 
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 realvalued 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/.)
