Math.Polynomial.VectorSpace

Same general interface as Math.Polynomial, but using AdditiveGroup, VectorSpace, etc., instead of Num where sensible.

Documentation

data Endianness

Constructors

BE

Big-Endian (head is highest-order term)

LE

Little-Endian (head is const term)

Instances

Bounded Endianness

Enum Endianness

Eq Endianness

Ord Endianness

Show Endianness

NFData Endianness

data Poly a

Instances

Functor Poly

(AdditiveGroup a, Eq a) => Eq (Poly a)

(Num a, Eq a) => Num (Poly a)

Show a => Show (Poly a)

NFData a => NFData (Poly a)

(Pretty a, Num a, Ord a) => Pretty (Poly a)

(RealFloat a, Pretty (Complex a)) => Pretty (Poly (Complex a))

(Eq a, VectorSpace a, AdditiveGroup (Scalar a), Eq (Scalar a)) => VectorSpace (Poly a)

AdditiveGroup a => AdditiveGroup (Poly a)

type Scalar (Poly a) = Scalar a

poly :: (Eq a, AdditiveGroup a) => Endianness -> [a] -> Poly a

polyDegree :: (Eq a, AdditiveGroup a) => Poly a -> Int

vPolyCoeffs :: (Eq a, AdditiveGroup a) => Endianness -> Poly a -> [a]

Get the coefficients of a a Poly in the specified order.

polyIsZero :: (Eq a, AdditiveGroup a) => Poly a -> Bool

polyIsOne :: (Num a, Eq a) => Poly a -> Bool

zero :: Poly a

The polynomial "0"

one :: (Num a, Eq a) => Poly a

The polynomial "1"

constPoly :: (Eq a, AdditiveGroup a) => a -> Poly a

Given some constant k, construct the polynomial whose value is constantly k.

x :: (Num a, Eq a) => Poly a

The polynomial (in x) "x"

scalePoly :: (Eq a, VectorSpace a, AdditiveGroup (Scalar a), Eq (Scalar a)) => Scalar a -> Poly a -> Poly a

Given some scalar s and a polynomial f, computes the polynomial g such that:

evalPoly g x = s * evalPoly f x

negatePoly :: (AdditiveGroup a, Eq a) => Poly a -> Poly a

Given some polynomial f, computes the polynomial g such that:

evalPoly g x = negate (evalPoly f x)

composePolyWith :: (AdditiveGroup a, Eq a) => (a -> a -> a) -> Poly a -> Poly a -> Poly a

composePoly f g constructs the polynomial h such that:

evalPoly h = evalPoly f . evalPoly g

This is a very expensive operation and, in general, returns a polynomial that is quite a bit more expensive to evaluate than f and g together (because it is of a much higher order than either). Unless your polynomials are quite small or you are quite certain you need the coefficients of the composed polynomial, it is recommended that you simply evaluate f and g and explicitly compose the resulting functions. This will usually be much more efficient.

addPoly :: (AdditiveGroup a, Eq a) => Poly a -> Poly a -> Poly a

Given polynomials f and g, computes the polynomial h such that:

evalPoly h x = evalPoly f x + evalPoly g x

sumPolys :: (AdditiveGroup a, Eq a) => [Poly a] -> Poly a

multPolyWith :: (AdditiveGroup a, Eq a) => (a -> a -> a) -> Poly a -> Poly a -> Poly a

Given polynomials f and g, computes the polynomial h such that:

evalPoly h x = evalPoly f x * evalPoly g x

powPolyWith :: (AdditiveGroup a, Eq a, Integral b) => a -> (a -> a -> a) -> Poly a -> b -> Poly a

Given a polynomial f and exponent n, computes the polynomial g such that:

evalPoly g x = evalPoly f x ^ n

quotRemPolyWith :: (AdditiveGroup a, Eq a) => (a -> a -> a) -> (a -> a -> a) -> Poly a -> Poly a -> (Poly a, Poly a)

Given polynomials a and b, with b not zero, computes polynomials q and r such that:

addPoly (multPoly q b) r == a

quotPolyWith :: (AdditiveGroup a, Eq a) => (a -> a -> a) -> (a -> a -> a) -> Poly a -> Poly a -> Poly a

remPolyWith :: (AdditiveGroup a, Eq a) => (a -> a -> a) -> (a -> a -> a) -> Poly a -> Poly a -> Poly a

evalPoly :: (VectorSpace a, Eq a, AdditiveGroup (Scalar a), Eq (Scalar a)) => Poly a -> Scalar a -> a

Evaluate a polynomial at a point or, equivalently, convert a polynomial to the function it represents. For example, evalPoly x = id and evalPoly (constPoly k) = const k.

evalPolyDeriv :: (VectorSpace a, Eq a) => Poly a -> Scalar a -> (a, a)

Evaluate a polynomial and its derivative (respectively) at a point.

evalPolyDerivs :: (VectorSpace a, Eq a, Num (Scalar a)) => Poly a -> Scalar a -> [a]

Evaluate a polynomial and all of its nonzero derivatives at a point. This is roughly equivalent to:

evalPolyDerivs p x = map (`evalPoly` x) (takeWhile (not . polyIsZero) (iterate polyDeriv p))

contractPoly :: (VectorSpace a, Eq a) => Poly a -> Scalar a -> (Poly a, a)

"Contract" a polynomial by attempting to divide out a root.

contractPoly p a returns (q,r) such that q*(x-a) + r == p

monicPolyWith :: (AdditiveGroup a, Eq a) => a -> (a -> a -> a) -> Poly a -> Poly a

Normalize a polynomial so that its highest-order coefficient is 1

gcdPolyWith :: (AdditiveGroup a, Eq a) => a -> (a -> a -> a) -> (a -> a -> a) -> Poly a -> Poly a -> Poly a

gcdPoly a b computes the highest order monic polynomial that is a divisor of both a and b. If both a and b are zero, the result is undefined.

polyDeriv :: (VectorSpace a, Eq a, Num (Scalar a)) => Poly a -> Poly a

Compute the derivative of a polynomial.

polyDerivs :: (VectorSpace a, Eq a, Num (Scalar a)) => Poly a -> [Poly a]

Compute all nonzero derivatives of a polynomial, starting with its "zero'th derivative", the original polynomial itself.

polyIntegral :: (VectorSpace a, Eq a, Fractional (Scalar a)) => Poly a -> Poly a

Compute the definite integral (from 0 to x) of a polynomial.