[SOLVED] Homework 5. amath 301 beginning scientific computing

Written Exercises
(7 points total)
Exercise 1. (Component Skill 5.1)
Solve the following least-squares problem:
⎛
⎜
⎝
1 0
0 1
1 1
⎞
⎟
⎠
(
x1
x2
) =
⎛
⎜
⎝
1
1
0
⎞
⎟
⎠
.
Show how you arrived at the normal equation. Once there, you can just report the solution of the
linear system.
Exercise 2. (Component Skill 5.2)
The following data represents the temperature T (in oC) in Seattle t hours after sunrise:
t T
0 0
1 0
2 2
What is the equation of the line that fits these data points and minimizes the least-squares error?
Show how your arrived at the normal equation. Once there, you can just report the solution of the
linear system.
Exercise 3. (Component Skill 5.3)
Suppose we sample y = sin2
(x) at x = 0, x = π/4, and x = π/2. We obtain the following data:
x y
0 0
π/4 1/2
π/2 1
a. Fit the data points above to the curve
y = a + b cos(2x).
Determine a and b that minimize the least-squares error. Show how you arrived at the normal
equation. Once there, you can just report the solution of the linear system.
b. The values of a and b you found in (a) should not surprise you. Give a one sentence explanation
for why you should not be surprised.
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Coding Exercises
(13 points total)
Exercise 1. (Component Skill 5.3)
The file salmon_data.mat contains the annual count of Chinook salmon taken at Bonneville on
the Columbia river from the years 1938 to 2020. In this problem, we will fit these data with various
polynomial curves to predict the salmon population in 2021.
a. Download the file salmon_data.mat from the Homework 5 page on Canvas. Make sure this
file is in the same folder as your hw5.m script. In the Exercise 1 section of your hw5.m script,
type
load(‘salmon_data.mat’);
This will automatically create two variables. The first variable is called salmon. This is
an 83 × 1 column vector of the annual salmon count taken at Bonneville. The second variable
is called year. This is an 83×1 column vector of the corresponding years for which the salmon
count was recorded.
Treating salmon as the y-variable and year as the x-variable, use the MATLAB built-in function polyfit to find the coefficients of the first-degree polynomial that best fits the salmon
data in the least-squares sense. Assign the row vector of coefficients of this first-degree polynomial to the variable A1. You may get MATLAB warnings about the polynomial
being badly conditioned. Normally, we’d heed these warnings, but let’s ignore them for
the purposes of this exercise.
Answer:
b. Use the MATLAB built-in function polyfit to find the coefficients of the third-degree polynomial that best fits the salmon data in the least-squares sense. Assign the row vector of
coefficients of this third-degree polynomial to the variable A2.
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Answer:
c. Use the MATLAB built-in function polyfit to find the coefficients of the fifth-degree polynomial that best fits the salmon data in the least-squares sense. Assign the row vector of
coefficients of this fifth-degree polynomial to the variable A3.
Answer:
d. Assuming (a)-(c) are correct, you now have three polynomials that approximate the count
of Chinook salmon as a function of the year: a linear polynomial, a cubic polynomial, and
a quintic polynomial. Call these polynomials p1, p3, and p5, respectively. Let’s test the
predictive power of these polynomials.
The number of Chinook salmon in 2021 was 489,523. For each of your three polynomials,
calculate the relative error between what your polynomial predicts is the salmon count in
2021 and what the true salmon count is. In other words, calculate
ϵ1 =
∣p1(2021) − 489523∣
489523
, ϵ2 =
∣p3(2021) − 489523∣
489523
, and ϵ3 =
∣p5(2021) − 489523∣
489523
.
Use the MATLAB built-in function polyval to evaluate your polynomials. Assign the variable A4 to the 3 × 1 row vector [ϵ1, ϵ2, ϵ3].
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Answer:
Exercise 2. (Component Skill 5.3)
The amount of CO2 in the atmosphere is measured at the Mauna Loa observatory in Hawaii in
parts per million (ppm). The file CO2_data.mat contains the monthly average CO2 concentrations
from March 1958 to December 20201. A plot of the data is shown below. Clearly, there is an overall
increasing trend in CO2 concentrations as well as small seasonal oscillations. In this exercise, we fit
the data to an exponential curve to capture the overall increasing trend in CO2 concentrations.
Figure 1: Monthly average concentration of atmospheric CO2 in ppm as a function of time
a. Download the file CO2_data.mat from the Homework 5 page on Canvas. Make sure his file is
in the same folder as your hw5.m script. In the Exercise 2 section of your hw5.m script, type
load(‘CO2_data.mat’);
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This will automatically create two variables. The first variable is called CO2. This is a 753 × 1
column vector of monthly averaged CO2 measurements (in parts per million). The second
variable is called year. This is a 753 × 1 column vector of time in years since January 1958.
These times correspond to when the CO2 measurements were taken.
Treating CO2 as the y-variable and year as the x-variable, suppose we fit the data with
an exponential curve of the form
y = aerx + b.
Create a function handle that takes in a 3×1 column vector whose first component is a, second
component is r, and third component is b and returns the least-squares error of the exponential fit to the data. Use the MATLAB built-in function fminsearch to find the values of a, r,
and b that minimize the least-squares error. Use the initial guess [30; 0.03; 300]. Assign the
variable A5 to the column vector [a; r; b].
Answer:
b. Repeat (a) but using the L1 error instead of the least-squares error. Assign the variable A6 to
the column vector [a; r; b].
Answer:
c. Let’s plot our results from (a) and (b). Do the following in order:
• Plot the CO2 data in black. Use ‘.’ for the ‘Marker’ option and ‘none’ for the
‘LineStyle’ option. Set the domain to 0 ≤ x ≤ 63. Set the range to 300 ≤ y ≤ 420.
• Superimpose your exponential curve from (a) to the plot of the CO2 data. Be sure to
sample this curve with enough points to appear continuous. Plot this exponential curve
in red. Set the ‘LineWidth’ option to 2.
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• Superimpose your exponential curve from (b) to the plot of the CO2 data and your
exponential curve from (a). Be sure to sample this curve with enough points to appear
continuous. Plot this exponential curve in blue. Set the ‘LineWidth’ option to 2. Set
the ‘LineStyle’ option to ‘–’.
• Add a legend that distinguishes your two exponential curves. Label your exponential
curve from (a) ‘LSE’ for “least-squares error.” Label your exponential curve from (b)
‘L1E’ for “L1 error.” Move the legend to the northwest corner of the plot.
• Create a x-axis label ‘Years since 1958’. Create a y-axis label ‘CO_2 Concentration
(ppm)’. Set the font size of both labels to 16. Create a title ‘CO_2 Concentration vs.
Time’. Set the font size of the title to 20.
• Turn on the grid and box.
Answer:
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Exercise 3. (Component Skill 5.3)
Hanging from one of Sally’s cat towers is a spring with a small toy connected at its end. When the
toy is vertically displaced, it oscillates up and down until eventually settling back to equilibrium
because of friction and air resistance. Sally is very interested in the behavior of her oscillating toy
and has taken careful measurements of its displacement as a function of time. These measurements
can be found in the file spring_data.mat.
a. Download the file spring_data.mat from the Homework 5 page on Canvas. Make sure his
file is in the same folder as your hw5.m script. In the Exercise 3 section of your hw5.m script,
type
load(‘spring_data.mat’);
This will automatically create two variables. The first variable is called deltaz. This is
a 51 × 1 column vector of vertical displacements in centimeters. The second variable is called
time. This is a 51 × 1 column vector of time in seconds. These times correspond to the
measured displacements.
Sally has a suspicion that her toy’s maximum displacement below its equilibrium position
happens at time t = π/2 seconds. She’d like to obtain the value of this displacement to know
where to strike during play time. Unfortunately for Sally, she has no measurements of displacement at exactly t = π/2 seconds. Her closest measurements are at t = 1.4 seconds and
t = 1.6 seconds. Thus, she’s going to need to interpolate.
Using the MATLAB built-in function interp1, help Sally estimate the value of the toy displacement at t = π/2 seconds using a cubic spline interpolation. Assign the value of this
displacement to the variable A7.
Answer:
b. Help Sally create a plot of her data and cubic spline interpolation. Do the following in order:
• Plot the cubic spline interpolation of the displacement data in blue. Use the MATLAB
built-in function interp1 with sample points t = 0:0.01:10 to construct this cubic
spline interpolation. Adjust the ‘LineWidth’ option to 2. Set the domain to 0 ≤ x ≤ 10.
Set the range to −1 ≤ y ≤ 1.
• Superimpose the displacement data to your cubic spline plot. Plot the displacement
data in black. Use ‘.’ for the ‘Marker’ option and ‘none’ for the ‘LineStyle’ option.
Adjust the ‘MarkerSize’ option to 15.
• Create a x-axis label ‘Time (s)’. Create a y-axis label ‘Vertical Displacement
(cm)’. Set the font size of both labels to 16. Create a title ‘Vertical Displacement vs.
Time’. Set the font size of the title to 20.
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• Turn on the grid and box.
Answer: