Polynomial division

Numerical division can be split into two variants: floor division (numpoly.floor_divide()) and true division (numpoly.true_divide()):

>>> dividend = 7
>>> divisor = 2
>>> quotient_true = numpy.true_divide(dividend, divisor)
>>> quotient_true
3.5
>>> quotient_floor = numpy.floor_divide(dividend, divisor)
>>> quotient_floor
3

The discrepancy between the two can be captured by a remainder (numpoly.remainder()), which allow us to more formally define them as follows:

>>> remainder = numpy.remainder(dividend, divisor)
>>> remainder
1
>>> dividend == quotient_floor*divisor+remainder
True
>>> dividend == quotient_true*divisor
True

In the case of polynomials, neither true nor floor division is supported like this. Instead it support its own kind of polynomial division numpoly.poly_divide(). Polynomial division falls back to behave like floor division for all constants, as it does not round values:

>>> q0, q1 = numpoly.variable(2)
>>> dividend = q0**2+q1
>>> divisor = q0-1
>>> quotient = numpoly.poly_divide(dividend, divisor)
>>> quotient
polynomial(q0+1.0)

However, like floor division, it can still have remainders using numpoly.poly_remainder(). For example:

>>> remainder = numpoly.poly_remainder(dividend, divisor)
>>> remainder
polynomial(q1+1.0)
>>> dividend == quotient*divisor+remainder
True

In numpy, the “Python syntactic sugar” operators have the following behavior:

• / is used for true division numpoly.true_divide().

• // is used for floor division numpoly.floor_divide().

• % is used for remainder numpoly.remainder().

• divmod is used for floor division and remainder in combination to save computational cost through numpoly.divmod().

In numpoly, which takes precedence if any of the values are of numpoly.ndpoly objects, take the following behavior:

• / is used for polynomial division numpoly.poly_divide(), which is not compatible with numpy.

• // is still used for floor division numpoly.floor_divide() which is compatible with numpy, and only works if divisor is a constant.

• % is used for polynomial remainder numpoly.poly_remainder(), which is not backwards compatible.

• divmod uses numpoly.poly_divmod() which is used to save computation cost by doing numpoly.poly_divide() and numpoly.remainder() at the same time.