Create faster Fibonacci function for n > 100 in MATLAB / octave

Question

I have a function that tells me the nth number in a Fibonacci sequence. The problem is it becomes very slow when trying to find larger numbers in the Fibonacci sequence doe

Answer 1

One simple way to speed up the recursive implementation of a Fibonacci function is to realize that, substituting f(n-1) by its definition,

f(n) = f(n-1) + f(n-2)
     = f(n-2) + f(n-3) + f(n-2)
     = 2*f(n-2) + f(n-3)

This simple transformation greatly reduces the number of steps taken to compute a number in the series.

If we start with OP's code, slightly corrected:

function result = fibonacci(n)
switch n
case 0
   result = 0;
case 1
   result = 1;
case 2
   result = 1;
case 3
   result = 2;
otherwise
   result = fibonacci(n-2) + fibonacci(n-1);
end

And apply our transformation:

function result = fibonacci_fast(n)
switch n
case 0
   result = 0;
case 1
   result = 1;
case 2
   result = 1;
case 3
   result = 2;
otherwise
   result = fibonacci_fast(n-3) + 2*fibonacci_fast(n-2);
end

Then we see a 30x speed improvement for computing the 20th number in the series (using Octave):

>> tic; for ii=1:100, fibonacci(20); end; toc
Elapsed time is 12.4393 seconds.
>> tic; for ii=1:100, fibonacci_fast(20); end; toc
Elapsed time is 0.448623 seconds.

Of course Rashid's non-recursive implementation is another 60x faster still: 0.00706792 seconds.