Science:Math Exam Resources/Courses/MATH307/April 2012/Question 07 (a)

MATH307 April 2012

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[hide] Question 07 (a)
Consider the MATLAB/Octave computation 1> A=[0 2 1; 1 0 0; 0 1 0] A = 0 2 1 1 0 0 0 1 0 2> [S D]=eig(A) S = 0.80902 -0.57735 0.30902 0.50000 0.57735 -0.50000 0.30902 -0.57735 0.80902 D = Diagonal Matrix 1.61803 0 0 0 -1.00000 0 0 0 -0.61803 (a) Write down a recursion relation for a sequence x₀, x₁, x₂, x₃, ... that you can analyse using this calculation. How many initial values do you have to specify to define the sequence?

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[show] Hint
Science:Math Exam Resources/Courses/MATH307/April 2012/Question 07 (a)/Hint 1

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[show] 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. We can actually infer all the information we need to answer this question from the matrix A that was given. Let us first rewrite the matrix A in the context of how it is used: ${\begin{bmatrix}x_{n+1}\\x_{n}\\x_{n-1}\end{bmatrix}}={\begin{bmatrix}0&2&1\\1&0&0\\0&1&0\end{bmatrix}}{\begin{bmatrix}x_{n}\\x_{n-1}\\x_{n-2}\end{bmatrix}}$ This equation yields the following three linear equations: ${\begin{aligned}x_{n+1}&=2x_{n-1}+x_{n-2}\\x_{n}&=x_{n}\\x_{n-1}&=x_{n-1}\end{aligned}}$ The first equation of this set of linear equations tells us the recursive relationship for this matrix and notice that to calculate the value for the $n^{th}$ value in the relationship where $n>2$ and $n$ is an integer, we only need the first three values to be initially stated. In summary, the recursive sequence that is inferred from A is: $\displaystyle x_{n+1}=2x_{n-1}+x_{n-2}$ and we would need the first three values of the sequence to calculate all the following values in the sequence ( $x_{0},x_{1},$ and $x_{3}$ ).