Computing Fibonacci

"The proof of this identity probably requires mathematical knowledge that is beyond my current capacity"

Not at all!

Let M = [[1 1] ; [1 0]], the matrix under discussion. What does M do to a column vector v = [x ; y] under the rules of matrix multiplication?

    M*v = [x+y ; x].

But what is this transformation, in terms of the input and output vectors? It's the same transformation as the Fibonacci transformation! We take [current, previous] --> [current + previous, current].

This tells us that multiplying the matrix n times will give us a matrix that gives the same result as applying the Fibonacci transformation n times:

    M * (M * v) = (M * M) v , 

etc. (Think of the left hand side as applying the transformation twice, one after the other, and the right hand side as applying once a single transformation that has the same effect as two Fibonacci transformations.)

Now this tells us that M^n [1 ; 0] (the n^th power of the matrix M, multiplied by the initial state vector Fib_1 = 1, Fib_0 = 0), equals [Fib_n ; Fib_{n-1}].

You should be able to work backwards from that to see that M^n must have the entries specified, since matrix multiplication is just a simple algebraic process.

I've probably made some sort of off-by-one error here. But that's the idea.

What this suggests is that any method of computing M^n will work to give you Fib_n. You could try repeated matrix multiplications, but why not an adaptation of the standard fast exponentiation algorithm? To compute M^k, either square M^(k/2) (if k is odd) or multiply M by M^(k-1) (if k is even). So M^13 would be

    M * M^12 = M * (M^6)^2 = M * ((M^3)^2) ^2 = M * ((M^2 * M)^2)^2, 

done in 5 multiplications instead of 12 the naive way

    (M * M * M * ... * M).

This is done in chapter 1, exercise 19 of SICP, although they never explicitly admit that the transformation under discussion is a linear transformation or write down its associated matrix. http://mitpress.mit.edu/sicp/full-text/book/book-Z-H-11.html...

By the way, this view of matrices--that they express transformations that can be applied to vectors, and that transformations that can be written down as matrices have special properties and can be manipulated and composed to form new transformations with the same properties--is why linear algebra will knock your socks off in the right hands. (By contrast, if all you're told is that a matrix is what we call it when you line numbers up in a pretty little row, you will begin to hate your math class.)

The naive recursive implementation + memoization would be a cool addition. :)

See also [1], which, however, does not provide a recursive version.

[1] http://stackoverflow.com/a/427810/48015