Important Note: Remember to include a published version of any Matlab ® function or script you write. Problem 1: Solving Equations (25 points) Consider a homogeneous spherical ball of radius r...

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Important Note: Remember to include a published version of any Matlab
®
function or script you
write.
Problem 1: Solving Equations (25 points)
Consider a homogeneous spherical ball of radius r submerged in water—see Figure 1. The ball and
water have densities ⇢b and ⇢w respectively. We wish to determine the distance d when the ball is
submerged in water.
At equilibrium, Archimedes’ principle states that the weight of the displaced water must be equal to
the weight of the ball.
(a) (6 points) Assuming the ball is submerged a distance d in the water, show that the integral
giving the volume of the submerged portion of the ball is
Vw =
Z d
0


r2 � (r � x)2

dx . (1)
(b) (3 points) Evaluate (by hand) the integral of Equation (1), thereby obtaining the volume of
the displaced water.
(c) (6 points) Apply Archimedes’ principle to obtain the equation to be solved for the distance d .
Write your answer in the form p(d) = 0 .
Hint: The volume of a sphere of radius r is V = 4⇡r3/3 .
For the remainder of the problem, we use the following normalized values:
r = 1 and
⇢b
⇢w
=
1
5
.
(d) (3 points) Plot the function p(d) for d 2 [�1.5 , 3.0] . Be sure to label your figure appropriately.
(e) (1 point) Suggest an integer as starting point for Newton’s method.
(f) (6 points) Use the function newtons.m on Canvas with the starting point of Part (e) to obtain
the depth d of the submerged ball to within 6 decimal places.
AAACJXicbVDLTgJBEJzFF+IL9OhlIzHxRHaNRo9ELx4hkUcCGzI728CE2ZnNTC8JIXyBV/0Gv8abMfHkrzgLHATsZDKVqq5Od4WJ4AY979vJbW3v7O7l9wsHh0fHJ8XSadOoVDNoMCWUbofUgOASGshRQDvRQONQQCscPWZ6awzacCWfcZJAENOB5H3OKFqqHvWKZa/izcvdBP4SlMmyar2Sc9yNFEtjkMgENabjewkGU6qRMwGzQjc1kFA2ogPoWChpDCaYzjeduZeWidy+0vZJdOfsX8eUxsZM4tB2xhSHZl3LyH81MGNgaKXss+OlwvmFK02ZDZUSZm1L7N8HUy6TFEGyxZL9VLio3CwyN+LaDhUTCyjT3N7psiHVlKENtmAT9Nfz2gTN64p/W/HqN+XqwzLLPDknF+SK+OSOVMkTqZEGYQTIC3klb8678+F8Ol+L1pyz9JyRlXJ+fgGurKUS
d
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r
x
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
Figure 1: Spherical ball submerged a depth d in water.
2
Problem 2: Interpolation (35 points)
1. (13 points) Lagrange Interpolating Polynomials
Write a Matlab
®
function that implements the Lagrange interpolating polynomial through the
points (x1, y1), (x2, y2), . . . , (xn, yn) .
• Your function should take as input two vectors x = (x1, x2, . . . , xn)T and y = (y1, y2, . . . , yn)T
containing the x� and y�coordinates of the data points respectively. It must also take as
input an array t = (t1, t2, . . . , tm)T at which the polynomial is to be evaluated.
• Your function should output the array p = (p1, p2, . . . , pm)T containing the values of the
Lagrange interpolating polynomial evaluated at the points t1, t2, . . . , tm respectively.
• As a way to test your code, use 100 points to plot, on the interval [�2, 3] , the Lagrange
interpolating polynomial through the points (�1, 6), (0, 4), (1, 0) , and (2, 0) . Use markers to
indicate the interpolation points. You should obtain the plot of Figure 2.
XXXXXXXXXX XXXXXXXXXX
-2
0
2
4
6
8
10
Figure 2: Interpolating polynomial through the points (�1, 6), (0, 4), (1, 0) , and (2, 0) .
2. (22 points) Consider the function f on the interval [�1, 1] given by
f(x) =
1
1 + 12x2
. (2)
We wish to (graphically) investigate the interpolation error En(x) = f(x) � Pn(x) as n ! 1 ,
where Pn is the Lagrange interpolating polynomial through n+ 1 equidistant points (including the
endpoints).
(a) (3 points) Plot the function f on the interval [�1, 1] . Be sure to label your figure appropriately.
3
(b) (9 points) Use your function from Part 1 to plot (on di↵erent figures) the Lagrange interpolating
polynomials P4 , P6 , P8 , P10 on the interval [�1, 1] using 1000 points. Include the function
f on each figure and use markers to indicate the interpolation points.
(c) (2 points) Comment on the convergence of thez interpolating polynomials Pn .
We now use the following n+ 1 points to obtain the Lagrange interpolating polynomial Qn :
xi = cos

2i� 1
n+ 1
· ⇡
2

, i = 1 , 2 , . . . , n XXXXXXXXXX)
(d) (6 points) Use your function from Part 1 to plot (on di↵erent figures) the Lagrange interpolating
polynomials Q4 , Q6 , Q8 , Q10 on the interval [�1, 1] using 1000 points. Include the function
f on each figure and use markers to indicate the interpolation points.
(e) (2 points) Comment on the convergence of the interpolating polynomials Qn .
Problem 3: Linear Regression and Least Squares (40 points)
An advantage of linear regression is its applicability to data with seemingly no linear relationship. As
an example, suppose we are given the data (x1, y1) , (x2, y2) , . . . , (xn, yn) and we wish to find the
“best” exponential curve of the form
y = B exp(Ax) (4)
that approximates the data.
(a) (3 points) Explain how the above nonlinear regression problem can be transformed into a linear
regression problem of the form
Y = ↵X + � . (5)
Be sure to specify how the (unknown) parameters ↵ and � relate to A and B and how the
variables X and Y relate to x and y .
Hint: Consider taking the natural logarithm of both sides of Equation (4).
In what follows, we consider the following (x, y) data:
(�1, 6.62) , (0, 2.78) , (1, 1.51) , (2, 1.23) , (3, 0.89) .
(b) (2 points) Use the result of Part (a) to obtain the (X, Y ) points corresponding to the above
data set. We will call these the linearized data.
(c) (3 points) Determine the inconsistent system that results from fitting the linear model of
Equation (5) to the linearized data. In other words, identify the matrix A and the vector b
such that Ax = b is an inconsistent system when we set x = (�, ↵)T .
(d) (5 points) Deduce the normal equations for the linearized problem and solve them for the
least-squares solution x⇤ using Matlab’s backslash (\) command. Remember to show your
work.
4
(e) (4 points) Find (by hand) the reduced QR factorization of A .
(f) (3 points) Use Matlab’s qr command to find the full QR factorization of A and verify that Q
is orthogonal.
(g) (6 points) Use the result of Part (f) (and Matlab’s \ command) to solve for the least squares
solution x⇤ to the inconsistent system of Part (c) and provide the corresponding least-squares
error. How does your answer compare to that of Part (d)?
(h) (6 points) For three vectors x of your choice, compute the Euclidean norm of the residual
Ax� b . How do these compare to the error computed in Part (g). Is this expected?
(i) (2 points) Deduce the sought-after parameters A and B .
(j) (3 points) Plot the linearized data and the best-fit linear model on the interval [�2, 4] . Label
your figure appropriately.
(k) (3 points) Plot the original data and the best-fit exponential curve on the interval [�2, 4] .
Label your figure appropriately.
5
Answered 3 days AfterMay 06, 2021

Solution

Shreyan Sanyal answered on May 08 2021
24 Votes

Solution to the MATLAB Toolbox problem
Although having the necessary software is your responsibility - I wish I could help you, but I cannot.
The Symbolic Math toolbox...

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