Copyright	(c) Masahiro Sakai 2012-2013
License	BSD-style
Maintainer	masahiro.sakai@gmail.com
Stability	provisional
Portability	non-portable
Safe Haskell	Safe-Inferred
Language	Haskell2010
Extensions	RankNTypes ExplicitForAll

ToySolver.Data.AlgebraicNumber.Real

Contents

Algebraic real type
Construction
Properties
Operations
Approximation
Misc

Description

Algebraic reals

Reference:

Why the concept of a field extension is a natural one http://www.dpmms.cam.ac.uk/~wtg10/galois.html

Synopsis

data AReal
realRoots :: UPolynomial Rational -> [AReal]
realRootsEx :: UPolynomial AReal -> [AReal]
minimalPolynomial :: AReal -> UPolynomial Rational
isolatingInterval :: AReal -> Interval Rational
isRational :: AReal -> Bool
isAlgebraicInteger :: AReal -> Bool
height :: AReal -> Integer
rootIndex :: AReal -> Int
nthRoot :: Integer -> AReal -> AReal
refineIsolatingInterval :: AReal -> AReal
approx :: AReal -> Rational -> Rational
approxInterval :: AReal -> Rational -> Interval Rational
simpARealPoly :: UPolynomial AReal -> UPolynomial Rational
goldenRatio :: AReal

Algebraic real type

data AReal Source #

Algebraic real numbers.

Instances

Instances details

Num AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods (+) :: AReal -> AReal -> AReal # (-) :: AReal -> AReal -> AReal # (*) :: AReal -> AReal -> AReal # negate :: AReal -> AReal # abs :: AReal -> AReal # signum :: AReal -> AReal # fromInteger :: Integer -> AReal #
Fractional AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods (/) :: AReal -> AReal -> AReal # recip :: AReal -> AReal # fromRational :: Rational -> AReal #
Real AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods toRational :: AReal -> Rational #
RealFrac AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods properFraction :: Integral b => AReal -> (b, AReal) # truncate :: Integral b => AReal -> b # round :: Integral b => AReal -> b # ceiling :: Integral b => AReal -> b # floor :: Integral b => AReal -> b #
Show AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods showsPrec :: Int -> AReal -> ShowS # show :: AReal -> String # showList :: [AReal] -> ShowS #
Eq AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods (==) :: AReal -> AReal -> Bool # (/=) :: AReal -> AReal -> Bool #
Ord AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods compare :: AReal -> AReal -> Ordering # (<) :: AReal -> AReal -> Bool # (<=) :: AReal -> AReal -> Bool # (>) :: AReal -> AReal -> Bool # (>=) :: AReal -> AReal -> Bool # max :: AReal -> AReal -> AReal # min :: AReal -> AReal -> AReal #
Pretty AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods pPrintPrec :: PrettyLevel -> Rational -> AReal -> Doc # pPrint :: AReal -> Doc # pPrintList :: PrettyLevel -> [AReal] -> Doc #
Degree AReal Source #	Degree of the algebraic number. If the algebraic number's `minimalPolynomial` has degree `n`, then the algebraic number is said to be degree `n`.
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods deg :: AReal -> Integer Source #
PrettyCoeff AReal Source #
Instance details Defined in ToySolver.Data.AlgebraicNumber.Real Methods pPrintCoeff :: PrettyLevel -> Rational -> AReal -> Doc Source # isNegativeCoeff :: AReal -> Bool Source #

Construction

realRoots :: UPolynomial Rational -> [AReal] Source #

Real roots of the polynomial in ascending order.

realRootsEx :: UPolynomial AReal -> [AReal] Source #

Real roots of the polynomial in ascending order.

Properties

minimalPolynomial :: AReal -> UPolynomial Rational Source #

The polynomial of which the algebraic number is root.

isolatingInterval :: AReal -> Interval Rational Source #

Isolating interval that separate the number from other roots of minimalPolynomial of it.

isRational :: AReal -> Bool Source #

Whether the algebraic number is a rational.

isAlgebraicInteger :: AReal -> Bool Source #

Whether the algebraic number is a root of a polynomial with integer coefficients with leading coefficient 1 (a monic polynomial).

height :: AReal -> Integer Source #

Height of the algebraic number.

The height of an algebraic number is the greatest absolute value of the coefficients of the irreducible and primitive polynomial with integral rational coefficients.

rootIndex :: AReal -> Int Source #

root index, satisfying

realRoots (minimalPolynomial a) !! rootIndex a == a

Operations

nthRoot :: Integer -> AReal -> AReal Source #

The nth root of a

refineIsolatingInterval :: AReal -> AReal Source #

Same algebraic real, but represented using finer grained isolatingInterval.

Approximation

approx Source #

Arguments

:: AReal	a
-> Rational	epsilon
-> Rational

Returns approximate rational value such that abs (a - approx a epsilon) <= epsilon.

approxInterval Source #

Arguments

:: AReal	a
-> Rational	epsilon
-> Interval Rational

Returns approximate interval such that width (approxInterval a epsilon) <= epsilon.

Misc

simpARealPoly :: UPolynomial AReal -> UPolynomial Rational Source #

goldenRatio :: AReal Source #

Golden ratio