ToySolver.Data.AlgebraicNumber.Real

Algebraic reals

Reference:

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

Algebraic real type

data AReal

Algebraic real numbers.

Instances

Eq AReal

Fractional AReal

Num AReal

Ord AReal

Real AReal

RealFrac AReal

Show AReal

Pretty AReal

PrettyCoeff AReal

Degree AReal

Degree of the algebraic number.

If the algebraic number's minimalPolynomial has degree n, then the algebraic number is said to be degree n.

Construction

realRoots :: UPolynomial Rational -> [AReal]

Real roots of the polynomial in ascending order.

realRootsEx :: UPolynomial AReal -> [AReal]

Real roots of the polynomial in ascending order.

Properties

minimalPolynomial :: AReal -> UPolynomial Rational

The polynomial of which the algebraic number is root.

isolatingInterval :: AReal -> Interval Rational

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

isRational :: AReal -> Bool

Whether the algebraic number is a rational.

isAlgebraicInteger :: AReal -> Bool

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

height :: AReal -> Integer

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

root index, satisfying

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

Operations

nthRoot :: Integer -> AReal -> AReal

The nth root of a

refineIsolatingInterval :: AReal -> AReal

Same algebraic real, but represented using finer grained isolatingInterval.

Approximation

approx

Arguments

:: AReal

a

-> Rational

epsilon

-> Rational

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

approxInterval

Arguments

:: AReal

a

-> Rational

epsilon

-> Interval Rational

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

Misc

simpARealPoly :: UPolynomial AReal -> UPolynomial Rational

goldenRatio :: AReal

Golden ratio