Effect of Asymmetric Finite Difference Formulas on the Orders of Central Difference Approximations for the Second Derivative of a Periodic Function

The traditional numerical computation of the first and higher derivatives of a given function f(x) of a single argument x by central differencing is known to involve aspects of both accuracy and precision. However, central difference formulas are useful only for interior points not for a certain number of end points belonging to a given grid of points. In order to get approximations of a desired derivative at all points, one has to use asymmetric difference formulas at points where central differencing doesn’t work. This must surely affect the accuracy and precision of the approximation. In this paper, we study the dependence of the orders of the five-point and the seven-point central difference formulas for the second derivative of f(x) on the oscillatory properties of this function and the value of the sampling period h in the case where it is necessary to use forward and backward formulas to approximate the derivative at some points belonging to a given grid of equally spaced points. As an illustrative example, we consider the case where f(x)=sin(αx).