Time for action – computing Fibonacci numbers

A matrix can represent the Fibonacci recurrence relation. We can express the calculation of Fibonacci numbers as a repeated matrix multiplication:

1. Create the Fibonacci matrix as follows:

   F = np.matrix([[1, 1], [1, 0]])
   print("F", F)

   The Fibonacci matrix appears as follows:

   F [[1 1]
     [1 0]]
   

2. Calculate the 8th Fibonacci number (ignoring 0), by subtracting 1 from 8 and taking the power of the matrix. The Fibonacci number then appears on the diagonal:

   print("8th Fibonacci", (F ** 7)[0, 0])

   The Fibonacci number is as follows:

   8th Fibonacci 21
   

3. The golden ratio formula, better known as Binet's formula, allows us to calculate Fibonacci numbers with a rounding step at the end. Calculate the first eight Fibonacci numbers:

   n = np.arange(1, 9)
   sqrt5 = np.sqrt(5)
   phi = (1 + sqrt5)/2
   fibonacci = np.rint((phi**n - (-1/phi)**n)/sqrt5)
   print("Fibonacci", fibonacci)

   The first eight Fibonacci numbers are as follows:

   Fibonacci [...