/ x; if n = 1 Power(x, n) = { Power(x^{2}, n/2); if n is even \ x * Power(x^{2}, (n1)/2); if n > 2 is odd
Compared to the ordinary method of multiplying x with itself n1 times, this algorithm uses only O(lg n) multiplications and therefore speeds up the computation of x^{n} tremendously, in much the same way that the "long multiplication" algorithm speeds up multiplication over the slower method of repeated addition.
The method works in every semigroup and is often used to compute powers of matrices, and, especially in cryptography, to compute powers in a ring of integers modulo q. It can also be used to compute integer powers in a group, using the rule Power(x, n) = (Power(x, n))^{1}.
This is an implementation of the above algorithm in the Ruby programming language. It doesn't use recursion, which increases the speed even further.
In most languages you'll need to replace result=1 with result=unit_matrix_of_the_same_size_as_x to get a matrix exponentiating algorithm. In Ruby, thanks to coercion, result is automatically upgraded to the appropriate type, so this function works with matrices as well as with integers and floats.
def power(x,n) result = 1 while (n != 0) # if n is odd, multiply result with x if ((n % 2) == 1) then result = result * x end x = x*x n = n/2 end return result end
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