
Linear Algebra Linear Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 11th, 2018, 02:09 PM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 173 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: 1,852 Thanks: 959 
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.


Tags 
arithmetic, complex, computing, eigen, series, values, vectors 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Eigen values for nxn matrix  didierB  Abstract Algebra  2  January 13th, 2013 04:29 PM 
QR factorization Eigen values  atee  Linear Algebra  1  February 11th, 2012 03:01 PM 
Computing values  guru123  Algebra  0  November 17th, 2011 08:11 AM 
Calculating Eigen Values error  Singularity  Linear Algebra  2  April 18th, 2010 01:49 PM 
Interval Arithmetic and Reliable Computing  YDVIPER  Computer Science  1  December 10th, 2007 05:14 AM 