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February 11th, 2018, 02:09 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 190 Thanks: 2  Using Eigen Values and Vectors for computing complex arithmetic series
The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, .... In general, $\displaystyle F_{k+2} = F_{k+1} + F_{k} $. Here is some background info, we derive the following matrix for computing $\displaystyle F_{100} $: $\displaystyle \mathbf{A} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} $ $\displaystyle \begin{bmatrix} F_{k+2} \\ F_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^{k} \begin{bmatrix} F_{k+1} \\ F_{k} \end{bmatrix} $ Using EigenValues, EigenVector matrix(S) and Diagonal matrix(D), we can compute the $\displaystyle F_{100}$. $\displaystyle \mathbf{A}^{k} = \mathbf{S} \mathbf{D}^{k} \mathbf{S}^{1} $ $\displaystyle \begin{bmatrix} F_{100} \\ F_{99} \end{bmatrix} = \mathbf{S} \mathbf{D}^{99} \mathbf{S}^{1} \begin{bmatrix} F_{2} \\ F_{1} \end{bmatrix} $ I have been using this technique to solve a number of standard arthmetric series and geometric series easily. Then I challenged myself and scouted online for some interesting recursive series. I stumbled the following: Hofstadter's QSequence $\displaystyle Q(n) = Q(n  Q(n  1)) + Q(n  Q(n  2)) \\ Q(1) = Q(2) = 1 \\ 1, 1, 2, 3, 3, 4, 5, 5, 6, 6, .... $ I have been working on this series for a couple of hours without any luck. Would anyone able to figure this out by using the same technique? Is it even possible to solve this problem with this technique alone? Last edited by zollen; February 11th, 2018 at 02:12 PM. 
February 11th, 2018, 08:10 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,100 Thanks: 1093 
the sequence relies on the nonlinear function $Q()$ and so you're not going to be able to generate this via a set of linear transformations.


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arithmetic, complex, computing, eigen, series, values, vectors 
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