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(a) Hand calculation: For the symbolic data on the left, derive the normal equations for solv- ing for the a, b, and c by the method of least squares. That is, define rı = 21- (ax1 +byj+ c), and similarly for r2, r3, and r4. Then let E = r? + r? + r? + rỉ, and take deriva- tives with respect to a, b, c. Hence derive a 3 x 3 system to solve for these unknowns. See pp. 2 5 of the Lecture 14 slides.

(a) Hand calculation: For the symbolic data on the left, derive the normal equations for solv- ing for the a, b, and c by the method of least squares. That is, define rı = 21- (ax1 +byj+ c), and similarly for r2, r3, and r4. Then let E = r? + r? + r? + rỉ, and take deriva- tives with respect to a, b, c. Hence derive a 3 x 3 system to solve for these unknowns. See pp. 2 5 of the Lecture 14 slides.

Advanced Engineering Mathematics
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

Consider the problem of fitting a plane
z = ax + by + c to data:

 

Please answer question a.

21 41 21 3. 12 2 1. T3 /3 23 一 24 0. 4. 201
Transcribed Image Text:21 41 21 3. 12 2 1. T3 /3 23 一 24 0. 4. 201
(a) Hand calculation: For the symbolic data on the left, derive the normal equations for solv- ing for the a, b, and c by the method of least squares. That is, define r¡ = 21-(ax1+byı+ c), and similarly for r2, r3, and r4. Then let E rř+ rž + rị + rỉ, and take deriva- %3D tives with respect to a, b, c. Hence derive a 3 x 3 system to solve for these unkmowns. See pp. 2 5 of the Lecture 14 slides. (b) For the numerical data on the right, solve the 3 x 3 system of part (a) with backslash. (c) Plot the plane along with the data on the same set of axes. Make sure all data points are visible. Show from two different view- points if necessary. (d) Also set up the problem as a 4 x 3 overde- termined system and solve with backslash. Confirm that the same solution as in part (b) is obtained.
Transcribed Image Text:(a) Hand calculation: For the symbolic data on the left, derive the normal equations for solv- ing for the a, b, and c by the method of least squares. That is, define r¡ = 21-(ax1+byı+ c), and similarly for r2, r3, and r4. Then let E rř+ rž + rị + rỉ, and take deriva- %3D tives with respect to a, b, c. Hence derive a 3 x 3 system to solve for these unkmowns. See pp. 2 5 of the Lecture 14 slides. (b) For the numerical data on the right, solve the 3 x 3 system of part (a) with backslash. (c) Plot the plane along with the data on the same set of axes. Make sure all data points are visible. Show from two different view- points if necessary. (d) Also set up the problem as a 4 x 3 overde- termined system and solve with backslash. Confirm that the same solution as in part (b) is obtained.
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