We have taken measurements of air humidity H as a function of time and we wish to fit a curve to the measurements. Let's assume that the measurements depend linearly on time. The following table contains the measurements. H34687 t12468 Let's fit a line to the data using the least-squares method using the Cholesky decomposition. Cholesky decomposition is computationally half as expensive compared with the LU decomposition, but it can only be computed for positive-definite matrices. The normal form for the least-squares problem is positive-definite so the decomposition is often handy when calculating least-squares problems. The following matrix equation was formed from the measurements. U= 2 4 6 Variables a and are the slope and the intercept of the regression line (H = at + B). By multiplying the matrix from both sides with the transpose of the coefficient matrix we end up with the normal form: y = [ 21 =A X = [a] 121 21 5 3 Let's compute the Cholesky decomposition for A. Input the upper triangular matrix U as an answer. Give only exact values. You can check that you have the correct decomposition by calculating UTU and making sure it equals A. Now solve the equation UT y = b for y. Give the exact answer. 8 139 ][²] = [¹28 Finally solve the equation Ux = y for x. Give the exact values here as well.
We have taken measurements of air humidity H as a function of time and we wish to fit a curve to the measurements. Let's assume that the measurements depend linearly on time. The following table contains the measurements. H34687 t12468 Let's fit a line to the data using the least-squares method using the Cholesky decomposition. Cholesky decomposition is computationally half as expensive compared with the LU decomposition, but it can only be computed for positive-definite matrices. The normal form for the least-squares problem is positive-definite so the decomposition is often handy when calculating least-squares problems. The following matrix equation was formed from the measurements. U= 2 4 6 Variables a and are the slope and the intercept of the regression line (H = at + B). By multiplying the matrix from both sides with the transpose of the coefficient matrix we end up with the normal form: y = [ 21 =A X = [a] 121 21 5 3 Let's compute the Cholesky decomposition for A. Input the upper triangular matrix U as an answer. Give only exact values. You can check that you have the correct decomposition by calculating UTU and making sure it equals A. Now solve the equation UT y = b for y. Give the exact answer. 8 139 ][²] = [¹28 Finally solve the equation Ux = y for x. Give the exact values here as well.
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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Question
![We have taken measurements of air humidity Has a function of time and we wish to fit a curve to the measurements. Let's
assume that the measurements depend linearly on time. The following table contains the measurements.
H34687
t 12468
Let's fit a line to the data using the least-squares method using the Cholesky decomposition. Cholesky decomposition is
computationally half as expensive compared with the LU decomposition, but it can only be computed for positive-definite
matrices. The normal form for the least-squares problem is positive-definite so the decomposition is often handy when
calculating least-squares problems. The following matrix equation was formed from the measurements.
1 1
=
4
6
8 1
Variables a and are the slope and the intercept of the regression line (H = at + B). By multiplying the matrix from both
sides with the transpose of the coefficient matrix we end up with the normal form:
y =
121
21
=A
X =
=
Now solve the equation UT y = b for y. Give the exact answer.
21 α
³¹] [8] =
5
3
Let's compute the Cholesky decomposition for A. Input the upper triangular matrix U as an answer. Give only exact values.
You can check that you have the correct decomposition by calculating UÃU and making sure it equals A.
6
8
139
28
Finally solve the equation Ux = y for x. Give the exact values here as well.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3802b3a4-f93d-4b7b-94b3-88a2c6183309%2F83d73fde-e9d4-4232-b0a7-d0a9a50e7820%2Flx4u9x6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:We have taken measurements of air humidity Has a function of time and we wish to fit a curve to the measurements. Let's
assume that the measurements depend linearly on time. The following table contains the measurements.
H34687
t 12468
Let's fit a line to the data using the least-squares method using the Cholesky decomposition. Cholesky decomposition is
computationally half as expensive compared with the LU decomposition, but it can only be computed for positive-definite
matrices. The normal form for the least-squares problem is positive-definite so the decomposition is often handy when
calculating least-squares problems. The following matrix equation was formed from the measurements.
1 1
=
4
6
8 1
Variables a and are the slope and the intercept of the regression line (H = at + B). By multiplying the matrix from both
sides with the transpose of the coefficient matrix we end up with the normal form:
y =
121
21
=A
X =
=
Now solve the equation UT y = b for y. Give the exact answer.
21 α
³¹] [8] =
5
3
Let's compute the Cholesky decomposition for A. Input the upper triangular matrix U as an answer. Give only exact values.
You can check that you have the correct decomposition by calculating UÃU and making sure it equals A.
6
8
139
28
Finally solve the equation Ux = y for x. Give the exact values here as well.
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