Algebra.Field.AlgebraicReal

Algebraic Real Numbers for exact computation

Since 0.4.0.0

Documentation

data Algebraic

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

Instances

Eq Algebraic

Methods

(==) :: Algebraic -> Algebraic -> Bool

(/=) :: Algebraic -> Algebraic -> Bool

Fractional Algebraic

Methods

(/) :: Algebraic -> Algebraic -> Algebraic

recip :: Algebraic -> Algebraic

fromRational :: Rational -> Algebraic

Num Algebraic

Methods

(+) :: Algebraic -> Algebraic -> Algebraic

(-) :: Algebraic -> Algebraic -> Algebraic

(*) :: Algebraic -> Algebraic -> Algebraic

negate :: Algebraic -> Algebraic

abs :: Algebraic -> Algebraic

signum :: Algebraic -> Algebraic

fromInteger :: Integer -> Algebraic

Ord Algebraic

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

Methods

showsPrec :: Int -> Algebraic -> ShowS

show :: Algebraic -> String

showList :: [Algebraic] -> ShowS

InvolutiveMultiplication Algebraic

Methods

adjoint :: Algebraic -> Algebraic

TriviallyInvolutive Algebraic

Commutative Algebraic

UnitNormalForm Algebraic

Methods

splitUnit :: Algebraic -> (Algebraic, Algebraic)

ZeroProductSemiring Algebraic

Ring Algebraic

Methods

fromInteger :: Integer -> Algebraic

Rig Algebraic

Methods

fromNatural :: Natural -> Algebraic

DecidableZero Algebraic

Methods

isZero :: Algebraic -> Bool

DecidableUnits Algebraic

Methods

recipUnit :: Algebraic -> Maybe Algebraic

isUnit :: Algebraic -> Bool

(^?) :: Integral n => Algebraic -> n -> Maybe Algebraic

DecidableAssociates Algebraic

Methods

isAssociate :: Algebraic -> Algebraic -> Bool

Division Algebraic

Methods

recip :: Algebraic -> Algebraic

(/) :: Algebraic -> Algebraic -> Algebraic

(\\) :: Algebraic -> Algebraic -> Algebraic

(^) :: Integral n => Algebraic -> n -> Algebraic

Unital Algebraic

Methods

one :: Algebraic

pow :: Algebraic -> Natural -> Algebraic

productWith :: Foldable f => (a -> Algebraic) -> f a -> Algebraic

Group Algebraic

Methods

(-) :: Algebraic -> Algebraic -> Algebraic

negate :: Algebraic -> Algebraic

subtract :: Algebraic -> Algebraic -> Algebraic

times :: Integral n => n -> Algebraic -> Algebraic

Multiplicative Algebraic

Methods

(*) :: Algebraic -> Algebraic -> Algebraic

pow1p :: Algebraic -> Natural -> Algebraic

productWith1 :: Foldable1 f => (a -> Algebraic) -> f a -> Algebraic

Semiring Algebraic

Monoidal Algebraic

Methods

zero :: Algebraic

sinnum :: Natural -> Algebraic -> Algebraic

sumWith :: Foldable f => (a -> Algebraic) -> f a -> Algebraic

Additive Algebraic

Methods

(+) :: Algebraic -> Algebraic -> Algebraic

sinnum1p :: Natural -> Algebraic -> Algebraic

sumWith1 :: Foldable1 f => (a -> Algebraic) -> f a -> Algebraic

Abelian Algebraic

LeftModule Integer Algebraic

Methods

(.*) :: Integer -> Algebraic -> Algebraic

LeftModule Natural Algebraic

Methods

(.*) :: Natural -> Algebraic -> Algebraic

RightModule Integer Algebraic

Methods

(*.) :: Algebraic -> Integer -> Algebraic

RightModule Natural Algebraic

Methods

(*.) :: Algebraic -> Natural -> Algebraic

LeftModule (Fraction Integer) Algebraic

Methods

(.*) :: Fraction Integer -> Algebraic -> Algebraic

LeftModule (Scalar (Fraction Integer)) Algebraic

Methods

(.*) :: Scalar (Fraction Integer) -> Algebraic -> Algebraic

RightModule (Fraction Integer) Algebraic

Methods

(*.) :: Algebraic -> Fraction Integer -> Algebraic

RightModule (Scalar (Fraction Integer)) Algebraic

Methods

(*.) :: Algebraic -> Scalar (Fraction Integer) -> Algebraic

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

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

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

See also nthRoot'.

nthRoot' :: Int -> Algebraic -> Algebraic

Unsafe version of nthRoot.

improve :: Algebraic -> Algebraic

improve r returns the same algebraic number, but with more tighter bounds.

approximate :: Rational -> Algebraic -> Rational

approximate eps r returns rational number r' close to r, with abs (r - r') < eps.

approxFractional :: Fractional r => Rational -> Algebraic -> r

Same as approximate, but returns Fractional value instead.

Equation solver

realRoots :: Unipol Rational -> [Algebraic]

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

complexRoots :: Unipol Rational -> [Complex Algebraic]

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

Constructors

Interval

Fields

• lower :: !r
• upper :: !r

Instances

Eq r => Eq (Interval r)

Methods

(==) :: Interval r -> Interval r -> Bool

(/=) :: Interval r -> Interval r -> Bool

Ord r => Ord (Interval r)

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)

Methods

showsPrec :: Int -> Interval r -> ShowS

show :: Interval r -> String

showList :: [Interval r] -> ShowS

(Ord r, Multiplicative r, Monoidal r) => Multiplicative (Interval r)

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)

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

Choose representative element of the given interval.

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

Test if the former interval includes the latter.

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

Takes intersection of two intervals.

Internal utility functions

strum :: Unipol Rational -> [Unipol Rational]

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

Pseudo resultant. should we expose this?

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