2nd-Order Centred Divided-Difference Formula
To determine the error for the 2nd-order centred divided-difference formula for the second derivative, we need only look at the two Taylor series approximations:
Add these two equations and transfer the 2f(x) to the left-hand side to get
If we divide both sides by h2 and make the approximation that:
then we can rearrange the equation as
Thus, the error is O(h2).
4th-Order Centred Divided-Difference Formula
A similar sum may be used to find the error of the 4th-order divided difference formula. If you add the linear combination -f(x + 2h) + 8 f(x + h) - 8 f(x - h) + f(x - 2h) of the 5th-order Taylor series approximations, then, after dividing by 12h, we are left with the error term:
If we divide through by -1/30 and factor out the h4, we get
Now, examining the contents of the parentheses, we note that the coefficients 2/3 - 1/6 - 1/6 + 2/3 = 1, and therefore, the contents of the parentheses is an approximation of the average of f(5)(x) on the interval [x - 2h, x + 2h], and thus, we may approximate the error by
Thus, the error is O(h4).
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.