The goal of this problem is to fit a trigonometric function of the form f(t) = co+c₁ cos(t) to the data points (0, 10.5), (,0.5), (, -7.5), (3, 0.5), using least squares. (a) The problem is equivalent to finding the least squares solution to the system Xc = y where X = (b) Find the coefficients of the best fit by finding the least squares solution to the system in part (a) Co = C₁ = and c = [C₁, C₂]T e
The goal of this problem is to fit a trigonometric function of the form f(t) = co+c₁ cos(t) to the data points (0, 10.5), (,0.5), (, -7.5), (3, 0.5), using least squares. (a) The problem is equivalent to finding the least squares solution to the system Xc = y where X = (b) Find the coefficients of the best fit by finding the least squares solution to the system in part (a) Co = C₁ = and c = [C₁, C₂]T e
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
The goal of this problem is to fit a trigonometric function of the form \( f(t) = c_0 + c_1 \cos(t) \) to the data points \((0, 10.5)\), \(\left(\frac{\pi}{2}, 0.5\right)\), \((\pi, -7.5)\), \(\left(\frac{3\pi}{2}, 0.5\right)\), using least squares.
#### Part (a)
The problem is equivalent to finding the least squares solution to the system \(Xc = y\) where
\[
X =
\begin{bmatrix}
1 & \cos(0) \\
1 & \cos\left(\frac{\pi}{2}\right) \\
1 & \cos(\pi) \\
1 & \cos\left(\frac{3\pi}{2}\right)
\end{bmatrix},
\quad
y =
\begin{bmatrix}
10.5 \\
0.5 \\
-7.5 \\
0.5
\end{bmatrix},
\quad \text{and} \quad
c =
\begin{bmatrix}
c_0 \\
c_1
\end{bmatrix}
\]
#### Part (b)
Find the coefficients of the best fit by finding the least squares solution to the system in part (a).
\[
c_0 = \hspace{2cm}
\]
\[
c_1 = \hspace{2cm}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2F6c8d525a-257e-4b80-9027-e0af1d3a16e6%2Fp67jph_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
The goal of this problem is to fit a trigonometric function of the form \( f(t) = c_0 + c_1 \cos(t) \) to the data points \((0, 10.5)\), \(\left(\frac{\pi}{2}, 0.5\right)\), \((\pi, -7.5)\), \(\left(\frac{3\pi}{2}, 0.5\right)\), using least squares.
#### Part (a)
The problem is equivalent to finding the least squares solution to the system \(Xc = y\) where
\[
X =
\begin{bmatrix}
1 & \cos(0) \\
1 & \cos\left(\frac{\pi}{2}\right) \\
1 & \cos(\pi) \\
1 & \cos\left(\frac{3\pi}{2}\right)
\end{bmatrix},
\quad
y =
\begin{bmatrix}
10.5 \\
0.5 \\
-7.5 \\
0.5
\end{bmatrix},
\quad \text{and} \quad
c =
\begin{bmatrix}
c_0 \\
c_1
\end{bmatrix}
\]
#### Part (b)
Find the coefficients of the best fit by finding the least squares solution to the system in part (a).
\[
c_0 = \hspace{2cm}
\]
\[
c_1 = \hspace{2cm}
\]
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