mathematics of computation volume 37,number 155 Summary: mathematics of computation volume 37,number 155 JULY 1981 Optimal Numerical Differentiation Using Three Function Evaluations By J. Marshall Ash and Roger L. Jones Abstract. Approximation off'(x) by a difference quotient of the form h~l{aj(x + bxh) + a2f(x + b2h) + a3f(x + b3h)] is found to be optimized for a wide class of real-valued functions by the surprisingly asymmetric choice of b = (¿,, b2,b3)= (1/V3 - 1, 1/V3 , 1/V3 + 1).The nearly opti- mal choice of b = (-2, 3, 6) is also discussed. 1. Introduction. The problem of best approximating the derivative of a function at a single point using two values of the function is "best" solved by using the difference quotient (1) d0(h) =---. We consider the same problem using three values of the function and arrive at three different solutions by interpreting "best" in five different ways. Our best difference quotients are (2) d,(h)-u-, f{x + h) + u2f{x + ich) + cofjx + to2/.) Collections: Mathematics