## Topic 12.4: Higher-Order Derivatives (Error Analysis)

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# 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 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).