How to explain this algorithm for calculating the power of a number?

Question

I am currently reading Skiena\'s \"The Algorithm Design Manual\".

He describes an algorithm for calculating the power of a number i.e. calculate a^n.

Answer 1

This formula a^n = ((a^n/2)^2), which you understand dictates recursive algorithm.

To get a^n you need first to calculate a^(n/2),
To get a^(n/2), you need to calculate a^((n/2)/2),
... and so on, until (n/2/2/...2) reaches 0, which is the termination condition of the recursion.

So, the recursive algorithm follows that formula exactly :

to get power(a,n) you first recursively calculate power(a,n/2) and then return the result adjusting for n being odd/even number.

There is also wikipedia article about this implementation.