Page 117 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

3. Numerical Linear Algebra 105

The problem is to obtain the value of the 100th element of the above
mentioned sequence. This is achieved using Eigen decomposition as
described below.

th
Let P k be the value of the k position of the sequence.

P k = P k-1 + P k-2 +P k-3

Let us define the matrix U k = P k+2
P k+1
P k
P k+3

U k+1 = P k+2
P k+1

U k+1 = 1 1 1 U K
1 0 0
0 1 0

U 0 = 2
1
0
Let the matrix A be 1 1 1
1 0 1
0 1 0

Represent the vector U 0 as the linear combinations of Eigen vectors of
the matrix A.
The Eigen values and the corresponding Eigen vectors are computed and
displayed below.

Eigen values = [λ 1 λ 2 λ 3 ]
= [1.8393 -0.4196 + 0.6063i -0.4196 - 0.6063i]

Eigen vectors arranged in columnwise =[ E1 E2 E3] =

-0.8503 -0.1412 - 0.3752i -0.1412 + 0.3752i

-0.4623 -0.3094 + 0.4471i -0.3094 - 0.4471i

-0.2514 0.7374 0.7374