Science:Math Exam Resources/Courses/MATH307/December 2010/Question 05 (a)

MATH307 December 2010

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Question 05 (a)
Define a sequence x0, x1, ... by the initial conditions x0 = a, x1 = b and x2 = c together with the recursion relation $\displaystyle x_{n+3}=x_{n+2}+x_{n+1}+x_{n}$ for n = 0, 1, 2, ... (a) Rewrite this recursion in matrix form Xn+1 = AXn for n = 0, 1, 2, ... for a sequence Xn of vectors, with an initial vector X0 and some matrix A.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
What will be the (minimal) number of entries in the vector Xn and what will be the size of the matrix A?

Hint 2
The vector Xn must contain xn+2, xn+1 and xn, which suggest that Xn has length 3, and that hence A is a 3×3 matrix.

Hint 3
To get to three equations we can always add trivial equations such as xn+2 = xn+2.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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. As suggested in the hint we first add two trivial equations to help us rewrite the recursion relation in matrix form: ${\begin{alignedat}{7}x_{n+3}&&\;=\;&&x_{n+2}&&\;+\;&&x_{n+1}&&\;+\;&&x_{n}&\\x_{n+2}&&\;=\;&&x_{n+2}&&\;\;&&&&\;\;&&&\\x_{n+1}&&\;=\;&&&&\;\;&&x_{n+1}&&\;\;&&&\end{alignedat}}$ This can be written as: ${\begin{bmatrix}x_{n+3}\\x_{n+2}\\x_{n+1}\end{bmatrix}}={\begin{bmatrix}1&1&1\\1&0&0\\0&1&0\end{bmatrix}}{\begin{bmatrix}x_{n+2}\\x_{n+1}\\x_{n}\end{bmatrix}}$ So define $A={\begin{bmatrix}1&1&1\\1&0&0\\0&1&0\end{bmatrix}}$ and $X_{n}={\begin{bmatrix}x_{n+2}\\x_{n+1}\\x_{n}\end{bmatrix}}$. Plugging in n = 0 we see that the initial vector X0 contains the initial conditions:$X_{0}={\begin{bmatrix}c\\b\\a\end{bmatrix}}$.

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