Safe Haskell | None |
---|---|
Language | Haskell98 |
- data Endianness
- data Poly a
- poly :: (Num a, Eq a) => Endianness -> [a] -> Poly a
- polyDegree :: (Num a, Eq a) => Poly a -> Int
- polyCoeffs :: (Num a, Eq a) => Endianness -> Poly a -> [a]
- polyIsZero :: (Num a, Eq a) => Poly a -> Bool
- polyIsOne :: (Num a, Eq a) => Poly a -> Bool
- zero :: Poly a
- one :: (Num a, Eq a) => Poly a
- constPoly :: (Num a, Eq a) => a -> Poly a
- x :: (Num a, Eq a) => Poly a
- scalePoly :: (Num a, Eq a) => a -> Poly a -> Poly a
- negatePoly :: (Num a, Eq a) => Poly a -> Poly a
- composePoly :: (Num a, Eq a) => Poly a -> Poly a -> Poly a
- addPoly :: (Num a, Eq a) => Poly a -> Poly a -> Poly a
- sumPolys :: (Num a, Eq a) => [Poly a] -> Poly a
- multPoly :: (Num a, Eq a) => Poly a -> Poly a -> Poly a
- powPoly :: (Num a, Eq a, Integral b) => Poly a -> b -> Poly a
- quotRemPoly :: (Fractional a, Eq a) => Poly a -> Poly a -> (Poly a, Poly a)
- quotPoly :: (Fractional a, Eq a) => Poly a -> Poly a -> Poly a
- remPoly :: (Fractional a, Eq a) => Poly a -> Poly a -> Poly a
- evalPoly :: (Num a, Eq a) => Poly a -> a -> a
- evalPolyDeriv :: (Num a, Eq a) => Poly a -> a -> (a, a)
- evalPolyDerivs :: (Num a, Eq a) => Poly a -> a -> [a]
- contractPoly :: (Num a, Eq a) => Poly a -> a -> (Poly a, a)
- monicPoly :: (Fractional a, Eq a) => Poly a -> Poly a
- gcdPoly :: (Fractional a, Eq a) => Poly a -> Poly a -> Poly a
- separateRoots :: (Fractional a, Eq a) => Poly a -> [Poly a]
- polyDeriv :: (Num a, Eq a) => Poly a -> Poly a
- polyDerivs :: (Num a, Eq a) => Poly a -> [Poly a]
- polyIntegral :: (Fractional a, Eq a) => Poly a -> Poly a
Documentation
data Endianness Source #
Functor Poly Source # | |
(AdditiveGroup a, Eq a) => Eq (Poly a) Source # | |
Show a => Show (Poly a) Source # | |
NFData a => NFData (Poly a) Source # | |
(Eq a, VectorSpace a, AdditiveGroup (Scalar a), Eq (Scalar a)) => VectorSpace (Poly a) Source # | |
AdditiveGroup a => AdditiveGroup (Poly a) Source # | |
type Scalar (Poly a) Source # | |
poly :: (Num a, Eq a) => Endianness -> [a] -> Poly a Source #
Make a Poly
from a list of coefficients using the specified coefficient order.
polyDegree :: (Num a, Eq a) => Poly a -> Int Source #
Get the degree of a a Poly
(the highest exponent with nonzero coefficient)
polyCoeffs :: (Num a, Eq a) => Endianness -> Poly a -> [a] Source #
Get the coefficients of a a Poly
in the specified order.
constPoly :: (Num a, Eq a) => a -> Poly a Source #
Given some constant k
, construct the polynomial whose value is
constantly k
.
scalePoly :: (Num a, Eq a) => a -> Poly a -> Poly a Source #
Given some scalar s
and a polynomial f
, computes the polynomial g
such that:
evalPoly g x = s * evalPoly f x
negatePoly :: (Num a, Eq a) => Poly a -> Poly a Source #
Given some polynomial f
, computes the polynomial g
such that:
evalPoly g x = negate (evalPoly f x)
composePoly :: (Num a, Eq a) => Poly a -> Poly a -> Poly a Source #
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 :: (Num a, Eq a) => Poly a -> Poly a -> Poly a Source #
Given polynomials f
and g
, computes the polynomial h
such that:
evalPoly h x = evalPoly f x + evalPoly g x
multPoly :: (Num a, Eq a) => Poly a -> Poly a -> Poly a Source #
Given polynomials f
and g
, computes the polynomial h
such that:
evalPoly h x = evalPoly f x * evalPoly g x
powPoly :: (Num a, Eq a, Integral b) => Poly a -> b -> Poly a Source #
Given a polynomial f
and exponent n
, computes the polynomial g
such that:
evalPoly g x = evalPoly f x ^ n
quotRemPoly :: (Fractional a, Eq a) => Poly a -> Poly a -> (Poly a, Poly a) Source #
Given polynomials a
and b
, with b
not zero
, computes polynomials
q
and r
such that:
addPoly (multPoly q b) r == a
evalPolyDeriv :: (Num a, Eq a) => Poly a -> a -> (a, a) Source #
Evaluate a polynomial and its derivative (respectively) at a point.
evalPolyDerivs :: (Num a, Eq a) => Poly a -> a -> [a] Source #
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 :: (Num a, Eq a) => Poly a -> a -> (Poly a, a) Source #
"Contract" a polynomial by attempting to divide out a root.
contractPoly p a
returns (q,r)
such that q*(x-a) + r == p
monicPoly :: (Fractional a, Eq a) => Poly a -> Poly a Source #
Normalize a polynomial so that its highest-order coefficient is 1
gcdPoly :: (Fractional a, Eq a) => Poly a -> Poly a -> Poly a Source #
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.
separateRoots :: (Fractional a, Eq a) => Poly a -> [Poly a] Source #
Separate a nonzero polynomial into a set of factors none of which have multiple roots, and the product of which is the original polynomial. Note that if division is not exact, it may fail to separate roots. Rational coefficients is a good idea.
Useful when applicable as a way to simplify root-finding problems.
polyDerivs :: (Num a, Eq a) => Poly a -> [Poly a] Source #
Compute all nonzero derivatives of a polynomial, starting with its "zero'th derivative", the original polynomial itself.
polyIntegral :: (Fractional a, Eq a) => Poly a -> Poly a Source #
Compute the definite integral (from 0 to x) of a polynomial.