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Section 2.6 Exponentiation Algorithm
We present an algorithm for computing a power of an integer. We call this algorithm the
Naive Exponentiation algorithm, since there is a more clever way of calculating powers which we will present with
Algorithm 15.22 .
Algorithm 2.44 . Naive Exponentiation for Integers.
Input:
An integer
\(b\) and a non-negative integer
\(n\)
Output:
if \(n=0\) then return \(1\)
In this algorithm, the number of steps in the sequence of computations,
\(n\text{,}\) is directly given as one of the inputs. We demonstrate this algorithm with a numerical example.
Example 2.45 . Computing \(5^3\) with Algorithm 2.44.
Input:
\(b=5\) and
\(n=3\)
1 . As
\(3\ne 0\) we continue with
step 2 .
4.a .
let \(c:=1\cdot 5=5\)
5 . As
\(i=1\) and
\(n=3\) the statement
\(i=n\) is false. We continue with
step 4.a .
4.a .
let \(c:=5\cdot 5=25\)
5 . As
\(i=2\) and
\(n=3\) the statement
\(i=n\) is false. We continue with
step 4.a .
4.a .
let \(c:=5\cdot 25=125\)
5 . As
\(i=3\) and
\(n=3\) we have
\(i=n\text{.}\) We continue with
step 6 .
6 . We return the value of
\(c\) which is 125.
We have computed
\(5^3=125\text{.}\)
Work through further the exponentiation algorithm with other input values in
Example 2.46 .
Example 2.46 . Exponentiation algorithm interactive.
Checkpoint 2.47 . Use Algorithm 2.44.
