Fibonacci

1
Fibonacci

• A non-recursive formula for the nth term in the
  Fibonacci sequence.

2
The rules(Rabbit-Pair Model)

• Put one pair of rabbits on an island.
• Each pair takes one month to mature.
• Each mature pair gives birth to one pair of
  rabbits every month.
• No rabbits die.
• After n months, how many pairs are there?

3
The sequence
Non-Recursive

• Recursive
• f01, f11
• fnfn-1fn-2 for ngt2
• 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ..
• To find the 1000th element requires knowing the
  preceeding 999 elements.

How do you derive this formula?
4
The golden ratio
x
1
x is to one as one is to x1
5
The Taylor Series

• This series is convergent by the root test over
  the interval
• We can use the Taylor series.

6
The Plan

• Another form for F(x)
• Find the Polynomial Roots
• Partial fractions
• Convert to series
• Compare the series to find an
• Simplify

7
Another form for F(x)
or
8
Finding roots
9
Partial Fractions
10
Convert to Series
11
Isolate equation
Rule there can be only one power Series
representation for a function, therefore
12
Simplify
13
Results for 1,000,000

• Recursive method
• 3,014,605 additions
• At one per second its 34 days, 21 hours
• 33.007 seconds in MatLab

• Tame Method
• 2.503 seconds in MatLab
• Faster?

At n10,000 there is a 13.18x decrease in
execution time.
14
Numerical approximation??
Nope!

• Approximations
• Polynomial
• Logarithmic
• Power
• Exponential