Science:Math Exam Resources/Courses/MATH221/December 2009/Question 03 (b)

MATH221 December 2009

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Question 03 (b)
Problem 3. Consider the traffic flow diagram: <INSERT PICTURE HERE> The system is clearly inconsistent because the total in flow does not equal total outflow. Find all least squares solutions to the system.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. See Q3 (a) for the flow diagram and system of equations. The least squares equation takes the form: $D^{T}D\mathbf {x} =D^{T}\mathbf {b}$ In general, $D^{T}D$, has a diagonal of the number of edges connected to the respective node and has off-diagonals entries of -1 if the nodes are connected and 0 if not. It turns out that all of the nodes are connected to 2 other nodes by some edge. Simplifying the least squares equation we have: ${\begin{bmatrix}2&-1&-1\\-1&2&-1\\-1&-1&2\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}450\\-450\\0\end{bmatrix}}$ Solving this system of equations through row reduction yields the following least squares solution: ${\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}}={\begin{bmatrix}1\\1\\1\end{bmatrix}}x_{3}+{\begin{bmatrix}150\\-150\\0\end{bmatrix}}$