Comments on: Fast exponentiation

By: Nemo

Nemo — Sat, 06 Feb 2010 17:42:02 +0000

@Jaime –

If it is an integer matrix, diagonalization might not be the best idea.

For example, the powers of the following matrix generate the Fibonacci numbers:

(1 1)
(1 0)

Using the “repeated squaring” technique, this matrix lets you compute the Nth Fibonacci number in logarithmic time using only integer arithmetic.

If you diagonalize this matrix, you will find the golden ratio and its reciprocal along the diagonal. Since those are irrational, they are not precisely representable on your computer. (Well, not with the usual floating-point representations, anyway…) That will lead to rounding errors when you exponentiate.

By: Modular Exponentiation « Numerical Recipes

Modular Exponentiation « Numerical Recipes — Tue, 19 May 2009 21:45:47 +0000

[...] introduce the really cool algorithm that this whole post really is about, binary exponentiation. In John D. Cook’s blog there is a great explanation on how it could be implemented iteratively, based on the binary [...]

By: Jaime

Jaime — Tue, 17 Mar 2009 23:58:35 +0000

Hmmmm… I had always thought that the standard procedure to calculate arbitrary power of a matrix was>a href=”http://mathworld.wolfram.com/MatrixDiagonalization.html” title=”diagonalization”>. That is, if you can write [A] as [B^-1][D][B], then [A^n] reduces to [B^-1][D^n][B], since all other [B^-1]s and [B]s cancel out. But of course diagonalizing a large matrix is equivalent to finding all roots of a very large polynomial… For very large exponents diagonalization has to be better, but do you have any hint of at what point both approaches break even?

By the way, you’ve got a great blog, John…

By: Gene

Gene — Thu, 11 Dec 2008 05:21:09 +0000

I ran across this package (Pari/GP) which has very large integer capabilities. You’ve probably heard of it or used it, but I found it fascinating. It seems to be well maintained. I used it for testing some encryption algorithms some time ago. I know it’s a little off topic, but some might find it interesting.

http://pari.math.u-bordeaux.fr/

I also want to point out that the newest “math coprocessor” architectures built into dual core Intel CPU’s use some fascinating optimizations for large integer arithmetic, similar to what you describe in your post.