Data.Poly.Multi.Laurent

Sparse multivariate Laurent polynomials.

Documentation

data MultiLaurent (v :: Type -> Type) (n :: Nat) (a :: Type)

Sparse Laurent polynomials of n variables with coefficients from a, backed by a Vector v (boxed, unboxed, storable, etc.).

Use the patterns X, Y, Z and the ^- operator for construction:

>>> (X + 1) + (Y^-1 - 1) :: VMultiLaurent 2 Integer
1 * X + 1 * Y^-1
>>> (X + 1) * (Z - X^-1) :: UMultiLaurent 3 Int
1 * X * Z + 1 * Z + (-1) + (-1) * X^-1

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

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

Due to being polymorphic by multiple axis, the performance of MultiLaurent crucially depends on specialisation of instances. Clients are strongly recommended to compile with ghc-options: -fspecialise-aggressively and suggested to enable -O2.

Since: 0.5.0.0

Instances

(Eq a, Ring a, GcdDomain a, Vector v (Vector 1 Word, a)) => GcdDomain (Laurent v a)

Defined in Data.Poly.Internal.Multi.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, KnownNat n, Vector v (Vector n Int, a), Vector v (Vector n Word, a)) => IsList (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Associated Types

type Item (MultiLaurent v n a)

Methods

fromList :: [Item (MultiLaurent v n a)] -> MultiLaurent v n a

fromListN :: Int -> [Item (MultiLaurent v n a)] -> MultiLaurent v n a

toList :: MultiLaurent v n a -> [Item (MultiLaurent v n a)]

(Eq a, Num a, KnownNat n, Vector v (Vector n Word, a)) => Num (MultiLaurent v n a)

Note that abs = id and signum = const 1.

Defined in Data.Poly.Internal.Multi.Laurent

Methods

(+) :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

(-) :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

(*) :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

negate :: MultiLaurent v n a -> MultiLaurent v n a

abs :: MultiLaurent v n a -> MultiLaurent v n a

signum :: MultiLaurent v n a -> MultiLaurent v n a

fromInteger :: Integer -> MultiLaurent v n a

(Show a, KnownNat n, Vector v (Vector n Word, a)) => Show (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Methods

showsPrec :: Int -> MultiLaurent v n a -> ShowS

show :: MultiLaurent v n a -> String

showList :: [MultiLaurent v n a] -> ShowS

NFData (v (Vector n Word, a)) => NFData (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Methods

rnf :: MultiLaurent v n a -> ()

Eq (v (Vector n Word, a)) => Eq (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Methods

(==) :: MultiLaurent v n a -> MultiLaurent v n a -> Bool

(/=) :: MultiLaurent v n a -> MultiLaurent v n a -> Bool

Ord (v (Vector n Word, a)) => Ord (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Methods

compare :: MultiLaurent v n a -> MultiLaurent v n a -> Ordering

(<) :: MultiLaurent v n a -> MultiLaurent v n a -> Bool

(<=) :: MultiLaurent v n a -> MultiLaurent v n a -> Bool

(>) :: MultiLaurent v n a -> MultiLaurent v n a -> Bool

(>=) :: MultiLaurent v n a -> MultiLaurent v n a -> Bool

max :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

min :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

(Eq a, Ring a, GcdDomain a, KnownNat n, forall (m :: Nat). KnownNat m => Vector v (Vector m Word, a), forall (m :: Nat). KnownNat m => Eq (v (Vector m Word, a))) => GcdDomain (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Methods

divide :: MultiLaurent v n a -> MultiLaurent v n a -> Maybe (MultiLaurent v n a)

gcd :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

lcm :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

coprime :: MultiLaurent v n a -> MultiLaurent v n a -> Bool

(Eq a, Ring a, KnownNat n, Vector v (Vector n Word, a)) => Ring (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Methods

negate :: MultiLaurent v n a -> MultiLaurent v n a

(Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => Semiring (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

Methods

plus :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

zero :: MultiLaurent v n a

times :: MultiLaurent v n a -> MultiLaurent v n a -> MultiLaurent v n a

one :: MultiLaurent v n a

fromNatural :: Natural -> MultiLaurent v n a

type Item (MultiLaurent v n a)

Defined in Data.Poly.Internal.Multi.Laurent

type Item (MultiLaurent v n a) = (Vector n Int, a)

type VMultiLaurent (n :: Nat) (a :: Type) = MultiLaurent Vector n a

Multivariate Laurent polynomials backed by boxed vectors.

Since: 0.5.0.0

type UMultiLaurent (n :: Nat) (a :: Type) = MultiLaurent Vector n a

Multivariate Laurent polynomials backed by unboxed vectors.

Since: 0.5.0.0

unMultiLaurent :: MultiLaurent v n a -> (Vector n Int, MultiPoly v n a)

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

>>> unMultiLaurent (2 * X + 1 :: UMultiLaurent 2 Int)
(Vector [0,0],2 * X + 1)
>>> unMultiLaurent (1 + 2 * X^-1 :: UMultiLaurent 2 Int)
(Vector [-1,0],1 * X + 2)
>>> unMultiLaurent (2 * X^2 + X :: UMultiLaurent 2 Int)
(Vector [1,0],2 * X + 1)
>>> unMultiLaurent (0 :: UMultiLaurent 2 Int)
(Vector [0,0],0)

Since: 0.5.0.0

toMultiLaurent :: (KnownNat n, Vector v (Vector n Word, a)) => Vector n Int -> MultiPoly v n a -> MultiLaurent v n a

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

>>> :set -XDataKinds
>>> import Data.Vector.Generic.Sized (fromTuple)
>>> toMultiLaurent (fromTuple (2, 0)) (2 * Data.Poly.Multi.X + 1) :: UMultiLaurent 2 Int
2 * X^3 + 1 * X^2
>>> toMultiLaurent (fromTuple (0, -2)) (2 * Data.Poly.Multi.X + 1) :: UMultiLaurent 2 Int
2 * X * Y^-2 + 1 * Y^-2

monomial :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => Vector n Int -> a -> MultiLaurent v n a

Create a monomial from a power and a coefficient.

Since: 0.5.0.0

scale :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => Vector n Int -> a -> MultiLaurent v n a -> MultiLaurent v n a

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

>>> :set -XDataKinds
>>> import Data.Vector.Generic.Sized (fromTuple)
>>> scale (fromTuple (1, 1)) 3 (X^-2 + Y) :: UMultiLaurent 2 Int
3 * X * Y^2 + 3 * X^-1 * Y

Since: 0.5.0.0

pattern X :: (Eq a, Semiring a, KnownNat n, 1 <= n, Vector v (Vector n Word, a)) => MultiLaurent v n a

Create a polynomial equal to the first variable.

Since: 0.5.0.0

pattern Y :: (Eq a, Semiring a, KnownNat n, 2 <= n, Vector v (Vector n Word, a)) => MultiLaurent v n a

Create a polynomial equal to the second variable.

Since: 0.5.0.0

pattern Z :: (Eq a, Semiring a, KnownNat n, 3 <= n, Vector v (Vector n Word, a)) => MultiLaurent v n a

Create a polynomial equal to the third variable.

Since: 0.5.0.0

(^-) :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a)) => MultiLaurent v n a -> Int -> MultiLaurent v n a

Used to construct monomials with negative powers.

This operator can be applied only to monomials with unit coefficients, but is instrumental to express Laurent polynomials in a mathematical fashion:

>>> X^-3 * Y^-1 :: UMultiLaurent 2 Int
1 * X^-3 * Y^-1
>>> 3 * X^-1 + 2 * (Y^2)^-2 :: UMultiLaurent 2 Int
2 * Y^-4 + 3 * X^-1

Since: 0.5.0.0

eval :: (Field a, Vector v (Vector n Word, a), Vector u a) => MultiLaurent v n a -> Vector u n a -> a

Evaluate the polynomial at a given point.

>>> :set -XDataKinds
>>> import Data.Vector.Generic.Sized (fromTuple)
>>> eval (X^2 + Y^-1 :: UMultiLaurent 2 Double) (fromTuple (3, 4) :: Data.Vector.Sized.Vector 2 Double)
9.25

Since: 0.5.0.0

subst :: (Eq a, Semiring a, KnownNat n, Vector v (Vector n Word, a), Vector w (Vector n Word, a)) => MultiPoly v n a -> Vector n (MultiLaurent w n a) -> MultiLaurent w n a

Substitute another polynomial instead of X.

>>> :set -XDataKinds
>>> import Data.Vector.Generic.Sized (fromTuple)
>>> import Data.Poly.Multi (UMultiPoly)
>>> subst (Data.Poly.Multi.X * Data.Poly.Multi.Y :: UMultiPoly 2 Int) (fromTuple (X + Y^-1, Y + X^-1 :: UMultiLaurent 2 Int))
1 * X * Y + 2 + 1 * X^-1 * Y^-1

Since: 0.5.0.0

deriv :: (Eq a, Ring a, KnownNat n, Vector v (Vector n Word, a)) => Finite n -> MultiLaurent v n a -> MultiLaurent v n a

Take the derivative of the polynomial with respect to the i-th variable.

>>> :set -XDataKinds
>>> deriv 0 (X^3 + 3 * Y) :: UMultiLaurent 2 Int
3 * X^2
>>> deriv 1 (X^3 + 3 * Y) :: UMultiLaurent 2 Int
3

Since: 0.5.0.0

segregate :: (KnownNat m, Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => MultiLaurent v (1 + m) a -> VLaurent (MultiLaurent v m a)

Interpret a multivariate Laurent polynomial over 1+m variables as a univariate Laurent polynomial, whose coefficients are multivariate Laurent polynomials over the last m variables.

Since: 0.5.0.0

unsegregate :: forall v m a. (KnownNat m, KnownNat (1 + m), Vector v (Vector (1 + m) Word, a), Vector v (Vector m Word, a)) => VLaurent (MultiLaurent v m a) -> MultiLaurent v (1 + m) a

Interpret a univariate Laurent polynomials, whose coefficients are multivariate Laurent polynomials over the first m variables, as a multivariate polynomial over 1+m variables.

Since: 0.5.0.0