Documentation

Represents a rational function.

Values can be constructed with fromPolynomial and fromPolynomials, while the numerator and denominator can be obtained by pattern matching with Function. Internally, common factors will be canceled out and the denominator will be monic.

>>> fromPolynomial $ power 1 :: Function IndexedPolynomial
Function (x) (1)
>>> fromPolynomials (power 2 + 1) (power 3 + 1) :: Function IndexedPolynomial
Function (x^2 + 1) (x^3 + 1)
>>> let (Function p q) = fromPolynomials (power 2) (power 1 + 1) :: Function IndexedPolynomial
>>> (p, q)
(x^2,x + 1)
>>> fromPolynomials (4 * power 2 - 4) (2 * power 1 - 2 :: IndexedPolynomial)
Function (2x + 2) (1)

The type is an instance of the Fractional type class, so values can be added, subtracted, multiplied, and divided. Their values can also be given as integer literals as well.

>>> let p = fromPolynomials (power 2 + 1) (power 3 + 1) :: Function IndexedPolynomial
>>> let q = fromPolynomials (power 1) (power 2 - 1) :: Function IndexedPolynomial
>>> p + q
Function (2x^3 + (-2)x^2 + 2x + (-1)) (x^4 + (-1)x^3 + x + (-1))
>>> p * q
Function (x^3 + x) (x^5 + (-1)x^3 + x^2 + (-1))
>>> 0 :: Function IndexedPolynomial
Function (0) (1)
>>> 5 :: Function IndexedPolynomial
Function (5) (1)

One could import this module with an alias so that Rational.Function can be used as the name of the type for rational functions.

>>> import Symtegration.Polynomial.Rational as Rational
>>> fromPolynomials (power 1) (power 2 + 1) :: Rational.Function IndexedPolynomial
Function (x) (x^2 + 1)

Returns an equivalent rational function from a given polynomial.

>>> fromPolynomial (power 2 + 1 :: IndexedPolynomial)
Function (x^2 + 1) (1)

Returns a rational function with the ratio of two polynomials.

>>> fromPolynomials (power 1) (power 2 + 1 :: IndexedPolynomial)
Function (x) (x^2 + 1)

Returns an equivalent polynomial from a rational function if possible.

It returns the numerator if the denominator is a constant.

>>> toPolynomial $ fromPolynomials (2 * power 1 + 2) (2 :: IndexedPolynomial)
Just x + 1

It returns nothing if the rational function is not equivalent to a polynomial.

>>> toPolynomial $ fromPolynomials 1 (power 1 :: IndexedPolynomial)
Nothing

Pattern synonym for matching the numerator and denominator of a rational function. This would be equivalent to a data constructor, except it cannot be used for constructing a rational function.

>>> let (Function p q) = fromPolynomials (power 2) (power 1 + 1) :: Function IndexedPolynomial
>>> p
x^2
>>> q
x + 1