

Approximation of $f’ (x)$ by a difference quotient of the form \[ {h^{ - 1}}[{a_1}f(x + {b_1}h) + {a_2}f(x + {b_2}h) + {a_3}f(x + {b_3}h)]\] is found to be optimized for a wide class of real-valued functions by the surprisingly asymmetric choice of ${\mathbf {b}} = ({b_1},{b_2},{b_3}) = (1/\sqrt 3 - 1,1/\sqrt 3 ,1/\sqrt 3 + 1)$. The nearly optimal choice of ${\mathbf {b}} = ( - 2,3,6)$ is also discussed.