TL;DR

Where I rediscover an old piece of code and remember something about Fibonacci numbers.

Updates: thanks to Andrew Solomon for pointing out that functions fib1 should actually ercurse to fib1 itself, not to fib. Thanks!

Fibonacci numbers?

If you don’t know what a Fibonacci number is, the Wikipedia page on them can be a good starting point. We will adopt the same convention as that page, i.e.:

\[F_0 = 0 \\ F_1 = 1 \\ F_n = F_{n - 2} + F_{n - 1} \quad (n > 1)\]

Fibonacci numbers are often used to explain some concepts in programming. I guess this comes from the fact that the generic formula is recursive (note the third line above, each item is defined in terms of the two previous ones), so it can be a good example for a recursive implementation.

Those who forget history are condemned to recode it

Let’s look at a naïve implementation in Perl:

sub fib1 ($n) {
    return $n if $n < 2;
    return fib1($n - 2) + fib1($n - 1);
}

This is pretty much a direct translation of the definition, so how come it is marked as naïve? Well… from a computational point of view, we’re calculating some of the items too many times. As an example, let’s look at what happens when we calculate $F_4$:

  F(4) --+-- F(3) --+-- F(2) --+-- F(1)
         |          |          |
         |          |          +-- F(0)
         |          |          
         |          +-- F(1)
         |
         +-- F(2) --+-- F(1)
                    |
                    +-- F(0)

We’re calling the function a lot of times, and calculating the same values over and over (e.g. $F_2$ is calculated twice).

So let’s remember

This is something that memoization can usually address pretty effectively, and Perl has a core module to do this, Memoize:

use Memoize;
memoize('fib1');
sub fib1 ($n) {
    return $n if $n < 2;
    return fib1($n - 2) + fib1($n - 1);
}

This dramatically reduces the number of calls, because it will reuse results as they are alredy available:

  F(4) --+-- F(3) --+-- F(2) --+-- F(1)
         |          |          |
         |          |          +-- F(0)
         |          |          
         |          +-- F(1)[cached]
         |
         +-- F(2)[cached]

This still requires calculating all values from $F_0$ up to $F_n$ though, so an iterative solution is probably even more readable at this point:

sub fib2 ($n) {
    my ($prev, $succ) = (0, 1);
    ($prev, $succ) = ($succ, $prev + $succ) while $n-- > 0;
    return $prev;
}

At each iteration, the successor is calculated as the currently available two values, and the curren value becomes the previous one. Maybe a tad idiomatic but I’d say it’s pretty readable even if you’re not much into Perl.

Where’s the blessing then?

Looking inside cglib, I noticed a module Fibonacci.pm. OK, I thought, I wonder why I wasted my time on this simHOLY COW!.

There’s a fibonacci_multiply sub. There’s a fibonacci_power sub. They are used to calculate the actual value for Fibonacci numbers by sub fibonacci_nth.

And then it hit me: Fibonacci numbers can be calculated with a suitable power of a basic matrix, as the Wikipedia page itself points out:

\[\left( \begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix} \right)^n = \left( \begin{matrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{matrix} \right)\]

So there you have it, I was blessed of forgetting about this cool fact… because I could discover it again!

So how does this help, you might ask?

Let’s consider how we can calculate the $n$-th power of an integer $i$. We might just apply the definition:

\[i^n = i \cdot i \cdot ... \cdot i\]

i.e. multiply $i$ by itself $n$ times, or (for example) observe that we can express $n$ in some fancy way and do some exponent manipulation. As an example, let’s consider $n = 8$; in this case:

\[i^8 = (i^4)^2 = ((i^2)^2)^2\]

which means that we can do much less than 7 multiplications:

\[i^2 = i \cdot i \\ i^4 = i^2 \cdot i^2 \\ i^8 = i^4 \cdot i^4\]

The same trick can be applied to the multiplications of the matrix for Fibonacci numbers, which leads to the code in Fibonacci.pm:

sub fibonacci_multiply {
   my ($x, $y) = @_;
   @$x = (
      $x->[0] * $y->[0] + $x->[1] * $y->[2],
      $x->[0] * $y->[1] + $x->[1] * $y->[3],
      $x->[2] * $y->[0] + $x->[3] * $y->[2],
      $x->[2] * $y->[1] + $x->[3] * $y->[3],
   );
} ## end sub _multiply

sub fibonacci_power {
   my ($q, $n, $q0) = (@_[0, 1], $_[2] || [@{$_[0]}]);
   return $q if $n < 2;
   fibonacci_power($q, int($n / 2), $q0);
   fibonacci_multiply($q, $q);
   fibonacci_multiply($q, $q0) if $n % 2;
   return $q;
} ## end sub _power

Here the matrix is represented by an array of four elements; this is OK in this case because the matrix is so little that we can code the multiplication explicitly in fibonacci_multiply.

The power function fibonacci_power applies our insight by recursing into the power calculation by always dividing the exponent by $2$, taking care to “account for the rest” in case $n$ is not a power of $2$. Annoying details, I know.

There’s more Perl to it

The module was a real (re)discovery anyway. Most probably I needed to calculate very big Fibonacci numbers, so bit that they did not fit in Perl’s basic integer type.

Easy to address: use Math::BigInt.

Well… not so fast. Using it means gaining a lot in terms of how big our integers can be, but losing in speed. I don’t want big numbers if I don’t need big numbers! Hence, there are two functions to calculate the $n$-th Fibonacci number: fibonacci_nth, for everyday needs, and fibonacci_nth_bi, for heavy duty needs. The latter is simple:

sub fibonacci_nth_bi {
   require Math::BigInt;
   return fibonacci_nth($_[0], Math::BigInt->new(1));
}

It calls the everyday one but provides… a specific definition for what we should consider for number $1$, i.e. what Math::BigInt considers for a $1$.

This also makes it clear why the argument unpacking in fibonacci_nth seems so overly complicated:

sub fibonacci_nth {
   my ($n, $one, $zero) = ($_[0], $_[1] || 1, ($_[1] || 1) - ($_[1] || 1));
   return
       $n < 1 ? $zero
     : $n < 3 ? $one
     :          fibonacci_power([$one, $one, $one, $zero], $n - 1)->[0];
} ## end sub nth

At first glance, you might think that I’ve gone mad by defining a value for $one and $zero, but now it’s clear! If I get passed what $1$ is, I’ll use it (via $one) and also define what $0$ is through it (this is easy: just subtract whatever we consider for $1$ from itself).

So, from now on we’re working only with Math::BigInt objects… and everything goes to its place!

Conclusions

I know I merely scratched the surface. There is so much about it, just look for it!