Safe Haskell	None
Language	Haskell2010

Algebra.Field.AlgebraicReal

Contents

Operations
Equation solver
Interval arithmetic
Internal utility functions

Description

Algebraic Real Numbers for exact computation

Since 0.4.0.0

Synopsis

data Algebraic
algebraic :: Unipol Rational -> Interval Rational -> Maybe Algebraic
nthRoot :: Int -> Algebraic -> Maybe Algebraic
nthRoot' :: Int -> Algebraic -> Algebraic
improve :: Algebraic -> Algebraic
approximate :: Rational -> Algebraic -> Rational
approxFractional :: Fractional r => Rational -> Algebraic -> r
realRoots :: Unipol Rational -> [Algebraic]
complexRoots :: Unipol Rational -> [Complex Algebraic]
data Interval r = Interval {
- lower :: !r
- upper :: !r
}
representative :: (Additive r, Division r, Num r) => Interval r -> r
includes :: Ord a => Interval a -> Interval a -> Bool
intersect :: (Monoidal a, Ord a) => Interval a -> Interval a -> Interval a
strum :: Unipol Rational -> [Unipol Rational]
presultant :: (Euclidean k, CoeffRing k) => Unipol k -> Unipol k -> k
factors :: Unipol Rational -> [Unipol Rational]
realPartPoly :: Unipol Rational -> Unipol Rational
imagPartPoly :: Unipol Rational -> Unipol Rational
sqFreePart :: (Eq r, Euclidean r, Division r) => Unipol r -> Unipol r

Documentation

data Algebraic Source #

Algebraic real numbers, which can be expressed as a root of a rational polynomial.

Instances

Instances details

Eq Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (==) :: Algebraic -> Algebraic -> Bool # (/=) :: Algebraic -> Algebraic -> Bool #
Fractional Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (/) :: Algebraic -> Algebraic -> Algebraic # recip :: Algebraic -> Algebraic # fromRational :: Rational -> Algebraic #
Num Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (+) :: Algebraic -> Algebraic -> Algebraic # (-) :: Algebraic -> Algebraic -> Algebraic # (*) :: Algebraic -> Algebraic -> Algebraic # negate :: Algebraic -> Algebraic # abs :: Algebraic -> Algebraic # signum :: Algebraic -> Algebraic # fromInteger :: Integer -> Algebraic #
Ord Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods compare :: Algebraic -> Algebraic -> Ordering # (<) :: Algebraic -> Algebraic -> Bool # (<=) :: Algebraic -> Algebraic -> Bool # (>) :: Algebraic -> Algebraic -> Bool # (>=) :: Algebraic -> Algebraic -> Bool # max :: Algebraic -> Algebraic -> Algebraic # min :: Algebraic -> Algebraic -> Algebraic #
Show Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods showsPrec :: Int -> Algebraic -> ShowS # show :: Algebraic -> String # showList :: [Algebraic] -> ShowS #
ZeroProductSemiring Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal
UnitNormalForm Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods splitUnit :: Algebraic -> (Algebraic, Algebraic) #
Ring Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods fromInteger :: Integer -> Algebraic
Rig Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods fromNatural :: Natural -> Algebraic #
DecidableZero Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods isZero :: Algebraic -> Bool #
DecidableUnits Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods recipUnit :: Algebraic -> Maybe Algebraic # isUnit :: Algebraic -> Bool # (^?) :: Integral n => Algebraic -> n -> Maybe Algebraic #
DecidableAssociates Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods isAssociate :: Algebraic -> Algebraic -> Bool #
Unital Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods one :: Algebraic # pow :: Algebraic -> Natural -> Algebraic # productWith :: Foldable f => (a -> Algebraic) -> f a -> Algebraic #
TriviallyInvolutive Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal
InvolutiveMultiplication Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods adjoint :: Algebraic -> Algebraic #
Division Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods recip :: Algebraic -> Algebraic # (/) :: Algebraic -> Algebraic -> Algebraic # (\\) :: Algebraic -> Algebraic -> Algebraic # (^) :: Integral n => Algebraic -> n -> Algebraic
Commutative Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal
Semiring Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal
Multiplicative Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (*) :: Algebraic -> Algebraic -> Algebraic # pow1p :: Algebraic -> Natural -> Algebraic # productWith1 :: Foldable1 f => (a -> Algebraic) -> f a -> Algebraic #
Monoidal Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods zero :: Algebraic # sinnum :: Natural -> Algebraic -> Algebraic # sumWith :: Foldable f => (a -> Algebraic) -> f a -> Algebraic #
Group Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (-) :: Algebraic -> Algebraic -> Algebraic # negate :: Algebraic -> Algebraic # subtract :: Algebraic -> Algebraic -> Algebraic # times :: Integral n => n -> Algebraic -> Algebraic #
Additive Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (+) :: Algebraic -> Algebraic -> Algebraic # sinnum1p :: Natural -> Algebraic -> Algebraic # sumWith1 :: Foldable1 f => (a -> Algebraic) -> f a -> Algebraic #
Abelian Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal
RightModule Integer Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (*.) :: Algebraic -> Integer -> Algebraic #
RightModule Natural Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (*.) :: Algebraic -> Natural -> Algebraic #
LeftModule Integer Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (.*) :: Integer -> Algebraic -> Algebraic #
LeftModule Natural Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (.*) :: Natural -> Algebraic -> Algebraic #
RightModule (Fraction Integer) Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (*.) :: Algebraic -> Fraction Integer -> Algebraic #
RightModule (Scalar (Fraction Integer)) Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (*.) :: Algebraic -> Scalar (Fraction Integer) -> Algebraic #
LeftModule (Fraction Integer) Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (.*) :: Fraction Integer -> Algebraic -> Algebraic #
LeftModule (Scalar (Fraction Integer)) Algebraic Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (.*) :: Scalar (Fraction Integer) -> Algebraic -> Algebraic #

algebraic :: Unipol Rational -> Interval Rational -> Maybe Algebraic Source #

Smart constructor. algebraic f i represents the unique root of rational polynomial f in the interval i. If no root is found, or more than one root belongs to the given interval, returns Nothing.

Operations

nthRoot :: Int -> Algebraic -> Maybe Algebraic Source #

nthRoot n r tries to computes n-th root of the given algebraic real r. It returns Nothing if it's undefined.

Equation solver

realRoots :: Unipol Rational -> [Algebraic] Source #

realRoots f finds all real roots of the rational polynomial f.

complexRoots :: Unipol Rational -> [Complex Algebraic] Source #

realRoots f finds all complex roots of the rational polynomial f.

CAUTION: This function currently comes with really naive implementation. Easy to explode.

Interval arithmetic

data Interval r Source #

Constructors

Interval
Fields lower :: !r upper :: !r

Instances

Instances details

Eq r => Eq (Interval r) Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (==) :: Interval r -> Interval r -> Bool # (/=) :: Interval r -> Interval r -> Bool #
Ord r => Ord (Interval r) Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods compare :: Interval r -> Interval r -> Ordering # (<) :: Interval r -> Interval r -> Bool # (<=) :: Interval r -> Interval r -> Bool # (>) :: Interval r -> Interval r -> Bool # (>=) :: Interval r -> Interval r -> Bool # max :: Interval r -> Interval r -> Interval r # min :: Interval r -> Interval r -> Interval r #
Show r => Show (Interval r) Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods showsPrec :: Int -> Interval r -> ShowS # show :: Interval r -> String # showList :: [Interval r] -> ShowS #
(Ord r, Multiplicative r, Monoidal r) => Multiplicative (Interval r) Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (*) :: Interval r -> Interval r -> Interval r # pow1p :: Interval r -> Natural -> Interval r # productWith1 :: Foldable1 f => (a -> Interval r) -> f a -> Interval r #
Group r => Additive (Interval r) Source #
Instance details Defined in Algebra.Field.AlgebraicReal Methods (+) :: Interval r -> Interval r -> Interval r # sinnum1p :: Natural -> Interval r -> Interval r # sumWith1 :: Foldable1 f => (a -> Interval r) -> f a -> Interval r #

representative :: (Additive r, Division r, Num r) => Interval r -> r Source #

Choose representative element of the given interval.

includes :: Ord a => Interval a -> Interval a -> Bool Source #

Test if the former interval includes the latter.

intersect :: (Monoidal a, Ord a) => Interval a -> Interval a -> Interval a Source #

Takes intersection of two intervals.

Internal utility functions

strum :: Unipol Rational -> [Unipol Rational] Source #

presultant :: (Euclidean k, CoeffRing k) => Unipol k -> Unipol k -> k Source #

Pseudo resultant. should we expose this?

factors :: Unipol Rational -> [Unipol Rational] Source #

realPartPoly :: Unipol Rational -> Unipol Rational Source #

imagPartPoly :: Unipol Rational -> Unipol Rational Source #

sqFreePart :: (Eq r, Euclidean r, Division r) => Unipol r -> Unipol r Source #