# 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:

and

Add these two equations and transfer the 2f(x) to the left-hand side to get

If we divide both sides by *h*^{2} and make the approximation that:

then we can rearrange the equation as

Thus, the error is **O**(*h*^{2}).

# 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* + 2*h*) + 8 f(*x* + *h*) - 8 f(*x* - *h*) + f(*x* - 2*h*) of
the 5th-order Taylor series approximations, then,
after dividing by 12*h*, we are left with the error term:

If we divide through by -1/30 and factor out the *h*^{4}, 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* - 2*h*, *x* + 2*h*], and thus, we may approximate the error by

Thus, the error is **O**(*h*^{4}).

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.