I want physics QAQ
特征方程学习笔记
Characteristic Equation Learning Note
(Mostly from other places….)
A great introduction of it is by Berkeley.
Our main theme of it is Characteristic Equations of Linear Recurrence Relations.
Primary
Recall from the Linear Recurrence Relations page that a linear recurrence relation of order k for the sequence $(fn){n=0}^{\infty} = (f_0,f_1,f_2,…)$ is a linear recurrence of the form:
$fn = \sum{i=1}^{k}aif{n-i} + F(n)$
For convenience , we just talk about the homogeneous recurrence which means $F(n) = 0$
For examble , $fn = f{n-1}+f_{n-2}$
We could shift items to the left.
$fn - f{n-1} - f_{n-2} = 0$
The Characteristic Equation is $x^2 - x - 1 = 0$
what is that mean ?
We could get $f_{n-1}$ by solving the equation.
$x = \frac{1 \pm \sqrt{5}}{2}$
And then we could write it as
$f_n = A(\frac{1 + \sqrt{5}}{2})^n + B(\frac{1 - \sqrt{5}}{2})^n ~(*)$
substitute $f_1$ and $f_2$ into $(*)$
We could get the formular of Fibonacci.
Why was that ?
Appearantly we need to understand it by principle of Characteristic Equations.
Advance
The first page of pdf has a clear explaination of it !
Actually all I want to say is in the essay , good luck !
(A little lazy today)