Numerical Differentiation

In this section, we describe methods of calculating the derivatives of a function f using only its values at (n+1) distinct points. We approximate the function by a polynomial and we assume that the derivatives of the function are approximately equal to the derivatives of the polynomial.

Recall that a polynomial of first degree passing through the points [x ₀, f(x ₀)] and [x ₁, f(x ₁)] is

(7.6 7)

Differentiating p ₁(x) with respect to x, we obtain

(7.7 1a)

(7.7 1b)

where h(=x ₁ ?x ₀) is the distance between the points x ₁ and x ₀.

A formula for f' (x ₀) is then given approximately by

(7.7 2)

The formula given by Equation (7.7 2) is known as the two-point formula since it involves two points x ₀ and x ₁ It is also known as the forward difference formula, as the derivative at x ₀ depends on the forward point x ₁.

From Equations (1.2 11, 12), we have

(7.7 3a)

where 0 ???1.

Equation (7.7 3a) may be written as

(7.7 3b)

By comparing Equations (7.7 2, 3b), we find that the error is of the order h.

The error can be reduced by taking h to be smaller, provided f" behaves well in the interval. Since f" is generally not known, we cannot estimate the error.

Similarly if we approximate f(x) by a polynomial of second degree p ₂