Homework Set #7, Math 575A
Part I: This part is the analytical part.
1. Which of the properties of a natural cubic spline does the following
function posess and which properties does it not?
f(x) = ((x+ 1) + (x+ 1)3x∈[−1,0]
4 + (x−1) + (x−1)3x∈(0,1]
2. Let f(x) = √1 + cos x.
(a) Prove fis a periodic function with periodicity of 2π;
(b) Find the trigonometric polynomial that interpolates fat x0=
0, x1=π.
3. Derive the following two formulas for approximating the third deriva-
tive. Which one is more accurate?
(a) f000(x)≈1
h3(f(x+ 3h)−3f(x+ 2h) + 3f(x+h)−f(x));
(b) f000(x)≈1
2h3(f(x+ 2h)−2f(x+h) + 2f(x−h)−f(x−2h)).
4. Derive a numerical differentiation formula of order O(h4) by applying
Richardson extrapolation to
f0(x) = 1
2h[f(x+h)−f(x−h)] −h2
6f000(x)−h4
120f(5)(x)− · · ·
Part II: This part is practical
1. Write a code that uses
1
12h(−f(x+ 2h) + 8f(x+h)−8f(x−h) + f(x−2h))
to approximate f0(x), where f(x) = tan−1xand x=√2. (Note
f0(√2) = 1
3.). Output k, h, r, e. Here kis the iteration index, his
the step size, ris the approximate value of f0(x) and eis the error,
i.e. e=|1
3−r|. From a nicely displayed table of the error, estimate
the order of the finite difference approximation you just coded, by
repeatedly changing the value of h.
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