MACM 316 COMPUTING ASSIGNMENT #4

MACM 316 COMPUTING ASSIGNMENT #4
Submit on Crowdmark by Tuesday, July 7, 2021, 11:59pm
Instruction
Upload a PDF file with two parts. Part one should include your typed report (your discussions,
data and figures). Part two should list your code. You will receive a Crowdmark link for
uploading your results.
Perturbation and eigenvalues
This computing assignment is about visualizing computed eigenvalues of some special matrices. In particular, we want to see how sensitive eigenvalues are with respect to perturbations
in the matrices. The focus is on setting up and conducting the experiments, and observing the
outcome without going too deep into the analysis.
1. Create perturbations.
a). Create an n × n random matrix B by the Matlab command rand such that each entry is a
number in the interval [−1, 1].
b). Create an n×n random orthonormal matrix Q by QR factorizing B. In your report, display
such a random orthonormal matrix Q with n ≥ 10.
c). Set  = 10−8
. Generate 100 random orthonormal matrices Q1, Q2, ..., Q100 and multiply the
matrices by . Store the resulted perturbations Q1, Q2, ..., Q100.
2. Jordan block and Limac¸on.
a). Create the following n × n matrix J (Jordan super block) in Matlab by the command diag
with n ≥ 50.
J =


0 1 0 · · · 0 0
0 0 1 · · · 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 · · · 1 0
0 0 0 · · · 0 1
0 0 0 · · · 0 0


b). Let A = 4J. Report the eigenvalues and condition number of A. Compute the eigenvalues
of A + Qi for the 100 perturbations created in Question 1. Note that these eigenvalues may
be complex numbers. Plot all the eigenvalues of the 100 matrices (i.e. A + Qi) on one plot by
the command plot(eigenvalues, ’.’). Report the plot and your observations.
c). Let A = 4J + 4J
2
. Report the eigenvalues and condition number of A. Compute the eigenvalues of A+Qi for the 100 perturbations created in Question 1. Note that these eigenvalues
may be complex numbers. Plot all the eigenvalues of the 100 matrices (i.e. A + Qi) on one
plot by the command plot(eigenvalues, ’.’). Report the plot and your observations.
1
MACM 316 COMPUTING ASSIGNMENT #4
3. Gauss-Seidel
a). Create the following n × n matrices S, U and L from the matrix J in Matlab with n ≥ 50.
J =


0 1 0 0 0 0
0 0 1 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0


, S =


−2 1 0 0 0 0
1 −2 1 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 1 −2 1 0
0 0 0 1 −2 1
0 0 0 0 1 −2


U = J =


0 1 0 0 0 0
0 0 1 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0


, L =


−2 0 0 0 0 0
1 −2 0 0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 1 −2 0 0
0 0 0 1 −2 0
0 0 0 0 1 −2


b). Let A = L
−1U. Compute the eigenvalues of A and plot the eigenvalues. Report the plot
of the eigenvalues and condition number of A. There is a formula for the exact eigenvalues
of A. It is known that the eigenvalues are real, and are all in the interval [0, 1]. What do you
observe? Is λ = 0 an eigenvalue of the matrix A?
c). Compute the eigenvalues of A + Qi for the 100 perturbations created in Question 1. Note
that these eigenvalues may be complex numbers. Plot all the eigenvalues of the 100 matrices
(i.e. A + Qi) on one plot by the command plot(eigenvalues, ’.’). Report the plot and
your observations.
2
MACM 316 COMPUTING ASSIGNMENT #4
4. Polynomial.
a). Generate n equally spaced points r1, r2, ..., rn on the interval [−2, 2] in Matlab with the
command linspace. Find the polynomial p(z) = c0 + c1z + · · · cn−1z
n−1 + z
n with r1, r2, ..., rn
as its roots by the Matlab command poly.
b). Create the following n × n matrix A, which is the companion matrix of the polynomial and
can be computed by Matlab command compan.
A =


−cn−1 −cn−2 −cn−3 · · · −c1 −c0
1 0 0 · · · 0 0
0 1 0 · · · 0 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0 · · · 0 0
0 0 0 · · · 1 0


c). Compute the eigenvalues of A and compare the eigenvalues with r1, r2, ..., rn. Report the
observations about eigenvalues and condition number of A. Compute the eigenvalues of A +
Qi for the 100 perturbations created in Question 1. Note that these eigenvalues may be
complex numbers. Plot all the eigenvalues of the 100 matrices (i.e. A + Qi) on one plot by the
command plot(eigenvalues, ’.’). Report the plot and your observations.
d). Use the following Matlab code to great a matrix B where r = [r1, r2, r3, ..., rn] are the vector
of values from c).
M = 2*rand(n) - 1; [W,R] = qr(M); B = W*diag(r)*W';
Compute the eigenvalues of B as well as those of B + Qi and B + (Qi + QT
i
), for ten different
random orthogonal matrices Qi from Question 1. The second perturbation is symmetric, so
the eigenvalues will remain real. Do not plot your results in this case. Instead, report by how
much the eigenvalues move under these perturbations. Compare the eigenvalues of A and
B. Do they have the same eigenvalues? What do you believe to be the reason for obtaining
different results with perturbations?
3