Data.Poly.Laurent

Laurent polynomials.

Documentation

data Laurent v a

Laurent polynomials of one variable with coefficients from a, backed by a Vector v (boxed, unboxed, storable, etc.).

Use pattern X and operator ^- for construction:

>>> (X + 1) + (X^-1 - 1) :: VLaurent Integer
1 * X + 0 + 1 * X^-1
>>> (X + 1) * (1 - X^-1) :: ULaurent Int
1 * X + 0 + (-1) * X^-1

Polynomials are stored normalized, without leading and trailing zero coefficients, so 0 * X + 1 + 0 * X^-1 equals to 1.

Ord instance does not make much sense mathematically, it is defined only for the sake of Set, Map, etc.

Instances

(Eq a, Eq (v a), Field a, Vector v a) => GcdDomain (LaurentOverField (Laurent v a))

Defined in Data.Poly.Laurent

Methods

divide :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> Maybe (LaurentOverField (Laurent v a))

gcd :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a)

lcm :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a)

coprime :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> Bool

Eq (v a) => Eq (Laurent v a)

Defined in Data.Poly.Laurent

Methods

(==) :: Laurent v a -> Laurent v a -> Bool

(/=) :: Laurent v a -> Laurent v a -> Bool

(Eq a, Num a, Vector v a) => Num (Laurent v a)

Note that abs = id and signum = const 1.

Defined in Data.Poly.Laurent

Methods

(+) :: Laurent v a -> Laurent v a -> Laurent v a

(-) :: Laurent v a -> Laurent v a -> Laurent v a

(*) :: Laurent v a -> Laurent v a -> Laurent v a

negate :: Laurent v a -> Laurent v a

abs :: Laurent v a -> Laurent v a

signum :: Laurent v a -> Laurent v a

fromInteger :: Integer -> Laurent v a

Ord (v a) => Ord (Laurent v a)

Defined in Data.Poly.Laurent

Methods

compare :: Laurent v a -> Laurent v a -> Ordering

(<) :: Laurent v a -> Laurent v a -> Bool

(<=) :: Laurent v a -> Laurent v a -> Bool

(>) :: Laurent v a -> Laurent v a -> Bool

(>=) :: Laurent v a -> Laurent v a -> Bool

max :: Laurent v a -> Laurent v a -> Laurent v a

min :: Laurent v a -> Laurent v a -> Laurent v a

(Show a, Vector v a) => Show (Laurent v a)

Defined in Data.Poly.Laurent

Methods

showsPrec :: Int -> Laurent v a -> ShowS

show :: Laurent v a -> String

showList :: [Laurent v a] -> ShowS

NFData (v a) => NFData (Laurent v a)

Defined in Data.Poly.Laurent

Methods

rnf :: Laurent v a -> ()

(Eq a, Ring a, GcdDomain a, Eq (v a), Vector v a) => GcdDomain (Laurent v a)

Consider using LaurentOverField wrapper, which provides a much faster implementation of gcd for polynomials over Field.

Defined in Data.Poly.Laurent

Methods

divide :: Laurent v a -> Laurent v a -> Maybe (Laurent v a)

gcd :: Laurent v a -> Laurent v a -> Laurent v a

lcm :: Laurent v a -> Laurent v a -> Laurent v a

coprime :: Laurent v a -> Laurent v a -> Bool

(Eq a, Semiring a, Vector v a) => Semiring (Laurent v a)

Defined in Data.Poly.Laurent

Methods

plus :: Laurent v a -> Laurent v a -> Laurent v a

zero :: Laurent v a

times :: Laurent v a -> Laurent v a -> Laurent v a

one :: Laurent v a

fromNatural :: Natural -> Laurent v a

(Eq a, Ring a, Vector v a) => Ring (Laurent v a)

Defined in Data.Poly.Laurent

Methods

negate :: Laurent v a -> Laurent v a

type VLaurent = Laurent Vector

Laurent polynomials backed by boxed vectors.

type ULaurent = Laurent Vector

Laurent polynomials backed by unboxed vectors.

unLaurent :: Laurent v a -> (Int, Poly v a)

Deconstruct a Laurent polynomial into an offset (largest possible) and a regular polynomial.

>>> unLaurent (2 * X + 1 :: ULaurent Int)
(0,2 * X + 1)
>>> unLaurent (1 + 2 * X^-1 :: ULaurent Int)
(-1,1 * X + 2)
>>> unLaurent (2 * X^2 + X :: ULaurent Int)
(1,2 * X + 1)
>>> unLaurent (0 :: ULaurent Int)
(0,0)

toLaurent :: (Eq a, Semiring a, Vector v a) => Int -> Poly v a -> Laurent v a

Construct Laurent polynomial from an offset and a regular polynomial. One can imagine it as scale', but allowing negative offsets.

>>> toLaurent 2 (2 * Data.Poly.X + 1) :: ULaurent Int
2 * X^3 + 1 * X^2
>>> toLaurent (-2) (2 * Data.Poly.X + 1) :: ULaurent Int
2 * X^-1 + 1 * X^-2

leading :: Vector v a => Laurent v a -> Maybe (Int, a)

Return a leading power and coefficient of a non-zero polynomial.

>>> leading ((2 * X + 1) * (2 * X^2 - 1) :: ULaurent Int)
Just (3,4)
>>> leading (0 :: ULaurent Int)
Nothing

monomial :: (Eq a, Semiring a, Vector v a) => Int -> a -> Laurent v a

Create a monomial from a power and a coefficient.

scale :: (Eq a, Semiring a, Vector v a) => Int -> a -> Laurent v a -> Laurent v a

Multiply a polynomial by a monomial, expressed as a power and a coefficient.

>>> scale 2 3 (X^2 + 1) :: ULaurent Int
3 * X^4 + 0 * X^3 + 3 * X^2 + 0 * X + 0

pattern X :: (Eq a, Semiring a, Vector v a, Eq (v a)) => Laurent v a

Create an identity polynomial.

(^-) :: (Eq a, Semiring a, Vector v a, Eq (v a)) => Laurent v a -> Int -> Laurent v a

This operator can be applied only to X, but is instrumental to express Laurent polynomials in mathematical fashion:

>>> X + 2 + 3 * X^-1 :: ULaurent Int
1 * X + 2 + 3 * X^(-1)

eval :: (Field a, Vector v a) => Laurent v a -> a -> a

Evaluate at a given point.

>>> eval (X^2 + 1 :: ULaurent Int) 3
10

subst :: (Eq a, Semiring a, Vector v a, Vector w a) => Poly v a -> Laurent w a -> Laurent w a

Substitute another polynomial instead of X.

>>> subst (X^2 + 1 :: UPoly Int) (X + 1 :: ULaurent Int)
1 * X^2 + 2 * X + 2

deriv :: (Eq a, Ring a, Vector v a) => Laurent v a -> Laurent v a

Take a derivative.

>>> deriv (X^3 + 3 * X) :: ULaurent Int
3 * X^2 + 0 * X + 3

newtype LaurentOverField laurent

Wrapper for Laurent polynomials over Field, providing a faster GcdDomain instance.

Constructors

LaurentOverField

Fields

• unLaurentOverField :: laurent

Instances

Eq laurent => Eq (LaurentOverField laurent)

Defined in Data.Poly.Laurent

Methods

(==) :: LaurentOverField laurent -> LaurentOverField laurent -> Bool

(/=) :: LaurentOverField laurent -> LaurentOverField laurent -> Bool

Num laurent => Num (LaurentOverField laurent)

Defined in Data.Poly.Laurent

Methods

(+) :: LaurentOverField laurent -> LaurentOverField laurent -> LaurentOverField laurent

(-) :: LaurentOverField laurent -> LaurentOverField laurent -> LaurentOverField laurent

(*) :: LaurentOverField laurent -> LaurentOverField laurent -> LaurentOverField laurent

negate :: LaurentOverField laurent -> LaurentOverField laurent

abs :: LaurentOverField laurent -> LaurentOverField laurent

signum :: LaurentOverField laurent -> LaurentOverField laurent

fromInteger :: Integer -> LaurentOverField laurent

Ord laurent => Ord (LaurentOverField laurent)

Defined in Data.Poly.Laurent

Methods

compare :: LaurentOverField laurent -> LaurentOverField laurent -> Ordering

(<) :: LaurentOverField laurent -> LaurentOverField laurent -> Bool

(<=) :: LaurentOverField laurent -> LaurentOverField laurent -> Bool

(>) :: LaurentOverField laurent -> LaurentOverField laurent -> Bool

(>=) :: LaurentOverField laurent -> LaurentOverField laurent -> Bool

max :: LaurentOverField laurent -> LaurentOverField laurent -> LaurentOverField laurent

min :: LaurentOverField laurent -> LaurentOverField laurent -> LaurentOverField laurent

Show laurent => Show (LaurentOverField laurent)

Defined in Data.Poly.Laurent

Methods

showsPrec :: Int -> LaurentOverField laurent -> ShowS

show :: LaurentOverField laurent -> String

showList :: [LaurentOverField laurent] -> ShowS

NFData laurent => NFData (LaurentOverField laurent)

Defined in Data.Poly.Laurent

Methods

rnf :: LaurentOverField laurent -> ()

(Eq a, Eq (v a), Field a, Vector v a) => GcdDomain (LaurentOverField (Laurent v a))

Defined in Data.Poly.Laurent

Methods

divide :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> Maybe (LaurentOverField (Laurent v a))

gcd :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a)

lcm :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a)

coprime :: LaurentOverField (Laurent v a) -> LaurentOverField (Laurent v a) -> Bool

Semiring laurent => Semiring (LaurentOverField laurent)

Defined in Data.Poly.Laurent

Methods

plus :: LaurentOverField laurent -> LaurentOverField laurent -> LaurentOverField laurent

zero :: LaurentOverField laurent

times :: LaurentOverField laurent -> LaurentOverField laurent -> LaurentOverField laurent

one :: LaurentOverField laurent

fromNatural :: Natural -> LaurentOverField laurent

Ring laurent => Ring (LaurentOverField laurent)

Defined in Data.Poly.Laurent

Methods

negate :: LaurentOverField laurent -> LaurentOverField laurent