AMSC/CMSC 460
Quiz 5
,
Fall 2007
Show all work. You may leave arithmetic expressions in any form that
a calculator could evaluate. By putting your name on this paper, you agree
to abide by the university’s code of academic integrity in completing the
quiz. Use no books, calculators, cellphones, communication with others,
scratchpaper, etc.
Name
1. In the last in-class exercise, you experimented with taking the discrete
Fourier transform (or discrete cosine transform) of the sunspot data and
then zeroing out some small components.
1a. (5) What do the components that are large in magnitude tell you about
sunspots?
1b. (5) Give two reasons for zeroing out small components of the transform
of a series of data like that for sunspots.
1
−4
6
Change in solution = x − xe
Residual r = (A)x − b
x 10
0.1
0.08
4
0.06
0.04
2
(x−xe)2
r2
0.02
0
0
−0.02
−2
−0.04
−0.06
−4
−0.08
−6
−6
−4
−2
0
r1
2
4
−0.1
−0.1
6
−0.08
−4
x 10
−0.06
−0.04
−0.02
0
(x−xe)1
0.02
0.04
2. (10) Consider a linear system Ax = b where A is 2 × 2. Suppose we
perturb the problem, by changing it to (A + E)xe = b, where kEk1 /kAk1 =
10−4 ≡ δ. We repeat this experiment 10,000 times with different values of E
(all with kEk1 /kAk1 ≈ δ) to get the figures above. Assuming that kxk1 = 1,
use the following fact to estimate the condition number κ of A:
kx − xe k
κ(A)
≤
δ.
kxk
1 − κ(A)δ
2
0.06
0.08
0.1