Covers material from Chapter 6, Sections 6.1 and 6.2.
1. Compute the central difference formula and the cubic difference formula (equations (3) and (10) in Section 6.1 in the text) for f'(x) for f(x)=x^5, h=0.05, and x=2. Find the differences between these estimates and the true value of f'(2). Compare the actual errors with the error estimates for the two formulas (equations (4) and (11) in Section 6.1 in the text) with the higher-order derivatives of f(x) in the error estimates replaced by their maximum values on the interval [0,3].
2. Section 6.1, p328. Exercise 16.
3. Section 6.2, p342. Exercises 3, 7, 14.
4. The central-difference formula
D_2f(x,h) = ( f(x+h) - 2f(x) + f(x-h) ) / h^2
= f''(x) + ch^2f''''(x) + O(h^4)
can be derived from Taylor's formula. Note that this implies a
similar formula for D_2f(x,2h). Use Richardson's extrapolation to derive a
difference formula for f''(x) depending on f(x+kh) for k=-2,-1,0,1,2 that
is correct to O(h^4). Compare your result with the O(h^4) formula for
f''(x) in Table 6.4 (page 333) in the text.
Last modified March 17, 1998