Show any work for the questions in the spaces provided. Make sure that
Matlab is pointed at the folder where you have downloaded the data and
script files for each problem. Also, make sure that you send me an email
containing your changes matlab files.
Please label your attachment clearly.(100 points total)
Name:
Math 307
Takehome Exam
(20 pts.) 1. (a) Setup the linear system corresponding to Problem #8 on page 87.
(b) Which of the four voltage sources most dramatically effects the current for the fifth
loop, I5 .
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(20 pts.) 2. I promised that you would see an application of a Singular Value Decomposition for
image compression. Consider that promise fulfilled. The Mars rovers Opportunity and
Spirit each have five cameras that need to send images back to earth. Their batteries are
recharged by solar panels, but the dust blowing around on Mars has limited their effectiveness over the years. Thus, it is critical that they conduct their operation consuming
as little energy as possible.
(a) To this end, the engineers limit the amount of energy needed to send data back
to earth by compressing the images taken by the rovers using a singular value
decomposition (see notes in Problem 2 folder on blackboard). This techniques
enables the rovers to send only a subset the singular values (full set of singular
values contained in the file Sing val mat.dat) for each image since the matrices
U and V (contained in the files U mat.dat and V mat.dat) will be the same for
every image. The file SVD image.m is a matlab script that implements this idea
using these files. The constructed image and the actual image in the folder look
nothing alike since the constructed image only used one singular value.
QUESTION: How many singular values should be used in order to generate a good
image while minimizing the amount of data sent? Support your conclussion by
altering line 41 in the provided script.
(HINT: The matlab command A(1:5,1:5) accesses the upper 5 × 5 block of the
matrix A.)
(b) The matrix multiplication on line 44 uses far more computations than necessary
given the block matrix multiplication that is being calculate. Using your conclussion
above, alter this command in such a way that these unnecessary computations are
no longer performed.
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(20 pts.) 3. As stated in the notes: Both the spotted owl and the rat populations go extinct when
the spotted owls consume 20% of the rat population per month. It was also stated
that
0.50 0.40
λ1 = 0.9 and λ2 = 0.7 are the eigenvalues for the matrix A =
, which
−0.20 1.10
defines the corresponding linear difference equation.
(a) Suppose that the spotted owls only consume 10.4% of the rat population per month.
What happens to both populations over time? What are the eigenvalues for the
corresponding matrix?
(b) Suppose that the spotted owls only consume 12.5% of the rat population per month.
What happens to both populations over time? What are the eigenvalues for the
corresponding matrix?
(c) Consider the largest eigenvalue in each of the three cases. What condition on the
largest eigenvalue appears to yield a stable population for both species over time?
(d) The eigenvalues are the roots of the quadratic equation Det(λI − A) = 0. That is,
Det(λI − A) = aλ2 + bλ + c for some scalars a, b, c ∈ R. Suppose the spotted owls
still consume 20% of the rat population, what would the value a1,2 = 0.4 have to
change to in order to yield a stable population for both species.
(e) Would you suspect that this is within the margin of error for the experiments used
to obtain that value? Why?
(f) Almost all biological models are a simplified version of the actual system that
they are trying to approximate. Sometimes the models do a very good job of
approximating the system, sometimes the assumptions used to simplify the model
leave out a very critical detail. Suppose that over a six month period we notice
the the owl population is growing very slowly, but the rat population growing very
quickly. What parameters in the original model, stated in the notes, would you
suspect are flawed? Why?
3
(20 pts.) 4. A tridiagonal matrix can be used to estimate the unsteady conduction of heat in a rod
when the temperature at four evenly spaced points p1 , p2 , p3 , and p4 on the rod change
with time. Let ∆t denote the time between sucessive measurments, and suppose that
for k = 1, 2, . . ., a vector tk ∈ R4 lists the temperatures at time k∆t. If the temperature
at the two ends of the rod are maintained at 0◦ C, then the temperature vectors satisfy
the equation A · tk+1 = tk , (k = 0, 1, . . .) where


(1 + 2C)
−C
 −C

(1 + 2C)
−C

A=

−C
(1 + 2C)
−C 
−C
(1 + 2C)
and C denotes a constant that gives the physical characteristics of the rod.
(a) Find the LU factorization of A when C = 1. (NOTE: L and U should both be
bidiagonal matrices.)
(b) Use the LU factorization of A to find the temperature distributions t1 , t2 , t3 , and t4
when t0 = (10, 15, 15, 10)T .
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(20 pts.) 5. Consider the network represented in the PDF file located in the files for Question 5. The
values in the nodes represent the supply (positive values) or demand (negative values)
for each node. The values on each edge represent the cost to send one unit along that
edge.
(a) Suppose that the corporation that you work for uses the central node as its processing center. Therefore, any packages that are being shipped MUST route through
the central node. Determine a feasible flow pattern for the network under these
circumstances. Make sure that your label your nodes and edges clearly.
(b) In order to increase their profitability, your company has now decided to open two
more processing centers (represented by the upper and lower central nodes). Would
you recommend that they open one, both, or neither of the proposed processing
centers? What is the optimal distribution (i.e. minimizes the shipping costs) for
this network?
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