Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

7.2. RICHARDSON EXTRAPOLATION 93
Exercises
(7.1) Derive the approximation
f ′ (x) ≈
4f(x + h) − 3f(x) − f(x − 2h)
6h
using Taylor’s Theorem.
(a) Assuming that f(x) has bounded derivatives, give the accuracy of the above approximation.
Your answer should be something like O ( h ?) .
(b) Let f(x) = x 3 . Approximate f ′ (0) with this approximation, using h = 1 4 .
(7.2) Let f(x) be an analytic function, i.e., one which is infinitely differentiable. Let ψ(h) be the
centered difference approximation to the first derivative:
ψ(h) =
f(x + h) − f(x − h)
2h
(a) Show that ψ(h) = f ′ (x) + h2
3! f ′′′ (x) + h4
5! f (5) (x) + h6
7! f (7) (x) + . . .
(b) Show that
8 (ψ(h) − ψ(h/2))
h 2 = f ′′′ (x) + O ( h 2) .
(7.3) Derive the approximation
f ′ (x) ≈
4f(x + 3h) + 5f(x) − 9f(x − 2h)
30h
using Taylor’s Theorem.
(a) What order approximation is this? (Assume f(x) has bounded derivatives of arbitrary
order.)
(b) Use this formula to approximate f ′ (0), where f(x) = x 4 , and h = 0.1
(7.4) Suppose you want to know quantity Q, and can approximate it with some formula, say φ(h),
which depends on parameter h, and such that φ(h) = Q + a 1 h + a 2 h 2 + a 3 h 3 + a 4 h 4 + . . .
Find some linear combination of φ(h) and φ(−h) which is a O ( h 2) approximation to Q.
(7.5) Assuming that φ(h) = Q + a 2 h 2 + a 4 h 4 + a 6 h 6 . . ., find some combination of φ(h), φ(h/3)
which is a O ( h 4) approximation to Q.
(7.6) Let λ be some number in (0, 1). Assuming that φ(h) = Q + a 2 h 2 + a 4 h 4 + a 6 h 6 . . ., find some
combination of φ(h), φ(λh) which is a O ( h 4) approximation to Q. To make the constant
associated with the h 4 term small in magnitude, what should you do with λ? Is this practical?
Note that the method of Richardson Extrapolation that we considered used the value λ = 1/2.
(7.7) Assuming that φ(h) = Q + a 2 h 2 + a 4 h 4 + a 6 h 6 . . ., find some combination of φ(h), φ(h/4)
which is a O ( h 4) approximation to Q.
(7.8) Suppose you have some great computational approximation to the quantity Q such that
ψ(h) = Q + a 3 h 3 + a 6 h 6 + a 9 h 9 . . . Can you find some combination of ψ(h), ψ(h/2) which is
a O ( h 6) approximation to Q?
(7.9) Complete the following Richardson’s Extrapolation Table, assuming the first column consists
of values D(n, 0) for n = 0, 1, 2:
n\m 0 1 2
0 2
1 1.5 ?
2 1.25 ? ?
(See equation 7.4 if you’ve forgotten the definitions.)