Generic implementation of powering

This implements powering of arbitrary objects using a square-and-multiply algorithm.

sage.arith.power.generic_power(a, n)

Return \(a^n\).

If \(n\) is negative, return \((1/a)^(-n)\).

INPUT:

• a – any object supporting multiplication (and division if n < 0)

• n – any integer (in the duck typing sense)

EXAMPLES:

sage: from sage.arith.power import generic_power
sage: generic_power(int(12), int(0))
1
sage: generic_power(int(0), int(100))
0
sage: generic_power(Integer(10), Integer(0))
1
sage: generic_power(Integer(0), Integer(23))
0
sage: sum([generic_power(2,i) for i in range(17)]) #test all 4-bit combinations
131071
sage: F = Zmod(5)
sage: a = generic_power(F(2), 5); a
2
sage: a.parent() is F
True
sage: a = generic_power(F(1), 2)
sage: a.parent() is F
True

sage: generic_power(int(5), 0)
1
sage: generic_power(2, 5/4)
Traceback (most recent call last):
...
NotImplementedError: non-integral exponents not supported

sage: class SymbolicMul(str):
....:     def __mul__(self, other):
....:         s = "({}*{})".format(self, other)
....:         return type(self)(s)
sage: x = SymbolicMul("x")
sage: print(generic_power(x, 7))
(((x*x)*(x*x))*((x*x)*x))