Matrix feature value => matrix power => general formula of the generalized Fibonacci Series

As we all know, the Fibonacci series are often involved in computers. In this paper, we use linear algebra to obtain the general formula of the generalized Fibonacci series.

In the broad sense, the definition of the Fibonacci series (custom, may not be rigorous enough, please advise) is as follows:

And a and B meet a ^ 2 + 4b> 0 (the reason is that the denominator is greater than 0)

 

The recurrence formula shows that:

(1)

 

 

 

 

 

 

 

 

 

 

OK! There are two inspirations for writing this blog:

1. we can also use the difference equation method to obtain the interpass formula, but I haven't learned it yet. I found that it is also possible to write the recursive relationship in the form of a matrix and then find the interpass, but it is difficult to construct this relationship, for example, type 1.

2. in a general algorithm, almost all the power of the matrix is obtained by the bipartite method. However, we do not know that there are ki linear-independent feature vectors under λ I (Ki is the repeated Number of λ I ), we can use a method similar to the diagonal shape, which is a pediatrics in advanced mathematics. Although the previous steps are a bit complex, they are used to calculate the power of the Matrix directly by means of "sharpening without mistake.

 

Update:

The recurrence formula found today can be simplified:

This makes it easier to calculate!

 

Update:

Do you want to be more exciting? Okay, just order n!

From:
Two blind spots of thinking and a general efficient solution to the extended series in Fibonacci data computing

 

 

Updating ing

 

Appendix: previous image of the Fibonacci sequence: fib and Exponential Explosion