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Unformatted text preview: MAD 4401 Quiz 4
James Keesling 1 Linear Ordinary Diﬀerential Equations Problem 1.1. Solve the following systems of ordinary diﬀerential equations using matrix
methods. d2 x
= 4x
dt2
d2 x
dx
−3·
+ 2x = 0
dt2
dt 2 Matrix Norms and Condition Number Problem 2.1. Compute the operator norm for the following matrix. Do this for the maximum norm and for the sum norm. 1
2
0
−1 7
0
−1 100
0
−32 0
17 0
0
−320 67
0
32 10
3
−7
0
5 −2 1 −13 −7
2
1 −25
2
5
−720
0
51
3
Problem 2.2. Determine the condition number for the matrix in Problem 2.1 for both the
maximum norm and for the sum norm.
Problem 2.3. Compute the condition number for the Hilbert matrix of size n × n for
n = 5, 10, and 15.
Problem 2.4. Compute the inverse of the Hilbert matrix for n = 4. Do this exactly.
Problem 2.5. Compute the inverse of the Hilbert matrix for n = 8 exactly. Convert
this Hilbert matrix to ﬂoating point entries and compute the inverse. Compare these two
answers. What does the condition number for this matrix say about this comparison? 1 ...
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 Summer '08
 Kyung
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