Discrete Dynamical Fibonacci Edward Early. Fibonacci Numbers F 0 = 0, F 1 = 1, F n = F n-1 +F n-2 for n > 1 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,

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Presentation transcript:

Discrete Dynamical Fibonacci Edward Early

Fibonacci Numbers F 0 = 0, F 1 = 1, F n = F n-1 +F n-2 for n > 1 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Differential Equations inspired by undetermined coefficients F n = F n-1 +F n-2 characteristic polynomial x 2 -x-1 formula initial conditions F 0 = 0, F 1 = 1 give C 1 and C 2 BIG DRAWBACK: students were already taking undetermined coefficients on faith (and this application not in book)

Linear Algebra discrete dynamical systems where λ 1 and λ 2 are the (distinct) eigenvalues of the 2×2 matrix A with eigenvectors v 1 and v 2, respectively, and x 0 =c 1 v 1 +c 2 v 2 (Section 5.6 of Lay’s book)

Linear Algebra Meets Fibonacci Let and Thus A n x 0 has top entry F n

Linear Algebra Meets Fibonacci Let and det(A-λI) = λ 2 -λ-1 eigenvaluesand eigenvectors and

Linear Algebra Meets Fibonacci Let and top entry

Caveat If then… ugly enough to scare off most students!