Question 8.6: Simply Eq. (8.2.33) for evenly spaced data.
Simply Eq. (8.2.33) for evenly spaced data.
f[x_3, x_2, x_1, x_0] = \frac{f[x_3, x_2, x_1] − f[x_2, x_1, x_0]}{x_3 − x_0} (8.2.33)
We can begin by applying Eq. (8.2.52) to get convenient expressions for the two second divided differences,
f[x_3, x_2, x_0]=\frac{f(x_3)-2f(x_2)+f(x_1)}{2h^2} (8.2.53)
f[x_2, x_1, x_0]=\frac{f(x_2)-2f(x_1)+f(x_0)}{2h^2} (8.2.54)
Substituting the latter into Eq. (8.2.33) gives
b_1 = f[x_1, x_0] (8.2.23)
f[x_3, x_2, x_1, x_0] =\frac{\left[f(x_3)-2f(x_2)+f(x_1)\right]-\left[f(x_2)-2f(x_1)+f(x_0)\right] }{6h^3} (8.2.55)
which we can simplify to
f[x_3, x_2, x_1, x_0]=\frac{f(x_3)-3f(x_2)+3f(x_1)-f(x_0)}{3!h^3} (8.2.56)
In the latter, note the factorial in the denominator. For evenly spaced data, the higher-order formulas are reasonably compact. For arbitrarily spaced data, which was the case in Example 8.2, the general formulas get to be very cumbersome.