61. 62. 63. 64. X f(x) X f(x) X f(x) X 式 f(x) -1 0 1 2 0 224 -10 13 0224 -10 23 0224 1 2 3 224 -1 0
61. 62. 63. 64. X f(x) X f(x) X f(x) X 式 f(x) -1 0 1 2 0 224 -10 13 0224 -10 23 0224 1 2 3 224 -1 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.5: Systems Of Linear Equations In More Than Two Variables
Problem 43E
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Question
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Parts 61-64 please… please use the second image as reference for the work that needs to be done
Transcribed Image Text:Problem IV Find a linear polynomial f(x) which is the best least squares fit to the
following data.
X -1 0 1 2
f(x)
0 224
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
X -1 0 1 3
224
f(x) 0
X -1
H
f(x)
X
f(x)
X
f(x)
X
f(x)
X
f(x)
X
f(x)
X
f(x)
X
f(x)
0
023
22 4
-1
1 2 3
0 224
-201 2
0 22 4
-2 -1 02
0 2 24
-1 012
1 223
-1 0 1 3
1 223
-1 023
1 22 3
-1 1 2 3
1 2 2 3
71.
72.
73.
74.
75.
76.
77.
78.
79.
80.
X -1 012
f(x) 4 220
X -1 013
4 220
f(x)
X
f(x)
Xx
f(x)
X
f(x)
X
f(x)
X
f(x)
-1 023
4 220
X
-2 -1 02
f(x) 4 2 20
X
f(x)
-1
1 2 3
4 2 20
-2 012
4
20
-1 0 1 2
3 22 1
-1 0 1 3
3 22 1
-1 023
3 22 1
X
-1
f(x) 3
1 23
21
![Problem 6. Find a linear polynomial which is the best least
squares fit to the following data:
We are looking for a function f(x) = C₁ + C₂x, where C₁, C₂
are unknown coefficients. The data of the problem give rise
to an overdetermined system of linear equations in variables c₁
and C₂:
A =
This system is inconsistent.
X -2 -1 0 1 2
f(x) -3 -2 1 2 5
We can represent the system as a matrix equation Ac = y,
where
1
-2 -1 0
1 1 1
1
1
1
1
1
1
1
2
-2
-1
0
1
2
C₁2C₂ = -3,
C₁-C₂ = -2,
C₁ = 1,
C₁ + C₂ = 2,
C₁ + 2C₂ = 5.
The least squares solution c of the above system is a solution
of the normal system AT Ac = ATy:
1
1 -2
1
C =
1 1 1 1 1
(0-44])
-2 -1 0 1 2
1 1
C1
C₂
-3
-2
-0
y =
1
2
5
1 2
(5) (2) (3)
=
C₁ = 3/5
C₂ = 2
-3
-2
1
2
Thus the function f(x) = 3 + 2x is the best least squares fit
to the above data among linear polynomials.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b19493d-2e0b-4edb-977e-29215d3bb0dd%2Fae84c256-311c-44d4-ae03-3eb038d6825c%2F9poffl5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 6. Find a linear polynomial which is the best least
squares fit to the following data:
We are looking for a function f(x) = C₁ + C₂x, where C₁, C₂
are unknown coefficients. The data of the problem give rise
to an overdetermined system of linear equations in variables c₁
and C₂:
A =
This system is inconsistent.
X -2 -1 0 1 2
f(x) -3 -2 1 2 5
We can represent the system as a matrix equation Ac = y,
where
1
-2 -1 0
1 1 1
1
1
1
1
1
1
1
2
-2
-1
0
1
2
C₁2C₂ = -3,
C₁-C₂ = -2,
C₁ = 1,
C₁ + C₂ = 2,
C₁ + 2C₂ = 5.
The least squares solution c of the above system is a solution
of the normal system AT Ac = ATy:
1
1 -2
1
C =
1 1 1 1 1
(0-44])
-2 -1 0 1 2
1 1
C1
C₂
-3
-2
-0
y =
1
2
5
1 2
(5) (2) (3)
=
C₁ = 3/5
C₂ = 2
-3
-2
1
2
Thus the function f(x) = 3 + 2x is the best least squares fit
to the above data among linear polynomials.
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