Derivatives Using Newton's Backward Difference Formula
This method is used to derive derivative of a numerical function f(x) using newton's backward difference formula.
Derivatives Using Newton’s Backward Difference Formula :
Consider the data (xi, f(xi)) given at equispaced points xi = x0 ih, where h is the step length. The Newton’s backward difference formula is given by
......................1.1
Let x be any point near xn. Let x – xn = sh. Then, the formula simplifies as
......................1.2
Note that,
The magnitudes of the successive terms on the right hand side become smaller and smaller.
Differentiating (1.2) with respect to x, we get
..............................1.3
At x = xn, we get s = 0. Hence, we obtain the approximation to the first derivative f ′(xn) as
..............................1.4
At x = xn-1, we have xn-1 = xn – h = xn sh. We obtain s = – 1. Hence, the approximation to the first derivative f ′(xn–1) is given by
...............................1.5
Differentiating (1.3) with respect to x again, we get
..............................1.6
At x = xn, that is, at s = 0, we obtain the approximation to the second derivative f ″(x) as
........................1.7